Modelling gas migration in compacted bentonite: GAMBIT ... · MODELLING GAS MIGRATION IN COMPACTED BENTONITE: GAMBIT CLUB PHASE 2 FINAL REPORT T1iv1stelma-Abstract This report describes

POSIVA 2001-02

Modelling gas migration in compacted bentonite:

GAMBIT Club Phase 2 Final Report A report produced for the members of the GAMBIT Club

POSIVA OY

B. T. Swift A.R. Hoch

W.R. Rodwell

AEA Technology

United Kingdom

January 2001

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Jukka-Pekka Salo Posiva Oy T oolonkatu 4 FIN - 00100 Helsinki Finland

18 January 2001

Dear Jukka-Pekka

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REFORMATTING THE GAMBIT CLUB PHASE 2 REPORT TO POSIVA STYLE

May I first wish you, belatedly, a happy New Year. I hope you had an enjoyable Christmas and New Year holiday.

You will no doubt be relieved to hear that we have today received the replacement fuser roll for our colour printer so we have at last been able to print a master copy of the reformatted GAMBIT Club Phase 2 Final report for you. This is enclosed, along with a hard copy of the abstract that I have already sent to you in electronic form. I hope the manuscript is OK; please let us know if there are any problems. You will see that Ben Swift did manage to incorporate the figures into the body of the text, as preferred in the POSIVA report style. The formatting is intended for double-sided printing.

Now that we have sent the reformatted report to you, I will ask our accounts department to send an invoice for payment for the work to POSIV A.

Again, I am very sorry that it has taken us so long to deliver this reformatted report.

Best wishes,

William Rodwell

I J.-l/11711

~ ~·

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and Wales, lll1111her 311'J5H62

TekiJa(t)- Author(s)

Posiva-raportti - Posiva Report

Posiva Oy Toolonkatu 4, FIN-00100 HELSINKI, FINLAND Puh. (09) 2280 30 -lnt. Tel. +358 9 2280 30

T01meks1anta]a(t)- CommiSSioned by

B.T. Swift, A.R. Hoch and W.R. Rodwell AEA Technology, United Kingdom Posiva Oy

Nimeke- Title

Raportin tunnus - Report code

POSIVA 2001-02 Julkaisuaika - Date

January 2001

MODELLING GAS MIGRATION IN COMPACTED BENTONITE: GAMBIT CLUB PHASE 2 FINAL REPORT

T1iv1stelma- Abstract

This report describes the second phase of a programme of work to develop a computational model of gas migration through highly compacted bentonite.

Experimental data that have appeared since the earlier report are reviewed for the additional information they might provide on the mechanism of gas migration in bentonite. Experiments carried out by Horseman and Harrington (British Geological Survey) continued to provide the main data sets used in model evaluation.

The earlier work (POSIV A Report 98-08) had resulted in a preliminary model of gas migration whose main features are gas invasion by microcrack propagation, and dilation of the pathways formed with increasing gas pressure. New work was carried out to further explore the capabilities of this model. In addition, a feature was added to the model to simulate gas pathway creation by water displacement rather than crack propagation.

The development of a new alternative gas migration model is described. This is based on a volumeaveraged representation of gas migration rather than on a description of flow in discrete pathways. Evaluation of this alternative model showed that it can produce similar agreement with experimental results to the other models examined.

The implications of flow geometry, confming conditions and flow boundary conditions on gas migration behaviour in bentonite are reviewed.

Proposals are made for the development of the new model into a tool for simulating gas migration through a bentonite buffer around a waste canister, and for possible enhancements to the model that might remove some of its currently perceived deficiencies.

Avamsanat- Keywords

bentonite, gas migration, modelling, repository

ISBN ISSN ISBN 951-652-103-7 ISSN 1239-3096

Sivumaara - Number of pages Kiell - Language 151 English

TekiJa(t)- Author(s)

Posiva-raportti - Posiva Report

Posiva Oy Toolonkatu 4, FIN-00100 HELSINKI, FINLAND Puh. (09) 2280 30 -lnt. Tel. +358 9 2280 30

T01meks1anta]a(t)- CommiSSioned by

B.T. Swift, A.R. Hochja W.R. Rodwell AEA Technology, United Kingdom Posiva Oy

Nimeke- Title

Raportin tunnus - Report code

POSIV A 2001-02 Julkaisuaika - Date

Tammikuu 2001

KAASUN KULKEUTUMISEN MALLINTAMINEN KOMPAKTOIDUSSA BENTONIITISSA: VAIHEEN 2 LOPPURAPORTTI

T11v1stelma- Abstract

Tassa raportissa kuvataan ohjelmatyon toista vaihetta, jossa kehitetaan laskentamallia kaasun kulkeutumiselle kompaktoidun bentoniitin lavitse.

Saatua uutta kokeellista tietoa on arvioitu, jotta saataisiin lisatietoa kaasun kulkeutull1ismekanismeista bentoniitissa. Horsemanin ja Harringtonin (British Geological Survey) kokeista saadut tiedot muodostavat kuitenkin paaasiallisen tietoaineiston kulkeutumismalleja arvioitaessa.

Aikaisemmassa raportissa (Posiva-raportti 98-08) esitettiin kaasun kulkeutumisen alustava malli, j ossa kaasu kulkeutuu bentoniittiin mikrorakoilun kautta ja kulkureittien avaumiin vaikuttaa kaasunpaine. Tassa tyossa jatkettiin em. mallin arviointia. Taman lisaksi malliin lisattiin uusi piirre, joka mahdollistaa kaasun kulkureitin luomisen olettaen kaasun syrjayttavan bentoniitissa olevaa vetta.

Vaihtoehtoisen kaasunkulkeutumismallin kehitystyota kuvataan raportissa. Malli perustuu kaasun kulkeutumisen keskimaaraisilimiseen tarkastelutilavuudessa, eika siina kuvata yksittaisia kulkeutumisreitteja. Taman mallin tarkastelu osoitti, etta silla voidaan saada samanlaisia tuloksia kuin aikaisemmin tutkituilla malleilla.

Raportissa tarkastellaan myos virtausgeometrian seka fysikaalisten ja virtausreunaehtojen vaikutuksia kaasun kulkeutumisen kayttaytymiseen bentoniitissa.

Lisaksi tehdaan ehdotuksia uuden mallin kehittamiseksi kuvaamaan kaasun kulkeutumista loppusijoituskapselin ymparilla olevassa bentoniitissa. Myoskin ehdotetaan mallin parantamista, joka mahdollisesti korjaa havaittuja puutteita.

Ava~nsanat- Keywords

bentoniitti, kaasun kulkeutuminen, mallinnus, loppusijoitustilat

ISBN ISSN

ISBN 951-652-103-7 ISSN 1239-3096

Sivumaara - Number of pages Kieli - Language 151 Englanti

EXECUTIVE SUMMARY

This report is of work carried out during Phase 2 of the GAMBIT Club programme on gas migration in highly compacted water-saturated bentonite. The ultimate goal of this programme is the development of a computational model that (i) will adequately represent the principal features observed in experiments on gas migration through highly compacted bentonite, (ii) can be used to analyse and interpret experimental results, and (iii) will exist in a version suitable to assess the effects of bentonite barriers on the build-up of pressure and the escape of hydrogen gas from various disposal canister designs. The work on Phase 2 of the programme was designed to build on the Phase 1 results in pursuit of these aims.

A first step in the Phase 2 programme was to update the review of experimental data previously carried out to ensure that the most recent data were considered in the model development. New data that had become available included those from further SKBfunded work at BGS, and from Canadian, Japanese and French laboratories. The implications of these and previous data for the choice of approach to modelling gas migration in bentonite was discussed. The data from BGS (including the new data) remained the most detailed and useful for modelling purposes. These results continued to indicate a pronounced threshold pressure for gas entry into bentonite; it was not possible to reconcile this observation with other results which showed gas migration at very low excess gas pressures.

Some work was carried out to further explore the potential of the preliminary model that was developed in Phase 1 of the GAMBIT Club programme. The main features of that model were gas invasion by microcrack propagation, and dilation of the pathways formed with increasing gas pressure. The further work on this model included: looking for alternative mathematical relationships between pathway dilation and gas pressure to those developed in Phase 1 of the project; allowing for pathways with a range of dimensions to form instead of assuming that they were all identical; some testing of numerical performance; and various technical improvements to the numerical algorithms used in the model. The first of these did allow some overall improvement to the fits to experimental data that had been obtained in the Phase 1 work. Allowing multiple pathways did not, in the tests conducted, lead to significant improvements in fits to experiments.

An additional feature was added to the Phase 1 model to simulate gas pathway creation by water displacement rather than crack propagation. This capability was implemented because it is the mechanism favoured by some workers in the field to explain gas invasion of compacted bentonite. It has proved possible, using the model of gas pathway formation by displacement of water from pre-existing pathways, to obtain similar agreement with the BGS results to those obtained with the crack propagation model. However, this agreement disguises significant differences, mainly in the dimensions and numbers of pathways used in the two models. In the water

displacement model, the threshold pressure is determined by capillary pressure, and a very small capillary radius is needed to produce the high threshold pressures observed. If consistency is demanded between the dimensions of the pre-existing pathways and those supporting gas flow after breakthrough, this means that a very large number of pathways is required to support the observed gas flows.

To address a number of features not included in the Phase 1 model and its derivatives, a new, alternative, model has been developed in the Phase 2 programme. This allows for flow of water through the bentonite, some coupling between gas and water pressure (as reported in some experiments), a mechanism for resealing gas pathways after gas flow has ceased, and some simple stress changes to the clay as for example might occur on partial dehydration. An important consideration motivating the investigation of this alternative model was that it should be readily upscalable to simulate canister-sized systems in two- or three-dimensions, as it was considered that there might be some difficulties in similarly upscaling the crack propagation model. In the alternative model, gas porosity, and hence gas permeability, is created by compression of the water saturated clay, both through compression of the water in the clay, and through drainage of water from the clay. The latter is controlled by the swelling pressure curve of the clay and the permeability to water. The performance of the alternative model has been evaluated, and it has been shown, with suitable choices of parameters, to produce similar agreement with experimental results to those of the other models examined.

A fairly detailed review of the implications of flow geometry, confining conditions and flow boundary conditions on gas migration behaviour in bentonite has been carried out, particularly in an attempt to gain some insights that might help interpret the results of recent experiments carried out at BGS. One potentially significant observation is that all the theoretical analyses point to the creation of such a low gas porosity in the bentonite that the overall volume dilation in the clay will be very small and therefore the manor in which the clay is externally constrained (e.g. constant volume versus constant stress) will only have a small effect on gas migration behaviour.

Finally the three models that have been developed were compared and evaluated, in part on the basis of the use of the models to simulate a recent set of experimental data from BGS. Difficulties presented by deficiencies in current understanding of gas migration in bentonite and anomalies in some data were highlighted in this evaluation. On the basis of the evaluation, it was considered that the alternative model developed within the Phase 2 programme could be upscaled to provide the basis for a tool for simulating gas migration through a bentonite buffer around a waste canister, but that there were some possible further developments to the model that should first be explored to try and remove some deficiencies that the model was currently perceived to possess.

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TABLE OF CONTENTS

page

Abstract

Tiivistelma

EXECUTIVE SUMMARY

TABLE OF CONTENTS ................................................................................................ 1

1 INTRODUCTION .................................................................................................. 5

2 REVIEW OF RECENT EXPERIMENTAL DATA ................................................... 9 2.1 Canadian Data ............................................................................................ 9 2.2 French data ............................................................................................... 12 2.3 BGS Data .................................................................................................. 13

2.3.1 Summary of BGS Experiment Mx80-8 (Radial Flow) ........................ 13 2.3.2 Summary of BGS Experiment Mx80-9 (K0 Geometry) ...................... 18

2.4 Relationship between Experimental Conditions and Gas Generation from a Waste Canister ......................................................................................... 22

3 MODELLING APPROACHES IN RELATION TO THE EXPERIMENTAL DATA.25 3.1 Threshold pressure for gas flow ................................................................ 25 3.2 Post-breakthrough gas flow behaviour ...................................................... 26 3.3 Mechanism of gas pathway formation ....................................................... 27 3.4 Other observations and effects ................................................................. 29

4 FURTHER DEVELOPMENT OF THE PHASE 1 MODEL ................................... 31 4.1 Variation in the Functional Form for the Pathway Dilation Model.. ............. 31 4.2 Multiple-Pathway Facility ........................................................................... 37

4.2.1 Development .................................................................................... 37 4.2.2 Example Tests ................................................................................. 38

4.3 Improvements to and Analysis of the Numerical Performance of the Phase 1 Model ..................................................................................................... 39 4.3.1 Linear Equation Solver .................................................................... .40 4.3.2 Adjustments to the Crack Propagation Model .................................. 40 4.3.3 Numerical Performance of the Phase 1 Model ................................. 41

5 GAS MIGRATION VIA WATER DISPLACEMENT FROM PRE-EXISTING CHANNELS ........................................................................................................ 43 5.1 Model Development .................................................................................. 43

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5.1.1 Pathway creation Model .................................................................. .43 5.2 Example Tests .......................................................................................... 44

6 AN ALTERNATIVE MODELLING APPROACH ................................................. .49 6.1 Description of the Alternative Model .......................................................... 50

6.1.1 Model assumptions .......................................................................... 50

6.1.2 Initial and Boundary Conditions ........................................................ 51 6.2 Mathematical Formulation ......................................................................... 51

6.2.1 Boundary Conditions ........................................................................ 55 6.3 Parameters Required for The Alternative Model. ....................................... 56

6.4 Evaluation of the Alternative Model (GMCiayW) ........................................ 58 6.4.1 Parameter Variations ....................................................................... 56 6.4.2 Resolution Testing ........................................................................... 65 6.4.3 Modelling Experimental Results ....................................................... 66 6.4.4 Resealing Behaviour ........................................................................ 71

7 IMPLICATIONS OF FLOW GEOMETRY, CONFINING CONDITIONS, AND FLOW BOUNDARY CONDITIONS .................................................................... 73

7.1 The Alternative Modelling Approach .......................................................... 74

7 .1.1 Relation of the Alternative Model to the Macroscopic Phenomenological Approach to Soil Consolidation .......................... 7 4

7.1.2 Implications of a Capillary Model ...................................................... 76

7 .1.3 Boundary Conditions ........................................................................ 79

7.2 Alternative Geometries for Gas Migration Through Bentonite ................... 81 7.2.1 Discussion of Experiment MxS0-8 .................................................... 81 7 .2.2 Discussion of Experiment Mx80-9 .................................................... 88

8 SIMULATIONS OF GAS MIGRATION IN A K0 GEOMETRY: COMPARISON OF MODELS ............................................................................................................ 97 8.1 Modelling the K0 Geometry Hydraulic Tests .............................................. 97

8.1.1 Mx80-9 Stage 2 ................................................................................ 97 8.1.2 Mx80-9 Stage 4 ................................................................................ 98

8.2 Modelling the K0 Geometry Gas Migration Test History ........................... 1 00 8.2.1 Modelling Gas Migration via Crack propagation or by water

Displacement ................................................................................. 1 00 8.2.2 Modelling with the Alternative Model Developed in Phase 2

(GMCiayW) .................................................................................... 1 05 8.3 Discussion of Approaches to Modelling the BGS Experiments ................ 1 07

9 RECOMMENDATIONS FOR FUTURE WORK PROGRAMME ........................ 113

REFERENCES .......................................................................................................... 119

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ACKNOWLEDGEMENTS .......................................................................................... 122

APPENDIX 1: DEVELOPMENTS TO THE PHASE 1 MODEL. .................................. 123 A 1.1 Linear Equation Solver ............................................................................ 123 A 1.2 Adjustments to the Crack Propagation Model ......................................... 124 A 1.3 Numerical Performance ........................................................................... 125

A 1 .3.1 The Base Case ............................................................................ 126 A1.3.2 Temporal Discretisation ............................................................... 126 A 1.3.3 Grid Refinement ........................................................................... 126

APPENDIX 2: THEORY OF DEFORMATION AND CONSOLDATION IN UNSATURATED SOILS- A MACROSCOPIC PHENOMENOLOGICAL APPROACH ..................................................................................................... 131 A2.1 Deformation State Variables ................................................................... 131

A2.1.1 Continuity Requirement. ............................................................... 132 A2.1.2 Total Volume Change .................................................................. 132 A2.1.3 Water and Gas Volume Changes ................................................. 134

A2.2 Stress State Variables ............................................................................. 135 A2.3 Constitutive Relations .............................................................................. 136

A2.3.1 Clay Structure .............................................................................. 136 A2.3.2 Water Phase ................................................................................ 138 A2.3.3 Gas Phase ................................................................................... 138 A2.3.4 Hysteresis .................................................................................... 138

A2.4 Flow Laws ............................................................................................... 139 A2.4.1 Water Phase ................................................................................ 139 A2.4.2 Gas Phase ................................................................................... 139

A2.5 Coupled Formulation of Three-dimensional Consolidation ...................... 140 A2.5.1 Equilibrium Equations .................................................................. 140 A2.5.2 Water Phase ................................................................................ 141 A2.5.3 Gas Phase ................................................................................... 142 A2.5.4 Summary ..................................................................................... 143

A2.6 Formulation of One-dimensional Consolidation ....................................... 143 A2. 7 Constitutive Relations for Different Loading Conditions ........................... 143

A2.7.1 Isotropic Test ............................................................................... 144 A2.7.2 Constant Volume (Radial Flow) Test. ........................................... 144 A2.7.3 K0 Test ......................................................................................... 144

APPENDIX 3: STRESSES AROUND A SPHERICAL CAVITY IN A SATURATED CLAY ................................................................................................................ 146 A3.1 Stresses around a Spherical Cavity in a Saturated Clay ......................... 146

A3.2 Tensile Stress Fracture Criterion ............................................................. 149

APPENDIX 4: ESTIMATION OF THE EFFECTS OF COMPLIANCE IN MEASURING DEVICES ON TRANSIENT RESPONSES ....................................................... 150

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5

1 INTRODUCTION

This report describes the work carried out by AEA Technology as Phase 2 of the GAMBIT Club work programme on the modelling of gas migration in highly compacted, water-saturated, bentonite.

Phase 1 of the project was described by Nash et al.(1998), and consisted principally of the development of a preliminary one-dimensional computational model of gas migration through compacted bentonite that was designed to test conceptual models of the process. The main features of this model were that:

a) Gas invasion was assumed to occur by the formation of microfissures which could propagate when the gas pressure reached a critical value. Creation of gas pathways by fracture propagation was described in the model.

b) Dilation of the gas pathways formed by fracture propagation was assumed to occur in response to changes in gas pressure, as such behaviour was implicated in the results of experiments on gas invasion of samples confined by a constant isotropic stress. The gas permeability was a function of the pathway dimensions.

c) The behaviour of the clay itself was represented simply by its mechanical and fracture resistance properties; neither water movement nor the swelling properties of the clay were modelled.

The results produced by the model were evaluated by comparison with the experimental "histories" obtained by Horseman and Harrington ( 1997) from controlled flow-rate gas injection experiments consisting of cycles of varying flow rates. The model was able to reproduce qualitatively the trends seen over the duration of the histories examined, and to reproduce quantitatively substantial parts of the histories. An inability to provide quantitative matches to complete histories, however, suggested that there were limitations to the model formulation (although there were also issues of experimental reproducibility, which might impinge on the degree of agreement that could reasonably be expected from comparisons between theory and experiment).

The ultimate goal of the GAMBIT Club programme remains the development of a computational model that (i) will adequately represent the principal features observed in experiments on gas migration through highly compacted bentonite, (ii) can be used to analyse and interpret experimental results, and (iii) will exist in a version suitable to assess the effects of bentonite barriers on the build-up of pressure and the escape of hydrogen gas from various disposal canister designs. The work on Phase 2 of the programme was designed to build on the Phase 1 results in pursuit of these aims, considering where the Phase 1 model might be improved and where further or alternative developments were required.

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The agreed plans for the GAMBIT Club phase 2 work programme included the following elements:

i) Review and selection of experimental data. This task was to take cognisance of any new experimental data on compacted bentonite that had become available, in particular evaluating the data, and identifying its relevance to the GAMBIT Club work.

ii) Further evaluation of the model. The evaluation of the Phase 1 model that could be carried out during that phase of the programme was limited, and it was intended to extend the range of testing and evaluation of the preliminary model to further explore its capabilities and potential.

iii) Further model development and algorithm analysis. In this task a number of model development and performance evaluation issues identified in Phase 1 were to be addressed. These included the nature of the downstream boundary condition to be imposed in simulations of gas migration through water-saturated bentonite cores, the numerical performance of the model, and the treatment of variability in the characteristics of the gas pathways that are formed.

iv) Treatment of different stress fields and geometries. The possible effects of different stress boundary conditions and geometrical configurations in experiments on gas migration in bentonite were to be considered.

v) Implementation and analysis of a pathway propagation model based on preexisting channels. As noted above, in the preliminary model developed in Phase 1 of the Gambit Club programme, it was assumed that gas pathways were formed by micro fissuring of the clay (i.e. there were no pre-existing pathways present for the gas to follow). In this task, the alternative proposition that gas migration pathways are formed by displacement of water and/or clay gels from preexisting connected pathways was to be investigated. This was in recognition that the view is held in some quarters that water displacement could provide the main mechanism of gas migration in bentonite, and to explore whether modelling might provide any discrimination between the alternative mechanisms.

vi) Water displacement mechanisms and effects of gas migration on bentonite properties. An attempt was to be made in this task to address issues arising from the GAMBIT Club Phase 1 report about the extent of water displacement from

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compacted bentonite by invading gas and the effect that this might have on transport properties of the bentonite after gas migration has ceased. Even if gas does not displace water from pre-existing pathways, an additional stress imposed by the gas pressure could squeeze water from the clay fabric.

vii) Outline upscaling plans. In order to apply the models developed in Phases 1 and 2 of the GAMBIT Club project on the scale of bentonite buffers around real disposal canisters, the issue of upscaling the demonstration models needs to be addressed. This issue was to be discussed in the Phase 2 programme, and outline plans for the implementation of an uspscaled model in Phase 3 of the project was to be formulated.

The work on task (i) is described in Sections 2 and 3. Experimental data from Canadian and French sources and from the continuing programme at BGS are discussed in Section 2, and the implications of this data for the modelling approaches to be followed are considered in Section 3.

The work carried out on tasks (ii) and (iii) is summarised in Section 4, with some of the more technical details relating to task (iii) presented in Appendix Al. This work of task (ii) was focused largely on the investigation of alternative pathway dilation models to those considered in the Phase 1 work to see if they offer improved fits to some of the experimental results of Horseman and Harrington ( 1997) that were discussed in the Phase 1 work programme (Nash et al., 1998). The work on task (iii) has included a mixture of items comprising rather technical developments to improve aspects of the model or to address issues arising during its use, investigation of the numerical performance of the model, and the implementation of an extension to the model to allow the simultaneous presence of gas migration pathways with a range of dimensions.

The implementation of a model based on gas migration through preexisting channels [task (v) above] is described in Section 5.

Section 6 provides a description of a new modelling approach for simulating gas migration in bentonite. This new model provides an alternative to the Phase 1 model. It was developed for the following reasons:

a) To allow for water flow through the clay and to represent the swelling behaviour of the clay in a simple way.

b) To provide for the incorporation of a coupling between gas invasion and water flow (as for example implied by changes in pore water pressure, seen during some gas injection experiments).

c) To provide a framework in which the effects of gas migration on subsequent water flow and the resealing of pathways could be considered.

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d) To produce a model that could be more easily upscaled than the Phase 1 model to deal with canister-scale problems in two- or three-dimensions.

Points (b) and (c) address items of task (vi) of the work plan indicated above. Point (d) provides the basis for the upscaling of the model and for plans for future work that are discussed in more detail in Section 9 (task (vii)) of the work plan). The incorporation of the features noted in point (a) should also be considered for inclusion in an upscaled simulation model.

The effects of different geometries from the one-dimensional flow geometry adopted in the Phase 1 and 2 model developments, and of some alternative possible confinement conditions are discussed in Section 7 in fulfilment of task (iv) of the work plan, again providing material for the development of future plans in Section 9. Section 7 also discusses some theoretical background to various aspects of the modelling, drawing on more detailed theoretical material summarised in Appendices A2 and A3. Section 8 discusses the modelling the results of one of the new BGS experiments, that in the ~ geometry, using the three conceptual models that have been developed in Phases 1 and 2 of the GAMBIT Club programme; gas migration via crack propagation, gas migration via water displacement, and macroscopic modelling of gas-water behaviour in the bentonite.

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2 REVIEW OF RECENT EXPERIMENTAL DATA

Since the start of Phase 1 of the GAMBIT Club project, a few published studies of gas migration in clay barrier materials have appeared or come to the notice of project staff. These include work carried out in Canada (Gray et al., 1996; Graham et al., 1998; Hume, 1999), French work on Fo-Ca Clay (Galle, 1998), and work at BGS (Horseman and Harrington, unpublished] on tests in a constant volume radial flow geometry and on linear flow in a cylindrical ~~ geometry (radially constrained but with a constant axial stress). These studies are described in the following subsections.

2.1 Canadian Data

All the Canadian experiments were carried out with the clay compacted into an oedometer cell, with the clay dry density controlled by axial compaction of a known mass of clay and water. Experiments were carried out on bentonite and illite and on sand-bentonite and sand-illite mixtures. The dependence of gas breakthrough on dry density, water content and water saturation was studied. The preliminary work reported by Gray et al. ( 1996) was mostly carried out on illite, as the 10 MPa rating of the apparatus was found to be insufficient for much testing on bentonite buffer material.

Construction of a cell with a pressure rating of 62.7 MPa allowed more extensive testing of gas breakthrough pressures for bentonite (Graham et al., 1998). The range of parameters used in the experimental programme undertaken by Graham et al. ( 1998) is shown in Table 2-1.

The 100% saturated specimens were obtained by applying a water pressure of 0.2 - 1.0 MPa (and rarely, 5 MPa) at both ends of the specimen (note that in some of the early "saturated'' experiments, a saturation time of 24 hours at 0.2 MPa backpressure may not have been sufficient to fully saturate the specimen (Hume, 1999) ). Other specimens were tested in the "as compacted" condition. The details of the experiments on compacted bentonite are described more fully by Hume ( 1999). These are the most relevant of the Canadian experiments to the GAMBIT Club project. They were carried out with A vonlea bentonite.

Table 2-1. Materials Tested by Graham et al. ( 1998).

Soil type Dry density (Mg m·3) Water content(%) Saturation (%)

Illitic clay 1.85- 2.10 10- 16 67- 100

Bentonitic clay 0.6- 1.45 30.0-63.5 60- 100

Sand-illite 1.97- 2.30 5.4- 13.0 45- 100

Sand-bentonite 1.67 11.4- 20.4 50-89

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Breakthrough gas pressures were measured by incrementing the pressure every 5 minutes by 0.2 or 1.0 MP a, respectively, for the low or high pressure cell, until indications of gas breakthrough were observed. It should be noted that this is a very rapid rate of pressure increase compared with, for example, that occurring in the work of Horseman and Harrington ( 1997). In other experiments, a constant gas pressure was applied for an extended period of time, and any flow from the specimen monitored.

Tests were carried out in the low-pressure cell for specimens with target dry densities of 0.90- 1.45 Mg m-3

, but with no attempt to saturate the clay after compaction; that is, the saturation state was that resulting from the mixture that was compacted. Breakthrough pressures· were typically less than 1 MPa, except when the notional saturation was high, say >97% (although this value was variable), when breakthrough occurred at a pressure between 5 - 6 MPa, or had not occurred at pressures of about 9 MPa. Although the saturations were calculated to be 100% in some cases, results obtained when efforts were made to ensure saturation in tests with the high pressure cell suggest that this was not the case.

For the experiments using the high pressure cell, the efforts to resaturate the bentonite resulted in much higher gas breakthrough pressures. For example, in the final series of "best optimised" experiments, breakthrough was not achieved with pressures of about 50 MP a for specimens with target dry densities of 0.80 - 1.20 Mg m-3

•

In contrast, when a constant gas pressure of between 0.3 and 19.8 MPa was applied to specimens with a dry density of from 0.80 - 1.4 Mg m-3

, gas breakthrough occurred in every case if sufficient time was allowed. The trend was for the breakthrough time to decrease with the applied gas pressure, and to increase with the dry density. For some limited ranges, there was a good relationship between the breakthrough time and the inverse of the applied gas pressure.

Some experiments were carried out by Graham et al. ( 1998) using paraffin oil instead of water, to investigate the effect of a non-polar fluid on breakthrough pressure. When paraffin oil replaced water in a sample with a dry density of 1.15 Mg m-3 and a saturation of 98.8%, the breakthrough pressure fell from an average value of about 4

MPa, for similar water saturated specimens, to 0.2 MPa .. This result is qualitatively as expected, as the non-polar paraffin oil is not expected to develop the swelling properties of the clay.

An attempt (Hume, 1999) was made to interpret the results of the tests with water as the wetting phase using a simple capillary entry model of gas threshold pressures, or a model of piston-like displacement of water by gas to determine breakthrough times. Variants of these models with different choices for capillary radii or permeability model were discussed, but these are not fundamentally different. No satisfactory correlations with "breakthrough" pressures were established, and, indeed, it was doubtful whether a genuine threshold pressure for gas migration was observed because of the high rates at

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which the applied gas pressures were increased. The partial correlation of breakthrough times with inverse pressure in the constant pressure tests may be significant in suggesting a fluid displacement model of gas invasion.

In some tests, the state of the specimen after the test was examined and two observations of note were made:

a) Determination of the water content along the cores showed, surprisingly, that in some test series the water content at the gas inlet end of the core was higher than at the outlet end after gas injection.

b) In some tests the length of the sample had decreased after the gas test, suggesting that the application of gas pressure had compacted the core.

An explanation of observation (a) was offered in terms of variations in effective stress. A lower effective stress at the inlet end compared to the outlet induced water movement against the gas flow. This explanation depends on assumptions about the relationship between pore fluid pressure and gas pressure.

Relating this Canadian work to the GAMBIT Club approach is difficult.

