Release 11.4 July 2016 - Foster Wheeler · ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling ... ConnectFlow Command Reference Manual; ... design and evaluation

© Amec Foster Wheeler 2016.

ConnectFlow VerificationRelease 11.4

July 2016

ConnectFlow Verification

Ref AMEC/ENV/CONNECTFLOW/16

Title ConnectFlow

Confidentiality,

copyright &

reproduction

© Amec Foster Wheeler 2016.

All rights

This computer program is protected by copyright law and international treaties.

Save as permitted under the Copyright Designs and Patents Act 1988, or the terms of any licence granted by the copyright owner, no part of this program may be used,

Warning: Any unauthorised act, including reproduction or distribution of this program or any portion of it, may rdamages and criminal prosecution.

Contact Details Amec Foster Wheeler

Building 150

Harwell Campus

Didcot

Oxfordshire OX11 0QB

United Kingdom

Tel +44 (0) 1635 280300

Fax +44 (0) 1635 280301

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ConnectFlow Verification

Amec Foster Wheeler 2016.

All rights reserved.

This computer program is protected by copyright law and international

Save as permitted under the Copyright Designs and Patents Act 1988, or the terms of any licence granted by the copyright owner, no part of this program may be used, reproduced, distributed or modified in any way.

Warning: Any unauthorised act, including reproduction or distribution of this program or any portion of it, may result in both a civil claim for

mages and criminal prosecution.

Amec Foster Wheeler

Building 150

Campus

Oxfordshire OX11 0QB

United Kingdom

Tel +44 (0) 1635 280300

Fax +44 (0) 1635 280301

amecfw.com

Page i

This computer program is protected by copyright law and international

Save as permitted under the Copyright Designs and Patents Act 1988, or the terms of any licence granted by the copyright owner, no part of this

reproduced, distributed or modified in any way.

Warning: Any unauthorised act, including reproduction or distribution of esult in both a civil claim for


Executive Summary

ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling software that combines the NAMMU continuum porous medium (CPM) module and the NAPSAC discrete fracture network (DFN) module. ConnectFlow can be used very flexibly to model grouand porous media on a variety of scales.

The following documentation is available for ConnectFlow:

► ConnectFlow Technical Summary Document;

► ConnectFlow Command Reference Manual;

► ConnectFlow Verification Document

► ConnectFlow Installation and Running Guide.

The following documentation is available for NAMMU:

► NAMMU Technical Summary Document;

► NAMMU User Guide;

► NAMMU Command Reference Manual;

► NAMMU Installation and Running Guide.

The following documentation is ava

► NAPSAC Technical Summary Document;

► NAPSAC On-line Help;

► NAPSAC Installation and Running Guide.

This document, the Verification Document, provides information on the verbuilds confidence in its flow and transport models

COPYRIGHT AND OWNERSHIP OF ConnectFlow

The ConnectFlow program makes use of the TGSL subroutine library.

All rights to the TGSL

All documents describing the ConnectFlow program and TGSL subroutine library are protected by copyright and should not be reproduced in whole, or in part, without the

permission of Amec Foster Wheeler.

amecfw.com

Summary

ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling software that combines the NAMMU continuum porous medium (CPM) module and the NAPSAC discrete fracture network (DFN) module. ConnectFlow can be used very flexibly to model groundwater flow and transport in both fractured and porous media on a variety of scales.

The following documentation is available for ConnectFlow:

ConnectFlow Technical Summary Document;

ConnectFlow Command Reference Manual;

ConnectFlow Verification Document;

ConnectFlow Installation and Running Guide.

The following documentation is available for NAMMU:

NAMMU Technical Summary Document;

NAMMU Command Reference Manual;

NAMMU Installation and Running Guide.

The following documentation is available for NAPSAC:

NAPSAC Technical Summary Document;

NAPSAC Installation and Running Guide.

This document, the Verification Document, provides information on the verification of ConnectFlow, which builds confidence in its flow and transport models.

COPYRIGHT AND OWNERSHIP OF ConnectFlow


All rights to the TGSL subroutine library are owned by Amec Foster Wheeler.


permission of Amec Foster Wheeler.

Page ii

ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling software that combines the NAMMU continuum porous medium (CPM) module and the NAPSAC discrete fracture network (DFN)

ndwater flow and transport in both fractured

ification of ConnectFlow, which


subroutine library are owned by Amec Foster Wheeler.



Table of Contents

1. Introduction

2. Continuum Porous Media Verification2.1 Radial Steady State Flow

2.1.1 Overview

2.1.2 Problem Definition

2.1.3 Variations

2.1.4 Constrained mesh

2.1.5 Results

2.2 Steady Flow in Fractured Rock

2.2.1 Overview


2.2.3 Variations

2.2.4 Results

2.3 2D Steady Flow with Particle Tracks

2.3.1 Overview


2.3.3 Variations

2.3.4 Results

2.4 Transient Buoyant Flow

2.4.1 Overview


2.4.3 Results

2.5 Unsaturated Heat Transport

2.5.1 Overview


2.5.3 Results

2.6 1D Transient Unsaturated Flow

2.6.1 Overview


2.6.3 Variations

2.6.4 Results

2.7 Steady State Radial Flow to a Well with

2.7.1 Overview


2.7.3 Variations

2.7.4 Results

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Contents

Continuum Porous Media Verification Radial Steady State Flow

Problem Definition

d mesh

Steady Flow in Fractured Rock

Problem Definition

2D Steady Flow with Particle Tracks

Problem Definition

ant Flow

Problem Definition

Unsaturated Heat Transport

Problem Definition

nsaturated Flow

Problem Definition

Steady State Radial Flow to a Well with a Seepage Face Boundary Condition

Problem Definition

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a Seepage Face Boundary Condition 31

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2.8 Henry’s Salt Transport

2.8.1 Overview


2.8.3 Results

2.9 1D Rock Matrix Diffusion (RMD)

2.9.1 Overview


2.9.3 Variations

2.9.4 Results

2.10 1D Nuclide Transport with Sorption and Decay

2.10.1 Overview


2.10.3 Variations

2.10.4 Results

2.11 Mass Flux Calculations

2.11.1 Overview


2.11.3 Results

2.12 Reactive Transport

2.12.1 Overview


2.12.3 Variations

2.12.4 Results

3. Discrete Fracture Network Verification3.1 3D Fracture Distributions

3.1.1 Overview


3.1.3 Results

3.2 3D Fracture Connectivity

3.2.1 Overview


3.2.3 Results

3.3 3D Fracture Connectivity with a Power Law Size Distribution

3.3.1 Overview


3.3.3 Results

3.4 Upscaling from DFN to CPM

3.4.1 Overview


3.4.3 Variations

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Henry’s Salt Transport

Problem Definition

1D Rock Matrix Diffusion (RMD)

nition

1D Nuclide Transport with Sorption and Decay

ition

culations

Problem Definition

Problem Definition

Discrete Fracture Network Verification 3D Fracture Distributions

Problem Definition

3D Fracture Connectivity

Problem Definition

3D Fracture Connectivity with a Power Law Size Distribution

Problem Definition

Upscaling from DFN to CPM

Problem Definition

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3.4.4 Results

3.5 Radial Steady State Flow

3.5.1 Overview


3.5.3 Results

3.6 Three Fracture Intersections

3.6.1 Overview


3.6.3 Variations

3.6.4 Results


3.7.1 Overview


3.7.3 Variations

3.7.4 Results


3.8.1 Overview


3.8.3 Variations

3.8.4 Results

3.9 Salt Transport

3.9.1 Overview


3.9.3 Results

3.10 Salt Upconing

3.10.1 Overview


3.10.3 Results

3.11 Grouting of a fracture

3.11.1 Overview

3.11.2 Problem definition

3.11.3 Results


3.12.1 Overview


3.12.3 Results

3.13 Transient Salt Diffusion

3.13.1 Overview


3.13.3 Results

3.14 1D Advection of Salinity

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Radial Steady State Flow

Problem Definition

Three Fracture Intersections

Problem Definition

Fractured Rock

Problem Definition


Problem Definition

nition

Problem Definition

Grouting of a fracture – surface intersection

efinition

Grouting of a fracture – fracture intersection

Problem definition

Transient Salt Diffusion

efinition

1D Advection of Salinity

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3.14.1 Overview


3.14.3 Results

3.15 Transient Salt Upconing

3.15.1 Overview


3.15.3 Results

4. Combined DFN/CPM Verification4.1 Radial Steady State Flow

4.1.1 Overview


4.1.3 Variations

4.1.4 Results

4.2 Flow to Fracture Network

4.2.1 Overview


4.2.3 Results

4.3 2D Steady State Flow with Particle Tracks

4.3.1 Overview


4.3.3 Variations

4.3.4 Results

4.4 2D Variable Density Flow with Particle Tracks

4.4.1 Overview


4.4.3 Variations

4.4.4 Results

4.5 Henry’s Salt Transport Using Interpolated Density

4.5.1 Overview


4.5.3 Results

5. Further Testing 5.1 Continuum Porous Media

5.1.1 Verification

5.2 Discrete Fracture Network

5.2.1 Verification

5.3 Automated Testing

5.4 Peer Review

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Problem Definition

Transient Salt Upconing

efinition

Combined DFN/CPM Verification Radial Steady State Flow

Definition

Flow to Fracture Network

Problem Definition

2D Steady State Flow with Particle Tracks

Problem Definition

2D Variable Density Flow with Particle Tracks

Problem Definition

Henry’s Salt Transport Using Interpolated Density

Problem Definition

Continuum Porous Media

Discrete Fracture Network

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6. References

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1. Introduction

ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling software that combines the NAMMU continuum porous medium (CPM) module and the NAPSAC discrete fracture network (DFN) module. ConnectFlow can be used very flexibly to modeland porous media on a variety of scales.