First, it appears that the A vonlea bentonite used is much less compacted than the Mx -80 used by Horseman and Harrington ( 1997). The dry density of the A vonlea bentonite was between 0.6 and 1.4 Mg m-3

, compared to values of 1.57 to 1.68 Mg m-3 for the medium and high swelling specimens employed by Horseman and Harrington. This has implications for the swelling pressure that would be exhibited by the A vonlea bentonite samples. No data is provided for this, but it would be expected from correlations developed for bentonite swelling pressures that this would be small, perhaps a few hundred kPa or less. To the extent that gas entry pressures are related to swelling pressures, it may be the case that the gas pressures applied to the samples may always have been in excess of the entry pressures, and that the very high "breakthrough" pressures observed were a consequence of time dependence effects: the gas did not have time to propagate through the samples. Nevertheless it is surprising that, at some of the high gas pressures that were reached, breakthrough was not seen, since the modelling studies carried out in the GAMBIT Phase 1 and Phase 2 studies indicate that the creation of gas pathways by microfissuring should be rather fast once the threshold pressure has been exceeded (or this could mean that the parameters currently being used are not the most appropriate).

Secondly, it is not clear that the mechanism of gas transport will be the same in the A vonlea bentonite, at relatively low compactions, as in the highly compacted bentonite being addressed by the GAMBIT Club. It has been recognised previously that it is possible that the dominant transport mechanisms may vary with conditions.

12

A further consideration is the possibility that differences in compaction procedure (Proctor compaction in the Canadian work, compaction under a very high, 50 MPa, load in the BGS work) may also lead to differences in comparative behaviour (Horseman, private communication).

2.2 French data

The smectite rich Fo-Ca clay used in the experiments described by Galle ( 1998) was prepared to compaction conditions more akin to those so far assumed for the GAMBIT Club work. The dry densities of the samples ranged from 1.6 to 1.9 Mg m-3

• One value of swelling pressure was quoted: 13 MPa for clay compacted to a dry density of 1. 7 5 Mg m-3

• Clay samples with water saturations of between 7 4 and 100% were studied. The results reported mostly related to measurements of gas permeabilities, the values of which shown in Table 2-2 (in m2

) were obtained.

These results show the expected decrease in permeability with increasing density and decrease in gas permeability with increasing water saturation.

Galle distinguishes between gas entry pressure and gas rupture pressure in tests involving step-wise increases in applied gas pressure (1-2 days between increments?). For the saturated samples, these pressures were close together. The only rupture pressure quoted for a saturated sample was that for one with a dry density of 1.6 Mg m-3

,

for which its value was 4.3 MPa. This may be of a similar magnitude to the swelling pressure of the material being used.

In similar work, Tanai et al. ( 1997) compared gas breakthrough pressures measured in a variety of laboratories (including their own measurements on Kunigel VI bentonite and

Table 2-2. Range of Experiments Carried out by Galle ( 1998): Measured Permeabilities in m2

•

Water Dry Density (Mg m·3)

Saturation (%) 1.6 1.7 1.8 1.9

74 1.7±0.3 10- 16 2.8±0.7 10-17 5.8±1.4 10-18 3.1±0.5 10-19

82 9.2±2.6 10-18 3.4±0.7 10-18 2.7±0.4 10-20

90 1.7±0.6 10-18 1.5±0.1 1 o- 19

97 3.9±2.4 10-18 5.1±1.5 10-20

100 1.3±0.1 1 o- 18 *2.0 10- 19 *4.5 10-20 *2.5 10-22

*Estimated by extrapolation of trend

13

Fo-Ca clay) with swelling pressures, and found a good correlation.

2.3 BGS Data

The main experimental data used in the GAMBIT Club Phase 1 work was that obtained by Horseman and Harrington (1997) at BGS. This SKB funded programme has continued with work on saturated bentonite samples constrained in different ways to the isotropically stressed samples previously investigated. Two new geometries are being investigated:

a) a cylindrical, constant volume geometry (experiment Mx80-8), and

b) a ~~ geometry, in which the sample is contained in a constant radius rigid jacket, with floating end caps to which a constant axial load is supplied.

A summary description of these two experiments is provided in the following two subsections.

2.3.1 Summary of BGS Experiment Mx80-8 {Radial Flow)

In the cylindrical constant volume apparatus, gas is injected from a filter in the centre of the specimen at the end of a thin gas injection tube oriented along the axis of the cylinder. Three arrays of sink filters are located around the external surface of the bentonite. The mid-plane of the sample corresponds to the mid-plane of the middle array, and neighbouring arrays are separated by 21.4 mm. Each array consists of four filters at 90° to each other. Each filter has a diameter of 6.4 mm and is profiled to match the internal surface of the apparatus. As well as the injection gas pressure and the flow rate out of the system, porewater pressure and radial and axial stresses were monitored. This geometry was designed to constrain the gas flow paths so that the gas had to pass radially through the clay (i.e. the possibility of the gas reaching a sink filter by bypassing the clay is minimised).

The bentonite sample used in Experiment Mx80-8 had the geotechnical properties shown in Table 2-3.

The swelling pressure can be estimated from the relation given by Borgesson et al. (1996)

n = 1.963 SW 4 85 e· 2-1

k = 3.45 1 o-20 e4•28

where Tisw is the swelling pressure of the bentonite sample (MPa); e is the void ratio;

14

k is the intrinsic permeability of the sample (m2).

This implies that Ilsw = 8.5 MPa and k = 9.5 1U21 m2• Using the alternative relationship

given by Horseman and Harrington ( 1997)

2200 Ilsw = (

1 + e y-ss = 9.4 MPa 2-2

a value for the swelling pressure of Ilsw = 9.4 MPa is obtained.

The test consisted of a number of distinct stages. These are summarised in Table 2-4.

This Section is concerned only with the results from the gas injection stages (i.e. Stage 10 to Stage 13) of the experiment, which are plotted in Figure 2-1 below (Horseman and Harrington, unpublished). As in the previous work by Horseman and Harrington ( 1997), gas in an upstream volume is compressed at a fixed rate of volume change, and the gas pressure rises by isothermal compression until a peak excess pressure is reached. In this test, this occurs at an excess pressure of 18.5 MPa, at which point gas begins to flow from the radial sink filters. The gas pressure then decays to an approximately constant value (end of Stage 10). Turning off the pumps used to compress the gas in the upstream volume leads to a further transient in the upstream gas pressure (Stage 11). These pressure responses are qualitatively similar to those seen in the axial flow tests under isotropically stressed conditions. Reinstating the gas flow (Stage 12) at the original rate produces a second peak in the upstream gas pressure, although this peak is not as high as in the one in Stage 10, nor does it display the same pointed shape. A final shut in (Stage 13) produces a decay in pressure that parallels that of the first shut in.

The peaks in the gas pressure are associated with the start of the main outflows at the radial sink filters.

Table 2-3. Geotechnical Properties of the Bentonite Used in Experiment Mx80-8 (Horseman and Harrington, unpublished).

Mx80 bentonite type 'Medium Swelling'

Water content (wto/o) 26.7

Bulk density (kg m-3) 2016

Dry density (kg m-3) 1592

Void ratio 0.740

Initial saturation(%) 97.6

15

Table 2-4. Table of the Different Stages in Experiment Mx80-8 (taken from Horseman and Harrington, unpublished).

Stage Description

1 Hydration with injection pressure Pw 2 MPa and back-pressure 2 MPa




5 Hydration with injection pressure Pw 10 MPa and back-pressure 10MPa


7 Hydration with injection pressure Pw 10 MPa and back-pressure 10MPa



10 Gas constant flow with flow rate 375 JlL h- 1 and back-pressure 1 MPa

11 Gas 'shut-in' with flow rate 0 Jll h- 1 and back-pressure 1 MPa

12 Gas constant flow with flow rate 375 JlL h- 1 and back-pressure 1 MPa

13 Gas 'shut-in' with flow rate 0 JlL h- 1 and back-pressure 1 MPa

Figure 2-2 shows an expanded plot of the top graph from Figure 2-1, for the period around the first peak in the pressure profile, and illustrates a quite complex variation in the measured values of stress and porewater pressure.

Particular points noted by Horseman and Harrington (unpublished) in relation to these experimental results are:

a) The first signs of flow (precursor flows) occur at a low level ( -1 JlL hf') at a gas pressure of 13.7 MP a. These flows are too small to be visible in Figure 2-1. At the same time some increase in the radial stress is seen (Figure 2-2).

b) A small temporary drop in gas pressure coincides with a sharp increase in axial stress, which rises to about 18.4 MPa. Horseman and Harrington interpret this as indicating that a gas pathway has propagated to the vessel wall but has not connected to a sink filter.

16

20

injection gas pressure

axial stress

radial stress

15 porewater pressure

backpressure

7 0...

~ Q) 10 1-. ;:::s l71 l71 Q) 1-.

0...

5

oL-~-L~--~-L~--L-~-L~~~-L~--L-~-L--L-~-L~ 0.0 107

~ 3 10-8

l71 '-.....

l Q) _, ~ 2 10-8

~ 0

G::

1 10-8

1.0 107

Time [s]

flow rate into system

radial array no . 1 (above injection filter)

radial array no. 2 (plane of injection filter)

radial array no . 3 (below injection filter)

0 10-8 L-~-L~~~_u~~L-~-L~~~~~--L--L~--L-~-L~ 0 .0 107 1.0 107

Time [s]

Figure 2-1. Data from the BGS experiments in a Constant Volume Radial Flow Geometry (the flow rate above the injection plane drifted during the first 'shut-in' - the large flow event during this period does NOT actually exist).

17

20


axial stress

radial stress


~ 0...

6 Q) 10 1-. ;::j l7l fll Q) 1-.

0...

5

aLL~~~~~~~~~~~~~~~~~~~~~

3 .1 106 3 .2 106 3 .3 106 3.4 106 3 .5 106

Time [s]

Figure 2-2. Detail of the experimental response during the gas breakthrough phase.

c) The gas pressure continues to rise to a peak value of 19.4 MP a, at which point there is a rapid rise in outflow from the upper filter array (Figure 2-1 ), indicating that gas breakthrough to one or more of these filters has occurred. During this period, the radial stress rises (as does the pore water pressure) while the axial stress falls (Figure 2-2). The radial stress and porewater pressure subsequently pass through a maximum and begin to fall (Figure 2-1 ).

d) After gas breakthrough, the gas pressure declines along a spontaneous negative transient to a steady state value of 11.2 MPa. Erratic outflows are interpreted as indicating pathway instability.

e) When the upstream pumping rate is stopped, a further negative transient is seen, with the upstream pressure decaying to a value of 8.5 MPa. During the early stage of this transient, the radial stress rises.

f) In the second flow cycle, breakthrough is obtained at a pressure of 9.4 MP a, much lower than in the first cycle. Pathways develop sequentially to each of the three arrays of sink filters. This is considered to provide evidence of multiple pathways.

g) The proportion of the total flow conducted by each array of filters varies, showing changes in the gas pathways.

18

h) During the second cycle, it is the measured radial stress that rises to approach the gas pressure, not the measured axial stress, suggesting that the gas pathways have evolved between the first and second cycles.

i) The maximum gas pressure in the second cycle is 11.5 MPa (cf 19.4 MPa for the first cycle).

j) The upstream gas pressure to which the system evolves after the second shut in is 8.6 MPa, which is close to the value of 8.5 MPa from the first cycle.

The results of this experiment is discussed further in Sections 3. 3 and 7 .2.1.

2.3.2 Summary of BGS Experiment Mx80-9 (~ Geometry)

As indicated in the introduction to this section, in the Ko geometry used in the BGS gas migration experiment, the sample was radially constrained in a rigid metal sheath, but axially the sample was confined by a pair of floating end caps (sealed against the walls of the metal sheath with 0-rings) to which a constant axial stress was applied by immersion of the specimen holder in a constant pressure fluid. The sample had a diameter and a length of 5.09 cm. The axial stress was 10 MPa. Fluids were injected at one end and withdrawn from the other against a constant backpressure of 1 MPa. Injection was, as usual in the BGS experiments, by displacement from a reservoir by pumping water into the reservoir at a prescribed flow rate.

The experimental apparatus provided for continuous measurement of the pore water pressure through a filter placed in the metal sample sheath at the mid-plane of the sample, as well as the upstream gas pressure and downstream flow rates.

The geotechnical properties of the clay sample used in this experiment are shown in Table 2-5, and are very similar to those used in experiment Mx80-9.

Table 2-5. Geotechnical Properties of the Bentonite Used in Experiment Mx80-9 (Horseman and Harrington, unpublished).

Mx80 bentonite type 'Medium Swelling'

Water content ( wt%) 26.5

Bulk density (kg m-) 2020

Dry density (kg m-3) 1596

Void ratio 0.735

Initial saturation(%) 96.9

19

The experiment consisted of the stages shown in Table 2-6. Throughout the downstream back pressure was maintained at 1 MPa.

The transient data from the constant flow rate water injection test at 2.2 ~ hr-1 were interpreted using standard hydrogeological models (e.g. Marsily, 1986) to obtain values of the intrinsic hydraulic conductivity and specific storativity of the sample. The values obtained were 7 10-14 m s-1

, for the hydraulic conductivity (corresponding to a saturated permeability of 7 10-21 m2

) and a specific storativity of 3 10-4 m-1•

Figure 2-3 provides an overview of the results of Stages 4-8 of the Ka geometry gas injection experiments, showing the upstream gas pressure, mid-plane pore water pressure, and inflows and outflows. And expanded view of the results around the time of first gas breakthrough is shown in Figure 2-4.

Horseman and Harrington (unpublished) note the following features of the results obtained using the Ka geometry:

a) There was no apparent gas movement when gas pressure was held constant at 8.8 MPa for 87 days.

b) Breakthrough and peak pressures coincided at 10.0 MPa. This is equal to the axial total stress applied to the bentonite.

Table 2-6. Table of the Different Stages in Experiment Mx80-9 (taken from Horseman and Harrington, unpublished).

Stage Description

1 Equilibration with water applied at a constant pressure of 1 MPa.

2 Hydration using a constant flow rate of water of 1.0-2.2 J..LL hr-1•

3 Equilibration with water applied at a constant pressure of 1 MPa.

4 Injection of gas at a constant displacement rate of 375 ~ hr-\ except for a break of over 80 days during which the pressure was held at a constant value to test whether gas breakthrough would occur at a reduced pressure if sufficient time were allowed.

5 Injection of gas at a constant displacement rate of 180 J..LL hr-1•

6 Termination of gas flow (shut in) with the upstream gas pressure allowed to evolve passively.

7 Injection of gas at a constant displacement rate of 375 J..LL hr-1•

8 Termination of gas flow (shut in) with the upstream gas pressure allowed to evolve passively.

'0:' E-< 1Zl ...., eo lll

"'-..

10

8

2

2 10-8

'k 1 10-6 f-

0 10-8

0 .0 107

--

--

"" I L

20


backpressure

porewater pressure

Time [s]

I I


flow rate out of system

-

{"-

~

I u I. L I~ 11 ~ Time [s]

Figure 2-3. Measured Pressures and Flow Rates for Stages 4-8 of Experiment Mx80-9 (K0 Geometry) of Horseman and Harrington (unpublished).

+-' Cd

Ill .........

10

9

9 .0 106

21

_ _ injection gas pressure

9.2 108 9.4 106

Time [s]



11 10- 8

(I) +-' Cd

0:::

~ 0

~

0 10 -8 ~~U--L~~--L-~~-L~~--LL~~~~~~±=~~ 8 .8 106 9 .2 108 9.4 108 9.6 108 9 .8 108

Time [s]

Figure 2-4. Measured Pressures and Flow Rates Experiment Mx80-9 (K0 Geometry) of Horseman and Harrington (unpublished)- Period around First Gas Breakthrough.

22

c) The "shut-in" values to which the pressures converged after termination of upstream gas injection were 8.4 MPa for the first cycle and 8.1 MPa for the second cycle. These values are close to the value of -8.5 MPa obtained in the Mx80-8 experiment (Section 2.3.1).

d) Comparison with constant stress experiments (Horseman and Harrington, 1997) shows that peak height and shut-in pressure are both sensitive to containment boundary conditions.

The results of experiment Mx80-9 are further discussed in Sections 3.4 and 7 .2.2, and simulation of the results of the experiment is discussed in Section 8.

2.4 Relationship between Experimental Conditions and Gas Generation from a Waste Canister

Finally in this section, brief consideration is given to the relationship between the rates of gas flow that are involved in some laboratory experiments on gas migration in bentonite and the rates of gas generation that might pertain in a repository in which waste canisters are emplaced.

The conditions under which gas flow through bentonite might be established in a repository may well vary significantly with both repository concept and evolution scenario. For example, if the gas is produced within a canister as a consequence of water ingress through a defect hole in the canister, flow through the bentonite may radiate from a small source associated with gas escape through the hole. In contrast, for example, gas generation from the surface of a carbon steel canister may produce an extended source term and lead to gas penetration of the bentonite across the whole contact area between bentonite and canister wall. Gas escape from a hole in a canister may result in a similar gas migration pattern if the gas spreads out along the interface between bentonite and canister before penetrating the bentonite. Intermediate flow patterns can be envisaged. The flux through the bentonite will depend on the gas generation rate and the area through which the gas flows. The rate of pressure rise in the free gas phase before flow in the bentonite is established will depend not only on the gas generation rate, but also on the size of the gas-filled void into which the gas is produced and on the current pressure of the gas. If the gas is being released, for example, into a substantial gas-filled void within a canister, the pressure rise may be slower than if the same rate of gas production occurred from the surface of a canister in close contact with the bentonite buffer (negligible gas-filled void space).

In a laboratory experiment, gas flow through the bentonite is generally established in a different way to that in which it is expected to occur in a real repository. There is no gas generation term; instead the rise in upstream gas pressure is simulated either by imposed step changes in the gas pressure, or by compressing a fixed mass of gas by reducing its volume at a controlled rate. This means that comparison between experimental and

23

repository conditions is not straightforward. Some comparisons that are possible are made below.

In the experiments carried out at BGS, Horseman and Harrington ( 1997, unpublished) typically employ a rate of compression of the gas of 375 J..LL hr- 1 for the first stage of their experiments, during which the gas pressure rises, gas breakthrough occurs and post -breakthrough steady flow is established. Depending on the density of the bentonite, steady-state flow is established at an upstream gas pressure typically between 8 and 15 MPa. If, for the sake of this discussion, the steady-state upstream gas pressure corresponding to a volume flow rate of 375 J..LL hr- 1 is taken to be 10 MPa, then the gas mass generation rate that would produce the same flow rate would be 13.5 mol yr" 1

•

Experimental work carried out by other groups has usually involved more rapid rates of increase in pressure than those imposed at BGS.

If anaerobic corrosion of the surface of a carbon steel canister is represented by the reaction:

then the gas generation rate, q, per unit area in mol m·2yr" 1 is

4vp q=--

3M

where v is the corrosion rate (m yr"1),

2-3

2-4

p is the density of iron in the steel (kg m-3), which is taken to be the density

of iron, and M is the molar mass of iron (kg mor1

).

Various values have been reported for the anaerobic corrosion rate of carbon steel in the presence of bentonite or other clay. In the compilation produced by Rod well et al. ( 1999) values in the range 2-50 J..Lm yr" 1 are quoted. On the other hand, the rates inferred from experiments with iron in the presence of Boom Clay have been reported to be lower at between 0.02-0.2 J..Lm yr" 1 (Rodwell (ed), 2000). Taking a rate of 1 J..Lm yr" 1 as representative, with values of the density and molar mass of iron of 7.86 103 kg m-3 and 55.847 10-3 kg mor 1 respectively, gives a hydrogen generation rate of0.19 mol m·2yr" 1

•

It is interesting to compare the above gas generation rate from corrosion of a canister surface with other estimates of gas generation from single canisters. Canister designs vary with repository concept and waste type. Typically they would have diameters from 0.6-1.0 m and heights from 2-5 m. Taking the surface area therefore to be about 10m2

would give rise to a gas generation rate of -2 mol yr" 1 per canister. When radiolysis and other sources of gas generation are taken into account, as required for some disposal concepts, or if corrosion rates at the higher end of the range indicated above are

24

assumed, then the gas generation rate per canister may be higher. Vercoutere (to be published) reports estimates of gas production from a vitrified waste canister that fall from an initial value of -35 mol yf' to a long term value of -5 mol yf', with lower rates for canisters for other waste types. For corrosion of an internal cast iron insert as a result of water ingress through a defect in a copper canister, it is suggested that the maximum gas generation rate, at the time at which gas is expected to escape from the canister, might be -0.7 mol yf' per canister (SKB, 1999). The gas generation rate equivalent to the steady-state flow in the experiments of Horseman and Harrington (13.5 mol yf') is therefore in the range of what can be expected as the (probably maximum) rate from a single canister.

25

3 MODELLING APPROACHES IN RELATION TO THE EXPERIMENTAL DATA

In this section an attempt is made to draw together the results of the experiments that have been carried out on gas migration in compacted bentonite and to relate the assumptions made in the GAMBIT Club models to the experimental data. The support that can be found in the experimental data for the assumptions made and the uncertainties that remain, for example through apparent inconsistencies in the experimental data, are emphasised. Some of these considerations were addressed in the GAMBIT Club Phase 1 Report (Nash et al., 1998), but an overall summary was thought helpful in evaluating the modelling ideas proposed in this report.

3.1 Threshold pressure for gas flow

It has been consistently assumed in the GAMBIT Club programme that gas will not enter saturated compacted bentonite until its pressure exceeds a threshold value. On the basis of the work by Horseman and Harrington (1997) and others (e.g. Pusch et al., 1985), the threshold gas pressure has been identified with or at least related to the swelling pressure. This relationship was supported generally by the gas injection experiments carried out by Horseman and Harrington ( 1997) under isotropic stress conditions, although a possible precise equivalence between swelling pressure and gas entry pressure was slightly obscured by the unknown additional stress that may be imposed on the clay by the copper sheath containing the clay in most of these experiments. Horseman and Harrington ( 1997) estimate that this might be as much as 2 MPa. It is possible that gas may enter the clay at a threshold pressure that is somewhat less than the swelling pressure (e.g. Pusch et al., 1985). Hume (1999) contends that there is no entry pressure for A vonlea bentonite that is less compacted than the material used by Horseman and Harrington ( 1997, unpublished), as discussed in Section 2.1, but this result is not consistent with those of other groups. For example, Horseman and Harrington (unpublished) find no gas breakthrough after 87 days continuous application of a gas pressure of 8.8 MPa in their Mx80-9 experiment discussed in Section 2.3.2.

For gas injection from a small source in the centre of a constant-volume cylinder, with radially disposed sinks for fluid outflow, Horseman and Harrington ( 1997) find the threshold pressure for gas entry is much higher (excess pressure of -18.5 MP a) than the estimated swelling pressure ( -9 MPa), as discussed in Section 2.3. It is possible that this difference in behaviour is a consequence of the different geometry of this "radialflow" experiment compared with the isotropically stressed "linear-flow" experiments. (cf Sections 2.3.1 and 7.2.1)

26

3.2 Post-breakthrough gas flow behaviour

Most studies of gas migration in compacted bentonite have focused on the measurement of gas entry pressure, with only a few measurements made of post-breakthrough flow behaviour. The exception is the work of Horseman and Harrington (1997, unpublished), which has included extensive measurement of the post-breakthrough behaviour under controlled flow-rate conditions. Salient features of these results include:

a) The steady-state upstream pressure needed to sustain a given flow rate after breakthrough does not fall when the flow rate is reduce by as much as would be expected by application of a constant gas-permeability model to the flow.

b) As a corollary to (i), the shape of the transient in the upstream pressure (in the gas source vessel) as a consequence of a change in flow rate cannot be explained by a constant permeability model.

c) Gas flow persists at upstream pressures well below the original threshold pressure for gas entry.

d) Increase of gas flow after a period of post-breakthrough reduced-rate flow causes a sharp increase in upstream pressure, but to a value less than the original threshold pressure (and sometimes not much above the pressure observed for steady state flow at the increased flow rate, unless the confining stress is increased) in contrast to the peak significantly above this value seen for initial breakthrough.

e) Complete sealing of gas pathways seems to occur on (but only on) rehydration of the bentonite.

Points (a) and (b) imply that the gas permeability varies with gas pressure. It has been presumed in the GAMBIT Club work that this is a consequence of pathway dilation (supported by other evidence from gas migration in clay pastes (Horseman and Harrington, 1997; Donohew et al., 2000)), but shutting off of pathways {perhaps as a consequence of capillary effects) could also be involved. The difference between these two mechanisms may only be significant in as far as the concepts might be invoked to develop constitutive models of the dependence of gas permeability on gas pressure.

The fact that gas flow continues after the upstream gas pressure has dropped substantially below the threshold pressure, is difficult to explain, especially if pathway opening is analogous to hydrofracturing (see Section 3.3 below), with the gas pressure maintaining the crack opening. In this concept, lowering of the gas pressure below the threshold pathway opening pressure would allow the pathways to close, stopping flow. It appears that the creation of gas pathways causes some change to the clay that is not reversed by lowering the pressure. One possibility is that this is because of some

27

desaturation of the clay, either by "squeezing" water from the clay, or displacement of water from pre-existing channels or voids. This would be consistent with the fact that (point (e) above) rehydration of the clay does cause pathway sealing.

When the pumping rate into the upstream vessel in the experiments by Horseman and Harrington (1997, unpublished) is set to zero (termed "shut in" by these authors), the gas pressure in the upstream vessel declines as gas escapes from the vessel through the sample. However, as in the case of changes in flow rate discussed above, the pressure decline occurs more slowly than would be expected if the permeability were constant, and instead of dropping to the same value as the downstream backpressure, seems to tend towards a value much higher than this, although not generally actually reaching a constant value within the duration of the experiments. The pressure to which the upstream pressure "converges" has been termed the apparent shut in pressure by Horseman and Harrington ( 1997), who have taken it to correspond to a capillary pressure applied at the downstream boundary, and used it to determine the effective downstream boundary condition for the gas pressure. However, the experimental data does not distinguish between some sort of threshold pressure for pathway closure applied throughout the sample and a capillary pressure applied at the boundary. Note that the gas migration models incorporating pathway dilation that have been tried result in a predicted gas pressure profile through the sample in which most of the pressure drop occurs very close to the downstream end of the sample. This is because the gas pressure assumed at the downstream boundary would effectively cause the pathways to close on the boundary.

The physical nature of the gas flow across the downstream boundary is also uncertain. Large fluctuations in the downstream post-breakthrough fluid flow have been interpreted as indicating that the flow is intermittent in nature, with pathways continually opening and closing.

3.3 Mechanism of gas pathway formation

As discussed in The GAMBIT Club Phase 1 report, several mechanisms of gas transport through compacted bentonite have been proposed: gas transport though preexisting water- or gel-filled capillary-like channels by displacement of the water or gels, creation of gas pathways by micro-fissuring of the clay, and creation of gas pathways by macroscopic rupturing of the clay.

The consensus view taken in the GAMBIT Club has been that the most likely mechanism of gas migration in initially water-saturated compacted bentonite is through the creation of gas pathways by micro-fissuring. The facts adduced in support of this view include:

28

a) Estimation (Horseman and Harrington, 1997) of the thicknesses of water films in highly compacted bentonite suggest that these are to thin to admit gas flow, and any macropores containing water would be disconnected.

b) The fact that there is a threshold gas pressure which has to be exceeded before gas can enter the clay and which seems to be related to the swelling pressure of the clay, suggests that the gas pressure has to force apart clay particles to invade the clay. If the clay behaved as a conventional rigid porous medium, gas entry would be controlled by a constant capillary entry pressure. However, the swelling nature of the clay gives rise to the possibility that the capillary entry pressure, itself, may vary with the stress state of the clay. In fact it is tentatively suggested by Horseman and Harrington ( 1997) that the apparent capillary pressure for a swelling clay is greater than the difference between externally measured gas and water pressures by precisely the swelling pressure, so that perhaps this argument does not discriminate between gas migration mechanisms as much as it at first appears.

c) Experiments on clay pastes (Horseman and Harrington, 1997; Donohew et al., 2000) have shown that gas invasion of bentonite is associated with dilation of the clay.

d) The fluctuating nature of flow from the downstream end of the sample suggests that gas flow occurs through a small number of discrete pathways, as the involvement of a large number of pathways would smooth the flow fluctuations (unless the flow in separate pathways was coupled). If there were only a few pathways, these are likely to have to be too large to exhibit a capillary pressure that matches the entry pressure, if they are also to provide the required flow capacity.

e) Initially saturated samples of bentonite that have been subject to gas invasion show patterns of fissuring when removed from the sample holder and dried. However, these are not visible before drying, so it is not clear whether the drying process reveals microscopically wide fissures formed by the gas (and which might have closed when the gas pressure was removed), or whether the fissures are caused by the drying process.

Evidence for the formation of macroscopic fractures has been seen in a radial flow geometry experiment conducted by Horseman and Harrington (unpublished) (experiment Mx80-8 described in Section 2.3.1). Large entry pressures were seen, and a sharp change in the upstream pressure at a high value was interpreted as indicating a failure of the specimen. Changes in the stress and porewater pressure were seen at the same time, and, on removal of the sample from its containment it was found to be in two pieces, apparently split by a gas fracture. Similar fracturing was seen in experiments on

29

gas injection from small sources in Boom Clay under stress conditions simulated using a centrifuge (Rodwell, 2000).

Note that it appears that gas invasion of saturated bentonite involves little displacement of water, but the data available about this does not provide any discrimination between alternative gas transport mechanisms.

Some authors (e.g. Gray et al., 1996; Graham et al., 1998; Hume, 1999) contend that gas migration by displacement of water or gels from pre-existing capillaries is the most likely mechanism of gas invasion of compacted bentonite. The main argument for this seems to be that the stress conditions envisaged are not adequate to cause deformation or rupturing of the clay fabric. As discussed in Section 2, a proportionality was also found in some results between breakthrough time and the inverse of the applied gas pressure when a constant gas pressure was applied for an extended period of time. This was also taken to indicate gas displacement of water as the mechanism of gas invasion of saturated bentonite, since this is the behaviour expected for such a process.