ConnectFlow models have been used in the following applications:

► safety assessment calculation in support of radioactive waste disposal programmes;

► modelling in support of groundwater protection schemes;

► modelling of landfills;

► regional groundwater flow;

► aquifer contamination;

► site investigation;

► pump test simulation;

► tracer tests;

► saline intrusion;

► design and evaluation of remediation strategies.

Further details of the ConnectFlow’s capabilities can be found in [

The ConnectFlow software has been developed over a period of more than 20quality system that conforms to the international standards ISO 9001 and TickIT.

The purpose of this document is to present evidence that ConnectFlow is an appropriate tool to use for modelling groundwater flow and transport. This evidence takes the form of

► Comparison with analytical solutions.

► Comparison of results against other independently writte

Sections 2, 3 and 4 provide details of a set of verification test cases and associated results. In all cases the results are considered “good” in terms of agreement with the reference data.

A number of the test cases are re-boundary condition types, mesh topology and algorithmic choices.

Section 5 references further testing that has taken place on the ConnectFlow suite of software, which complements and extends the testing covered in the earlier sections.

amecfw.com

ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling software that combines the NAMMU continuum porous medium (CPM) module and the NAPSAC discrete fracture network (DFN) module. ConnectFlow can be used very flexibly to model groundwater flow and transport in both fractured and porous media on a variety of scales.

ConnectFlow models have been used in the following applications:

safety assessment calculation in support of radioactive waste disposal programmes;

port of groundwater protection schemes;

design and evaluation of remediation strategies.

the ConnectFlow’s capabilities can be found in [1] and [2].

The ConnectFlow software has been developed over a period of more than 20 years under a rigorous quality system that conforms to the international standards ISO 9001 and TickIT.

is document is to present evidence that ConnectFlow is an appropriate tool to use for modelling groundwater flow and transport. This evidence takes the form of

Comparison with analytical solutions.

Comparison of results against other independently written groundwater flow software.

provide details of a set of verification test cases and associated results. In all cases the results are considered “good” in terms of agreement with the reference data.

-used to extend the range of capabilities tested. These variations include boundary condition types, mesh topology and algorithmic choices.

references further testing that has taken place on the ConnectFlow suite of software, which complements and extends the testing covered in the earlier sections.

Page 1

ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling software that combines the NAMMU continuum porous medium (CPM) module and the NAPSAC discrete fracture network (DFN)

groundwater flow and transport in both fractured

safety assessment calculation in support of radioactive waste disposal programmes;

years under a rigorous

is document is to present evidence that ConnectFlow is an appropriate tool to use for

n groundwater flow software.

provide details of a set of verification test cases and associated results. In all cases

used to extend the range of capabilities tested. These variations include

references further testing that has taken place on the ConnectFlow suite of software, which


2. Continuum Porous Media Verification

A summary of the CPM test cases is given in

Case Title

2.1 Radial steady state flow.

2.2 Steady state flow in fractured rock.

2.3 2D Steady state flow with particle

tracks

2.4 Transient buoyant flow

2.5 Unsaturated heat transport

2.6 1D Unsaturated flow

2.7 Seepage face

2.8 Henry’s salt transport

2.9 1D Rock matrix diffusion.

2.10 1D Nuclide transport with sorption

and decay

2.11 3D mass conserving flux calculation

2.12 Reactive transport

Table 2-1 CPM verification tests

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Continuum Porous Media Verification

A summary of the CPM test cases is given in Table 2-1.

Overview

Radial steady state flow.

Steady state groundwater flow. Modelled using hexahedral, prismatic and constrained meshes. Boundary conditions include mass flux, constant pressure and point sinks.

Steady state flow in fractured rock.

Steady state groundwater flow. Variations include mesh aligned with fractures and a regular mesh with permeabilties modified via an imported fracture file.

2D Steady state flow with particle Steady state groundwater flow with varying permeability. Forward and backward particle tracks and conservative and regular particle tracks are generated.

Transient buoyant flow

Transient groundwater flow driven by buoyancy via the Boussinesq approximation. Decaying heat source and heat conduction through rock. Transient particle tracks are generated.

Unsaturated heat transport Transient unsaturated groundwater flow and heat transport.

turated flow Transient unsaturated groundwater flow. Tested using both Crank Nicholson and Gears Method time stepping.

Seepage face Steady state radial flow to a well with a seepage face boundary condition.

’s salt transport Steady state ground water flow. Density dependent on salinity.

1D Rock matrix diffusion.

Transport of salinity allowing for diffusion between fractured rock and the rock matrix. Tested using Crank Nicholson and Sequential Iteration solver options.

ort with sorption

Steady state groundwater flow with nuclide transport. Sorption and decay modelled individually and combined. Tested using Crank Nicholson and Fast Linear Transport options.

3D mass conserving flux calculation

Post processing for a CPM steady state ground water calculation. Calculation of mass fluxes through finite elements using Cordesalgorithm.

Reactive transport Multi-component solute transport with chemical reactions.