3.4 Other observations and effects

In a few experiments, attempts have been made to follow some system variables other than the fluid pressures at the boundaries across which flow occurs (Horseman and Harrington, 1997, unpublished). Some monitoring of porewater pressures and boundary stresses has been undertaken. Various responses are seen in these variables to the application of a gas pressure. In particular, the measurable porewater pressure and boundary stresses (in constrained directions) appear to rise with the applied gas pressure or when gas enters the sample. However, there are a number of anomalies in these measurements which raise questions about their interpretation. For example, in the recent experiments undertaken by Horseman and Harington (unpublished) and summarised in Section 2.3, the following features that are difficult to explain were noticed:

a) Some of the measured stresses were similar in value to the measured porewater pressures, whereas these would be expected to differ by the swelling pressure of the clay (cf Figures 2-1 and 2-2). In fact the stress measurements did not appear to reflect the swelling pressure at all in either the rehydration or gas injection stages of the BGS experiment Mx80-8 summarised in Section 2.3*.

b) The sudden rise in axial stress in experiment Mx80-8 seen in Figure 2-2 is attributed to the transfer of the gas pressure to the axial stress sensor following the creation of a radial fracture across the sample. If this is the case, it is not clear why this axial stress should rapidly decay while the gas pressure continues to rise, and does not continue to match the gas pressure.*

* Subsequent experimental investigation has confirmed that the stress sensors were not properly measuring the stresses in the clay during this experiment.

30

c) In the linear flow, ~ geometry, experiment, the mid-plane porewater pressure showed an initial rise in response to an applied gas pressure, but before apparent gas entry into the clay, probably as a result of water driven from the upstream end cap. However, this porewater pressure did not decay in the way expected to the back pressure value, stabilising at about 2.5 MPa above the back pressure value (cf Figure 2-3).

Given the association of breakthrough pressure with swelling pressure in most experiments, it is to be expected that the applied stress will affect the breakthrough pressure since an increase in confining stress will, if equilibrium is established, increase the swelling pressure. This is indeed found in experiments on isotropically confined bentonite (Horseman and Harrington, 1997). It has also been found that increasing the confining stress after a gas injection experiment, raises the breakthrough pressure required to initiate further gas flow (although not by as much as expected for a virgin specimen).

Given also that there is some evidence of clay dilation on gas entry, it might be expected that a difference in behaviour would be found in experiments carried out under constant stress conditions compared with those carried out under constant volume conditions ( cf, however, Section 7 .1.2). Unfortunately, there are no experiments reported that have been carried out under conditions that are the same except for the confinement conditions and that would allow this expectation to be tested. It may, however, be relevant that the highest entry pressures reported have been carried out under constant volume conditions.

31

4 FURTHER DEVELOPMENT OF THE PHASE 1 MODEL

Work that has been carried out to evaluate, extend and improve the computational model developed in Phase 1 of the GAMBIT Club programme is described in this section.

Recall, as background to the work on the Phase 1 model, that the model of gas migration in compacted bentonite developed in Phase 1 of the GAMBIT Club program (Nash et al., 1998) consisted of two stages: a pathway propagation stage in which elliptical cracks were driven through the clay by the gas pressure, and a continuous flow stage which occurred after gas breakthrough and involved flow of a connected gas phase through circular capillaries (it being assumed that some adjustment of the pathway shape from elliptical to circular took place). The apertures of the cracks before breakthrough and the radii of capillaries after breakthrough were functions of the gas pressure (i.e. the pathways dilated as the gas pressure increased).

The new work carried out on the Phase 1 model includes:

a) Further evaluation of the ability of the model to reproduce available experimental data. This work is described in Section 4.1, and involved investigation of different functional forms to describe the dilation of the gas pathways.

b) Extension of the model to allow for the existence of pathways with a range of cross-sectional dimensions, rather than only the single size allowed in the preliminary Phase 1 model. This work is described in Section 4.2.

c) Analysis of and improvement to the numerical performance of the Phase 1 model. This work included a number of changes to the equations describing the relationship between the pressures and dilations along the cracks; the incorporation of a more efficient linear equation solver into the computer program; implementation of additional controls to assist both the convergence of the velocity of propagating cracks and the convergence of the crack pressures and dilations, and analysis of the numerical performance of the computational model with respect to temporal and spatial discretisation. This work is summarised in Section 4.3, with additional technical details provided in Appendix A 1.

4.1 Variation in the Functional Form for the Pathway Dilation Model

Testing has shown that with the previously developed 'Phase 1' pathway dilation models used in the GAMBIT Club gas migration model (GMClay) (with gas pathway propagation modelled by crack formation, as described by Nash et al. (1998), or by displacement of water from capillaries, as described in Section 5) it is not possible to

32

accurately reproduce the whole of the upstream pressure transient seen in experiment Mx80-4A of Horseman and Harrington (1997) (for an example, see Figure 4-1). In particular, no tests performed with the previous pathway dilation models have been able to match the section of the pressure transient immediately after the first peak, while simultaneously maintaining a reasonable correspondence to the remainder of the pressure profile. No changes in the parameters for the pre-breakthrough pathway models have sufficient influence on the pressure transient after the first peak to be able to alter it significantly. Also, no reasonable changes in the parameters for the continuous pathway dilation model are able to change the shape of this part of the transient sufficiently.

In an attempt to model more closely the upstream pressure over the whole course of the experiment, alternative functional forms for the relationship between the pressure and pathway dilation for a continuous pathway have been considered, as described below. In each case the pathway geometry used was circular, unaltered from the Phase 1 continuous pathway model geometry.

The 'Phase 1' pressure-dilation relationship is an exponential dependence of the dilation on the pressure.

16.5

16

15.5

15

14.5

14

13.5

13

12.5

12

O.OE+OO 1.0E+06

- - - - - - simulation 1

- - - -simulation 2 1---experiment

~ ...... \

I ""\

' "'\

' ""'\

\ \

' I ' ...... I "'

-..... """'-....-

2.0E+06 3.0E+06

Time (s)

4.0E+06 5.0E+06 6.0E+06

Figure 4-1. Example Simulations from Nash et al. ( 1998) of Experiment Mx80-4A of Horseman and Harrington ( 1997)- Upstream Pressure versus Time.

33

l+v( ) -p-a roo =roe E

where r~

ro is the steady-state pathway radius (m), is the pathway radius at p = a (m),

v is Poisson' s ratio, E is Young's modulus (Pa), p is the pressure (Pa), a is the confining pressure (Pa).

4-1

Testing of the model had suggested that the main requirement of any alternative pressure-dilation relationship is that the rate of change of dilation with pressure is less rapid for large pressures (above the confining pressure), while remaining broadly similar for smaller pressures, compared to the relationship of Equation 4-1. A functional form that satisfies this requirement is one formed from a negative exponential

l+v( ) l+v( -- Po-a - p-a) e E -e E

4-2

where p0 is the pressure at which the steady-state dilation is zero (Pa).

The constants in the exponents ((1 +V)! E) have been chosen to follow the form of Equation 4-1, although they can be grouped together and regarded as single fitting parameters.

Testing using this continuous pathway model showed that it was possible to approximately match the first negative transient for the upstream pressure seen in the experiment without significantly affecting the fit at later times. The values used to achieve the 'best fit' (with comparisons to the current model where appropriate) are given in Table 4-1. Note that the half-width for the propagation stage has also been adjusted to ensure the peak pressure is well matched. A capillary pressure of 11MPa was also added to the downstream pressure boundary condition to ensure that the gas pressure remained above the closing pressure, p0• The pressure transient for this test is compared with the experiment in Figure 4-2.

To attempt to improve the fit further, a second alternative pressure-dilation relationship has been implemented. The functional form in this case is constructed from a hyperbolic function

1+v( ) 1+tanh -- p- Pm E

4-3

34

Table 4-1. Values used for the 'Best Fits' to Experiment Mx80-4A with Exponential, Negative Exponential and Hyperbolic Pressure-dilation Relationships in the Continuous Pathway Model.

Parameter Value for Value for negative Value for exponential exponential hyperbolic relationship relationship relationship

Time dependence 4.00 10-ss-1 2.00 10-5s-1 2.00 10-5s-1

constant, A

Poisson' s ratio, v 0.45 0.45 0.45

Young's modulus, E 3.30 106Pa 4.00 106Pa 1.50 106Pa

Equilibrium radius, r0 2.18 10-6m 1.45 10-6m 1.30 10-6m

Closing pressure, Po - 1.20 107Pa -

Pressure, Pm - - 1.35 107Pa

Crack half-width, c 2.00 10-6m 5.00 10-6m 3.00 10-6m

16.5 -,-------------------------,

"(?" ~

16.0

15.5

-~-w experiment

--simulation

~ 15.0 0 1-< g

~ ~ 0 !:1

~

14.5

14.0

13.5

13.0

12.5

12.0 -+-----+---+----+---+----+---~

O.OE+OO 1.0E+06 2.0E+06 3.0E+06 4.0E+06 5.0E+06 6.0E+06

Time (s)

Figure 4-2. Upstream Pressure Transient for Experiment Mx80-4A and the 'Best Fit' using a Negative Exponential Pressure-dilation Relationship in the Continuous Pathway Model.

35

where Pm is the pressure at which the change of dilation with pressure is a maximum (Pa); that is the gradient of r~ with respect to pressure has a maximum.

In Equation 4-3, the freedom to choose the pressure at which the change in dilation is most rapid has allowed an improved 'best fit' to the Mx80-4A experimental upstream pressure transient (Figure 4-3). The parameters used for this fit are also given in Table 4-1. Adjustment of the half-width for the propagating crack has again been used to optimise the fit to the peak pressure.

Plotting the graphs for Equations 4-1, 4-2 and 4-3 using the 'best fit' parameter values (Figure 4-4) suggests the form of relationship that would be required for a more exact fit to the experimental data.

The form required is indicative of a change in dilation behaviour above the confining pressure compared to below it. This seems plausible, although at present unexplained. Another feature to be noted is that it remains difficult to fit the shape of the long decline in upstream pressure after pumping has been stopped. Consideration has been given to the possibility that the form of this decline may result from a contribution from diffusion to gas transport after shut in, but this mechanism provided an insufficient flux to be significant.

(? ~

6 !!) 1-1

~ !!) 1-1 ~

~ b <ll

~

16.5

16.0

15.5

15.0

14.5

14.0

13.5

13.0

12.5

''"'""""'""experiment

--simulation

12.0 +----+---+----+---+----+----+-'

O.OE+OO 1.0E+06 2.0E+06 3.0E+06 4.0E+06 5.0E+06 6.0E+06

Time (s)

Figure 4-3. Upstream Pressure Transient for Experiment Mx80-4A and the 'Best Fit' using Hyperbolic Pressure-dilation Relationship in the Continuous Pathway Model.

36

2.5E-06 .---------------------,-r,--, /

.. //

2.0E-06 //

/ ]: 1.5E-06

Exponential -----------. /---------

---~1/ / s::

.9 ~ ~

Q LOE-06

S.OE-07 4111 Hyperbolic

O.OE+OO -*---1----+--+---+----+--+---+----+-----l

12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5

Pressure (MPa)

Figure 4-4. Comparison of Pressure-dilation Relationships from Equations 4-1, 4-2 and 4-3 using 'Best Fit' Parameters (Table 4-1) to Upstream Pressure Transient of Experiment Mx80-4A.

At present the relationships described by Equations 4-2 and 4-3 represent mathematical forms which are designed simply to fit the results from the example experiment. They have not been developed from any physical insights, as in the development of the Phase 1 model. Clearly it is desirable to try and maintain a relationship between mathematical models and physical understanding.

Since there are a number of aspects of the gas migration model model which have some degree of uncertainty associated with them, it is not reasonable at this stage to invest too much effort to fit a single experiment by choice of the functional form of the pressuredilation relationship alone, or to ascribe to much significance to the detail of the best fit obtained. This is especially so as the number of experimental test results available is limited and there is also variability amongst those that do exist. Adjustment of the Phase 1 model to improve fits to experiments has not therefore been pursued further; effort instead has been directed to the evaluation of alternative conceptual models of gas migration in bentonite (e.g. Sections 5 and 1).

37

4.2 Multiple-Pathway Facility

4.2.1 Development

At the end of Phase 1 of the GAMBIT Club project, the gas migration model model had been developed so that a number of cracks could be specified, provided they were all identical. As part of the development of the model during Phase 2 this multiplepathway model has been extended so that a number of pathways with varying descriptions can exist simultaneously.

The parameters that define each crack within the model are:

a) the initial length of the crack, b) the half width of the crack for the pathway propagation stage, c) the equilibrium radius of the crack for the continuous pathway stage.

In the extended model all three of these parameters can be varied between cracks, so any desired combination of cracks can be defined for a single simulation run. In addition there may be a number of copies of each crack (the number of copies being the same for each different crack) as was possible for the single crack in the Phase 1 model. The parameters describing the "elasticity" of cracks or continuous pathways and the time dependence of the dilation response are assumed to be material properties common to the whole specimen.

Allowing cracks with differing properties means that the propagation of each crack can occur at a different time, so that at any instant there may be some cracks that have not started to propagate, some that are propagating, and some that have reached breakthrough and exist as continuous pathways. Due to the coupling between the cracks, the advance of the model solution has to be calculated for all the cracks simultaneously.

To allow differing cracks to be modelled simultaneously, the equations for gas flow used for the Phase 1 model have been modified to include the coupling that exists between the pathways through their common upstream pressure. The conservation equation, as before (Nash et al., 1998), is

where p is the gas density (kg m-3),

u is the gas flux (m s-1),

us is the velocity of the boundary S of the volume element V (m s-1),

q is the mass source or sink of gas (kg m-3s-1),

t is time (s).

4-4

38

The discretised form of Equation 4-4 must now be applied across all grid blocks at the upstream end as one. Therefore in the extended model, for the upstream end, V is a volume element linked to all the cracks, and S the surface of V. The discretised form of Equation 4-4 can be applied one crack at a time to grid blocks not at the upstream end as before, since the only direct coupling between cracks is at the upstream end.

The non-linear equations obtained are solved, also as before, using a multi-dimensional Newton-Raphson iteration scheme. The resulting linear equations are solved in two parts to enable the use of the efficient banded matrix solver discussed in Section 4.3.1.

4.2.2 Example Tests

To demonstrate the effect of the use in the model of a number of pathways with a range of properties, several simulations have been performed, with different pathway properties being allowed to take a range of values in each simulation. The simulations performed and their results are discussed below.

Each of the multiple pathway simulations is a variation from a base case (a fit to the upstream pressure transient of experiment Mx80-4A of Horseman and Harrington ( 1997)) that has 100 identical cracks. In each of the variant cases the 100 cracks are divided into 5 sets of 20 cracks. One of the pathway properties is given a different value for each of the 5 sets. The values used in each case are given in Table 4-2.

For case M3 the upstream pressure is smaller than that of the base case during almost all of the continuous pathway stage, as shown in Figure 4-5. This is because the permeability of the core is larger for case M3 than for the base case at the same pressure. The larger permeability allows a greater gas flow, reducing the pressure, and therefore the permeability, more quickly, so reducing the flow to match that of the base case. Case M3 differs from the base case in the values selected for the equilibrium radii, and as this property controls the crack behaviour during the continuous pathway stage only; it should not affect behaviour before breakthrough, as indeed is observed to be the case.

For cases M 1 and M2 the inflow rate and upstream pressure transients show no significant differences from the base case. Both the initial crack length and the crack half-width affect the propagation of the crack. The initial length determines the pressure at which propagation begins, and the half-width determines the transit time for a given start time. Over the range of values of these properties used, the start time and duration of the propagation stage vary only slightly, so that there is no significant effect on the continuous pathway stage. Therefore, with the relatively very short duration of the propagation stage, no significant differences in the pressure transients are seen.

By choosing more widely varying values for, for example, the initial crack length, it is possible to produce a change in the continuous pathway stage. Case M4 (Figure 4-5) is

39

Table 4-2. Crack Properties for each of the Multiple Pathway Test Cases.

Test Crack Set Initial Length Half-width Equilibrium Number (m) (m) Radius (m)

Base case 1.00 10-5 2.00 10-6 2.18 10-6

M1 1 1.20 10-5 2.00 10-6 2.18 10-6

2 1.10 10-5 2.00 10-6 2.18 10-6

3 1.00 10-5 2.00 10-6 2.18 10-6

4 9.00 10-6 2.00 10-6 2.18 10-6

5 8.00 10-6 2.00 10-6 2.18 10-6

M2 1 1.00 10-5 4.00 10-6 2.18 10-6

2 1.00 10-5 3.00 10-6 2.18 10-6

3 1.00 10-5 2.00 10-6 2.18 10-6

4 1.00 10-5 1.00 10-6 2.18 10-6

5 1.00 10-5 5.00 10-7 2.18 10-6

M3 1 1.00 10-5 2.00 10-6 2.68 10-6

2 1.00 10-5 2.00 10-6 2.48 10-6

3 1.00 10-5 2.00 10-6 2.18 10-6

4 1.00 10-5 2.00 10-6 1.88 10-6

5 1.00 10-5 2.00 10-6 1.68 10-6

M4 1 1.00 10-5 2.00 10-6 2.18 10-6

2 5.00 10-6 2.00 10-6 2.18 10-6

3 2.00 10-6 2.00 10-6 2.18 10-6

4 1.00 10-6 2.00 10-6 2.18 10-6

5 5.00 10-7 2.00 10-6 2.18 10-6

an example where some of the initial crack lengths are so small that the maximum pressure reached is not sufficient to initiate propagation. Therefore only a fraction ( 40 out of 1 00) of the cracks create continuous pathways, giving a smaller effective gas permeability and a smaller gas flow compared with the base case at the same pressure. This reduces the rate of decrease of the pressure, so that the cracks that do provide gas pathways through the specimen remain more open, and a smaller number of pathways are able to support the post breakthrough flows, although at a higher pressure than observed experimentally.

4.3 Improvements to and Analysis of the Numerical Performance of the Phase 1 Model

A number of improvements have been made to the numerical algorithms used in the Phase 1 gas migration model, and an analysis of the numerical performance of the model with respect to temporal and spatial discretisation has been carried out. This work is summarised in Sections 4.3.1-4, with more complete technical details provided in Appendix A 1.

'C? 0..

40

16.5 ,...-----------------------,

16.0

15.5

"-~---~~Base case

--M3 --M4

6 15.0 ~ ::l ~ ~

0..

~ 0 ~ 4-1

~

14.5

14.0

13.5

13.0

12.5

12.0 -+------+----+-----+----+-----+----+-'

O.OE+OO 1.0E+06 2.0E+06 3.0E+06 4.0E+06 5.0E+06 6.0E+06

Time (s)

Figure 4-5. Upstream Pressure Transients for Multiple Pathway Test M3 and the Base Case.

4.3.1 Linear Equation Solver

In the Phase 1 model a simple direct linear equation solver, which took no account of the banded structure of the matrix arising from the iterative solution of the non-linear equations, was used (see Appendix Al.l). This has been replaced by a solver designed to take advantage of the banded structure of the matrix that is obtained. The resulting efficiency improvement was necessary to facilitate solution of the increased number of equations involved when the multiple pathway option described in Section 4.2 was invoked.

4.3.2 Adjustments to the Crack Propagation Model

Several small modifications were made to the numerical algorithm for the crack propagation stage of the of the Phase 1 model:

a) The assumption of a constant dilation within each grid block was replaced by one of a dilation that varies linearly between nodes.

b) The requirement that the crack aperture is zero at the crack tip was relaxed, so that the tip aperture is now determined by the gas pressure at the crack tip. This results in a blunt rather than a sharp tip to the crack.

41

c) The equations for gas flow during the crack propagation stage were modified to allow for the possibility that there could be a number of grid block boundaries behind the crack tip that move during a time step.

These changes improved the robustness of the model and removed some irregular changes with time in some of the fracture dilations at points near to the propagating crack tip.

The changes are described in more detail in Appendix A1.2.

4.3.3 Numerical Performance of the Phase 1 Model

The numerical performance of the Phase 1 model has been evaluated with respect to the level of time step or grid refinement required by running simulations with input data varying only in the value of the parameter controlling either the maximum time step size, the grid block size or the crack tip movement.

The choice of time step is only relevant to the continuous flow stage of the simulations. During the crack propagation stage, the required time step size is set automatically to control the convergence of the propagation algorithm, and is generally required to be much smaller than during the subsequent continuous flow stage.

This investigation confirmed that the maximum time step size and the grid refinement chosen for the simulations of the experiments of Horseman and Harrington ( 1997) within the Phase 1 programme (Nash et al., 1998) were adequate for the purpose of the calculations reported. They did exhibit some discretisation errors that could have been removed by slightly reducing the grid spacing (by around a factor of 4), and including a maximum time step size ( -103 s). However, the dicretisation errors were small compared with the total error in the fits to the experimental data.

Varying the ratio of maximum crack tip movement per time step to the grid block size highlighted an additional discretisation issue. The crack tip velocity during the propagation stage does not decrease monotonically, as expected. This is due to the spatial discretisation scheme, which involves the addition of grid blocks as the crack advances, rather than the level of refinement. The error was not significant when considering a simulation as a whole.

The effect was due to the wide variations in grid block size that resulted from the addition of new grid blocks, and caused the error in the tip velocity. An alternative discretisation was tried that used three expandable grid blocks at the crack tip, rather than two. This reduced the variation in grid block size, but was insufficient to prevent the effect. It is believed that the error could be removed by a discretisation scheme in which the propagation of the crack was represented by a uniform scaling of a constant number of grid blocks, which would then all be of the same size. Adding new grid

42

blocks would not then be required. However, it was not deemed worthwhile pursuing this rather technical issue further as it was considered more important within the Phase 2 programme to investigate alternative conceptual models of gas migration, in particular, that discussed in Section 1.

43

5 GAS MIGRATION VIA WATER DISPLACEMENT FROM PREEXISTING CHANNELS

5.1 Model Development

As already mentioned, the Phase 1 gas migration model consists of two phases: a crack propagation model, which operates prior to gas breakthrough in a crack, and a continuous gas flow model, which applies to a gas channel after breakthrough. The second part of the model, which allows for pathway dilation as a function of gas pressure, can be applied equally to a pathway created by water displacement from a preexisting channel as to a pathway created by crack propagation. (Note, in addition, that, from the point of view of the model structure, gas flow behind a propagating gas front is described using the same model, based on Darcy' s Law, as gas flow in a completed pathway (but with different permeability)).

To model gas migration through pre-existing channels requires a new model to replace the crack propagation model (which also allowed for crack dilation during propagation in the Phase 1 model). The new model needs to define the pathway aperture (or apertures) and provide a condition to control the velocity of the gas front, in place of the condition, derived from Griffith crack propagation theory, used to determine the appropriate crack tip velocity for a propagating crack. The developments required to model gas migration in pre-existing channels can be used in combination with the revised version of the flow model that allows multiple pathway sizes and was described in Section 4.2; the multiple pathway facility can therefore be used in this case as well.

5.1.1 Pathway creation Model

To model water displacement from a pre-existing pathway, it is assumed that the pathway is a constant aperture capillary, with a circular cross section. The cross sectional area and permeability for the capillary model are therefore given by the same equations as for the continuous pathway dilation model (Nash et al., 1998 (Section 3.2.4)). However, the relationship between pressure and dilation used for a continuous pathway is not used. The relationship is removed so that the aperture of the capillary remains fixed whatever the pressure in the capillary. This simplification is invoked in the present implementation since it would seem to be inconsistent to allow pathway dilation in the gas-filled part of the channel, but not in the water-filled part, and to allow the latter would require more substantial changes to the program. However, these are not obstacles in principle, and pathway dilation could be allowed during water displacement from pre-existing channels if there was evidence that this was a required feature.

GMClay models gas migration in a capillary using the same method as for a propagating crack, considering only the section of the capillary filled with gas. To do this a boundary condition at the gas-water interface is required. The water in the

44

capillary is assumed incompressible so that water at all distances along the capillary moves with the same velocity

k Pgi- Pc- Pd V = w llw L-xl

where k

J1w Pgi

Pc

Pd L

is the permeability (m2),

is the water viscosity (Pas), is the gas pressure at the gas-water interface (Pa), is the capillary pressure (Pa), is the downstream water pressure (Pa), is the length of the capillary (m), is the distance of the interface along the capillary (m).

The capillary pressure is given by

2y Pc=

r

where r r

is the surface tension between the gas and water (N m-1),

is the capillary radius (m).

5-1

5-2

The water velocity can then be used as the required boundary condition for the gas at the interface, since the gas and water at the interface must move with the same velocity. The correct interface advance must therefore be the advance resulting in the calculation of equal gas and water velocities at the interface. The interface gas velocity is determined iteratively in the same way as the crack tip velocity for a propagating crack, since the water velocity calculated depends on the interface movement. (Note that this approach was adopted for convenience in testing this pathway propagation model, as it minimised the changes required to the existing model. It is not necessarily the most efficient implementation of the model.)

5.2 Example Tests

Several tests were carried out using the capillary water-displacement model to model experiment Mx80-4A of Horseman and Harrington ( 1997), attempting to match the upstream pressure transients in the same way as for the crack propagation model in the base case test described in Sections 4.2.2 and 4.3.1. It was found that, by choosing a suitable value for the capillary radius, a simulation with a single set of identical capillaries could produce the same quality of fit to the pressure transient (Figure 5-1) as that of the base-case crack propagation test. The pressure drop after the initial peak could not be reproduced any more accurately using the capillary model for the prebreakthrough stage than using the crack propagation model. In both cases, the behaviour before breakthrough did not have a great deal of influence on the post-

45

16.5 ,.-----------------------,

'(? ~

16.0

15.5

~ 15.0 ~ Bl 14.5 "" £ ~ 14.0

~ ~ 13.5

~ 13.0

12.5

~~~-experiment

--simulation

12.0 -f----+----+------1-----+----+-----1-'

O.OE+OO 1.0E+06 2.0E+06 3.0E+06 4.0E+06 5.0E+06 6.0E+06

Time (s)

Figure 5-1. Fit of Upstream Pressure Transient from Test using Model Based on Preexisting Pathways with Experiment Mx80-4A (Horseman and Harrington, 1997).

breakthrough behaviour. The parameters used in the simulation with the waterdisplacement model are shown in Table 5-1 (fit 1).

To match the results of the crack propagation base case, the capillary radius required gave a cross section over 103 times smaller than the average cross section of the propagating crack. The gas in the capillary also migrated much more slowly, taking 1.5 105s compared with 35s for the crack propagation. However, the breakthrough times were only slightly different, the gas in the capillary reaching the outlet 4 103s earlier

Table 5-l. Model Parameter Values for Fits to Experiment Mx80-4A using GMClay with a Capillary Pre-breakthrough Pathway Model.

Model parameter Value for fit 1 Value for fit 2

Young's modulus 3.3 106 Pa 3.3 106 Pa

Capillary radius 1.047 10-8 m 1.047 10-8 m

Number of cracks 100 1.88 1011

Equilibrium radius 2.18 10-6 m 1.047 10-8 m

Time-dependence constant, 2 4.0 10-5 s-1 4.0 10-5 s-1

46

than in the propagating crack.

Although the results shown in Figure 5-1 suggest that the pre-existing pathway model is as capable of fitting this particular set of experimental data as the crack propagation model, there are significant concealed differences.

In order to obtain the correct breakthrough pressure, a small capillary radius has to be assumed for the pre-existing pathway before breakthrough. In this model, the threshold pressure is largely determined by the capillary pressure, so in the calculation of Figure 5-1 a capillary radius of 1.047 10-8 m was used for the propagation phase, to reproduce the value observed of the breakthrough pressure. The "equilibrium" radius (Equation 4-1) used post breakthrough was the same as used in the previous best-fit to the experimental data using the crack propagation model, that is, 2.18 10-6 m, which is inconsistent with the fixed radius before breakthrough (but note that the actual radius at breakthrough can be very different from this, and will readjust in accordance with the time dependence behaviour assumed).

If, as seems more reasonable, the post-breakthrough equilibrium radius is set to the same value as the fixed radius pre-breakthrough, very many more capillaries have to be introduced to match the observed post breakthrough gas permeability. In the calculations of Figure 5-1, 100 pathways were assumed present (before and after breakthrough). If the equilibrium radius after breakthrough is set to the same value as for the radius of the pre-existing pathway before breakthrough, 1.047 10-8 m, a total of about 1011 pathways are required (the pre-breakthrough behaviour is not sensitive to the number of pathways). The results obtained are shown in Figure 5-2 and the parameters used in the simulation are listed in Table 5-1 (fit 2). Note that in this particular case, there is additionally a rapid fall in pressure after breakthrough, as a consequence of the change in the capillary characteristics. This can be avoided by a compromise choice of a radius of about 5 10-8 m for the post-breakthrough equilibrium radius (with a prebreakthrough radius of 1.047 10-8 m) but the number of capillaries required is still large at about 1010

• Further tests, using capillaries with a range of radii, have so far not been able to improve on the fit to experiment Mx80-4A.

It seems therefore that the model of gas migration by displacement of water from preexisting pathways requires a quite different conceptual picture of the nature of the gas pathways from the crack propagation model. In the former case, a very large number of pathways seems demanded ( -1010 for the 4.9 cm diameter, highly compacted bentonite specimens), whereas in the latter, a relatively small number would suffice ( -100 in the current example), although the crack propagation model does not in principle exclude larger numbers of pathways. The large number of pathways in the water displacement model would result in very small average separations between pathways. Note, however that the large number of gas pathways required by the water-displacement model still only corresponds to a very small porosity.

"@' ~

47

16.5 -.------------------------,

16.0

15.5

'"'"""'"""" experiment

--simulation

6 15.0

13.0

12.5

12.0 +----+----+-----+-----+---+-----+-'

O.OE+OO 1.0E+06 2.0E+06 3.0E+06 4.0E+06 5.0E+06 6.0E+06

Time (s)

Figure S-2. Fit of Upstream Pressure Transient from Test Using Model Based on Preexisting Pathways (with Continuous Pathway Equilibrium Radius Matching Capillary Radius) with Experiment Mx80-4A.

It may be possible to account for the apparent large number of pathways. The number required in the model may not represent the true number of pathways if the radius required in the model does not represent the "average" radius of the pathways. In the water-displacement capillary model, a small value of the radius is required to produce a sufficiently high gas entry pressure. However, only a small portion of the pathway is required to have this small radius to prevent gas from migrating the full length of the capillary at a much lower pressure than observed in the experiment. The majority of the capillary may therefore have a much larger radius. If the constriction in the capillary was to occur over only a very small fraction of the total length of the capillary then the permeability after breakthrough may be only slightly smaller than that for a capillary with the larger radius along its full length. This is much larger than the permeability calculated from the small radius specified in the model for the pre-breakthrough stage, and therefore far fewer capillaries would be required to give the required total permeability.

In other words the large value for the apparent number of capillaries used in the model may be an indirect way of specifying that the capillaries are wider than the specified (minimum) radius over large parts of their length, rather than that there are actually a large number of capillaries present. Since the total permeability for the sample is given by an equation of the form

4 k = llJlcr

8

48

5-3

then the approximate effect on the number of pathways of an effective capillary radius larger than the specified value can easily be calculated. For fit 2 to experiment Mx80-4A this implies that to reduce the number of pathways back to the 100 used in fit 1, the effective capillary radius would have to be approximately 200 times the radius of the constriction (i.e. the radius specified for the post-breakthrough stage in fit 1).

49

6 AN ALTERNATIVE MODELLING APPROACH

Although the fracture propagation model, developed in the GAMBIT Club Phase 1 programme and further analysed in work reported here, provided an encouraging capability to reproduce the features seen in experimental work on gas migration in compacted bentonite, the model does not address all issues arising in connection with gas migration in bentonite; some of the missing features are relevant to developing understanding of the process, other are of potential importance in assessing performance as a buffer material in a radioactive waste repository. The preliminary model that had been formulated did not include:

a) The flow of water through the bentonite.

b) A coupling between the gas pressure and the water pressure as reported in some experiments (Horseman and Harrington, 1997, unpublished).

c) Resealing of the gas pathways after gas transport has finished.

d) Changes to the stress state of the clay, as for example produced by any dehydration of the clay that might occur.

It is also the case that a model that may be the most appropriate for developing and demonstrating understanding of a process, may not be the most appropriate for building a computational model to be used as· an assessment tool. For example, a difficulty that may be faced in upscaling a model based on the representation of the explicit propagation of fracture pathways, is how to define the direction of pathways. Standard arguments about fracture propagation would suggest that this would require the considerable additional complexity of representing the evolution of the tensor stress field within the bentonite. While it is possible that in some situations this might be necessary, it is considered here that if a viable simpler approach can be found, then this would be desirable. The results of the studies with the Phase 1 model suggest that if gas pathway propagation occurs via fissuring of the clay, then this step will be rapid once the gas pressure threshold required to initiate fissuring has been exceeded. As a consequence of this, it has already been suggested that it may not be necessary to accurately model the details of the pathway propagation for performance assessment, which is generally concerned with events occurring on much longer time scales.

Similarly, if gas migration is assumed to occur by displacement of water from capillarylike pathways, then upscaling of this to a two- or three-dimensional model may require that the explicit representation of the capillaries be replaced by an effective permeability type of model.

In the light of these considerations, an alternative to the Phase 1 model has been considered as part of the GAMBIT Club Phase 2 programme. The criteria for selection

50

of this model were that it should address the missing features of the Phase 1 model in some way, and it should be of a form that could be readily upscaled and applied in twoor three-dimensions in future developments, as well as in one-dimension. This model is described in this Section. In addition to describing the essential features of the gas migration process (a threshold pressure for gas entry, pathway dilation with gas pressure) it also attempts to represent features (a) to (d) listed above, although changes to the stress field are included in a very simple, isotropic fashion. The model also attempts to provide a physical basis for the fact that, once created, gas pathways remain open even when the gas pressure has fallen substantially below the initial threshold for gas entry. The implementation and application of the alternative model to onedimensional test cases is discussed in Section 6.4 and its use in simulating experiment Mx80-9 of Horseman and Harrington (unpublished) is described in Section 8.

6.1 Description of the Alternative Model

As already noted the alternative model (GMClayW - gas migration with water displacement) is not intended to explicitly model the detail of the individual gas pathways as in the Phase 1 model (GMClay). Instead it has been designed to represent the development and maintenance of gas porosity and the associated gas permeability by the effect of the applied gas pressure on the bentonite sample, as described in the next two subsections. In particular it allows a coupling between gas and water pressures through the swelling behaviour of the clay.

As with the Phase 1 model, the alternative model is formulated at present as a onedimensional model, however, as noted above, an important consideration in its formulation was the ease with which it could be extended to two and three dimensions.

6.1.1 Model assumptions

The following assumptions are made in the model formulation:

a) The volume of clay solids within the sample under study (and at the numerical level, within each computational grid block) is fixed by the initial conditions.

b) The volume of water in the sample may vary as a function of the stress in the clay in accordance with the swelling behaviour of the clay. For the purpose of the current implementation, it is assumed that there is no hysteresis in the swelling behaviour.

c) Water flow through the clay is modelled. The water permeability is currently assumed constant on the grounds that there is only a very small saturation change caused by gas migration. However, this is not an essential feature of the model; a variable water permeability could be included. The driving force for

51

the water flow is expressed in terms of the gradient of the external equilibrium water pressure as the experimentally observable measure of the water potential.

d) Gas-filled porosity can be created as a result of an applied gas pressure by compression of the water in the clay, displacement of water by consolidation, and by expansion of the sample volume. The last depends on the model options chosen; a constant volume condition or a dependence of total volume on internal stress can be selected. Gas can flow through the sample when a flowing gas porosity connects the two flowing faces of the sample.

e) The gas permeability depends on the gas-filled porosity.

f) When gas is present, it is currently assumed that the stress in the clay is equal to the gas pressure, although other relationships between stress and gas pressure could be adopted. The stress is taken to be isotropic.

6.1.2 Initial and Boundary Conditions

It is assumed that typical initial conditions will be of water-saturated clay of a specified dry density, in hydraulic equilibrium throughout with water at a specified external pressure, and exerting a stress determined by the swelling pressure (which depends on the specified dry density) and equilibrium water pressure. Alternative initial conditions are in principle possible, including an initial gas saturation, but have not been explored here.

To simulate the experiments undertaken by Horseman and Harrington ( 1997), provision is made in the model for gas to be injected into the sample by means of a 'syringe' that contains a volume of gas in contact with the inlet end. This volume is decreased at a set rate, increasing the pressure of the gas until the gas is able to enter the core. No water is allowed to flow across the inlet. Gas and water are allowed to flow out at the opposite (outlet) end, which is held at fixed gas and (external) water pressures. Water that has left the system at the outlet end is not allowed to re-enter the system, reflecting the gravity segregation that is expected to occur at the lower, outlet end of the sample.

Alternative boundary conditions can also be applied at the inlet. Constant (or linearly varying with time) gas and water pressures can be applied. Constant injection rates (including zero) of gas and water can also be applied. At any time any combination of one condition for the gas and one for the water may be used.

6.2 Mathematical Formulation

When both gas and water are present the state of the system is defined by specification of the gas and water pressures at all points, where the water pressure is the externally measurable equilibrium water pressure corresponding to the particular position.

52

Equations for the evolution of these pressures can be constructed from the conservation equations for gas and water by assuming the flows obey Darcy' s law. The equations for water and gas are respectively:

where Pw Pg lPwc l/>g Pw pg kw kg J.lw J.lg

is the water pressure (Pa), is the gas pressure (Pa), is the water porosity in the water-clay system considered separately, is the gas porosity, is the water density (kg m-3

),

is the gas density (kg m-), is the water permeability (m2

),

is the gas permeability (m2),

is the water viscosity (Pa s ), is the gas viscosity (Pas).

6-1

6-2

Note that the water-filled porosity, l/Jwn is the volume fraction of water in the water and clay alone, therefore the overall water-filled porosity is l/Jw = (1-l/>g)lPwc· Equations 6-1 and 6-2 are written in terms of the gas and water density, permeability and occupied porosity (fractional volume), so that equations relating these quantities, directly or indirectly, to the phase pressures are also required.

Relating the densities to the pressures is straightforward. The gas density is obtained from the Ideal Gas law

where Mg R T

is the molar mass of the gas (kg mor1),

is the gas constant (J mor 1K-1),

is the temperature (K).

For the water density, a standard compressibility model can be used.

where Pw0 is the reference water pressure (Pa), Pwo is the water density at the reference pressure (kg m-3

),

6-3

6-4

53

{3 is the water compressibility (Pa-').

To construct an equation relating the water-filled porosity to the pressures, the additional assumption that (where gas exists) the gas pressure is equal to the stress, a, on the water-clay system ( p g = a ) is invoked. Then the swelling pressure is

ITs=a-pw=Pg-Pw 6-5

But, from the fit to the experimental data of Borgesson et al. (1996) quoted by Horseman and Harrington ( 1997), the swelling pressure can be represented empirically to a good approximation by ( cf Equation 2-1)

6-6

where a, r are fitting parameters, e is the void ratio, defined as

e = lPwc 6-7 1-l/Jwc

The particular form of the empirical relationship of Equation 6-6 is not essential to the model; it could be replaced by any improved representation that becomes available, or by relationships appropriate to different types of bentonite if required.

Combining Equations 6-5, 6-6 and 6-7

_l _l

ar ar

rf>wc =(a- pJf +a~ = {p8

- Pw}f +af 6-8

Since the sum of the volume fractions of each of the components, solid, water, and clay, must be unity, then the gas porosity can be related to the pressures via the water porosity.

6-9

where l/Js is the solid volume fraction.

Provision has been made in the model to allow the volume of the clay to vary, particularly, for example, to dilate in response to gas invasion, as might occur in a gas migration experiment under constant stress confinement. A constant volume constraint is imposed by specifying a zero volume change. Since the solid is assumed

54

incompressible, the volume of solid will remain constant and variation of the sample volume will be reflected by variation in <l>s in the constraint Equation 6-9. This variation is assumed to be a function of the internal stress only.

The solid volume fraction can be related to the total volume by

where V is the total volume (m3),

V0 is the total volume under reference (e.g. initial) conditions (m3),

</>so is the solid volume fraction at the reference total volume.

6-10

The variation in <l>s is in fact represented by a dependence of total volume on the stress in the sample. In the work carried out so far, the functional form chosen for this dependence has had the general form:

<1 ~ (1-8~0

(1-8~0 <<1 < (1+8~0 <1;;::: (1+8~0

where o;1 is a reference stress (at which the volume is V0) (Pa), 8 is a small increment (i.e. 10-4

),

6-11

a 1, a2 are constants that define the linear rates of change of the relative total volume with the stress above and below the reference stress (Pa-1

),

c~, b1, b2, b3, c2 are determined automatically to ensure continuity between the three sections of the function defined by Equation 6-11.

This functional form provides for two linear segments, one for <1 ~ (1- 8 Xr 0 with gradient a1, and one for <1 ;;::: (1 + 8 XJ 0 with gradient a2. These two segments are joined over the interval (1- 8 Xr 0 < <1 < (1 + 8 Xr 0 containing the reference stress and in which the total volume is defined by a quadratic relationship to the stress. This is intended to connect the two linear relationships whilst allowing the total volume and its derivative with respect to the stress to remain continuous. The provision of different linear relationships for different stress ranges was suggested by the work on capillary dilation models discussed in Section 4.1, and by the thought that dilation on gas entry might occur more readily from an assumed initial equilibrium stress state, than compression might occur if the internal stress were reduced. However, the latter assumption is speculative. Setting a 1 = a2 = 0 gives a constant volume condition.

The gas is only expected to take up a small fraction of the volume of the system ( </>g < 0.01 ), so the fraction of water will be approximately constant. It is therefore reasonable to make the approximation that the water permeability is constant.

55

6-12

This assumption could be replaced by a constitutive relationship relating water permeability to other system variables if required.

Finally, the gas permeability is assumed to be a function of the gas porosity depending on the geometry of the gas flow paths. For example, the following gas permeability is derived from the permeability obtained assuming flow along capillaries.

6-13

where l/>gc is the critical gas porosity below which the gas permeability is zero, ne is the number density of capillaries (m-2

).

Alternatives to Equation 6-13 for prescribing the gas permeability have also been tried; one of these is the exponential form

6-14

where kg0 is a constant (m2),

A. is a constant.

Where no gas is present the system can be defined by the water pressure alone using Equation 6-1 with lPg = 0. However, it is convenient to define the stress to replace the gas pressure. In this case Equation 6-2 is replaced with the first relationship in Equation 6-5, but the void ratio (given by Equation 6-7) defining the swelling pressure is now defined in terms of the solid volume fraction as

1 e=--l

lf>s

6.2.1 Boundary Conditions

6-15

The type of boundary conditions used in the alternative model have been listed in Section 6.1.2. The mathematical implementation of the constant pressure and constant flow rate conditions is straightforward, and requires no further comment.

When gas is injected from a syringe pump at a constant piston displacement rate, as in the BGS experiments (Horseman and Harrington, 1997, unpublished), the boundary conditions on Pw and pg are also relatively straightforward to define.

56

For the water phase, there is no flow across the upstream boundary, so it seems natural to assume that

while for the gas phase

_l_aPg +-1 dVr =~kg Vp I P at V dt V J.l g z=O

g z=O r r g

where Vr A

is the volume of the gas reservoir, a known function of time; is the cross-sectional area of the clay sample.

6-16

6-17

In the experiments carried out at BGS, dV ,ldt is a piecewise constant function of time, determined by a pumping rate that is constant over specified time intervals.

6.3 Parameters Required for The Alternative Model

Table 6-1 summarises the parameters required for the specification of the alternative gas migration (GMClayW) model developed within Phase 2 of the GAMBIT Club project.

Other parameters define the physical geometry of the sample, the initial conditions and the applied boundary conditions. The parameters in Table 6-1 fall into three groups:

a) Established physical properties in principle obtainable from the literature. These include f.lw, J.l

8, Mg, {3, and the Pwo' Pwo pair, although it should be borne in mind

that the properties of water in thin films may differ from those of free water.

b) Parameters that may in principle be determined directly by experimental measurement or are fixed by the experimental conditions. These include T, et>~,

a, y, kw. In practice not all these parameters may be provided by independent measurement, or may be subject to sufficient uncertainty to affect results obtained with the model.

c) The remaining parameters are essentially model fitting parameters which may be adjusted to try and reproduce the experimental results, although the values chosen should be assessed for physical reasonableness.

57

Table 6-1. Main Parameters for the Alternative Gas Migration Model.

Symbol Explanation Comment

Water properties

kw Water permeability (m2) Treated as constant

f3 Water compressibility (Pa- 1) Also requires reference water

density, pwO, at reference water

pressure Pwo·

Jlw Water viscosity (Pa s) Treated as constant

Gas properties

Mg Molar mass of the gas. Constant used in determining gas density. (Temperature, T, and Gas Constant, R, also required).

Jlg Gas viscosity (Pa s) Treated as constant

Gas permeability models

l/Jgc Critical gas occupied porosity Gas occupied porosity below which no gas flows.

- capillary model

ne Number density of capillaries (m-2) See Equation 6-13

-"exponential" model

kg0, A- Gas permeability model parameters See Equation 6-14

Clay properties

l/Jso Initial volume fraction occupied by clay solids.

a,y Parameters for the clay swelling Equation 6-6. See Equation 2-1 for pressure model. values used.

al' a2, O"o Parameters for the clay volume a! and a2 are the gradients of dilation model volume change with internal stress

at values of stress greater and less than 0"0 (Equation 6-11 ).

58

6.4 Evaluation of the Alternative Model {GMCiayW)

GMClayW has been run with a number of test data sets to evaluate various aspects of the model.

a) Tests in which the value of just one parameter was varied have been performed to show the effect that each of these parameter values has on the results of a simulation.

b) The time step size and grid spacing have been varied to determine what resolutions in space and time are required to produce results with insignificant discretisation inaccuracies.

c) Using the results of (a) and (b), a number of attempts have been made to model an experiment (Mx80-4A of Horseman and Harrington (1997)), maximising the fit to different parts of the experiment in each case.

d) Further fits to the experiment have also been attempted by using alternative expressions for the gas permeability in terms of the gas porosity.

Each of these sets of tests is described separately in the following sections.

6.4.1 Parameter Variations

Of all the input parameters required for a GMClayW run there are a number for which a definite value is not known, or for which the value may vary from one situation to another. These parameters are the number density constant used to calculate the gas permeability, the critical gas porosity, the solid porosity, the constant water permeability and the water compressibility (water compressibility is not strictly a parameter that would be expected to change significantly in the different circumstances considered, but allowing it to vary was one way of representing possible porosity changes from other causes, for example overall volume dilation, especially before the option to allow volume change was implemented). Before simulating any real situation with the model it is useful to first understand the effect that each of the variable parameters has on the model results. To do this a number of preliminary tests have been completed in which one parameter at a time has been varied. Comparing these tests to the base case run (with no variations) shows the effect of varying each of the parameters.

The base case from which the variations have been made has been chosen to loosely fit experiment Mx80-4A, and the parameters for this case are shown in Table 6-2 (the base case used in Subsection 6.4.1.6 was slightly different from that used in the other subsections that follow). The results of this series of tests have been used to obtain closer fits to this experiment; these are described in Section 6.4.3 below.

59

6.4.1.1 Variation of number density of capillaries in gas permeability model

Figure 6-1 shows the effect on the inlet gas pressure of varying the number density of capillaries assumed in the gas permeability model of Equation 6-13. An increase in the number density means that a given porosity created by the gas provides a lower permeability, so that gas migrates more slowly from the inlet to the outlet, forcing the gas pressure higher. After breakthrough, it can be seen that a larger number density

Table 6-2. Model Parameter Values for Parameter Variations Base Case.

Model parameter Base case value

Number density of capillaries 1.0 lOll m-2

Critical gas porosity 1.0 10"6

Water permeability 5.0 10-21 m2

Initial volume fraction of solid 0.6033

Volume parameter, a1 O.OPa·•

Volume parameter, a2 O.OPa·•

Water compressibility, {3 5 10-10 Pa·•

17.0

-c:? i 16.0

....., V 14.0 ]

13.0

12.0 4-----+-----+----+----+----+--==------+--~

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

- 1e11 - 1e10 - 5e11

Figure 6-1. Inlet Gas Pressure For Variations of the Number Density Constant used to Calculate the Gas Permeability.

60

results in a more rapid drop in pressure, so that the pressure eventually reaches a lower value with a higher capillary density. It appears that although the gas permeability remains lower through most of the core, the gas permeability close to the outlet becomes higher for the larger number density, and it is the permeability near the outlet (where it is lowest) which controls the gas flow and the rate of change of pressure.

6.4.1.2 Variation of critical gas porosity

Figure 6-2 shows the effect on the inlet gas pressure of varying the critical gas porosity in the gas permeability model of Equation 6-13. An increase in the critical gas porosity from the base case value of 1 1 o-6 to 1 1 o-3 causes the gas migration up to breakthrough to occur more slowly since more gas must enter each section of the core before the gas permeability becomes non-zero. This results in a higher peak inlet gas pressure. The higher pressure reached before breakthrough leads to a more rapid pressure decline after breakthrough, but following this the pressure remains consistently above that for the base case for the rest of the run. With the higher value critical gas saturation, the creation of the extra gas-filled porosity needed to produce a given permeability is evidently more difficult than creating a similar permeability in the base case, presumably because of the additional water that would need to be ejected during the periods of high gas pressure, so that the average gas permeability in the sample remains higher. Note that the cases with a critical gas porosity of zero and 1 10-6 are indistinguishable in the figure.

18.0

~ 17.0

6 ~ 16.0

"' "' ~ A.. ..... V

] 14.0

13.0

12.0 --t-----+----+----+-----+----+-----t-1

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 S.OE+6 6.0E+6

Time (s)

- 1e-6 - 1e-3 - 0

Figure 6-2. Inlet Gas Pressure For Variations of the Critical Gas Porosity (the cases with critical porosities of zero and 1 1 o-6 are indistinguishable in this figure).

61

6.4.1.3 Variation of volume fraction of clay solids present

Figure 6-3 shows the effect on the inlet gas pressure of varying the fraction of the core volume occupied by clay solids. The solid volume fraction controls the minimum difference between the gas and water pressures required for gas to enter the core. A larger solid volume fraction implies a larger gas entry pressure (above the backpressure ), thus increasing the inlet gas pressure at which gas begins to flow into the core. After the initial peak in the gas pressure, the pressure transient appears to be similar in shape to that for a lower solid volume fraction, but displaced by a pressure equal to the change in the gas entry pressure implied by the change in solid volume fraction.

6.4.1.4 Variation of the water permeability

Figure 6-4 shows the effect on the inlet gas pressure of varying the water permeability. A higher water permeability allows the water to be displaced and the gas to flow into the core more easily. Therefore the inlet gas pressure has a lower initial peak and a more rapid decrease towards a steady-state pressure. Also, since more water is able to flow out of the core during the period (up to 1.2 l06s) when the water pressure in the core is greater than the boundary water pressure, there is less water (and more gas) in the core for the remainder of the run. This implies a higher gas permeability, so the remaining inlet gas pressure transients also show more rapid decreases in pressure with the higher water permeability values.

17.0 --r-----..-------------------,

16.5

~ 15.5

6 15.0

~ :; 14.5 ~

p. 14.0 ..... Q)

] 13.5

13.0

12.5

12.0 -+----+----+----+-----+--_,::,=-+-----1-1

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

- 0.6033 - 0.5980 - 0.6000 - 0.6018

Figure 6-3. Inlet Gas Pressure For Variations of the Solid Volume Fraction.

62

17.0

~

~ 16.0

~ ~ 15.0 ~

0.. .... V 14.0 ]

13.0

12.0 +-----+----+----+----t---+----+'

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 S.OE+6 6.0E+6

Time (s)

- Se-21 - Se-22 - 2e-21 - Se-20

Figure 6-4. Inlet Gas Pressure For Variations of the Water Permeability.

6.4.1.5 Variation of water compressibility

Figure 6-5 shows the effect on the inlet gas pressure of varying the water compressibility. With a higher water compressibility the gas is initially able to flow into the core more easily, so reducing the peak gas pressure reached. This is because additional porosity is created for the gas by a greater compression of the water. However, since more of the gas porosity is created by compression of the water and less by outflow of water from the core, there is a larger mass of water left in the core. As the pressure drops this water recovers a greater proportion of its original volume, decreasing the gas porosity further than in the lower water compressibility case. The resulting lower gas permeability causes less rapid decreases in gas pressure, and higher steadystate pressures for each of the stages after breakthrough.

6.4.1.6 Effect of allowing variation in the total volume

To demonstrate the effects of allowing the total volume to vary with the stress state, a number of tests with variations of the volume change parameters of Equation 6-11, from base case values of zero have been performed. A fit to experiment Mx80-4A of Horseman and Harrington ( 1997) assuming a constant volume constraint was again chosen as the base case but was a slightly different case to that assumed in Subsections 6.4.1.1-6.4.1.5. Recall that the model of Equation 6-11 allows, through the specification of two parameters, for a different volume change behaviour at stresses

63

17.0

~ i 16.0

~ ~ 15.0 ~ ~ ..... V 14.0 ]

13.0

12.0 --t----+----+-=--...J..._-+--~---+----+.J

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 S.OE+6 6.0E+6

Time (s)

- Se-10 - 1e-10 - 2.5e-9

Figure 6-5. Inlet Gas Pressure For Variations of the Water Compressibility.

above and below a reference stress value, taken by default to correspond to the initial stress.

The simplest volume variation is a single linear relationship to the stress over its whole range (i.e. both parameters have the same value). Figure 6-6 shows the effect on the inlet gas pressure of allowing the volume to vary in this simple way for a range of values of the parameters. The volume changes allow the gas porosity (and permeability) to vary more rapidly as the pressure varies than when the total volume is fixed. The larger the volume change with stress the less the pressure needs to vary to create the gas permeability required to just support the flow. Recall that when gas is present the stress is set equal to the gas pressure. A variable volume also results in a lower initial peak inlet gas pressure, because once the gas enters the core the core can expand so that the gas porosity reaches its critical value more rapidly, allowing a more rapid breakthrough and earlier peak in the pressure.

Independent variation of both volume change parameters gives more complex volume variation behaviour, but the behaviour follows the pattern of the simple case of the preceding paragraph, with account taken of the stress ranges (i.e. gas pressures) over which each parameter applies.

At this point it is useful to better understand what values the volume change parameters, a1 and a2, could take. First the assumptions are made that the gas flows through equally sized capillaries regularly spaced throughout the core, and that the gas occupies only a

64

17.0 .--------------------.,

16.5

16.0

~ 15.5

6 15.0 ~ ~ 14.5 ~

0-t 14.0 ...... V

] 13.5

13.0

12.5

12.0 +----+----+----+-----+-----+-----+J

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

- 0.0 - 1e-9 - 1e-10