Page 2

Continuum Porous Media Verification

Steady state groundwater flow. Modelled using hexahedral, prismatic and constrained meshes. Boundary conditions include mass flux, constant

Steady state groundwater flow. Variations include mesh aligned with fractures and a regular mesh

permeabilties modified via an imported

Steady state groundwater flow with varying permeability. Forward and backward particle tracks and conservative and regular particle

t groundwater flow driven by buoyancy via the Boussinesq approximation. Decaying heat source and heat conduction through rock. Transient particle tracks are generated.

Transient unsaturated groundwater flow and heat

Transient unsaturated groundwater flow. Tested using both Crank Nicholson and Gears Method

Steady state radial flow to a well with a seepage

Steady state ground water flow. Density

diffusion between fractured rock and the rock matrix. Tested using Crank Nicholson and Sequential

Steady state groundwater flow with nuclide transport. Sorption and decay modelled individually and combined. Tested using Crank Nicholson and Fast Linear Transport options.

Post processing for a CPM steady state ground water calculation. Calculation of mass fluxes through finite elements using Cordes-Kinselbach

component solute transport with chemical



2.1.1 Overview

This case models steady groundwater flow in a 2D disk where water is removed from the centre at a constant rate and the outer disk boundary is maintained at a constant head.

The example has a simple analytical solution, which is used to verify range of mesh topologies.


The problem definition and solution is a variant of that from Feftra [

Figure 2-1 Schematic of problem definition

Symbol

h

Q

K

R

d

r

Table 2-2 Input parameters

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The example has a simple analytical solution, which is used to verify the ground water flow solution on a

The problem definition and solution is a variant of that from Feftra [3] and de Marsily [

lem definition

Parameter Value

Head at disk circumference 0 m

Outflow from disk 1.0E-7 m

Hydraulic Conductivity 1.0E-8 m/s

Radius of disk 2000 m

Thickness of disk 1 m

Radial distance from axis 0-2000 m

Page 3

This case models steady groundwater flow in a 2D disk where water is removed from the centre at a

the ground water flow solution on a

] and de Marsily [4].

Value

0 m

7 m3/s

8 m/s

2000 m

1 m

2000 m


2.1.3 Variations

2.1.3.1 Mass flux boundary condition

The modelled domain consists of a 15 degree sector, which is truncated at r = 1 m where a mass flux boundary condition is applied. The mesh consists of a line of 3Drefined towards the centre of the domain.

Figure 2-2 Hexahedral mesh with sector truncated close to origin

2.1.3.2 Point sink and prism element

The modelled domain consists of a 15 degree sector, which is composed of a line of hexahedral elements except at the origin where a prism element is used. The outflow is modelled using point sinks at the two vertices on the axis.

Figure 2-3 Hexahedral mesh with prism at origin

2.1.4 Constrained mesh

The modelled domain consists of a 15 degree sector, which is truncated at r = 1 m, where a mass flux boundary is applied. The mesh consists of a line of 3D hexahedral elements for r = 1 m to r mesh is then refined into two cells circumferentially from r = 1000 m to r = 2000 m. The meshes are joined using ConnectFlow constraints. This is an advanced technique that allows grids of different densities to be connected together.

Figure 2-4 Hexahedral constrained mesh

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Mass flux boundary condition

The modelled domain consists of a 15 degree sector, which is truncated at r = 1 m where a mass flux boundary condition is applied. The mesh consists of a line of 3D hexahedral elements, with the spacing refined towards the centre of the domain.

Hexahedral mesh with sector truncated close to origin

Point sink and prism element

of a 15 degree sector, which is composed of a line of hexahedral elements except at the origin where a prism element is used. The outflow is modelled using point sinks at the two

Hexahedral mesh with prism at origin

The modelled domain consists of a 15 degree sector, which is truncated at r = 1 m, where a mass flux boundary is applied. The mesh consists of a line of 3D hexahedral elements for r = 1 m to r mesh is then refined into two cells circumferentially from r = 1000 m to r = 2000 m. The meshes are joined using ConnectFlow constraints. This is an advanced technique that allows grids of different densities to be

Hexahedral constrained mesh

Page 4

The modelled domain consists of a 15 degree sector, which is truncated at r = 1 m where a mass flux hexahedral elements, with the spacing

of a 15 degree sector, which is composed of a line of hexahedral elements except at the origin where a prism element is used. The outflow is modelled using point sinks at the two

The modelled domain consists of a 15 degree sector, which is truncated at r = 1 m, where a mass flux boundary is applied. The mesh consists of a line of 3D hexahedral elements for r = 1 m to r = 1000 m. The mesh is then refined into two cells circumferentially from r = 1000 m to r = 2000 m. The meshes are joined using ConnectFlow constraints. This is an advanced technique that allows grids of different densities to be


2.1.5 Results

The analytical solution is given by

)ln(2

)()(r

R

Kd

QRhrh

π−=

The results from ConnectFlow show very good agreement with the analytical solution in Figure 2-6. The solution for the constrained mesh in

Figure 2-5 Mass flux boundary condition

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Radius [m]

He

ad

[m

]

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The results from ConnectFlow show very good agreement with the analytical solution in . The solution for the constrained mesh in Figure 2-7 is a little less accurate as expected.