Figure 6-6. Inlet Gas Pressure For Variations of the Volume Parameter Applied over the Full Stress Range.

small fraction of the total volume. Then, a crude estimate of the volume dilation can be obtained by applying results given by Timoshenko and Goodier (1951) for one such capillary and its associated surrounding volume, and multiplying by the number of capillaries. This gives the following relationship

6-18

where v is Poisson's ratio, E is Young's modulus (Pa).

Suitable values for the elastic constants in Equation 6-18 are thought to be v • 0.45 and E- 108Pa (Daeman and Ran, 1996). So for </Jg- 10-4 the volume change parameters would be -10-12

• However, as shown in Figure 6-6 a much larger value is required to be significant. An alternative estimate for the Young's modulus is that required in the GMClay model to fit experimental data (Nash et al., 1998), this implies E- 106Pa. In this case the volume change parameters would be -10-10

• Even this only represents a small effect on the inlet gas pressure due to the volume change. Therefore it is likely that the variation of the volume does not have a significant effect on the experiments considered here.

65

6.4.2 Resolution Testing

To determine the significance of numerical effects due to the discretisation applied in the model, a number of tests with the core represented by different numbers/sizes of grid blocks and with different maximum time step sizes have been run. It is assumed that the numerical effects caused by a given resolution are insignificant if increasing the resolution causes no observable difference in the inlet gas pressure results produced.

Variations in the maximum time step size and the grid blocks are made from the same base case as defined in Section 6.4.1.

6.4.2.1 Time step size

Figure 6-7 shows the effect of varying the maximum time step size. Reducing the maximum time step size from 2 104s to 5 103s causes a slightly more rapid reduction in the gas pressure. However, reducing the maximum time step size further, to 103s, causes no further noticeable change in the gas pressure. Therefore, using time steps no larger than 5 103s will ensure that numerical errors due to the time discretisation are not significant.

6.4.2.2 Spatial discretisation

Figure 6-8 shows the effect of varying the size and number of grid blocks. Increasing the number of grid blocks from 10 equal sized blocks to 20, then 100 equal sized blocks

16.5

-- 16.0 ~

0..

6 15.5

~ ~ 15.0 ~

~ 14.5 Q.)

] 14.0

13.5

13.0 +-+---+----+----+----+---+----+'

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

- 1000s - 5000s - 20000s

Figure 6-7. Inlet Gas Pressure for Variations of Maximum Time Step Size.

66

17.0 --.----...-------------------,

16.5

16.0

~ 15.5

6 15.0

~ ~ 14.5 ~ ~ 14.0 .....,

11)

] 13.5

13.0

12.5

12.0 -t----1-----+----t----1------+------+-'

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

- 10 equal - 20 equal - 19 varying - 100 equal

Figure 6-8. Inlet Gas Pressure for Variations of Grid Block Discretisation.

causes a more rapid reduction in the gas pressure. However, discretising from the inlet end with blocks as in the 10 block case and from the outlet end with the smaller blocks of the 100 block case such that there are 19 grid blocks in total, gives a result for the inlet gas pressure close to that for 100 equal sized grid blocks. The finer discretisation near to the outlet results in a smaller numerical error for the same total number of grid blocks. This is because there is a much larger change in pressure near to the outlet than in the rest of the core, so a finer discretisation is required to represent the rapidly changing pressure gradients across this region to a similar accuracy to the rest of the core. The 19 grid block discretisation described above results in only a small numerical error. This error could be reduced further by increasing the number of grid blocks; however, for the simulations described in the following sections, 19 grid blocks was adequate and had the advantage of requiring little computing time for each run.

6.4.3 Modelling Experimental Results

6.4.3.1 Evolution of inlet pressure

Several attempts have been made to model experiment Mx80-4A of Horseman and Harrington (1997) using the newly developed gas migration model, GMClayW. As with the Phase 1 model (Nash et al., 1998), it has proved difficult to reproduce the experimental data for the inlet gas pressure over the whole experimental history using a single input data set. However (as with the Phase 1 model) it has been possible to match most sections of the experimental data over a number of model runs that used

67

different input values. These 'best fits' are shown in Figure 6-9, and are each described below. The parameters used in these simulations are shown in Table 6-3.

The values for fit 1 have been chosen to fit as closely as possible the initial peak in the inlet gas pressure and the following negative transient. Fit 1 also represents the best fit to the second transient (the first shut-in period when the syringe was held at a constant volume). However, the fit to the shut-in is less close, as this period has proved difficult to fit accurately with any choice of input values. Note that during the period of the initial pumping rate, fit 1 is indistinguishable from the experimental curve, and during the first shut in it coincides with fit 2 (in both cases within plotting accuracy).

The values for fit 2 have been chosen to fit the period on either side of the second peak in inlet gas pressure, from the end of the first shut-in until the next change in pumping rate. This choice of input values also gives a similar fit to the first shut-in as fit 1. To produce the fit to this period of the experiment lower gas and water permeabilities were used than for fit 1, as well as a lower critical gas porosity. The lower permeabilities were required to give a more rapid rise in pressure to its second peak, and a slower response of the gas to the pressure increase, so that the pressure overshot its steady-state value and subsequently reduced as the gas flow increased. The critical gas porosity was only reduced in order to get a better approximate fit to the first peak in pressure, since the decreases in permeabilities had the effect of increasing the first pressure peak. The reduction of the critical gas porosity had only a small effect on the inlet gas pressure around the period of its second peak.

'(? Pot

6 ~ <ll <ll <1) 1-1

Pot ..... <1)

]

17.0

16.5

16.0

15.5

15.0

14.5

14.0

13.5

13.0

12.5

12.0

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

-experiment - fit 1 - fit 2 - fit 3

Figure 6-9. Fits to the Inlet Pressure of Experiment Mx80-4A of Horseman and Harrington ( 1997).

68

Table 6.3. Model Parameter Values for Fits to Experiment Mx80-4A using GMClayW.

Model parameter Value for fit 1 Value for fit 2 Value for fit 3

Number density of capillaries 6.5 1010 m-2 3.0 1011 m-2 2.0 1010 m-2

Critical gas porosity 2.0 10-4 1.0 10-6 6.0 10-4

Water permeability 2.5 10-21 m2 1.2 10-21 m2 5.0 10-21 m2

Initial volume fraction of solid 0.5995 0.5983 0.5990

Volume parameter, a 1 O.OPa-1 O.OPa-1 O.OPa-1

Volume parameter, a 2 O.OPa-1 O.OPa-1 O.OPa-1

Water compressibility, {3 5 10-10 Pa-1 5 10-10 Pa-1 5 10-10 Pa-1

The values for fit 3 have been chosen to fit the inlet gas pressure transients over the remainder of the experiment (from 3.1 106s). An almost exact fit was possible to the three negative transients corresponding to non-zero pumping rates, with the exception of the pressure at the start of the first. However, as with the first shut-in, an exact match was not possible to the second shut-in transient. To produce the fit to the final period of the experiment higher gas and water permeabilities and a higher critical gas porosity than used in either fit 1 or fit 2 were required. The higher water permeability allowed more water to flow out during the periods of largest pumping rate (before the period of the experiment being fitted), so that more gas could flow in, increasing the gas permeability during the first negative transient. The larger gas permeability then allowed the inlet gas pressure to reduce further than for the other two fits, so that the pressure was lower at the beginning of the period being targeted for fit 3. By this time the two opposing effects on the gas permeability had balanced, giving a similar permeability and therefore similarly shaped pressure transients to the other two fits, but at a lower pressure due to the lower starting value. However, for the second shut-in at the end of the history, the pressure dropped less rapidly for fit 3 than for the other fits.

6.4.3.2 Apparent permeability variation and gas outflow rates

It is also useful to compare the permeabilities and gas outflow rates given by the model with those in the experiment. As an example these comparisons are shown for fit 1 in Figures 6-10 and 6-11. The permeabilities are plotted against net mean effective stress, which is defined as

6-19

where a is the confining stress (Pa), pg in is the gas pressure at the inlet (Pa),

69

pg out is the gas pressure at the outlet (Pa) (including an apparent capillary pressure (Horseman and Harrington, 1997), which for experiment Mx80-4A was taken to be 11 MPa).

The permeability for fit 1 agrees well with the experiment. This is expected since the inlet gas pressures agree well, and it is the permeability that determines the shape of the pressure transients. In both the experiment and fit 1 the permeability initially increases from zero at a net mean effective stress of approximatly 2.0MPa. It then levels off and reduces slightly as the pressure decreases (and the net mean effective stress increases) toward steady state. Up to a mean effective stress of about 2.3 MPa, the simulation agrees well with experiment. With the initiation of the shut-in the permeability and pressure decrease to a new steady state (at Ge.ff• 3.2MPa), but the graph does not follow the same line as in the experiment, which at this stage is following the upper branch of the "hysteresis" loop. As pumping is resumed the permeability increases, again following a slightly different route to the experiment. These differences are not altogether clear from Figure 6-10, but it suffices here to be aware that the details of the graph of the simulation results at effective stresses above about 2.3 MPa differ from those of the graph constructed from experimental observations. It appears that the hysteresis loop is not correctly represented; this of course follows directly from the mismatches shown in Figure 6-9 for the corresponding period of the experiment.

2.5E-20

2.0E-20

N' s 1.5E-20 c

i-S ~

Q)

§ l.OE-20 Q)

~

5.0E-21

O.OE+OO

1.5 2.0 2.5 3.0 3.5 4.0

Net Mean Effective Stress (MPa)

-experiment - fit1

Figure 6-10. Permeability against Net Mean Effective Stress for Experiment Mx80-4A and Fit 1.

-,..... I en

)_ V .... ~

~ ~ 0

r:I:

70

3.0E-8

2.5E-8

2.0E-8

1.5£-8

1.0E-8

S.OE-9

O.OE+O +---L_+-...:.:::::=:::::~=:iii/!.~1----+~~!!!!llooofl----+l

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 S.OE+6 6.0E+6

Time (s)

-experiment - fit1

Figure 6-11. Gas Outflow Rates for Experiment Mx80-4A and Fit 1.

The gas outflow rates from the experiment and fit 1 (Figure 6-11) do agree well. There is only a small difference, the modelled flow rate being consistently slightly larger than in the experiment.

6.4.3.3 Variation in pore water pressures

A final comparison that it is useful to make here is that between the gas and water pressures at the same point in the simulation. This comparison is shown in Figure 6-12 for the pressures at the inlet end of the core in fit 1. Since the volume is constant, in the limit of zero gas porosity the difference between the gas and water pressures is defined by the solid volume fraction (by Equations 6-5 - 6-9) as 13.9MPa. As the gas porosity increases the difference between the pressures increases. However, as shown the difference does not change much (by less than 0.2MPa), since the gas porosity remains small at all times.

6.4.3.4 Use of an alternative gas-permeability model

For all the tests performed above it has been assumed that the gas permeability is related to the gas porosity by Equation 6-13, which is based on the assumption that the gas permeability can be represented by a bundle of capillaries.

It is not known whether this is a suitable definition of the gas permeability, therefore tests using an alternative definition given by Equation 6-14.

71

3.0 17.0

2.5 16.5

2.0 16.0 '(:?

'(:? ~ 1.5 15.5 6 ~

1.0 15.0 6 V

~ ~ "-' 0.5 14.5 "-' "-' V "-' ~ V ~ 0.0 14.0 ~

~ ~ V

~ ...... ~ -0.5 13.5 0 ~

-1.0 13.0

-1.5 12.5

-2.0 12.0

O.OE+O 1.0E+6 2.0E+6 3.0E+6 4.0E+6 5.0E+6 6.0E+6

Time (s)

- water - gas

Figure 6-12. Comparison of Water and Gas Pressures at the Inlet End of the Core for Fit 1.

Using the alternative definition for the gas permeability, simulations have again been carried in an attempt to fit the inlet gas pressure variation for experiment Mx80-4A. These results have been compared with those given in the previous section for the original definition of the gas permeability. This has shown that, by choosing the appropriate input values for the model, almost identical results can be achieved using the two functions for the gas permeability. It was also found that the additional variable parameter in Equation 6-14 has not increased the flexibility of the results that can be produced in a way that allows a better fit to experiment Mx80-4A to be obtained.

6.4.4 Resealing Behaviour

An additional test that has been performed with the GMClayW model involved a demonstration of resealing of the clay after gas migration has occurred through it. For convenience the parameters used for fit 1 to experiment Mx80-4A were used for the gas migration stage of the test, except that the critical gas porosity was set to zero so that potentially all gas could later be removed from the core. This was followed by a resealing stage when water at 1MPa was applied at both ends of the core over a period of 3.9 106s (45 days). This allowed water lost from the core during gas migration to be replaced, and forced out gas remaining in the core at the end of the gas migration stage. The gas migration test was then repeated with the same parameters, except that the initial conditions were those existing at the end of the resealing stage.

72

Over the period of the resealing the gas porosity was reduced by at least a factor of t 00 throughout the core. By the end of the resealing stage the remaining gas was flowing out of the core at a very low rate, since the gas permeability at the outlet had reduced to less than t0-30m2 (from -t0-23m2

). The repeat of the gas migration test then showed no discernible difference from the original test in terms of the inlet gas pressure. This demonstrates that the GMClayW model is capable of representing resealing of the clay, with the clay returned to sufficiently close to its original state that the history has no effect on subsequent gas migration. It also follows that the resealing by the mechanism involved, which is essentially re-swelling of the clay, does not leave any preferential pathways for groundwater flow and transport along the previously gas-filled channels.

73

7 IMPLICATIONS OF FLOW GEOMETRY, CONFINING CONDITIONS, AND FLOW BOUNDARY CONDITIONS

The original BGS experiments (Horseman and Harrington, 1997) were carried out for a linear flow geometry, and used a 'constant stress' boundary condition to confine the sample.

The question arises as to whether a 'constant stress' boundary condition is representative of the situation likely to be experienced by buffer material around a waste canister; for example, expansion of the bentonite may be constrained by the surrounding rock.

In addition, recent results obtained by BGS are for:

a) a constant volume radial flow geometry, with flow from an injection filter at the centre of a cylinder to sinks located on the cylinder boundary. (This reduces the possibility of flow between the sample and its confining jacket, thereby bypassing the sample.)

b) a ~~ flow geometry, in which the sample is constrained in a radial direction, but is subject to a constant stress in the axial direction.

It therefore is necessary to consider the effects of different geometrical configurations on the flow of gas through bentonite, and the implications of imposing a 'zero strain' boundary condition as an alternative to a 'constant stress' boundary condition.

As background to such considerations, the relationship between the modelling approach introduced in Section 1 to standard theories of drained consolidation is discussed first in Section 7 .1. The theory on which this discussion draws is summarised in Appendix A 1. Next, in Subsection 7 .1.2, the volume dilation when gas migrates through bentonite is estimated, and used to enquire how significant the difference between a constant stress and constant volume confining condition is likely to be.

Section 7.2 discusses the implications of the geometry used in the recent radial flow experiment of Horseman and Harrington (unpublished), and attempts to provide some interpretation of the experimental results. The difference in geometrical configuration between the Ka geometry and the earlier constant isotropic stress experiments is clearly not as marked as that between the constant volume radial flow geometry and the earlier experiments, as at least the flow geometry remains linear. However, the Ka experiment does provide a new comprehensive data set that is amenable to modelling using the various one-dimensional computer models that have been developed in the GAMBIT Phase 1 and Phase 2 programmes. Application of these models to this new data set is discussed in Section 8.

74

7.1 The Alternative Modelling Approach

Appendix A 1 provides a summary of the standard macroscopic phenomenological theory of drained consolidation, as formulated for an unsaturated soil. In the following subsection, the assumptions of the alternative model of gas migration in bentonite, presented in Section 1, are related to this theory. This is instructive, because it clarifies the assumptions that underlie the 'alternative model', and because it indicates how those assumptions may be generalised to different geometries and confining conditions.

7 .1.1 Relation of the Alternative Model to the Macroscopic Phenomenological Approach to Soil Consolidation

The two fluids (i.e. water and gas) in an unsaturated clay are able to flow. Therefore, two independent partial differential equations are required to solve for the pore-water and pore-gas pressures with respect to time. These partial differential equations must satisfy continuity of the water and gas phases respectively.

7.1.1.1 Water Phase

Consider a reference element of unsaturated clay, with water and gas flow during onedimensional consolidation. The net flux of water through the element (generalising Equation A2-29 to the case of compressible flow) is

7-1

or

7-2

After breakthrough, the alternative modelling approach uses constitutive relations (see Appendix A2.3) based on the following assumptions:

a) the water is compressible (Equation 6-4 ), so that

7-3

where f3w is the compressibility of water.

b) the total volume of the sample (at least in the simplest version of the model) is constant, so that

75

de =0 V

7-4

where ev is the volumetric strain, and is equal to the difference between the volumes of the voids in the element before and after deformation, L1 Vv, referenced to the initial volume of the element, Vo.

(The implications of this constitutive relation are discussed in Section A2.7 .2.)

c) there is a relation between swelling pressure, Ils (identified with pg- Pw), and void ratio, e (Equation 6-6). Differentiating this relation, gives (cf Equation A2-44, a special case of Equation A2-22)

where r

1 Vw d{pg- Pw)

Y Vo Pg- Pw

is a constant.

d) the permeability to water is constant (Equation 6-12), so that

dk =0 w

Hence, Equation 7-2 becomes

7.1.1.2 Gas Phase

Similarly, the equation describing the net flux of gas through the element is

a ( vg) ( kg ) -PT- =V· p -Vp at g v. g , g 0 f"'g

The constitutive relations Equations 7-3-7-6 have to be supplemented by

a) the Ideal Gas Law, which relates the gas density, pg, to the gas pressure,pg

7-5

7-6

7-7

7-8

7-9

where Mg R T

76

is the molar mass of the gas, is the gas constant, is the temperature.

b) the continuity equation, which (at least in the simplest constant volume version of the model) implies that (cf Equation A2-23)

7-10

c) the assumption that the permeability to gas varies with gas porosity (Equation 6-13), so that

dk =2 fk:d(vg) g V~' V0

7-11

Then

It should be emphasised that the constitutive relations used above are not the most general that could be written down. In particular, the assumptions (Equations 7-4 and 7-5) that

7-13

are special cases of Equations A2-17 and A2-23, and may be generalised.

7 .1.2 Implications of a Capillary Model

In its simplest form, the alternative model (Section 6.1) assumes that the sample volume is constant, i.e.

77

7-14

This subsection investigates the validity of this assumption, using a simple model (a capillary model) for the gas saturation in bentonite.

Recall that for a capillary model

4 k =Nrcr

g 8

V _2_ = Nrc r 2

Vo

where N r

is the number of capillaries per unit volume; is the capillary radius;

and so, rearranging

7-15

7-16

Now, for BGS experiment Mx80-4, kg- 10-21 m2• A further assumption is needed to

solve Equation 7-16. Possibly additional conditions that might be invoked could be:

a) While there is no data on the density N of gas flowing pathways, the intermittent gas outflow observed during the experiments suggests that there can not be too many pathways through the sample. The area of the sample is -2 10-3 m\ and an upper limit on the number of pathways may be 103

, implying

7-17

(Note, however, that both the new alternative model of Section 1 and the model of gas migration by water displacement (Section 5) seem to require much larger numbers of capillaries to represent the required gas permeability behaviour in terms of a capillary geometry for the gas pathways.)

b) Alternatively, the maximum gas entry pressure ( pg- Pw ) -15 MPa for this experiment. If gas entry is controlled by capillary pressure, then the YoungLaplace Equation implies that

2y r=(Pg-Pw)

> 10-8 m

78

7-18

where r is the interfacial tension between water and gas, taken to be 0.072Nm-1

•

Using this value in Equation 7-16 implies that the volume occupied by gas is

7-19

c) The simulations that have been carried out all predict very low average gas saturations, generally less than 10-3

, and often less than 10-4•

The change in water volume

is likely to be similar in size (unless the clay sample consolidates).

For such small changes in the water and gas saturation in the clay, the sample may be regarded as being at constant volume. (e.g. in a sample initially 10-1 m long a strain of 10-4 corresponds to a length change of only 10 J..Lm, and, using the compressibility of water, 5 10-10 Pa-1

, a stress change of 2 105 Pa.) In this case, the appropriate constitutive equations (see Appendix A2.7.2) are

dev =0

dVw wd{__ ) --=1(2 \Pg-Pw

Vo 7-20

dVg dVw --=---

Vo Vo

These are precisely the equations (Equations 7-4, 7-5 and 7-10) that form the basis of the alternative model, provided that the following identification is made

7-21

In order to complete the system of equations in the alternative model, it is necessary to specify the boundary conditions.

79

7.1.3 Boundary Conditions

The upstream boundary determines the pressure Pgi at which gas first enters the bentonite. The experimental results of Horseman and Harrington ( 1997, unpublished) suggest that this pressure satisfies

7-22

where Q is a geometrical factor (e.g. 1 for a linear geometry, and 2 for a cylindrical or spherical geometry);

ITs is the swelling pressure of the clay.

The downstream boundary conditions on Pw and pg are more difficult to specify.

In the case of Pw, the obvious boundary condition is

Pw = Pb 7-23

where Pb is the applied backpressure (1 MPa).

However, it is possible that gas in the backpressure circuit after breakthrough may prevent water from flowing back into the bentonite sample at its downstream end, and in that case the appropriate boundary condition would be

Pw = Pb ·r dpw 0 1 --< dx

dp w = 0 otherwise dx

7-24

A problem with both these possible boundary conditions is that in Experiments Mx80-8 and Mx80-9 the mid-plane pore-water pressure tends to an equilibrium value (-2.5 MP a) that is significantly above the backpressure. Currently there is no explanation of this observation, but the simplest conclusions are that either the water pressure at the downstream end of the sample is increased by r above the backpressure

7-25

or the downstream boundary has become isolated from the backpressure circuit (although no explanation for this is apparent), in which case the boundary condition would be

dpw =0 dx

7-26

80

In the case of pg, the downstream boundary condition is more complicated: intermittent gas outflow is observed. It is not practicable to simulate this outflow in detail; rather a pragmatic approach is adopted in which only the average outflow is modelled. Noting that there is no outflow for gas pressures less than a 'shut-in' pressure, Pco, the boundary condition is taken to be

Pg = Pco 7-27

This boundary condition, when combined with the postulate that gas permeability is a strongly varying function of the pressure, has interesting implications for the pressure and gas permeability profiles. At steady state the pressure profile is not linear, but decreases sharply towards the downstream boundary, where the pressure, and hence gas permeability, is smallest. Figure 7-1 shows a typical gas permeability profile from a simulation of Experiment Mx80-4A (Fit 1 of Figure 6-9 at the point of the first injection rate change); the sharp decrease at the downstream end of the sample is a feature of all simulations. It follows that a significant part of the resistance to gas flow derives from a small region of the clay sample close to the downstream boundary.

This is a potential source of concern in that it appears that the significant features of the calculated behaviour may be occurring over a short region of the sample in which the detailed behaviour has been approximated by the use of a representative average gas pressure boundary condition. In reality, the detailed physics at the downstream boundary must be complicated, involving, as it appears, to the opening and closing of

15.30

15.25

15.20 LE-19 N'"'

'(? ~ 15.15 6

C1) 15.10 ~

g c

1.E-20 ~ ~ C1)

en 15.05 C1)

1-< ~

§ C1)

~

15.00 1.E-21

14.95

14.90 +-----.------.-----..,.----r----..L..f- 1.E-22

0.00 0.01 0.02 0.03 0.04 0.05

Position (m)

- Pressure - Permeability

Figure 7-1. Gas Permeability and Pressure as a Function of Position along the Clay Sample, from a Simulation of Experiment Mx80-4A (Fit 1 of Figure 6-9 ).

81

gas pathways. However, the fact that most of the pressure drop occurs near the downstream end of the sample means that changing the gas pressure applied at the downstream boundary only affects the behaviour near the end of the core; behaviour in the rest of the sample remains little changed. Since the precise choice of value for the downstream boundary gas pressure has been found not to significantly affect simulation results, this gives some reassurance that the use of an effective gas-pressure value to represent the time averaged behaviour at the boundary may also be reasonable. No practicable alternative has been devised.

7.2 Alternative Geometries for Gas Migration Through Bentonite

The main experimental data used in the GAMBIT Club work so far is that obtained by Horseman and Harrington (1997, unpublished) at BGS. The earlier BGS work was on specimens confined by an isotropic stress (Horseman and Harrington, 1997), but two new configurations have been recently investigated:

a) a constant volume geometry (Experiment Mx80-8);

b) and a ~ geometry (Experiment Mx80-9).

The details of these two experiments were summarised in Section 2.3. In the following subsections, an attempt is made to evaluate the experiments in the light of the conceptual models considered in the GAMBIT Club programme and the theoretical considerations presented in this section and in Appendices A1and Al.

7.2.1 Discussion of Experiment MxS0-8

The details of experiment Mx80-8 are summarised in Section 2.3.1. This Section attempts to interpret the significant features during the gas injection stages of this experiment (i.e. Stage 10 to Stage 13, listed in Table 2-4) that are plotted in Figure 2-1.

The first significant feature of the results, at 3.204 106 s, is a rapid increase in the axial stress from 1.6 MPa to 18.4 MPa, which occurs simultaneously with a small drop in the

gas pressure. This is followed by gas breakthrough at 3.218 106 s (Figure 7-2, which reproduces Figure 2-2 for convenience).

This rise in the axial stress has been attributed to the sudden formation of a fracture cutting across the bentonite sample. The hypothesis assumes that the fracture fills with gas at a pressure similar to the injection pressure, and that this gas then deforms the clay so that it exerts a load on the axial pressure transducer.

However, the volume of gas that flows into the fracture can be estimated from the small drop in gas pressure using Boyle' s Law, and is about 0.3 cm3

• Assuming that the area of the fracture equals the cross-sectional area of the apparatus, its aperture must be greater

82

20


15

'";' 0...

~ Q) 10 $.... ;:::l Ul Ul Q) $....

0...

5

3.3 106

Time [s]

axial stress

radial stress

porewater pressure

3 .5 106

Figure 7-2. Detail of the experimental response during the gas breakthrough phase.

than 100 J.lm. It is difficult to imagine that a fracture with such a large aperture could reseal, and isolate the gas pathway from the injection filter. But, if that does not happen, how is it possible for the measurement in the axial stress transducer to fall while the injection gas pressure continues to increase?*

An alternative hypothesis is that a gas pathway has formed that connects the injection filter to the end cap of the apparatus. The axial pressure transducer is responding to gas flowing through this pathway. The sequence of events is:

a) gas flows from the injection filter to the end cap of the apparatus;

b) the axial pressure transducer (which previously has been measuring the porewater pressure) responds to the gas;

c) the gas pathway closes, isolating the gas at the end cap;

d) the clay consolidates, thereby expanding the gas occupied volume at the end cap, and as a result the measurement of 'axial stress' decreases;

*Subsequent investigation has in fact confirmed that the stress sensors in this experiment were not properly measuring the stresses in the clay.

83

e) water is 'squeezed' out of the clay, and in response both the pore-water pressure and radial stress measurements increase (albeit on different time-scales).

In support of this hypothesis, it is noted that the consolidation constant is (Biot, 1941)

C =kw 3(1-v) f.l w 0(1 +V)

where kw is the permeability to water (9.4 10-21 m2, see Section 2.3.1)

J.1 w is the water viscosity; v is Poisson's ratio (-0.4 (Borgesson et al, 1996));

7-28

0 is the compressibility of the porous matrix (a value of 3 1 o-s Pa-1 has been measured for a similar sample, see Section 7 .2.2).

The timescale over which the clay consolidates through a distance x = 10-3 m (this distance corresponds to an increase by about an order of magnitude in the volume occupied by the initial pulse of gas that invades the clay, as estimated above) is

x2 t =c 4C 7-29

= 6.2102 s

and seems plausible when compared to the axial stress response.

A further point is that the pore-water pressure and radial stress responses appear to be measuring the same quantity, but on different timescales. Could the difference between the pore-water pressure and radial stress responses be due to compliance in the measurement of the pore-water pressure? It is possible to estimate the effect of pipe work in the pore-water pressure circuit.

First, the flow of water out of the clay and into the pore-water pressure circuit can be estimated from the analytical solution for radial flow from a semi-infinite region to a disk (Carslaw and Jaeger, 1959)

7-30

where Q is the flow to the pipe in the pore-water pressure circuit (m3s-1); r is the radius of the 'disk' (if the disk is taken to be the pipe, r- 5 10-4 m;

but if it were taken to be the filter, r = 3.2 10-3 m); 11JJw is the difference between the pore-water pressure in the clay and the

pressure in the pore-water pressure circuit ( < 8 MPa);

84

Taking r- 5 10-4 m, Q is less than 1.5 10-13 m3 s-1•

Next, consider this flow being injected into a pipe of radius rand length I. The pipe has permeability

r2 k=-

p 8 7-31

=3.110-8 m2 forapipeofdiameter10-3 m

and hydraulic diffusivity

kp D =--"---

P f.l w f3 w 7-32

= 6.3 104 m2s-1 for a pipe of diameter 10-3 m

The solution of the transient flow equation gives for the pressure at the end of the pipe (Carslaw and Jaeger, 1959)

Qt 1 Q f.l { 1 2 00

{-It [ DPn2

n2 t]} p(x=l)=-2---+--21-2:!_ ----2 L-2-exp- 2

nr I f3w nr kP 6 n n=I n I

Qt 1

~ n r 2 1 f3w 7-33

= 380~ Pa I

It follows, for example, that at the end of a pipe 0.5 m long the pressure will increase by 6 MP a in 8 103 s. This timescale, based on an assumption of compliance in the measurement of the pore-water pressure, appears a little short to explain the experimentally observed response.

However, Equation 7-33 implies also that the timescale for the pore-water pressure response should be larger than the timescale for the radial stress response, in proportion to the ratio of water volumes in the two circuits. This (weaker) observation is not clearly unreasonable when compared to the data.

Gas breakthrough is first observed at 3.218 106 s in the sink filter array above the injection point. This event does not affect the pore-water pressure, axial stress or radial stress measurements.

Considering the stresses around a spherical cavity in an isotropic saturated clay, it would be expected that hydraulic fissuring should occur when the excess gas pressure

85

exceeds twice the average stress in the clay (Appendix A3). This stress, at equilibrium, is the sum of the swelling pressure and the external water pressure (backpressure). The peak injection pressure is 19.44 MPa, which is indeed not very different from twice the swelling pressure plus backpressure (Ils - 9.4 MPa, backpressure =1 MPa). This agreement is suggestive of tensile fissuring as the mechanism of gas invasion of this sample. However, this is a subtly different argument to that used to explain gas invasion in the linear flow geometries. For these there is an implicit (explicit in the case of crack propagation) assumption that gas can enter the surface of the clay through imperfections or crevices that provide the entrance to incipient pathways, and that this fluid pressure can then dilate and extend the crevices; the argument is not that the application of pressure on the surface of the clay causes a tangential stress which fractures the clay. The latter is the argument in the case of the high pressure required for gas invasion in the radial geometry. It must be assumed that the small radius opening round the gas source in this geometry is relatively free of imperfections that would allow fluid penetration and fracture propagation at lower pressures. This is perhaps plausible for a small carefully machined opening. It may be significant in this respect that subsequent gas entry occurs at a much lower pressure than initial entry.

After breakthrough, the injection pressure falls rapidly to an approximately constant value, and the flow in the radial sink filters increases, in a way similar to that observed in previous experiments (Horseman and Harrington, 1997). Turning off the pump used to compress the gas in the injection volume leads to a further transient in the upstream gas pressure (Figure 7-3). These pressure responses are qualitatively similar to those seen in axial flow tests under isotropically stressed conditions.

The only anomaly during this stage of the experiment, is an increase in the radial stress by about 1 MPa above porewater pressure (Figure 7-3) that seems to occur in response to 'shut-in' at 4.503 106 s. A possible explanation of this observation is that the clay structure shifts in response to 'shut-in', and exerts an additional load on the radial stress transducer, which is maintained at about 1 MP a for the duration of the experiment. Such a possibility is a speculative one which is difficult to evaluate; however, more convincing explanations are equally elusive.

Note also that the mid-plane pore-water pressure decreases to an equilibrium value of about 2.1 MPa. This is significantly above the applied backpressure of 1 MPa, and implies that equilibrium is not re-established between the clay pore-water pressure and the backpressure (cf Experiment Mx80-9, Section 7.2.2).

At 1.048 106 s, the injection volume was reduced again at a rate of 375 JlL hr-1• Stage 12

of the experimental history is shown in Figure 7-4.

The injection pressure rises slowly in accordance with Boyle's Law.

86

20


axial stress

radial stress


backpressure

7 0..

~ Q) 10 s... ;:l Tll. l7.l Q) s...

0..

5

0~~~--~~~~~~--~~~--~~--~~~--~~--~~~ 0.0 106 4.0 106 6.0 106 10.0 106

Time [s]

5 10-6


radial array no. 1

4 10-8 (above injection filter)

radial array no. 2

'0:' (plane of injection filter) E-< ID. ......,

10-8 radial array no. 3

«l 3 Tll. (below injection filter)

........... t')

~ Q) ......,

10-8 «l 2 et! ;r; 0

f;:;

1 10-8

4.0 106 6.0 106

Time [s]

Figure 7-3. Stages 10 and 11 of the experimental history for Mx80-8.

'P:' E-< ID .... tO

Tll ........... (')

~ Q) .... tO

0:: ;:: 0 ~

87

20


axial stress

radial stress


backpressure

':;' 0..

~ Q) 10 ~ ::l Tll Tll V ~

0..

5

oL-~~~-L-L~~--L-L-~~~_L_L~~--L-~~~-L-L~~

10.4 106 10.6 106

5 10-8

4 10-8

3 10-8

2 10-8

1 10-8

10.8 106 11.0 106

Time [s]


radial array no . 1 (above injection filter)

radial array no. 2 (plane of injection filter)

radial array no. 3 (below injection filter)

0 10-8 L_LL~~~~~~=±~~=d~~~r=~4-~~~_L~~~~ 10.4 106 10.6 106 11.0 106 11.6 106

Time [s]

Figure 7-4. Stage 12 of the experimental history for Mx80-8.

88

At 1.076 107 s, when the injection pressure is 10.9 MPa, there is a sudden breakthrough event and gas starts to flow out of the sample. In contrast to the first breakthrough Stage, when gas flowed out of the radial array above the injection filter, the flow is now to the radial array below the injection filter. That is, the system of gas pathways has evolved. The fact that the breakthrough pressure is much lower than for the first Stage implies that the pathways have not fully resealed: indeed, it is likely that only those parts of the pathways close to the downstream boundary have resealed.

Simultaneously, a gas pathway connects with the radial stress transducer, which responds by immediately registering a pressure similar to the injection pressure. For the duration of this Stage of the experiment, the radial stress measurement tracks the injection pressure, implying that the connection stays open. A second 'event', which is associated with an increase in the axial stress and (more slowly) the pore-water pressure, occurs at 1.095 107 s. This 'event' might be the formation of a major fracture across the clay that causes consolidation.

The system of gas pathways continues to develop during the experiment. Initially, the flow is exclusively to the radial array below the injection filter. However, this flow peaks at 1.087 107 s, then declines and is replaced by flows to the remaining sink filters .

The shut-in response is typical. However, note that the mid-plane pore-water pressure decreases to an equilibrium value of about 2.6 MPa. This is significantly above the applied backpressure of 1 MPa.

7.2.2 Discussion of Experiment Mx80-9

The details of experiment Mx80-9 are summarised in Section 2.3.2. This Section attempts to interpret the significant features during the gas injection stages of this experiment (i.e. Stage 4 to Stage 8, listed in Table 2-6), that are plotted in Figure 2-3. However, the hydraulic data have interesting implications for the modelling, and therefore are discussed first.

7 .2.2.1 Hydraulic Data (Stage 2)

Prior to gas injection, a water injection test was carried out with a constant flow rate of 2.2 J1L hr-1

• The transient data were interpreted by Horseman and Harrington (unpublished) using the standard hydrogeological model (Marsily, 1986), to obtain values of the intrinsic hydraulic conductivity (7 10-14 m s-1

) and specific storativity (Ss = 3 1 o-4 m-1

) of the clay sample.

Noting (Marsily, 1986) that

7-34

where lf>w f3w f3s E>

89

is the porosity of the clay; is the compressibility of water (5 10-10 Pa-1);

is the compressibility of the solid grains (fis << f3w); is the compressibility of the porous matrix.

it follows that the dominant contribution to Ss comes from E>, and

From the definition of E> (Marsily, 1986)

and also

Now, in the alternative model (Section 6.2 above) it is assumed that

l+e r a- Pw

Comparing Equations 7-36 apd 7-38, it is evident that in the alternative model

where cf>w

r is the porosity occupied by water in the clay-water system (0.424); is 4.85;

a-Pw is the conventional effective stress ( ..... 9 MP a).

7-35

7-36

7-37

7-38

7-39

In particular, for Experiment Mx80-9, these numbers mean that the alternative model assumes

E> = 9.7 10-9 Pa -I 7-40

which is a factor 3less than the experimental value (Equation 7-35).

(As an aside, it should be noted that the derivation of Equation 7-34 allows for the motion of the porous medium. If this motion is neglected, as it is in the alternative model, then the value of E> in Equation 7-40 should be multiplied by a factor 1 - l/Jw.)

90

7 .2.2.2 Gas Data

The history of the gas injection test is shown in Figure 7-5 (which reproduces Figure 2-3 for convenience).

As the volume of the inlet gas reservoir is reduced, the injection pressure increases in accordance with Boyle's Law. After 1.3 106 s, the injection pressure is held fixed at 8.8 MPa for a further 7.6 106 s. During this time, the experimental response shows the following features:

a) the mid-plane pore-water pressure increases to 4.1 MPa at 2.4 106 s, before gradually declining to an equilibrium value of about 2.5 MPa;

b) corresponding to this increase in the pore-water pressure, there is a very small outflow from the downstream end of the sample;

c) there is no gas breakthrough.

The most plausible explanation for the response seen in the mid-plane pore-water pressure is that a small volume of residual water has been forced out of the inlet end cap and into the sample.

The peak in the mid-plane pore-water pressure occurs at 2.4 106 s, and this is comparable to the theoretical timescale for a pressure pulse to propagate from the inlet to the mid-plane. Assuming, as a first approximation, that Pw at the inlet increases as a linear function of time, then the mid-plane pore-water pressure is (Carslaw and Jaeger, 1959)

7-41

where I is the sample length (5.09 10-2 m); Pw0 is the initial pore-water pressure (1 MPa);

dp gi is the (assumed) constant rate of increase. of inlet pore-water pressure dt

(MPa s-1);

K is the hydraulic conductivity (7 10-14 m s-1)

t is the time (s); i2erfc is a repeated integral of the complementary error function (Abramowitz

and Stegun, 1965).

Approximating the rate of increase of the inlet pore-water pressure by the average increase in injection pressure

91

10

injection gas pressure 8

backpressure

porewater pressure

2

0~~--~~--~--~~--~~--~--~~--~--~~~ 0.0 107

f/1 ........... 11 10-8

Cl) ..... «l

0::

~ 0 ~

-

--------

~ . I

Time [s]

I I



-

{"-

~

l .. . ll I. l ·~ ~-Time [s]

Figure 7-5. Measured Pressures and Flow Rates for Stages 4 - 8 of Experiment Mx80-9 (K0 Geometry) of Horseman and Harrington (unpublished).

dpgi ~ (8.8-1.0)/1.3 106 MPa s-1

dt

92

7-42

leads to the prediction that at t = 1.3 106 s the mid-plane pore-water pressure should be about 2.1 MPa (cf experimental value 1.9 MPa), and at t = 2.4 106 s should be about 4.6 MPa (cf experimental value 4.1 MPa). These results suggest that water indeed is being forced into the sample.

At this Stage it is argued that there is no gas breakthrough, and the diffusive flow of gas can be shown to be negligible. For example, in 'the case of this experiment, the steady state diffusive flow would be

Hp . Q=AD __ g_z

e l 7-43

where A is the area of the sample (2 10-3 m2);

De is the effective diffusion coefficient of gas through the sample (estimated to be about 4 10-12 m2s-1);

H is Henry's Constant for the gas (3.79 10-6 moles m-3 Pa-1 in the case of helium);

Pgi is the injection pressure (8.8 MPa); l is the sample length (5.09 10-2 m).

It follows that

Q = 5.2 10-12 moles s- 1

= 5.210-3 JlL hr-1 (at inlet conditions) 7-44

and so this is insignificant. Any measured outflow must therefore be water. A crude integration of the experimental outflow from the sample suggests that this volume of water is about 6.6 cm3

• This volume of water is comparable to the pore volume of the sintered disk at the inlet ( -5 cm3

) , and so possibly arises from desaturation of this disk.

The final observation in connection with this stage of the experiment is that the midplane pore-water pressure decreases to an equilibrium value of about 2.5 MPa. This is significantly above the applied backpressure of 1 MPa, and implies that the clay porewater pressure does not return to equilibrium with the backpressure ( cf Experiment Mx80-8, Section 7 .2.1 ).

After 8.9 106 s, the injection pressure is again increased. Gas breakthrough occurs when the injection pressure has reached 10 MPa at 9.0 106 s. This appears to be the usual breakthrough criterion

where pgi

Pwb

fls

is the injection pressure; is the backpressure;

93

is the swelling pressure of the clay ( -9 MPa);

7-45

The history of this Stage of the gas injection test is shown in Figure 7-6 (which reproduces Figure 2-4 for convenience).

The post-breakthrough behaviour is similar to that observed in earlier experiments; e.g. there is a sharp negative transient in the injection gas pressure, and the negative transient does not monotonically approach the steady-state condition but there are clear signs of overshooting.

The second gas breakthrough is shown in Figure 7-7, and is also typical; e.g. breakthrough occurs at a lower injection gas pressure than for the primary breakthrough.

A particularly significant feature of this experiment is that there was no gas breakthrough when the inlet gas pressure was kept fixed at 8.8 MPa for 88 days. This appears to be in direct contradiction with Canadian results (Hume, 1999) which showed gas flow at all injection pressures given sufficient time (cf Section 2.1).

94

10

__ injection gas pressure

9

8~~~~~~--~~~~~~--~~~~~~--~~~

07 E-< ID ..... a::l

m ...........

8.8 106

l1 10-6

9.2 106 9.4 106

Time [s]



9.2 106 9.4 106

Time [s]

Figure 7-6. Measured Pressures and Flow Rates for Experiment MxB0-9 (K0

Geometry) of Horseman and Harrington (unpublished) - Period around First Gas Breakthrough.

95

10

__ injection gas pressure

9

e~~~~~~~~~~~~~~~~~~~~~~~~

_, aj

lll ...........

2 10-8

'k 1 10-8

Q) _, aj

0::

~ 0 ~

1.0 107 1.1 107 1.2 107

Time [s]



1.3 107 1.4 107

0 10-8 UL~~~=c~~~~~~~~~~~~~~~~~~~ 1.0 107 1.1 107 1.2 107 1.3 107 1.4 107

Time [s]

Figure 7-7. Measured Pressures and Flow Rates for Experiment Mx80-9 (K0

Geometry) of Horseman and Harrington (unpublished) - Period around Second Gas Breakthrough.

97

8 SIMULATIONS OF GAS MIGRATION IN A K0 GEOMETRY: COMPARISON OF MODELS

Most of the testing of the models implemented for the GAMBIT project has been performed by attempting to reproduce the results from experiment Mx80-4A of Horseman and Harrington (1997). The recent addition of Mx80-9 (see Section 2.3.2) to the available experimental data sets provides a further opportunity to test the models for a case of linear flow, and, since this experiment was performed under slightly different conditions, the experimental results obtained may also reveal whether they are sensitive to the confining constraints imposed on the sample.

Data is also available from the hydraulic tests carried out prior to gas injection on the sample constrained in the Ko geometry. Interpretation of these tests could in principle provide information on the constitutive relationships required in the alternative model to predict the behaviour of the saturated clay. Modelling of the hydraulic tests is discussed in Section 8.1, and that of the gas injection histories in Section 8.2.

8.1 Modelling the ~ Geometry Hydraulic Tests

8.1.1 Mx80-9 Stage 2

As noted in Section 2.3.2, Stage 2 of experiment Mx80-9 involved injecting water into the sample at a constant rate of 2.2 J!L hf '. This has been modelled using GMClayW assuming that the sample was initially saturated with water in equilibrium at a measurable water pressure of 1MPa throughout. Other parameters for the model were calculated from the intrinsic hydraulic conductivity and specific storativity evaluated by standard analysis of the hydraulic tests by Horseman and Harrington (unpublished) (see Section 2.3.2). The water permeability used was 6 10-21m2 (including a correction indicated below) and the compressibility used was 7.5 10-8Pa-' . The compressibility was calculated from the specific storativity, Ss, of the sample by assuming that the storativity could be entirely represented by water compressibility, so that (Marsily, 1986)

8-1

with Pw • 1000 kg m3 and lf>w • 0.4. In Equation 8-1 it is assumed that the solid is incompressible and that the sample volume is constant. It is reasonable to assume that the solid is incompressible since it is much less compressible than the water. Although in practice the pore volume will not remain constant, a similar effect to the volume variation can be produced in the model by using a suitably increased value for the water compressibility. However, this also affects the water density in the flow terms (Equation 6-2) and so a small correction to the water permeability was made to compensate for the average error introduced in the water density.

98

Figure 8-1 shows a comparison of the results from the model and those from the experiment. There is reasonable agreement in the inlet pressure, although the increase is slightly more rapid initially for the model before quickly becoming slightly less rapid. The mid-plane pressures do not agree well. The mid-plane pressure in the experiment was initially about zero instead of at the expected equilibrium value of lMPa, suggesting that the sample was not fully equilibrated before the water injection commenced. The mid-plane pressure also responded much more rapidly to the injection in the model than in the experiment. The significant difference between the outflow rates is that in the experiment the outflow rate rises above the injection rate after about 5 l06s, whereas in the model it approaches the injection rate as would be expected. It is believed that the outflow rate became larger than the injection rate in the experiment because of a small leak from the surrounding pressurising fluid into the sample (Harrington, private communication).

8.1.2 MxS0-9 Stage 4

During the first part of Stage 4 of experiment Mx80-9 (before the inlet pressure is increased above 8.8MPa) it is believed that no gas enters the sample. However, as shown by the experimental data in Figure 8-2, there is a small outflow that initially increases to a peak at -2 l06s then drops away slowly. There is also a similar trend in the mid-plane pore-water pressure. One possible explanation for this is that it is caused by water in the end cap being displaced by the gas and forced into the sample. This possibility was examined theoretically in Section 7 .2.2.2 above. Here the hypothesis

3.0

2.5 ~ ~

6, 2.0

~ <I) 1.5 ~ ~

1.0

0.5

0.0 ~--~--,-----r------,----,-----..,--...L-

O.OE+O 2.0E+6 4.0E+6 6.0E+6 8.0E+6 1.0E+7

Time (s)

- Inlet Pressure (model) - Inlet Pressure (experiment)

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

- Mid-plane Pressure (model) - Mid-plane Pressure (experiment) - Flow Rate (model) - Flow Rate (experiment)

~ ~ ~

11) ..... ~

~ ~ 0

r:r:

Figure 8-1. Comparison of Modelled Results with Experiment Mx80-9 Stage 2 Hydraulic Test Results.

~ ~

~ ll)

~ "' "' ~ ~

99

9

8

7

6

5

4

3

2 -· 1 _...,..

0 ./ .....:;._--..,..--------,..---::__--,.------

O.OE+O 2.0E+6 4.0E+6

Time (s)

6.0E+6 8.0E+6

7

6

5 ...... --~

4 ~ ..:;;. ll)

~ 3 ~

~ 0

2 il:

1

0

- Mid-plane Pressure (Model) - Mid-plane Pressure (Experiment) - Flow Rate (Model) - Flow Rate (Experiment)

Figure 8-2. Comparison of Experiment Mx80-9 Stage 4 and Modelling of Water Injection under Similar Conditions.

has been tested by modelling the effect of a small flow of water into the sample under the relevant conditions using GMClayW.

The simulation was performed using the same values for the water permeability and compressibility as used for modelling Stage 2 of the experiment (Section 8.1.1 ). The sample was again assumed to be initially saturated with water at 1MPa. The inlet boundary condition was initially that water was present at a pressure of 1MPa. This pressure was increased linearly at a rate of 6 Pa s-1 to 8.8MPa after 1.3 106s. This is an approximation to the behaviour of the inlet gas pressure over the same period, and should be a reasonable representation of the water pressure since capillary pressures outside the sample should be small. From a time of 1.3 l06s a no-flow condition for water was applied. No inflow of gas was allowed at any time.

The results of the simulation are plotted against the experimental results in Figure 8-2. The mid-plane pore-water pressure in the experiment began above lMPa, suggesting the water was not in equilibrium initially. It is also greater than given by the model over most of the period, although the response to the injection is slightly more rapid in the model resulting in a mid-plane pore-water pressure above that in the experiment for a short time, and a broader peak. This may be due to the approximation made in the inlet boundary condition, giving an increase in the injection pressure that is too rapid initially. The fact that the mid-plane pore water pressure in the experiment does not return to its equilibrium value, as expected, is not understood. The flow rates out of the

100

sample differ significantly mainly in the times of their maxima. The outflow rate given by the model peaks at a slightly later time than in the experiment. This may be due to the approximation made by representing volume changes of the sample by using a larger water compressibility and corrected water permeability.

These simulations of the early part of Stage 4 of experiment Mx80-9 confirm the theoretical considerations of Section 7 .2.2.2 above in showing that the forcing of residual water from the end cap could provide a plausible explanation of the experimental results. Exact agreement would not be expected because of the uncertainty in the volume of residual water that could be present in the end cap, the anomalies noted in the experiment, which the model could not reproduce, and the approximations made in the modelling.

8.2 Modelling the K0 Geometry Gas Migration Test History

Modelling of the gas injection stage of experiment Mx80-9 using the two principle options now available in the GM Clay computer model is discussed Section 8.2.1; that using the alternative model implemented in the GMClayW computer code is discussed in Section 8.2.2.

8.2.1 Modelling Gas Migration via Crack propagation or by water Displacement

The Phase 1 computer model, GMClay, extended as discussed in Section 5, allows the use of two possible pathway models for the pre-breakthrough stage. It can either be assumed that the gas advances through the clay by creating fractures (crack propagation), or that the gas displaces water from existing capillaries. Tests have been performed using both these options. The multiple pathway facility, which allows gas flow through a range of different cracks simultaneously, has not been used, since the use of this facility in the modelling of experiment Mx80-4A did not significantly improve the results obtained (see Section 4.2).

GMClay is designed to model experiments similar to Mx80-4A, which have been performed under constant isotropic external stress (confining pressure) conditions. Although Mx80-9 was carried out under the constraints of a Ko configuration rather than under such constant stress conditions, it is assumed that it is reasonable to approximate the conditions of this experiment as ones of constant stress. The value of the confining pressure used in the fits described below is 9 .83MPa. This value falls within the range predicted for the swelling pressure of the sample (cf. Section 2.3). The confining pressure is a parameter that affects the dilation of the gas flow paths in the models in the GMClay program.

101

8.2.1.1 Crack Propagation

Two fits to the inlet gas pressure of Mx80-9 using the crack propagation pathway model option are shown in Figure 8-3. The pumping rate at which the upstream gas volume was reduced is also shown to indicate the points at which this was changed and thus identify the relationships between the transient responses seen in the upstream pressure and these changes. The parameters used in the simulations are shown in Table 8-1. Fit 1 attempts to reproduce the inlet gas pressure behaviour up to the end of the first shut-in (9.853 106s), while fit 2 concentrates on the period after the first shut-in. It is not possible to fit both these sections of the experimental results simultaneously with this model because, for a given pumping rate, the inlet gas pressure will always reach the same steady-state value in the model and this does not appear to be the case in the experiment. In Mx80-9 the pumping rate is the same after both pressure peaks. After the second peak it is easy to see that the steady-state pressure is approximately 9.2MPa. After the first, largest, peak the pressure drops to a minimum and then increases, and although it does not approach its steady state value it seems unlikely that the steadystate pressure would be greater than 9.0MPa.

In both fit 1 and fit 2 the critical stress intensity and initial crack length have been chosen so that breakthrough coincides with the peak in the inlet gas pressure as in the experiment. The half-width is chosen so that at breakthrough the permeability is much larger than required to support the gas flow. When accompanied by an appropriate time

~ ~

6 ~ C/J C/J <1) 1-< ~ ~ <1)

]

10.0

9.8

9.6

9.4

9.2

9.0

8.8

8.6

8.4

8.2

8.0

~--------------------------------------------------------------------~ 400

350

- 300 :;

250 ], <1)

200 ~ ~

150 .~ Q.,.

100 £ so

+---~----_, ________________ ~------~----+--------------+ 0

8.90E+6 9.40E+6 9.90E+6

Time (s)

1.04E+7 1.09E+7

- pressure (fit 1) -pressure (experiment) - pressure (fit 2) - pumping rate

Figure 8-3. Fits to the Inlet Gas Pressure of Experiment Mx80-9 using GM Clay with a Crack Propagation Pre-breakthrough Pathway Model.

102

Table 8-1. Model Parameter Values for Fits to Experiment Mx80-9 using GM Clay with a Crack Propagation Pre-breakthrough Pathway Model.

Model parameter Value for fit 1 Value for fit 2

Young's modulus 1.03 106 Pa 9.5 105 Pa

Crack half-width 8.0 10-6 m 8.0 10-6 m

Initial crack length 1.0 10-5 m 1.0 10-5 m

Critical stress intensity 1.0 103 Pa m 1/2 1.0 103 Pa m112

Number of cracks 100 100

Equilibrium radius 7.0 10-6 m 5.5 10-6 m

Time-dependence constant, A, 1.0 10-4 s-1 5.0 10-5 s-1

dependence (Nash et al., 1998) in the post-breakthrough behaviour, this results in the rapid pressure drop required immediately after breakthrough. Note that with no time dependence in the model the half-width would not affect the pressure after breakthrough, but the overshoots seen in the experimentally observed pressure transients suggest that a time dependence does exist. The half-width also determines the transit time for the propagating cracks to traverse the sample. However, the choice of halfwidth required to produce a sufficiently rapid pressure drop also ensures rapid crack propagation so that there are no significant secondary effects of this choice on the results obtained.

For fit 1 the equilibrium radius and Young's modulus are chosen in combination so that the steady-state pressures for the two pumping rates have the approximate values expected, and the decay of the pressure for the shut-in occurs at approximately the correct rate. The equilibrium radius is also chosen to give a large permeability so that the rapid pressure drop after breakthrough continues at the required rate to the minimum pressure. The same effects can be achieved by altering the number of cracks as by altering the equilibrium radius; this allows a choice of equilibrium radius giving a similar size of pathway at the confining pressure before and after breakthrough. Finally, the time dependence is set so that the overshoot of the pressure after breakthrough reaches the same minimum pressure as in the experiment.

The difference between the fit 1 and experimental inlet gas pressures at around 9.1 106s is due to the way in which the permeability changes with the pressure after the initial overshoot. The pressure recovers toward its steady-state value more rapidly in the model than in the experiment and therefore becomes too large by the time the pumping rate is altered. An alternative pathway dilation model may improve this (see Section 4.1). In the experiment there is also an overshoot in the inlet pressure for the second

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pumping rate. It is not been possible to reproduce this with any of the models. This is because the upstream pressure at the time the flow rate is reduced from 375 to 180 J.1L hr-1 seems already to be at or below the steady state value for the lower rate, so there would have to be a long time lag in the response of the permeability to changes in pressure to reproduce this behaviour. Introducing such a long time lag leads to other deficiencies in the model fit.

For fit 2, a slightly smaller equilibrium radius for the post-breakthrough flow is used than for fit 1, although it is still large enough to reproduce the rapid pressure drop after breakthrough. The smaller equilibrium radius in combination with a slightly smaller Young's modulus reduces the permeability more rapidly as the pressure drops, resulting in the higher steady-state pressure that matches that after the second pressure peak in the experiment at about 9.96 106 s. This has the consequence that fit 2 is a poor match to the inlet gas pressure of the experiment from after the first pressure minimum (9.029 106 s) until after the end of the first shut-in (9.853 106 s). The remaining model parameter, that controlling the time lag between changes in pressure and the corresponding changes in permeability, is chosen so that the value of the pressure for the second main peak matches the experiment. However, there remains some discrepancy between the model and experiment from the second peak to the steady state before the second shut-in, when the time dependence causes an oscillation in the pressure. This is due to the same problem as found in providing the match over the period of the first pumping rate in fit 1, that of obtaining an appropriate combination of the variation of steady-state permeability with gas pressure, and the time lag in the response of the permeability to pressure changes.

8.2.1.2 Water Displacement from Capillaries

Two fits to the inlet gas pressure of Mx80-9 obtained using the option in the computer model to allow the gas pathways to be created by displacement of water from capillaries are shown in Figure 8-4. The parameters used in the simulations are shown in Table 8-2. In fit 3, the parameters that control the post-breakthrough behaviour took the same values as in fit 1, while in fit 4 an attempt was made to reproduce the fit 3 results using a much smaller value for equilibrium radius, which is one of the parameters determining the post-breakthrough behaviour.

The inlet gas pressure curve for fit 3 is almost identical to fit 1. This is because they only differ in their pre-breakthrough models, and the capillary radius for fit 3 has been chosen to give breakthrough at the same pressure as fit 1 (and the experiment).

As found previously with the fits to experiment Mx80-4A there is an apparent inconsistency in the parameters used for fit 3. The capillary radius for pathway propagation is less than a hundredth of the size of the post-breakthrough radius at the confining pressure. This can be rectified by reducing the equilibrium radius and compensating for this by increasing the number of capillaries. However, the number of

104

capillaries required is then -1 Ou, and there is also some change to the pressure drop after breakthrough with such a large number of capillaries as shown by fit 4.

No results are shown of attempts, using the water displacement pathway propagation model, to specifically match the results from the second gas injection cycle from 9.853 106s because for this stage of the experiment there is no difference between this model and the one discussed in Section 8.2.1.1; the results, for example, of fit 2 of Figure 8-3 are equally applicable to the water displacement model for this part of the experiment.

Table 8-2. Model Parameter Values for Fits to Experiment Mx80-9 using GM Clay with a Water-displacement Pre-breakthrough Gas Invasion Model.

Model parameter

Young's modulus

Capillary radius

Number of cracks

Equilibrium radius

Time-dependence constant, A

10.0

9.8

9.6 ~ 9.4 0..

6 9.2 Q.)

~ 9.0 V) V) Q.) 1-<

0.. 8.8 ..... Q.)

] 8.6

8.4

8.2

Value for fit 3

1.03 106 Pa

1.808 10·8 m

100

7.0 10·6 m

1.0 10·4 s·1

Value for fit 4

1.03 106 Pa

1.808 10·8 m

1.0 1010

7.0 10·8 m

1.0 10·4 s·1

350

300 ... i 250 ~

50

8.0 +---~---4--------~----~--+--------+ 0

8.90E+6 9.40E+6

- pressure (fit 3) - pressure (fit 4)

9.90E+6

Time (s)

1.04E+7

-pressure (experiment) - pumping rate

1.09E+7

Figure 8-4. Fits to the Inlet Gas Pressure of Experiment Mx80-9 using GM Clay with a Water-displacement Pre-breakthrough Gas Invasion Model.

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8.2.2 Modelling with the Alternative Model Developed in Phase 2 {GMCiayW)

Two attempts at reproducing the history of the inlet gas pressure of Mx80-9 using the alternative macroscopic model implemented in GMClayW are shown in Figure 8-5. The parameters used in these simulations are shown in Table 8-3. Fit 5 attempts to match the pressure curve over the first set of decreasing pumping rates (8.886 l06s to 9.853 l06s), and fit 6 attempts to match the pressure curve from the end of this period onward.

Both fits represent the initial pressure before the first peak satisfactorily. However, this is just the isothermal compression of the gas prior to entry into the sample, and the only issue in trying to fit this is matching the peak pressure reached.

Fit 5 is not a good fit to the inlet gas pressure of the experiment over the intended period after the first breakthrough. There are two deficiencies in the model pressure profile in this region:

a) the pressure is consistently too high, and

b) the model fails to represent "overshoots" seen in the experimental results, where, following breakthrough or the first change in flow rate, the upstream gas pressure falls below the steady-state value of the upstream gas pressure for the particular flow rate.

10.0 400

9.8 350 9.6

~ 9.4 300 -:;

0..

6 250 ~ 9.2 _.:; Cl) Cl) a 9.0 200 ~ <I) ~ ~ M 0.. 8.8 150 .s .... Cl) 0.. ] 8.6 §

100 0.. 8.4

8.2 50

8.0 0

8.90E+6 9.40E+6 9.90E+6 1.04E+7 1.09E+7

Time (s)

- pressure (fit 5) - pressure (experiment) - pressure (fit 6) - pumping rate

Figure 8-5. Fits to the Inlet Gas Pressure of Experiment Mx80-9 using GMClayW.

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Table 8-3. Model Parameter Values for Fits to Experiment MxB0-9 using GMClayW.

Model parameter Value for tit 5 Value for tit 6

Number density of capillaries 3.0 108 m -2 1.0 1010 m-2

Critical gas porosity 2.0 10-3 1.4 10-4

Water permeability 5.0 10-20 m2 3.0 10-21 m2

Initial volume fraction of solid 0.5705 0.5719

Volume parameter, a1 O.OPa-1 O.OPa-1

Volume parameter, a2 O.OPa-1 O.OPa-1

Water compressibility, f3 5 10-10 Pa-1 5 10-10 Pa-1

The first is due to the value for the volume fraction of clay solid used in the calculations. In the actual experiment there was a period, not shown in Figure 8-5, ending at 8.886 106s during which the inlet gas pressure was held at a constant value of 8.77MPa. During this period no gas flow out of the core was observed. Due to the length of the period, and the constancy of the gas pressure, it is therefore assumed that no significant quantities of gas can enter the core at this gas pressure (except perhaps in small quantities by diffusion (see Section 7.2.2)). Since there was a water back pressure of 1 MPa, this implies that the difference of 7. 77MPa between the gas and water pressures that existed during this period must be exceeded before gas will enter the core. In the model this is interpreted as the minimum value taken by the swelling pressure which, assuming a constant total volume, is defined by the (constant) solid volume fraction, or equivalently the void ratio of the saturated system (cf Equation 6-6). The initial solid volume fraction for fit 5 was chosen to give a swelling pressure of 7 .77MPa. In fact, the void ratio given for this sample by Horseman and Harrington (unpublished) is 0.735 (Table 2-5), which corresponds to a swelling pressure of 8.7 MPa; i.e. the gas entry pressure is predicted to be greater than 9. 7 MPa, as observed. The choice of solid volume fraction dictated by these considerations makes it difficult to match the pressure seen after initial breakthrough in the experiment.

The values of the water permeability and number density of capillaries for fit 5 have been chosen to reproduce as closely as possible the shape (rather than absolute value) of the pressure transients for the second pumping rate and the first shut-in (zero pumping rate) from the experiment (for details of the experimental stages, see Table 2-6). Given the high value of the inlet gas pressure at the first change of pumping rate for fit 5 compared with the experiment, the focus was on matching the relative changes after this point, rather than the absolute values measured. The values chosen for these two parameters does also produce the rapid pressure drop after breakthrough seen in the

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experiment. The value of the first peak in the pressure was refined by the choice of value for the critical gas porosity.

The adjustable time lag that was built into the Phase 1 model, in part to facilitate the modelling of overshoots of the type seen in this experiment (Nash et al., 1998), is not present in this alternative model Without an adjustable time dependence in the model, it is difficult to produce the overshoots in the pressure response seen in the experiments. Reducing the water permeability significantly can produce a similar result, but in the case of fit 5 a reduction in the water permeability has not been made, since this would reduce the rate at which the pressure drops after breakthrough as well as reducing the total pressure drop. In fit 6 described below, a reduction in water permeability did lead to a reverse type of overshoot at the start of the second pressure peak that followed the reinstatement of the flow rate at 375 JlL hr-\ with the pressure rising above the steady state value.