Mass flux boundary condition

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

Page 5

The results from ConnectFlow show very good agreement with the analytical solution in Figure 2-5 and is a little less accurate as expected.

ConnectFlow

Analytical Solution


Figure 2-6 Point sink and prism element

Figure 2-7 Constrained mesh

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Head

[m

]

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Head

[m

]

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Point sink and prism element

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

Page 6

ConnectFlow

Analytical Solution

ConnectFlow

Analytical Solution



2.2.1 Overview

This case is taken from Level 1 of the international HYDROCOIN project flow codes [5]. It models steady state flow in a twocontains two inclined fractures which intersect one another at depth, and have a higher permeability than the surrounding rock.

The topography has been made simple so that it consists of two valleys located where the fracture zones meet the surface. To simplify the problem definition, the shape of the surface is described by straight lines. Although the surface topography is symmetric, the flow is influenced by the asymmetry of the fracture zones.

This problem is based on an idealized version of the hydrogeological conditions encountered at a potential site for a deep repository in Swedish bedrock. A detailed threeseparate study [6].


Figure 2-8 depicts the modelled domain.

Figure 2-8 Fractured rock

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This case is taken from Level 1 of the international HYDROCOIN project for verification of groundwater ]. It models steady state flow in a two-dimensional vertical slice of fractured rock. The rock

contains two inclined fractures which intersect one another at depth, and have a higher permeability than

The topography has been made simple so that it consists of two valleys located where the fracture zones meet the surface. To simplify the problem definition, the shape of the surface is described by straight lines.

hy is symmetric, the flow is influenced by the asymmetry of the fracture

This problem is based on an idealized version of the hydrogeological conditions encountered at a potential site for a deep repository in Swedish bedrock. A detailed three-dimensional model of this was made in a

depicts the modelled domain.

Page 7

for verification of groundwater dimensional vertical slice of fractured rock. The rock

contains two inclined fractures which intersect one another at depth, and have a higher permeability than


hy is symmetric, the flow is influenced by the asymmetry of the fracture

This problem is based on an idealized version of the hydrogeological conditions encountered at a potential sional model of this was made in a


Symbol

Kr Hydraulic conductivity of rock

Kf

φ


2.2.3 Variations

2.2.3.1 Multiple Element Types

In this variation the region is meshed using a mix of hexahedral (CB27) and prismatic elements as shown in Figure 2-9.

Figure 2-9 Mesh around fractures

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Parameter Value

Hydraulic conductivity of rock 1.0E-8 m/s

Hydraulic conductivity of fracture

1.0E-6 m/s

Porosity 0.03

In this variation the region is meshed using a mix of hexahedral (CB27) and prismatic elements as shown

Page 8

Value

8 m/s

6 m/s

0.03

In this variation the region is meshed using a mix of hexahedral (CB27) and prismatic elements as shown


2.2.3.2 Hexahedral Elements

In this variant hexahedral elements only are used, and then the element permeabilties are modified according to the imported fractures. It was found that in this scenario the lower ora more accurate solution than the CB27 elements for a given mesh resolution. This is likely to be due to the rapid change in permeabilties that are not aligned with the grid.

The results presented here are for mesh with 160x80 elemethis approach is illustrated in Figure

Figure 2-10 Hexahedral elements

2.2.4 Results

The results presented here compare head profile at a height of y=show excellent agreement with the HYDROCOIN study. The hexahedral element results have a head profile that is close to the HYDROCOIN results, but with slightly higher heads across the range.

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In this variant hexahedral elements only are used, and then the element permeabilties are modified according to the imported fractures. It was found that in this scenario the lower order CB08 elements gave a more accurate solution than the CB27 elements for a given mesh resolution. This is likely to be due to the rapid change in permeabilties that are not aligned with the grid.

The results presented here are for mesh with 160x80 elements. The representation of the fractures using Figure 2-10.

The results presented here compare head profile at a height of y=-200. The multiple element type results show excellent agreement with the HYDROCOIN study. The hexahedral element results have a head

he HYDROCOIN results, but with slightly higher heads across the range.

Page 9

In this variant hexahedral elements only are used, and then the element permeabilties are modified der CB08 elements gave

a more accurate solution than the CB27 elements for a given mesh resolution. This is likely to be due to

nts. The representation of the fractures using

200. The multiple element type results show excellent agreement with the HYDROCOIN study. The hexahedral element results have a head

he HYDROCOIN results, but with slightly higher heads across the range.


Figure 2-11 Head at height -200m (multiple element types)

Figure 2-12 Head at height -200m (hexahedral element mesh)

In addition, the ConnectFlow results were compared against the Feftra base case results reported in [The differences in head between the two codes being less than 1% for the multiple element type mesh and less than 2% for the hexahedral mesh (relative to the head variation in the surface boundary condition).

100

105

110

115

120

125

0 400

Distance [m]

Head

[m

]

100

105

110

115

120

125

0 400

Distance [m]

Head

[m

]

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200m (multiple element types)

200m (hexahedral element mesh)

In addition, the ConnectFlow results were compared against the Feftra base case results reported in [rences in head between the two codes being less than 1% for the multiple element type mesh and

less than 2% for the hexahedral mesh (relative to the head variation in the surface boundary condition).

800 1200 1600

Distance [m]

ConnectFlow

Hydrocoin

800 1200 1600

Distance [m]

ConnectFlow

Hydrocoin

Page 10

In addition, the ConnectFlow results were compared against the Feftra base case results reported in [3]. rences in head between the two codes being less than 1% for the multiple element type mesh and

less than 2% for the hexahedral mesh (relative to the head variation in the surface boundary condition).

ConnectFlow

Hydrocoin

ConnectFlow

Hydrocoin


A particle track starting from position (100,as illustrated in Figure 2-13.

Figure 2-13 Particle track from (100

In Figure 2-14 the ConnectFlow particle track is overlaid on the geometry, with the track colouring representing elapsed time. The total time of the track is 5% hig

Figure 2-14 Particle track overlaid on geometry

-700

-600

-500

-400

-300

-200

-100

0

100

200

0 200 400

Y m

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A particle track starting from position (100,-200) also showed good agreement with the Feftra base case,

Particle track from (100,-200) with multiple element types

the ConnectFlow particle track is overlaid on the geometry, with the track colouring otal time of the track is 5% higher than the Feftra base case.

Particle track overlaid on geometry

600 800 1000 1200 1400

X m

Page 11

showed good agreement with the Feftra base case,

the ConnectFlow particle track is overlaid on the geometry, with the track colouring her than the Feftra base case.

ConnectFlow

Feftra


2.3 2D Steady Flow with Particle Tracks

2.3.1 Overview

This case is taken from Level 3 of the flow codes [7]. It models steady state flow in a tworegion of higher permeability.

The case has a non-uniform analytical solution antracking.


The analytical solution assumes an infinite domain for the low permeability region. The original HYDROCOIN setup had a disk radius acalculated and analytically prescribed flow fields.

In the ConnectFlow results presented here both the flow field and particle tracks are calculated. From initial tests it was found that a larger outer region was required in order to appropriatanalytical solution.


Symbol

Lx

Ly Vertical outer region distance

a

P1

P0

ko

ki

φ

ρ

µ


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2D Steady Flow with Particle Tracks

This case is taken from Level 3 of the international HYDROCOIN project for verification of groundwater ]. It models steady state flow in a two-dimensional vertical slice of rock, containing a circular

uniform analytical solution and is used in the HYDROCOIN study to test particle

solution assumes an infinite domain for the low permeability region. The original HYDROCOIN setup had a disk radius a = 10m, an outer region Lx = 50m, Ly = 30m ancalculated and analytically prescribed flow fields.

In the ConnectFlow results presented here both the flow field and particle tracks are calculated. From initial tests it was found that a larger outer region was required in order to appropriat

Schematic of problem definition

Parameter Value

Upstream and downstream distances

250 m

Vertical outer region distance 240 m

Radius of inner disk 10 m

Upstream pressure 2.5E5 Pa

Downstream pressure -2.5E5 Pa

Permeability of outer region 1.0E-15 m

Permeability of inner region 1.0E-13 m

Porosity 0.1

Density 1000 kg/m

Viscosity 1.0E-3 Pa.s

Page 12

international HYDROCOIN project for verification of groundwater dimensional vertical slice of rock, containing a circular

d is used in the HYDROCOIN study to test particle

solution assumes an infinite domain for the low permeability region. The original 10m, an outer region Lx = 50m, Ly = 30m and used both

In the ConnectFlow results presented here both the flow field and particle tracks are calculated. From initial tests it was found that a larger outer region was required in order to appropriately model the

Value

250 m

240 m

10 m

2.5E5 Pa

2.5E5 Pa

15 m2

13 m2

0.1

1000 kg/m3

3 Pa.s


Eight particle tracks are released 50 m upstream of the disk centre and at Y values of 10, 12, 14, 16, 18, 20, 22 and 24 m.

The analytical solution for the pathlines is given in the HYDROCOIN report [

)1/()(

)(2

2

oi

oi

kk

kk

r

aoyy

+

−+= for r > a

o

oi

k

kk

oyy2

)( += for r < a

Where r is the distance from the center of the disk and ya long distance away from the origin.

2.3.3 Variations

2.3.3.1 Wrapped Mesh

In this variation, the mesh is modelled to wrap around the cylinder. A higher quality mesh is generated, but some vertices are surrounded by 3 elements and others by 5. A mesh of around 3000 elements was used.

Figure 2-16 Wrapped mesh

The mesh topology does not support the mass conserving particle tracking method, so just the regular particle tracking approach was used.