The pumping rate after each peak in the inlet gas pressure (at 8.994 106 s and 9.945 106 s) is the same (375 JlL hr-1

). If the water saturation change between the two cycles is small it would be expected that the steady-state pressure would be the same for the two stages at this flow rate. This is indeed found to be the case in the modelling studies. Steady state is not reached for the pumping rate after first breakthrough in the experiment before the rate is changed, so the steady state pressure is not known for this part of the cycle. As already noted, it does not seem likely to be as high as for the later period of pumping at the same rate, or, if it is, the pressure has evidently fallen much further below the steady-state value after breakthrough than has been observed in other experiments (Horseman and Harrington, 1997). However, although it was difficult to match the behaviour of the experiment after the first peak, this is not the case when fitting the pressure profile found in the second cycle of gas injection, which is well matched by fit 6.

In addition to reproducing the steady-state inlet gas pressure at 1.0198 107 s the water permeability in fit 6 has been chosen to reproduce the second peak in pressure and the subsequent approach to the steady state. The pressure transient for the second shut-in in the experiment has also been matched by the choice of the number density of capillaries.

8.3 Discussion of Approaches to Modelling the BGS Experiments

Three different basic approaches to the simulation of gas migration in compacted bentonite have now been investigated in the GAMBIT Club programme:

a) The Phase 1 crack propagation model, with pathway dimensions which dilate with increasing gas pressure, and the facility for allowing some time delay between gas pressure variation and the concomitant change in pathway size.

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b) The extension (Section 5) to the Phase 1 model to allow for pathway propagation by displacement of water rather than crack propagation; after breakthrough, this model is the same as model (a).

c) The alternative model (Section 1) in which pathway propagation is not explicitly represented; rather, the introduction of gas porosity and hence gas permeability is modelled in terms of the drained response of the clay to the application of gas pressure.

These models have been used to model a number of data sets provided from the SKBfunded programme at BGS (Horseman and Harrington, 1997, unpublished), in particular all of them have been applied to experiments Mx80-4A and Mx80-9. Two questions that can now be asked are:

a) Can one model be shown to be superior to any other for modelling these experiments, and if so what are the characteristics that make it so?

b) Do the modelling results provide any discrimination between different mechanisms that have been proposed to explain gas migration in bentonite?

For experiment Mx80-4A, representative results are shown in Figures 4-1, 5-1, and 6-9 for models (a), (b), and (c) above, respectively. For experiment Mx80-9, the corresponding Figures are 8-3, 8-4, and 8-5.

Both these experiments involve two cycles of gas injection (at sometimes a series of different rates) followed by a shut in during which the upstream pressure evolved passively with no pumping. For both experiments it has proved difficult to match both of these cycles equally well with a common set of parameters. This suggests that there is some evolution of the system between the cycle that follows initial breakthrough and the second cycle, possibly through relaxation of changes associated with the breakthrough event itself. Part of the purpose of model (c), which was developed in this Phase of the GAMBIT Club work programme, was to investigate whether displacement of water from the sample might provide the required evolution of the system, although, at least from the evidence of the manually adjusted fits, there is no clear difference in this respect between the three models.

Mx80-4A appears to present less of a challenge to the models in that, recognising that the models do not fit both cycles of the experiment equally well, they nevertheless can be claimed to produce reasonably satisfactory representations of the experiments.

For experiment Mx80-9, the difference between the first and second cycles of the experiment is more marked. The first cycle shows overshoots involving the pressure passing through a minimum following both the initial gas breakthrough and the subsequent halving of the pumping rate; if the steady-state associated with the initial

109

pumping rate is the same as that observed for the same rate in the second cycle (as generally found in earlier experiments (Horseman and Harrington, 1997)) then these overshoots are much larger than seen in previous experiments. Models (a) and (b), with the facility to explicitly incorporate a time lag between changes in gas pressure and the consequent changes in permeability, are better able to represent these marked transients. Although the incorporation of this time lag was deemed to be required to explain overshoots of the type seen in experiment Mx80-9 and to explain the hysteresis seen in the effective gas permeability evaluated from the experimental data (see, for example, Figure 6-11 ), explanation of the physical processes that might cause the time lag were a matter for conjecture. It required, for example, the assumption of a time dependent readjustment of pathway shapes after breakthrough or an element of creep in the response of the clay to pressure changes. Model (c) does allow for some time dependence in the response of the clay permeability to changes in gas pressure: the immediate initial response to an increase in the gas pressure would be to compress the clay (primarily through compressing the pore water), but a slower response would be produced by expulsion of water, further consolidating the clay, and simultaneously increasing the gas porosity if in a constant volume cell. This mechanism, however, only allows gas porosity to reduce by the small amount resulting from water expansion when the pressure is reduced (assuming that there is no water available to re-enter the clay to replace that removed, and in any case this is a comparatively slow process), so that this model appears not to be able to produce the minima in the pressure profile seen in experiment Mx80-9.

In the experiments on bentonite pastes reported by Horseman and Harrington ( 1997), gas injection produced a response in measurements of the pore water pressure in the clay. The same was true of experiments on Boom Clay also carried out by Horseman and Harrington (see Rodwell, 2000), and, on apparent gas entry into the sample, in the radial flow experiment Mx80-8 of Horseman and Harrington (unpublished) described in Section 2.3.1. However, in experiment Mx80-9, gas entry into the clay produced no change in the measured mid-plane pore water pressure (see Figure 2-3), although responses were seen in hydraulic tests and prior to gas entry (cf Section 8.1).

In interpreting the experimental results, a particular difficulty that remains is the characterisation of the nature of the gas flow pathways that are created, together with the mechanism by which they are formed. In remains unclear why pathways, if formed by fracturing, do not reseal when the gas pressure falls well below the initial gas breakthrough pressure. The alternative model described in Section 1 attempts to account for this in terms of the effect of the removal of some water from the clay, and this does indeed prevent the gas-filled porosity disappearing until replacement water is provided. This approach simultaneously provides an explanation of the resealing behaviour. In the crack propagation model, the failure to reseal is accounted for in terms of a change of pathway shape from an initial elliptical shape to a circular shape more resistant to closure. In the model of gas migration by displacement of water from pre-existing pathways, the issue of pathway closure does not arise in the same way, as

110

the pathways are assumed to be permanently present (although capable of dilation). The difficulty with the last model is reconciling the need for very small pathways to produce a capillary entry pressure large enough to match the observed threshold pressure for gas entry, while at the same time providing a credible account of the post breakthrough gasfilled porosity and gas permeability (cf Section 5.2).

In all the gas migration models considered, the gas porosity associated with the gas permeability needed to support the applied gas flow, is very small. For example, in the simulations of experiment Mx80-9 described in Section 8.2, the effective gas porosity, at the approximate steady state for the second period of injection at 375 ~ hr-' (cf Figures 8-3- 8-5), was about 5 10-8

- 6 10-7 for fit 2 with the crack propagation model, about 5 1 o-4

- 6 1 o-3 for fit 4 using the displacement of water model, and about 1 1 o-4 -

6 10-4 for fit 6 using the alternative GMClayW model. The different gas-filled porosities for fits 2 and 4 are a result of the vastly different numbers of capillaries assumed (Tables 8-1 and 8-2), the large number being chosen in fit 4 in order to have small capillaries with a large capillary entry pressure. For fit 6, the assumption of a critical gas porosity of 10-4 means that it is only the porosity in excess of this that contributes to the permeability (Equation 6-13). If the pathways are like capillaries, then the gas-filled porosity needed to provide a given permeability will increase inversely with the capillary radius. A larger capillary size (e.g. -1 J.Lm radius in fit 2) gives a more reasonable number of capillaries but results in gas porosity that is almost negligible. If the gas pathways remain more crack like, then the numbers required to produce a given permeability might be smaller, because the cross-sectional area per pathway might be larger than for a capillary. For example, 3 cracks which are 100 J.Lm by 1 J.Lm would provide a permeability of aboutl0-20 m2 in a 5 cm diameter core, but only a porosity of about 2 10-7

•

Experimental data on the number, dimensions, and distribution (including tortuosity) of gas migration pathways would be valuable in helping to evaluate different gas migration mechanisms.

The very small gas porosity needed to support gas migration means that:

a) The volume dilation or compression of the clay (depending on boundary conditions) required to accommodate the gas flow is correspondingly small.

b) The stress/strain boundary conditions would be expected to have only a small effect on flow behaviour (see Section? .1.3).

c) The quantities of water expelled from the clay, either directly or indirectly, will be small.

The parameters needed for the numerical models that have been developed vary in the extent to which they are at least in principle well defined physical quantities (even if in

111

practice some adjustment is made to them to improve the fits to experiment) as opposed to simply being treated as fitting parameters.

For the crack propagation model developed in Phase 1 of the GAMBIT Club programme, a discussion was provided by Nash et al. (1998) of the relationship between the values used for parameters that were amenable to experimental determination and published values of those parameters. It suffices here to note that the parameters used in applying the model to experiment Mx80-9 (Section 8.2.1) were similar to those used by Nash et al. (1998).

For the model based on the assumption that gas pathways are formed by displacement of water from pre-existing channels (Section 5.2), there is no change, compared to the model based on crack propagation, in the parameters required to describe the postbreakthrough behaviour. The new parameter required here is the specification of the pre-breakthrough capillary radius that controls the gas entry pressure. As discussed in the paragraphs above, and in Section 5.2, this requires very small capillaries which then must be present in numbers which seem unreasonably large to describe the postbreakthrough permeability. Possible rationalisation of this dichotomy between the requirements of pre- and post-breakthrough behaviour have been offered in terms of constrictions in the pathways that control the entry pressure (Section 5.2) and alternative pathway shapes (above), but these are speculative.

Horseman and Harrington ( 1997) estimate that the average interlayer spacing for Mx80 bentonite at a compaction corresponding to a swelling pressure of -1 OMPa will contain about two layers of water molecules ( -0.5 nm). Clearly gas invasion of such small spaces by a capillary displacement mechanism is not feasible as the capillary entry pressure calculated using the Young-Laplace equation would be -300 MPa. It is also interesting to consider whether in fact, for highly compacted bentonite, it is appropriate to use this relationship, in which it is effectively assumed that the pressure of the water in the interlayers is at the externally measured equilibrium water pressure. Since it is postulated that it is this interlayer water that generates the swelling pressure, it must be assumed that the water pressure (or at least a component of this pressure if it has tensor properties) in these layers must be of similar magnitude to the swelling pressure, and not at the externally measured value. It is generally assumed that the structure of the clay fabric will also give rise to the presence of water in larger interstack spacings. Horseman and Harrington ( 1997) estimate that the average thickness of the water films in a clay with a swelling pressure of around -10 MP a will be -3-4 nm. This is still significantly smaller than the spacing of -15 nm required to give a capillary entry pressure of 10 MP a, although it is perhaps plausible that the distribution of inters tack spacings may result in some that are of the required magnitude. Water in inters tack spaces may be at a pressure not significantly different (in relation to the swelling pressure) from the external equilibrium water pressure.

112

The alternative macroscopic model introduced in Section 1 requires a different set of parameters to the other two models; in particular it requires some hydrogeological properties of the clay. In the fits to the experiments (Mx80-4A and Mx80-9) it was always assumed that the swelling behaviour of the clay was given by Equation 6-6. The water compressibility was also assumed, in fitting the gas migration test results, to have its literature value of -5 10-10 Pa. Parameters for which estimates were available were the initial volume fraction of solid and the water permeability (7 10-21 m2 for Mx80-9 (Horseman and Harrington, unpublished)), although these were adjusted to improve the matches between simulated and observed results, In the case of the solid volume fraction, the adjustments were less than 5 1 o-3

, and in the case of the water permeability less than an order of magnitude (and in most cases significantly smaller than this). These departures from the experimental estimates are probably slightly larger than the measurement uncertainty in these parameters (although it should be remembered that there is uncertainty in those parameters that have not been adjusted as well). The critical porosity and number of capillaries are essentially fitting parameters. These have been discussed above.

In order to resolve some of the uncertainties in the mechanism of gas migration in bentonite and to constrain more of the parameters used in the models (as well as to provide discrimination in model selection), it would be very valuable to have experimental data on the nature of gas migration in compacted bentonite (numbers and dimensions), the total gas porosity created, and the volume of water displaced.

113

9 RECOMMENDATIONS FOR FUTURE WORK PROGRAMME

The provisional intention for the planned Phase 3 of the GAMBIT Club programme was the translation of ideas developed and tested within Phases 1 and 2 of the programme into a computational tool that could be applied at the scale of deposition holes (or similar disposal configurations) for simulating the migration through bentonite buffers of gas produced in or from canisters for high-level waste. This scale up process requires a number of decisions about how it should be carried forward. In particular, it needs to be decided whether the concepts that have been developed provide a satisfactory description of gas migration in compacted bentonite that can form the basis of the assessment tool, or whether further developments are required, and, if the concepts are viewed as satisfactory, which elements of the preliminary evaluation models should be included in the final model. Issues of practicality such as the availability of suitable data and computational complexity may influence this choice.

It is evident that none of the conceptual models that have been explored in Phase 1 and 2 of the GAMBIT Club programme provide a thoroughly quantitative description of the detailed experimental data provided by Horseman and Harrington (1997, unpublished). This indicates that there may be some deficiency in the modelling approaches, but it is not clear to what extent this is due to the deliberate approximations that have been made, or whether there is a significant fundamental feature of the behaviour of the clay that is missing from the models. The shortage of data covering a range of different conditions makes this assessment difficult, especially when results from different sources appear not to be consistent, and there is significant variability amongst data from a single source. On the other hand it has proved possible using the models that have been developed to provide a reasonable representation of all the features seen in the experimental profiles.

One feature of all the conceptual models that have been examined is that only simplified treatments of the mechanical response of the clay to the stresses induced by the applied gas pressure have been investigated. The treatments included elastic dilation of a crack in the crack propagation model, the response of pathway dimensions to changes of gas pressure post breakthrough in the models with explicit representation of pathways, and the volumetric swelling behaviour of the clay in the macroscopic alternative model. The time dependence of the response to pressure changes investigated in the Phase 1 work, may perhaps be interpreted as a representation of creep in the clay. No attempt to implement a detailed constitutive model of the mechanical behaviour of the clay has been made for a number of reasons. First, it would introduce enormously increased complexity into the model, when really this is not justified by the understanding that currently exists of the effect of the stress field in clays on gas migration behaviour, beyond the simple concepts that have been investigated (e.g. that there is a relationship between entry pressure and swelling pressure). Secondly, it seems unlikely that in practical applications of any model of gas migration in bentonite, the detailed knowledge of the stress state of the clay needed to use the model would be generally

114

available, as this would depend on the evolution of the buffer within a repository. For completeness, however, an indication of the traditional theories that could be used to provide a framework for representing stress-strain behaviour in a model of bentonite in an unsaturated state has been provided in the Appendices to this report. The fact that in the modelling studies, gas migration in bentonite involves only the creation of very small gas porosities suggests that the mechanical perturbation of the clay and hence of its stress state would be small, and therefore that it is reasonable to treat the mechanical response of the clay in an approximate, volumetrically averaged way, as is done, in particular, in the alternative model developed in Phase 2 of the programme.

Two issue to be considered in the upscaling of a prototype model to a canister-scale model are (a) the nature of the interface between the gas source volume and the bentonite buffer, and (b) whether the geometry of the canister-buffer system affects gas entry pressures or gas migration behaviour.

If gas is generated within a canister and escapes from a small hole in the canister, the gas could accumulate locally around that hole (perhaps creating a small depression in the bentonite buffer) and enter the bentonite only from that local accumulation when its pressure reached the gas entry pressure. Alternatively, the gas could spread out from the defect region along the interface between the canister and the bentonite, forming an extended source over a large part of the canister surface. This may only occur when the gas pressure reaches a level sufficient to force the bentonite back from the canister surface. Whether, as the gas pressure rises (presumably to a value similar to the stress exerted by the bentonite on the canister), the gas would first develop pathways between the canister and the bentonite or through the bentonite itself has not been definitively established. A similar issue has been raised in discussion of the experiments carried out by Horseman and Harrington ( 1997), and in this case it has been argued that the route followed by the gas is through the bentonite. If the gas is generated by corrosion of a carbon steel canister overpack, then an extended source may be produced directly. To address these two conditions, it seems that an upscaled model should be capable of performing simulations in cylindrical or spherical coordinates.

It is assumed for the present that the gas entry pressure and gas migration behaviour would be the same for gas from a cylindrical canister as found in laboratory experiments in linear flow geometries. Some question is raised about this by the results of the radial flow experiment Mx80-8 of Horseman and Harrington (unpublished) (see Sections 2.3.1 and 7 .2.1 ), for which a very high initial gas entry pressure was observed; this has been rationalised here in terms of the shape of the gas source term. Even if the gas pressure is higher for gas entry via induced fractures that are formed perpendicular to the curvature of a surface with a concave inwards curvature (because of the form of the stress field around such a geometry), in a cylindrical geometry it should also be possible to form radial fractures which are initiated at normal gas entry pressures.

115

The precise mechanism by which gas traverses water saturated, highly compacted, bentonite remains a matter of contention, as discussed in this report and the Phase 1 report (Nash et al., 1998). The GAMBIT Club has favoured the mechanism of gas pathway formation by crack propagation, and the recent Mx80-8 radial flow experiment of Horseman and Harrington (unpublished) shows evidence of discrete events that may be associated with fracture formation. However, while modelling explicit crack propagation in one direction has been shown to be feasible (Nash et al., 1998), the upscaling of the model that was developed to three dimensions is more difficult because of the expected dependence of the direction of crack propagation on the (unknown) stress distribution and because of the very limited knowledge of the morphology of gasinduced fractures in bentonite. As has been previously discussed, the simulation studies have shown that the crack propagation phase in gas invasion of bentonite (if this is the mechanism by which it occurs) is likely to be a fast process relative to the timescales over which gas generation might occur. It therefore should not be important to represent the crack propagation phase explicitly in an assessment of gas migration in bentonite.

At this point it may be helpful to reiterate the important observed features of gas migration in compacted bentonite that it is considered a satisfactory assessment model should reflect:

a) There is a threshold pressure below which gas does not enter the bentonite, except at a very low rate by diffusion (note, however the contrary evidence to this presented in Section 2.1).

b) Both gas entry to the clay and gas flow after breakthrough results in very little desaturation of the clay.

c) The gas permeability after breakthrough appears to be a function of the gas pressure, decreasing with decreasing gas pressure.

d) Once gas pathways have been created, gas continues to flow at pressures significantly below the threshold pressure required for initial invasion of the clay.

e) Gas pathways will reseal given the right circumstances, the most important of which seems to be the passage of water through the clay. Given the likely significance of flow of water into the clay for resealing, possibly through reswelling, it seems desirable that the model should also allow for water flow and include an element of coupling between water and gas flows (as seen in some experiments, although not yet properly understood).

The model developed in the Phase 1 programme and its derivatives discussed in this report do not deal with the resealing, although the alternative model of Section 1 does.

116

Given that the alternative model introduced in Section 1 is the only one of the models discussed to address all the issues listed above, it seems the most obvious choice to use as the basis of an assessment tool. It also has the advantage (as part of the reason for developing it) that there is nothing in the conceptual model that would limit its extension to two- or three-dimensions. However, there are some concerns about some aspects of the alternative model:

a) The representation of gas pathway propagation in the model honours the relationship, accepted within the GAMBIT Club, between swelling pressure and gas entry pressure, but may not be accurate in its representation of the gas-filled porosity created, only allowing for the creation of this porosity by "squeezing" of water from the clay. There is no allowance for the possibility of some direct displacement of water, and some of the model parameters (e.g. the critical gas porosity) are not well constrained.

b) The current permeability model (dependence of permeability on porosity) based on a bundle of capillaries approach seems to lead to an unreasonably high density of capillaries (but there is only indirect experimental evidence to suggest that only a low density of pathways is involved), and may not properly represent the nature of the gas pathways that are actually formed.

c) The model does not include the explicit time dependence that was incorporated into the other two models considered; this proved useful in representing some of the hysteresis-like features of the experimental results. However, justification of this time dependence in terms of verified physical mechanisms is not available.

If the alternative model of Section 1 is adopted as the basis of the assessment tool, it is worth considering what improvements to the model might be made to address some of these issues.

A possible extension to the model that might address some of these issues is the introduction of a dual permeability feature to the model. The fact that the clay may have a more complex fabric than the homogeneous structure assumed in all the models investigated so far was discussed in Section 8.3. In particular, it may be useful to represent the distinction between interlayer water and interstack water mentioned in Section 8.3. The main contribution to the permeability would come from the part of the clay structure containing inters tack water, but the swelling behaviour would be controlled by the interlayer water. Gas would only flow through the interstack spaces. The water in the interstack spaces might have an actual pressure essentially the same as that of the external equilibrium water pressure, whereas that of the interstack water would be treated in the same way as in the existing model. In order to reflect the impermeability of the clay to gas until a threshold pressure was reached, it would have to be assumed that the interstack spaces were not connected until the gas pressure reached the threshold value (typically the swelling pressure). The opening of these

117

pathways to gas would have to be a feature of a revised gas permeability model. These ideas would need to be investigated more deeply to determine whether they would significantly improve the existing model. It should be possible to implement the proposed extension in such a way that the resulting program could be used to model gas invasion by direct displacement of water without any pathway dilation, as well as via a pathway opening mechanism. The proposed extension may help to address points (a) and (b) above.

There is no apparent reason why an explicit time delay between changes in gas pressure and changes in gas permeability should not be incorporated into the model (point (c) above) in a way similar to the incorporation of such a feature into the Phase 1 model. The physical justification of such a feature, however, is not clear at present; presumably it would be a consequence of some sort of creep in the mechanical behaviour of the clay.

It is recommended that the incorporation of these features into the alternative model of Section 1 be investigated, and if they improve the model satisfactorily then the model should be upscaled to allow calculations to be carried out on the scale of single waste canisters and their surrounding buffer material.

It is, however, likely, that the strength of the conceptual foundations on which the model is based will remain in doubt until more specific experimental data on the nature of gas migration pathways in bentonite become available, as discussed in Section 8.3.

118

119

REFERENCES

Abramowitz, M., and Stegun, LA., 1965. Handbook of Mathematical Functions. With Formulas, Graphs and Mathematical Tables. Dover.

Borgesson, L., Karnland, 0., and Johanesson, L-E., 1996. Modelling of the Physical Behaviour of Clay Barriers Close to Water Saturation. Eng. Geol., 41, 127-144.

Biot, M.A., 1941. General Theory of Three-Dimensional Consolidation. J. Appl. Phys., 26, 182.

Carslaw, H.S., and Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford University Press.

Daeman, J., and Ran, C., 1996. Bentonite as a Waste Isolation Plant Shaft Sealing Material. Albuquerque, NM: Sandia National Laboratories Report SAND96-1968.

Donohew, A., Horseman, S.T., and Harrington, J., 2000. Gas Entry into Unconfined Clay Pastes at Water Contents between the Liquid and Plastic Limits. In press.

Dullien, F .A.L., 1992. Porous Media: Fluid Transport and Pore Structure - 2nd Edition. San Diego, CA: Academic Press.

Fredlund, D.G. and Morgenstem, N.R., 1977. Stress State Variables for Unsaturated Soils. ASCE J. Geotech. Eng. Div. GT5, 103,447.

Fredlund, D.G. and Rahardjo, H., 1993. Soil Mechanics for Unsaturated Soils. New York: Wiley.

Galle, C., 1998. Migration des Gaz et Pression de Rupture dans une Argile Compactee Destinee a la Barriere Ouvragee d'un Stockage Profond. Bull. Soc. Geol., 169 (5), 675-680.

Graham, J., Gray, M., Halayko, K.G., Hume, H., Kirkham, T., and Oscarson, D., 1998. Gas Breakthrough Pressures in Compacted Illite and Bentonite. Preprint supplied by G. V ercoutere.

Gray, M.N., Kirkham, T.L., Wan, A.W.-L., and Graham, J., 1996. On the Gasbreakthrough Resistance of Engineered Clay Barrier Materials Proposed for Use in Nuclear Fuel Waste Disposal. Presented at the CNS Int. Conf. on Deep Geological Disposal of Radioactive Waste, Winnipeg, Manitoba, 16-19 Sept. 1996. Canadian Nuclear Society.

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Horseman, S.T., and Harrington, J.F., 1997. Study of Gas Migration in Mx80 Buffer Bentonite. BGS Internal Report WE/97/7 to SKB.

Hume, H.B., 1999. Gas Breakthrough in Compacted Avonlea Bentonite. MSc Thesis, University of Manitoba, Canada.

LAPACK User's Guide, 1994. Second Edition. SIAM.

Marsily, G. de, 1986. Quantitative Hydrogeology: Ground water Hydrogeology for Engineers. San Diego and London: Academic Press.

Nash, P.J., Swift, B.T., Goodfield, M., and Rodwell, W.R., 1998. Modelling Gas Migration in Compacted Bentonite. AEA Technology, AEAT-2882 (also issued as POSIV A Report 98-08).

Pusch, R., Ranhagen, L., and Nilsson, K., 1985. Gas Migration through MX-80 Bentonite, Final Report. Nagra Technical Report 85-36.

Rice, J.R. and Cleary, M.P., 1976. Some Basic Stress Diffusion Solutions for FluidSaturated Elastic Porous Media with Compressible Constituents. Rev. Geophys., 14, 227.

Rodwell, W.R. (ed), 2000. Research into Gas Generation and Migration in Radioactive Waste Repository Systems (PROGRESS Project): Final Report. European Commission Report EUR 19133 EN.

Rodwell, W.R., Harris, A.W., Horseman, S.T., Lalieux, Ph., Muller, W., Ortiz Amaya, L., and Pruess, K., 1999. Gas Migration through Engineered and Geological Barriers for a Deep Repository for Radioactive Waste. A Joint EC/NEA Status Report published by the European Commission, European Commission Report EUR 19122 EN.

SKB, 1999. Deep Repository for Spent Nuclear Fuel: SR 97- Post Closure Safety. SKB Technical Report TR-99-06.

Tanai, K., Kanno, T., and Galle, C., 1997. Experimental Study of Gas Permeabilities and Breakthrough Pressures in Clays. In: Proceedings of the Scientific Basis of Nuclear Waste Management Conference Number XX, Boston, USA, December, 1996. Material Research Society.

Terzaghi, K., 1943. Theoretical Soil Mechanics. New York: Wiley.

Timoshenko, S.P., and Goodier, J.N., 1951. Theory of Elasticity: Second Edition. McGraw-Hill.

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Vercoutere, G., to be published. ANDRA's Strategy for the Treatment of Gas-Related Issues. In: Proceedings of an EC/NEA Workshop on Gas Generation, Accumulation and Migration in Underground Repository Systems for Radioactive Waste: Safety Relevant Issues, Reims, France, 26-28 June, 2000. NEA.

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ACKNOWLEDGEMENTS

The work reported here was funded by the members of the GAMBIT Club: SKB, ANDRA, Nagra, Posiva Oy, JNC and ENRESA. The authors are grateful to these organisations for their encouragement of the work.

The authors are also grateful to Steve Horseman for making available to them. unpublished data from the radial and ~ geometry flow experiments carried out at BGS by himself and Jon Harrington on behalf of SKB.

123

A1 APPENDIX 1: DEVELOPMENTS TO THE PHASE 1 MODEL

A 1.1 Linear Equation Solver

The Phase 1 model used a simple solver based on Gaussian elimination to calculate solutions for the sets of linear equations resulting from the multi-dimensional NewtonRaphson iteration scheme. For Phase 1, this solver was sufficient as less than 100 equations were being solved on each occasion. However, during Phase 2, as the complexity of the model and the number of equations have increased (it is expected that up to 10 000 equations may be required with the fully implemented multiple pathway facility), the efficiency of the simple solver has become unacceptable. Therefore, the linear solve algorithm has been developed to use a new solver. The solver currently implemented is a banded matrix solver from LAPACK (LAPACK User's Guide, 1994). To implement this solver, some reformatting of data has been necessary in the C++ code so that it can be passed into the FORTRAN LAPACK routine.

In addition, with the use of the multiple pathway facility the complexity of the set of linear equations has increased. The linear equations can be represented as

Al-l

where J1, J2 are the Jacobian elements representing the derivatives of the discretised conservation equation for the "inlet" block,

J3 is the vector of Jacobian elements representing the derivatives of the remaining equations with respect to the inlet block pressure,