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Eight particle tracks are released 50 m upstream of the disk centre and at Y values of 10, 12, 14, 16, 18,

analytical solution for the pathlines is given in the HYDROCOIN report [7] as

distance from the center of the disk and y0 is a constant representing the height of the track a long distance away from the origin.


The mesh topology does not support the mass conserving particle tracking method, so just the regular particle tracking approach was used.

Page 13


is a constant representing the height of the track


The mesh topology does not support the mass conserving particle tracking method, so just the regular


2.3.3.2 Regular Mesh (Distorted Elements)

In this variation a regular mesh is used where elemdistorted mesh inside the cylindrical region. A mesh of around 6000 elements was used. The mesh is finer in this case, as the refinement of the cylinder propagates to the boundaries.

Figure 2-17 Regular mesh

Both regular particle tracking and the mass conserving method were used.

2.3.4 Results

The calculated particle tracks were within 1% of the analytical solution, both in terms of location at each point and in terms of overall travel from x =

Calculation method

Wrapped mesh, regular tracks

Regular mesh, regular tracks

Regular mesh, mass conserving tracks

Table 2-5 Particle tracking results

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Regular Mesh (Distorted Elements)

In this variation a regular mesh is used where elements always have 4 neighbours, which results in a distorted mesh inside the cylindrical region. A mesh of around 6000 elements was used. The mesh is finer in this case, as the refinement of the cylinder propagates to the boundaries.

Both regular particle tracking and the mass conserving method were used.

The calculated particle tracks were within 1% of the analytical solution, both in terms of location at each in terms of overall travel from x = -50m to x = +50m.

% Error in Location % Error in Travel Time

0.96% 0.44%

0.41% 0.17%

0.27% 0.20%

Page 14

ents always have 4 neighbours, which results in a distorted mesh inside the cylindrical region. A mesh of around 6000 elements was used. The mesh is finer

The calculated particle tracks were within 1% of the analytical solution, both in terms of location at each

% Error in Travel Time

0.44%

0.17%

0.20%


Backward particle tracks from x = 50m to x = analytical values.

Calculation method

Wrapped mesh, regular tracks

Regular mesh, regular tracks

Regular mesh, mass conserving tracks

Table 2-6 Backward particle tracking results

Figure 2-18 Particle tracks for wrapped mesh

In addition, the volumetric flow rate through the inner disk was calculated for variation 1, using the “calculate conserved mass flux” option.

The flow through the disk from [7] has a constant velocity in the X direction of

the values in Table 2-4, this gives a flow rate of 1.9801E1.9834E-7 m3/s which is within 1% of the analytical solution.

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Backward particle tracks from x = 50m to x = -50m were also calculated and were again within 1% of the

% Error in Location % Error in Travel Time

0.93% 0.45%

0.41% 0.17%

0.27% 0.21%

Backward particle tracking results

Particle tracks for wrapped mesh

In addition, the volumetric flow rate through the inner disk was calculated for variation 1, using the “calculate conserved mass flux” option.

] has a constant velocity in the X direction of ( i

o

k

k

+

, this gives a flow rate of 1.9801E-7 m3/s. The calculated ConnectFlow value is /s which is within 1% of the analytical solution.

Page 15

50m were also calculated and were again within 1% of the

ror in Travel Time

0.45%

0.17%

0.21%

In addition, the volumetric flow rate through the inner disk was calculated for variation 1, using the

)( )

µϕxo

io

L

PP

k

k

2

01−

+. For

/s. The calculated ConnectFlow value is


2.4 Transient Buoyant Flow

2.4.1 Overview

This case is taken from Level 1 of theflows [5]. It models the flows arising from an exponentially decaying heat source and has an asolution.

This type of problem is relevant when considering the disposal of heat emitting radioactive waste, where buoyancy induced flows can last for thousands of years.



The test problem models transient heat flow through the rock only and ignores advection of heat. The viscosity is taken to be constant and the density variation is only applied to the buoyancyequations.

The modelled domain consists of a thin one cell thick segment of the sphere. The analytical solution assumes an unbounded region of surrounding rock. Following some initial test runs, a surrounding region of 12000 m was selected as having a minimal impact on the solution.

A relatively fine mesh of 26000 elements was used. Coarser meshes of around 5000 elements give good results for the temperature and pressure profiles but have larger errors on particle tracking positions and travel times.

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Transient Buoyant Flow

This case is taken from Level 1 of the international HYDROCOIN project for the verification of groundwater ]. It models the flows arising from an exponentially decaying heat source and has an a

This type of problem is relevant when considering the disposal of heat emitting radioactive waste, where buoyancy induced flows can last for thousands of years.


The test problem models transient heat flow through the rock only and ignores advection of heat. The viscosity is taken to be constant and the density variation is only applied to the buoyancy

The modelled domain consists of a thin one cell thick segment of the sphere. The analytical solution assumes an unbounded region of surrounding rock. Following some initial test runs, a surrounding region

having a minimal impact on the solution.