J4 is the matrix of Jacobian elements representing the remaining derivatives of the equations,

x1, x2 are the changes to the independent variables to be calculated, r 1, r2 are the residuals of the discretised equations.

With multiple pathways the Jacobian entries, J1, J2 and J3, do not conform to the banded structure of the remainder of the Jacobian, J4, required by the banded matrix solver, because the inlet end block covers all the pathways. To enable the banded matrix solver to be used in this case the equations are therefore rewritten and solved in two parts:

Al-2

and

Al-3

Now the solution for x1 and x2 can be calculated from the solutions for y and z, since

124

Al-4

and

Al-5

A 1.2 Adjustments to the Crack Propagation Model

In the Phase 1 model discontinuities were observed in the dilation near the crack tip during propagation as the length of the final grid blocks varied and new blocks were added. The two modifications described below have been made to the discretisation in the Phase 2 model in order to eliminate these discontinuities.

The discretisation used in the Phase 1 model assumed that the dilation and pressure were constant within each grid block (with boundary values calculated as averages between grid block values). In the latest model it is assumed that the dilation varies linearly between values calculated at the nodes that define the grid blocks (pressure varying as before). The dilation for each grid block is then calculated from the node values as the average dilation over the length of the grid block.

A further contributing factor to the discontinuities was that the crack was closed at the tip. Since the tip dilation was independent of the tip pressure, as the tip advanced and a new grid block was added, this forced the pressure to vary discontinuously in time to account for the change in the pressure-dilation relationship. Requiring the crack to be closed at the tip is an unnecessary restriction on the dilation, since the boundary condition applied to the gas flow equation ensures that there is no gas flow across the crack tip for any value of the tip dilation. Therefore, in the current model the dilation at the crack tip is determined by applying the same pressure-dilation relationship as for the other dilations. This means that the tip of the crack will be blunt rather than pointed.

The special treatment of the final grid block at the downstream end of a continuous crack, described by Nash et al. (1998) has also been removed, as it is no longer necessary. The dilation near to the downstream boundary in a continuous crack is now calculated in the same way as the dilation for the rest of the crack.

A further adjustment has been incorporated in the equations for gas flow in the Phase 2 model to fully account for the movement of block boundaries. The details of this adjustment are given below.

The crack propagation model is described using a variable number of nodes at which the pressure and dilation are defined, with the final node, located at the crack tip, allowed to move as the tip advances. The grid-block boundaries are defined to be at the midpoints between these nodes. Therefore as the tip advances the boundary behind the tip will

125

also move with a velocity half that of the tip. This boundary velocity has been incorporated into the Phase 1 model. However, as the tip advances, new nodes are added behind it, so that the distance between the tip node and the next nearest node to the tip is never more than some fixed value. This means that it is possible for several boundaries behind the tip to have moved during a time step as the tip moves from one position to another.

An example case where several boundaries near the crack tip are moving during a time step is shown in Figure A 1-1. As the crack tip, T, moves forward from its position at time index n, boundary X initially also moves (with half the velocity ofT). When the spacing between T and the previous node reaches the standard spacing a new block is formed, and the new boundary, Y, initially coincident with T, then moves (with half the velocity of T) until T reaches its position at time index n+ 1, X remaining stationary. This must be represented in the model by appropriate adjustments to the equations for gas flow during the time step, to account for the boundary velocity in the calculation of flow across each boundary. The current model contains all the adjustments required.

A 1.3 Numerical Performance

The numerical performance of the model has been evaluated by performing a number of tests, using variations in the time step or grid refinement. The aim was to determine the levels of refinement required to ensure results reported by the model are sufficiently accurate. The tests involved running simulations with input data varying only in the value of a parameter controlling either the time step size, grid block size or end

End of crack at time index n

· I I X T

End of crack at time index n+ 1

• • • H

X y T

New grid block

Figure Al-l. Grid Arrangement Around the Crack Tip for Two Consecutive Time Indices.

126

boundary movement. The results of the simulations were compared to find any differences that indicated inaccuracies arising from the discretisation used.

A1.3.1 The Base Case

The base case used for all the tests is the simulation constructed to match the inlet pressure transient of experiment Mx80-4A of Horseman and Harrington ( 1997). It contains 100 identical cracks, discretised into 126 grid blocks each (for cracks traversing the entire clay sample), with a maximum allowed time step of 2.5 103s.

A 1.3.2 Temporal Discretisation

Investigation of the effect of maximum time-step size concentrated on the continuous pathway stage of the simulation, as during the propagation stage the time steps are chosen automatically by the program (they are not affected by the maximum time-step parameter) to achieve convergence, and are generally very short (typically of order 10-3s) compared with later in the simulation. (In general the maximum time step length is only taken during periods when there is no other restriction on the length of the time step.)

A set of simulations with maximum time step lengths of 106s, 104s and 103s, but otherwise with the same input data as the base case, were run. The four transients for the pressure at the upstream end of the sample (around the second peak in the pressure) are shown in Figure A1-2. This shows the period over which there was greatest disagreement between the pressures given by each of the simulations. As finer discretisations are used, the pressure curves converge toward the presumed accurate result (within the model assumptions). The results from the simulations using maximum time step lengths of 2.5 103s and 103s are virtually indistinguishable, and are assumed to represent this accurate result. The differences between these two coincident curves and the other two pressure transients show the inaccuracies resulting from insufficient refinement in the temporal discretisation; the larger the value of the maximum time step length used, the greater the inaccuracy in the pressures calculated.

The results of this test show that a maximum time step length of 2.5 103s or less should produce simulation results that are not affected by errors due to the temporal discretisation. It should be noted that the propagation of each crack occurs over a period much shorter than this maximum time step length, so that much shorter time steps are required during this period to produce an accuracy in the results similar to that in the rest of the simulation.

A 1.3.3 Grid Refinement

Simulations with 26 grid blocks and 1251 grid blocks, and otherwise with the same input data as the base case, were run. The three transients for the pressure at the upstream end of the sample (around the second peak in the pressure) are shown in

127

15.2 .------------------------,

~

15.18

15.16

~ 15.14

~ 15.12 ~ ~ 15.1

~ 15.08 ~ ~

:3- 15.06

15.04

Maximum Titre Step

--- -1000000s ....... 10000s - ·- · -2500s --1000s

15.02

15+--~~4-----+---~-----~----~

2.7£+06 2.8£+06 2.9£+06 3.0£+06 3.1£+06 3.2£+06

Time (s)

Figure Al-2. Upstream Pressure Transients between 2.7 Jds and 3.2 106s for the Temporal Discretisation Test Simulations.

Figure A1-3. This shows the period over which there was greatest disagreement between the pressures given by each of the simulations. As finer discretisations are used the pressure curves converge toward the accurate result. In this case, the results from the simulations using 126 grid blocks and 1251 grid blocks are virtually indistinguishable, and are assumed to represent this accurate result. There is a small difference between these two curves and that for the 26-grid-block simulation, indicating some discretisation errors at the coarsest level of grid refinement.

The results of this test show that using -100 or more grid blocks (a block spacing of -0.5 mm or less) should produce simulation results that are not affected by errors due to the grid refinement.

A further discretisation error occurs in the crack tip propagation velocity as the tip moves through the clay. This velocity is expected to reduce as the crack tip advances, but substantial variation from a monotonic decrease is seen. Figure A1-4 (a, b, c) shows the tip velocity at each step during propagation for the three cases described above. The errors in the tip velocity are thought to be due to the use of a non-uniform grid to represent the crack. Where the tip grid block has a similar length to the other grid blocks the errors due to the non-uniformity are small, but where the tip grid block is much smaller than the other grid blocks the errors can become much larger.

128

15.2 --..----------------------,

15.18 Number of Nodes

....... 26 15.16 -.-.-126

--1251 c:? ~ 15.14

B 15.12 :::l ~

~ 15.1

~ 15.08 !i ::3- 15.06

15.04

15.02

15+---~-,_----r----r-------+~--~

2.7E+06 2.8E+06 2.9E+06 3.0E+06 3.1E+06 3.2E+06

Time (s)

Figure Al-3. Upstream Pressure Transients between 2.7 Jds and 3.2 Jdsfor the Grid Refinement Test Simulations.

The examples in Figure A 1-4 appear to demonstrate that increasing the grid refinement reduces the variations in the velocity. However, the maximum tip advance in each case was the same (1.96 10-4m). With 1251 grid blocks this allows the addition of several new grid blocks for each advance of the tip, so that very small tip block lengths can be avoided. But with 26 grid blocks the tip can only advance a small fraction of a block length in each step, so that periodically (when the crack length is just greater than a whole number of standard block lengths) a very small tip block length must be used. This also explains the repeating pattern seen in the velocity in Figure A1-4(a). Figure A1-4(d) shows that errors in the velocity are not simply due to insufficient grid refinement, since in this case the greatest refinement ( 1251 grid blocks) is again used, but with a much smaller tip advance in each step (3.92 10-6m); the result is large errors in the velocity.

In the simulations studied here, differences in the propagation stages have an insignificant effect on the rest of the simulation results, so that the inaccuracy in the tip propagation velocity is not considered important. However, to correct this problem, modifications to the discretisation scheme used near the tip of a propagating crack have been considered.

One modification that would be expected to reduce the errors in the tip velocity is the use of two moving nodes, resulting in three variable length blocks, at the tip end of the

129

crack. This modification would mean that no block could have a length less than a quarter of the length of the largest block at any time. Tests have been performed using this alternative discretisation, but the results showed similar errors to the original discretisation. It is believed that the errors in this case are largely due to the addition of nodes within the body of a crack as it propagates rather than differing grid block sizes. This type of node addition is a necessary feature of this type of discretisation, but the discontinuous changes in block lengths cause discontinuous changes in the pressures and dilations.

0.0030 0.0030

0.0025 0.0025

~ 0.0020

! ~ 0.0020

! -~ 0.0015 0

~

>. ·13 0.0015 0

~ 0.0010 0.0010

0.0005 0.0005

0.0000 10 15 20 25 30 35 0 10 15 20 25 30 35

Time (s) Time (s)

(a) 26 nodes (b) 126 nodes

0.0030 0.0030

0.0025 0.0025

~ 0.0020

! ~ 0.0020

! -~ 0.0015 g ~

-~ 0.0015 0

~ 0.0010 0.0010

0.0005 0.0005

0.0000 0.0000 0 10 15 20 25 30 35 0 10 15 20 25 30 35

Time (s) Time (s)

(c) 1251 nodes (d) 1251 nodes, reduced tip advance per step

Figure Al-4. Crack Tip Velocity During Propagtion Stage against Time from Start of Crack Propagation for (a, b, c) the Grid Refinement Test Simulations, (d) a Repeat of (c) with a smaller Tip Advance for each Step.

130

To avoid both of the errors in the tip propagation velocity described above would require all grid blocks to be included from the start of a simulation, and then all to expand equally from the same starting size.

131

A2 APPENDIX 2: THEORY OF DEFORMATION AND CONSOLDATION IN UNSATURATED SOILS A MACROSCOPIC PHENOMENOLOGICAL APPROACH

This Appendix provides a summary of the macroscopic phenomenological approach to deformation and consolidation in unsaturated soils. It provides a background framework to discuss the migration of gas through bentonite clay. This approach is successful in saturated soil mechanics, and can be readily 'upscaled'.

Continuum mechanics defines the following terms:

a) State variables: nonmaterial variables required for the characterisation of a system;

b) Deformation state variables: nonmaterial variables required for the characterisation of the deformation from an initial state;

c) Stress state variables: nonmaterial variables required for the characterisation of the stress condition;

d) Constitutive relations: single-valued equations expressing the relationship between state variables. (A stress versus strain relationship is a constitutive relation that describes the mechanical behaviour of a material. The material properties involved may be an elastic modulus and a Poisson's ratio. The ideal gas equation relates pressure to density and temperature. The gas constant is the material property. Idealised constitutive equations are well established for a perfectly elastic solid, a non-viscous fluid and a Newtonian viscous fluid.)

A2.1 Deformation State Variables

Unsaturated clay can be visualised as a mixture with two phases that come to equilibrium under applied stress gradients (i.e. soil particles and 'contractile skin'), and two phases that flow under applied stress gradients (i.e. water and gas).

Volume changes in the clay are expressed in terms of deformations or relative movements of these phases.

Water

Clay particle

Contractile skin (Water-gas interface)

132

dy

dx

Figure A2-1. An element of unsaturated clay (after Fredlund and Rahardjo, 1993).

A2.1.1 Continuity Requirement

The deformation state variables must satisfy the continuity requirement of multiphase continuum mechanics; that is, the total volume change of the clay element should be equal to the sum of the volume changes associated with each phase. If the soil particles are incompressible, continuity implies

where Vo Vv Vw Vg Vc

is the initial volume of an unsaturated clay element; is the volume of voids; is the volume of water; is the volume of gas; is the volume of 'contractile skin' (i.e. the water-gas interface).

A2-1

Assuming that the 'contractile skin' is only a few molecular layers thick, the continuity requirement reduces to

~Vv ~Vw ~Vg --=--+-- A2-2

Vo Vo Vo

It follows that the volume changes associated with any two of the three variables must be measured, while the third volume change can be computed.

A2.1.2 Total Volume Change

'Total volume change' refers to the volume change of the clay structure.

133

Consider an element of unsaturated clay, defined with respect to a fixed mass of clay particles, that has infinitesimal dimensions dx, dy and dz in the x-, y- and z- directions respectively. In response to an applied stress gradient, the element will undergo a translation of u, v and w from its original x-, y- and z- co-ordinate position and deform. The deformation will consist of a change in length, and a rotation of the element sides with respect to each other.

The changes in length in the x-, y- and z- directions are

au dx au dy and au dz ax ' ay ' az

Defining normal strain, e, as a change in length per unit length, the normal strains of the clay structure in the x-, y- and z- directions can be expressed as

au £=

X ax

av e =-Y ay

aw e =

z az

where ex is the normal strain in the x-direction; e y is the normal strain in the y-direction; e z is the normal strain in the z-direction.

A2-3

The distortion of the element is expressed in terms of a shear strain, y, which is defined as the change (in radians) of the original right angle between two axes. The shear strain components of a three-dimensional element are

au av r =-+xy ay ax

av aw Yyz=az+ay

aw au Yzx = ax + az

where r xy is the shear strain in the z-plane ( r xy = r yx);

r yz is the shear strain in the x-plane ( r yz = r zy);

r zx is the shear strain in they-plane ( r zx = r xz).

A2-4

The normal and shear strains of the clay structure can be combined as a deformation tensor

134

1 1 ex 2rxy 2rxz 1 1

A2-5 2Yyx £Y 2ryz 1 1 2rzx 2Yzy ez

The sum of the normal strain components is just the volumetric strain

A2-6

where £v is the volumetric strain.

The volumetric strain is equal to the difference between the volumes of the voids in the element before and after deformation, L1 Vv, referenced to the initial volume of the element, Vo.

~Vv £ =-

V V. 0

A2-7

The volumetric strain, £v, may be used as a deformation state variable for the clay structure. It defines the volume change in the clay structure resulting from deformation.

A2.1.3 Water and Gas Volume Changes

The unsaturated clay element is also used to describe net changes in the fluid volumes (i.e. water and gas phases).

The change in the fluid volume is defined as the difference between the fluid volume in the deformed and the undeformed element. The deformation variable can be written as

A2-8

for the water phase, and

A2-9

for the gas phase.

135

A2.2 Stress State Variables

The mechanical behaviour of a clay (i.e. volume change and shear strength) is determined by its state of stress. The stress state variables should be independent of the physical properties of the clay.

The stress state variable used to describe the behaviour of a saturated soil is the 'effective stress'

(J- Pw A2-10

where a is the total normal stress; Pw is the pore-water pressure;

It is more difficult to establish satisfactory stress state variables for an unsaturated clay. Fredlund and Morgenstem ( 1977) present a theoretical analysis, based on multi phase continuum mechanics, of stress in an unsaturated soil. The unsaturated soil is considered as a four-phase system (i.e. soil particles, the 'contractile skin' or water-gas interface, water and gas). The soil particles are assumed incompressible, and the soil is treated as though it were chemically inert.

The equilibrium equation for the soil structure depends on a set of three independent normal stresses

a- Pg

Pg -pw

Pg

where pg is the pore-gas pressure.

A2-11

If the soil particles are incompressible, pg can be eliminated. The complete stress state for an unsaturated soil can therefore be written in terms of two independent tensors

and

0

Pg- Pw

0

A2-12

A2-13

136

A2.3 Constitutive Relations

Constitutive relations that incorporate soil properties in the form of coefficients link the stress state and deformation state variables.

There are three constitutive relations for an unsaturated clay: one for the clay structure, one for the water phase and one for the gas phase. In each constitutive equation, the deformation state variables are the total, water, or gas volume change, while the stress state variables are (a- pg) and (pg- Pw ). Biot (1941) suggested similar constitutive relations for a saturated soil.

A2.3.1 Clay Structure

The constitutive equation for the clay structure is derived by assuming that the clay behaves as an isotropic, linear elastic material. (This assumption is acceptable in an incremental manner.) The constitutive relations are developed in a semi-empirical manner as an extension of the elasticity formulation used for saturated soils (Biot, 1941). The constitutive relations associated with normal strains in the x-, y- and zdirections are

A2-14

where E is the modulus of elasticity for the soil structure; v is Poisson' s ratio; H is the modulus of elasticity for the soil structure with respect to a change

in matric suction (pg- Pw ).

The constitutive relations associated with shear deformations are

A2-15

where G is the shear modulus.

137

An incremental procedure, using small increments of stress and strain, can be used to apply the linear elastic formulation to a non-linear stress versus strain curve. The elastic moduli E and H may then vary from one increment to another.

The clay structure constitutive relations associated with the normal strains, when written in an incremental form, are

dex =_!_d(ax- Pg)-~d(ay +Gz -2pg)+-1

d(pg- Pw) E E H

dey =_!_d(ay- Pg)-~d(ax +Gz -2pg)+-1

d(pg- Pw) E E H

A2-16

dez =_!_d(az- Pg)-~d(ax +ay -2pg)+-1

d(pg- Pw) E E H

A change in the volumetric strain of the clay for each increment, dev, can be obtained by summing the changes in normal strains in the x-, y- and z- directions

dev = dex + dey + dez

= 3(l-;v )d(amean- Pg )+ ~ d(pg- Pw)

where O"mean is the average total normal stress (i.e. ( O"x + O"y + O"z )/3).

Rearranging Equation A2-16

d(a x- Pg )= 2G(dex +adev)- f3 d(pg- Pw)

d(ay- Pg)=2G(dey +adev)- {3d(pg- Pw)

d(az- Pg )= 2G(dez +adev)- {3 d(pg- Pw)

where

G= E 2(1 +V)

V a=--

1-2v

{3 = E H(l-2v)

A2-17

A2-18

A2-19

A2-20

A2-21

138

A2.3.2 Water Phase

The water phase constitutive equation defines the change in volume of water present in a clay element for any change in the total, pore-water and pore-gas pressures. The water itself is assumed to be incompressible, and the equation accounts for the net inflow or outflow from the element. The constitutive equation is formulated as a linear combination of the stress state variable changes, and has the incremental form

dV w 3 f ) 1 f __ ) --=-d\Gmean-Pg +--d\Pg-Pw ~ Ew Hw

A2-22

where Ew is the water volumetric modulus associated with a change in

( Gmean - Pg ); Hw is the water volumetric modulus associated with a change in ( pg-Pw ).

A2.3.3 Gas Phase

The change in the volume of air in an element can be computed as the difference between the clay structure and water volume changes, since the continuity requirement implies

A2-23

A2.3.4 Hysteresis

Hysteresis is associated with the phases that behave as a solid (e.g. the clay structure and the 'contractile skin' or water-gas interface) and is the main cause of nonuniqueness of a constitutive relationship under loading and unloading conditions. The clay structure hysteresis during loading and unloading of a saturated clay is reflected in the different compression and rebound curves. Hysteresis associated with the 'contractile skin' can be visualised from the drying and wetting curves.

A reversal in the direction of deformation results in different constitutive surfaces. Therefore, the volumetric deformation coefficients associated with a decreasing volume would be different from those associated with an increasing volume. However, the uniqueness of the volume increase and decrease surfaces can be verified independently.

139

A2.4 Flow Laws

A2.4.1 Water Phase

The flow of water in a saturated clay is usually described using Darcy's Law. Darcy postulated that the rate of water flow through a soil mass was proportional to the hydraulic head gradient

kw n q =--vp w f..lw w

where qw kw f..lw

Pw

is the flow rate of water; is the coefficient of permeability with respect to the water phase; is the dynamic viscosity of water; is the pressure associated with the pore-water.

A2-24

In an unsaturated clay the coefficient of permeability will be a function of two volumemass properties, e.g.

where S

e

is the degree of saturation ( S = ~: }

is the void ratio (e = Vv J. Vo -Vv

A2-25

In a saturated clay the coefficient of permeability is just a function of the void ratio.

A2.4.2 Gas Phase

Darcy' s equation for the gas phase is

kg q =--Vp

g J..lg g

is the flow rate of gas; is the coefficient of permeability with respect to the gas phase; is the dynamic viscosity of gas; is the pressure associated with the pore-gas.

A2-26

As the saturation increases, the gas phase becomes occluded, and gas flow is reduced to a diffusion process through the pore-water.

140

A2.5 Coupled Formulation of Three-dimensional Consolidation

A rigorous formulation of consolidation requires that the equilibrium equations be coupled with the continuity equations. This method was proposed by Biot (1941) to analyse the consolidation of a soil.

The assumptions used in this derivation are similar to those proposed by Terzaghi (1943) and Biot (1941) for saturated soils. An outline of the assumptions used is as follows:

a) The coefficients of the volume change for the clay element (i.e. E, v, H, Ew, Hw) remain constant during the consolidation process. It is possible, however, to make these coefficients a function of the stress state.

b) The strains during consolidation are small.

c) The clay particles and the pore-water are incompressible.

d) The gas phase is continuous. However, if the gas phase is occluded, the coefficient of permeability with respect to the gas phase approaches the diffusivity of gas through water.

e) The coefficients of permeability with respect to the water and gas phases are assumed to be functions of the volume-mass properties (i.e. S, e) of the clay.

t) The effects of gas diffusing through water, gas dissolving in water and the movement of water vapour are ignored.

In a three-dimensional consolidation problem, there are five unknowns of deformation to be solved: the displacements in the x-, y- and z- directions (i.e. u, v and w

respectively) and the water and gas volume changes (i.e. dVw and dVg). These five unknowns are obtained from three equilibrium equations and two continuity equations (i.e. water and gas phase continuities). The equations are outlined below.

A2.5.1 Equilibrium Equations

The stress state for an unsaturated clay element should satisfy the following equilibrium conditions

aa X + ar yx + ar zx = 0 ax ay az ar xy aa y ar zy --+--+--=0 ax ay az ar xz + ar yz + aa z = 0 ax ay az

141

A2-27

Substituting Equation A2-18 into Equation A2-27 gives the following form for the equilibrium equations

GV2u+_!!_ Ot:v _ f3 a(pg- Pw) + Opg =O 1-2v ax ax ax

GV2v+_!!_ Ot:v _ f3 a(pg- Pw) + () Pg = O 1-2v ay ay ay A2-28

GVzw+_!!_ Ot:v- {3 a(pg- pJ + Opg =0 1-2v az az az

These are three equations with five unknowns: u, v, w, Pw and Pg· Darcy's Laws governing the flow of water and gas in the clay are needed to complete the system.

A2.5.2 Water Phase

The continuity equation for the water phase is obtained by equating the net flux of water per unit volume of the clay to the divergence of the flow rate, as described by Darcy's Law. Thus,

A2-29

The water phase constitutive relation, Equation A2-22, defines the water volume change in a clay element caused by changes in the mean normal stress d( C1mean- Pg ), and the matric suction d( pg- Pw ). Differentiating with respect to time

A2-30

The coefficients of volume change are assumed constant during consolidation. In other words, the coefficients are not a function of time, but a function only of the stress state variables.

Equations A2-29 and A2-30 can now be equated to give

142

A2-31

or

ae a (p ) k ( k ) ICw_v +Kw- -p =~V2p +V~ ·Vp 1 at 2 at g w w 11 w

Jlw w A2-32

where

A2-33

A2.5.3 Gas Phase

The continuity equation for the gas phase is obtained from the net mass rate of gas flow, as described by Darcy's Law, and the time derivative of the gas phase constitutive equation. Hence,

A2-34

or

A2-35

where

A2-36

143

A2.5.4 Summary

The five unknowns (e.g. u, v, w, Pw and pg) in a coupled analysis for two-phase flow through an unsaturated clay are obtained from the simultaneous solution of Equations A2-28, A2-31 and A2-34.

These equations can be generalised to include additional effects, such as the water compressibility etc.

A2.6 Formulation of One-dimensional Consolidation

In one-dimension (i.e. the clay sample is confined laterally in a rigid sheath) there are just three unknowns: w, Pw and Pg· These unknowns are obtained from the equilibrium equation

and the two continuity equations, i.e. water phase

and gas phase

{1-S)e 1 apg

1+e Pg at

A2. 7 Constitutive Relations for Different Loading Conditions

A2-37

A2-38

A2-39

The above constitutive relations (Section A2.3) have been formulated for a general, three-dimensional loading in the x-, y- and z- directions. The formulation can also be applied to special loading conditions. The constitutive equations for each loading condition can be derived from the general formulation, and are presented in the following sections.

144

A2.7.1 Isotropic Test

For isotropic loading, the total stress increments in the three directions are equal (i.e. da x =daY= da z = da). No shear stress is developed in the clay. The clay structure constitutive equation for isotropic loading is

A2-40

where a is the total isotropic stress.

The clay will undergo equal deformation in all directions (or isotropic compression) provided that the clay properties are isotropic.

The constitutive equation for the water phase can be derived from Equation A2-22

dVw 3 ~ ) 1 (p ) -=-da-p +-d -p V. E g H g w 0 w w

A2-41

A2.7.2 Constant Volume (Radial Flow} Test

The clay structure constitutive equation, assuming a constant volume, is

de =de =de =0 X y Z A2-42

Hence

A2-43

The constitutive equation for the water phase becomes

A2-44

A2.7.3 ~Test

For Ko-loading, a total stress increment of da z is applied in the vertical direction, while the clay is not allowed to expand laterally (i.e. de x =de y = 0 ). This loading condition occurs during one-dimensional consolidation, where deformation is allowed only in the vertical direction.

The strain conditions during Ko-loading can be used in Equations A2-16. The volumetric strain for the clay structure can then be written as follows

145

de =(l+v)(l-2v)dra _ )+ (l+v) df __ _ ) v E(l-v) ~ z Pg H(l-v) \Pg Pw

A2-45

The water phase constitutive relation is obtained in a similar manner

A2-46

146

A3 APPENDIX 3: STRESSES AROUND A SPHERICAL CAVITY IN A SATURATED CLAY

A3.1 Stresses around a Spherical Cavity in a Saturated Clay

Consider a spherical cavity of radius a in a saturated clay, subjected to total radial stress arr = aR and fluid pressure p = Po on its boundary r = a.

Stress equilibrium requires that

G V2u + ____!!____ aev - fJ a Pw = 0 1-2V ax ax

G V2v + ____!!____ aev - fJ a Pw = 0 1-2v ay ay

GV2w+____!!____ aev - fJ a Pw = 0 1-2V az az

while the flow of water is assumed to satisfy the differential equation (i.e. V kw= 0)

If the strains are derivable from a purely radial displacement, so that

u = ,;(r )x v = ,;(r) y

w = ,;(r )z

then

and also

A3-1

A3-2

A3-3

A3-4

147

A3-5

which implies that

A3-6

In this case, the equilibrium equations simplify to give

A3-7

The boundary conditions at r = oo (i.e. Sv = 0 and Pw = 0) mean thatf(t) = 0. Using this expression in Equation A3-2,

where c is the 'coefficient of consolidation'.

A3-9

It follows that

a (r-aJ Pw = Po -erfc ,.-;-:-; r v4ct

A3-10

In spherical co-ordinates, the equations of equilibrium are

148

aa 2a, -ae -all, --'+ ., =0 ar r A3-11

a 8 -a'I'=O

while, using Equations A2-17 and A3-7,

( 1- 2v J( a r + a (J + a 'I' J 3 £ =3 -- -p +-p

V E 3 w H w

A3-12 (1+v)

= H(1-v)Pw

Solving from this last pair of equations gives

3 r ( 1 J a a 1 r -a , a, =a R - 3 - 21] p0 - 3 J r erfc TA::: dr

r r a v4ct

a, a (r-aJ a 8 =a'l' =---1]p0 -erfc TA::: 2 r v4ct

A3-13

where

3 E 1] = --+ --:------:-

2 H(1-v) A3-14

In the region near the wall of the cavity where the applied fluid pressure has penetrated (i.e. ( r- a )2 <<et; air= 1 ), the pore pressure is p0, but the radial total stress is still that applied, so

a a e = a 'I' = - 2R -1] Po A3-15

Pw = Po

It is proposed that fracture occurs when the maximum effective tensile stresses at the surface of the cavity reach the tensile strength -a 0. The expression for the effective stresses, a ij, appears to be described by the classical effective stress law, namely

A3-16

149

A3.2 Tensile Stress Fracture Criterion

The tensile stress fracture criterion is, e.g.

A3-17

where a; are the principal values of the effective stress tensor, a ij.

For the purposes of fracture analysis, Equation A3-15 gives

(jr = (j R- Po

- - (jR (1 ) CFe =a"'=--- +1] Po

2

A3-18

In the case p0 = 0, it follows that the internal stress required to cause fracture is

A3-19

The same conclusion follows from analysis of a cylindrical geometry.

150

A4 APPENDIX 4: ESTIMATION OF THE EFFECTS OF COMPLIANCE IN MEASURING DEVICES ON TRANSIENT RESPONSES

The time scale of transient responses in pore water pressure measurements can be affected by the compliance in the measuring device. It is possible to estimate the effect of pipe work in the pore-water pressure circuit on response times. First, the flow of water out of the clay and into the pore-water pressure circuit can be estimated from the analytical solution for flow to a flat circular disk in a semi-infinite porous medium (Carslaw and Jaeger, 1959)

A4-1

where Q is the flow to the pipe in the pore-water pressure circuit (m3 s-1);

kw is the permeability to water of the clay (m2);

f.1w is the viscosity of water (Pas); r is the radius of the disk (m); fl.pw is the difference between the pore-water pressure in the clay and the

pressure in the pore-water pressure circuit (less than 8 MPa);

Taking the disk radius to be the same order of magnitude as the internal radius of a small bore pipe (0.5 10-3 m) gives a value of Q that is less than 1.5 10-13 m3 s-1

• Consider this flow as being injected into a pipe of radius r and length I. The pipe has permeability

2

k =!___ p 8

= 3.1 10-8 m2 for a pipe of diameter 10-3 m

and hydraulic diffusivity

kp

A4-2

D =-~-P f.lw Pw A4-3

= 6.3 104 m 2s-1 for a pipe of diameter 10-3 m

The solution of the transient flow equation gives (Carslaw and Jaeger, 1959)

151

Q t 1 Q J.lw { 1 2 oo ( -1 t [ D P n 2

7!2

t ]} p(x=I)=-2---+--2I- ----2 :L,-2-exp- 2 1! r I Pw 1! r k P 6 1! n=l n I

Qt 1 ::::::::---- A4-4

1lr2 I Pw

= 380! Pa I

It follows that at the end of a pipe 0.5 m long the pressure will increase by 6 MPa in 8 103 s. This timescale, based on an assumption of compliance in the measurement of the pore-water pressure, appears a little short to explain the experimentally observed responses in experiment Mx80-8 of Horseman and Harrington (unpublished) that are discussed in Section 7 .2.1 above.

LIST OF REPORTS 1 (1)

POSIV A REPORTS 2001, situation 1/2001

POSIV A 2001-01

POSIVA 2001-02

Geochemical modelling of groundwater evolution and residence time at the Hastholmen site Petteri Pitkiinen, Ari Luukkonen VTT Communities and Infrastructure Paula Ruotsalainen Fintact Oy Hilkka Leino-Forsman, Ulla Vuorinen VTT Chemical Technology January 2001 ISBN 951-652-102-9

Modelling gas migration in compacted bentonite: GAMBIT Club Phase 2 Finat report - A report produced for the members of the GAMBIT Club B.T. Swift, A.R. Hoch, W.R. Rodwell AEA Technology, United Kingdom January 2001 ISBN 951-652-103-7

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Modelling gas migration in compacted bentonite: GAMBIT ... · MODELLING GAS MIGRATION IN COMPACTED BENTONITE: GAMBIT CLUB PHASE 2 FINAL REPORT T1iv1stelma-Abstract This report describes