A relatively fine mesh of 26000 elements was used. Coarser meshes of around 5000 elements give good results for the temperature and pressure profiles but have larger errors on particle tracking positions and

Page 16

international HYDROCOIN project for the verification of groundwater ]. It models the flows arising from an exponentially decaying heat source and has an analytical flow

This type of problem is relevant when considering the disposal of heat emitting radioactive waste, where

The test problem models transient heat flow through the rock only and ignores advection of heat. The viscosity is taken to be constant and the density variation is only applied to the buoyancy term of the

The modelled domain consists of a thin one cell thick segment of the sphere. The analytical solution assumes an unbounded region of surrounding rock. Following some initial test runs, a surrounding region

A relatively fine mesh of 26000 elements was used. Coarser meshes of around 5000 elements give good results for the temperature and pressure profiles but have larger errors on particle tracking positions and


Symbol

W0

λ

rρ

C

rΓ

k

φ

Ss

ρ

µ

β


2.4.3 Results

The comparison of results includes temperature profiles transient particle tracks Figure 2-22

The results are all within 7% of the analytical solution, including the particle travel times.

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Parameter Value

Initial power output 250 MW

Decay constant in heat source 7.3215E

Rock density 2.6E3 kg/m

Rock specific heat 8.79E2 J/kg K

Rock thermal conductivity 2.51 W/m K

Permeability 1.0E

Porosity 1.0E

Specific storage coefficient 2.0E

Density 992.2 kg/m

Viscosity 6.529E

Expansion coefficient of water 3.85E

The comparison of results includes temperature profiles Figure 2-20, pressure profiles 22 and Figure 2-23.

of the analytical solution, including the particle travel times.

Page 17

Value

250 MW

7.3215E-10 1/s

2.6E3 kg/m3

8.79E2 J/kg K

2.51 W/m K

1.0E-16 m2

1.0E-4

2.0E-6 1/m

992.2 kg/m3

6.529E-4 Pa.s

3.85E-4 1/K

, pressure profiles Figure 2-21 and

of the analytical solution, including the particle travel times.


Figure 2-20 Vertical temperature rise along the vertical sphere centreline

Figure 2-21 Vertical pressure rise along the vertical sphere centreline

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0 200

Radius [m]

Tem

pera

ture

Ris

e [

K]

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 200

Radius [m]

Pre

ssu

re R

ise [

Pa]

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Vertical temperature rise along the vertical sphere centreline

Vertical pressure rise along the vertical sphere centreline

400 600 800

Radius [m]

ConnectFlow t=50

Analytical t=50

ConnectFlow t=100

Analytical t=100

ConnectFlow t=500

Analytical t=500

ConnectFlow t=1000

Analytical t=1000

400 600 800

Radius [m]

ConnectFlow t=50

Analytical t=50

ConnectFlow t=100

Analytical t=100

ConnectFlow t=500

Analytical t=500

ConnectFlow t=1000

Analytical t=1000

Page 18

ConnectFlow t=50

Analytical t=50

ConnectFlow t=100

Analytical t=100

ConnectFlow t=500

Analytical t=500

ConnectFlow t=1000

Analytical t=1000

ConnectFlow t=50

Analytical t=50

ConnectFlow t=100

Analytical t=100

ConnectFlow t=500

Analytical t=500

ConnectFlow t=1000

Analytical t=1000


Figure 2-22 Particle tracks originating from z = 0 starting at t = 100 years

Figure 2-23 Particle tracks originating from z = 0 starting at t = 1000 years

-400

-200

0

200

400

600

800

1000

0 500

Radius [m]

z [

m]

-400

-200

0

200

400

600

800

1000

0 500

Radius [m]

z [

m]

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Particle tracks originating from z = 0 starting at t = 100 years

Particle tracks originating from z = 0 starting at t = 1000 years

500 1000

Radius [m]

ConnectFlow r = 125 m

Analytic r = 125 m


Analytic r = 250 m

Sphere

1000

Radius [m]


Analytic r = 125 m


Analytic r = 250 m

Sphere

Page 19


Analytic r = 125 m


Analytic r = 250 m


Analytic r = 125 m


Analytic r = 250 m


2.5 Unsaturated Heat Transport

2.5.1 Overview

In this section, the base-case modelSystems (EBS) Task Force [ 8 ] is investigated. A model, including the bentonite buffer and the host rock. The canister is not explicitly representedheat source term is applied on a canister buffer interface. Emphasis of the analysis is othe temperature and saturation with time along fixed positions on the bentonite buffer.


The deposition hole is based on the KBSand its features are schematically illustrated in

Figure 2-24 Dimensions of the computational domain (dark grey is the host rock, light grey is the bentonite yellow is the area of canister.

The modelling process is subdivided into two phases (i.e. Phase 1 and Phase 2). Phase 1 hydraulic calculation and considers an open deposition hole (i.e. with the canister and bentoniteDirichlet boundary conditions (BCs) (i.e. atmospheric pressure) and on the upper anis applied along the axis of symmetry and at the initial condition (IC) throughout the rest of the state is reached. Phase 1 providescoupled Thermo – hydraulic calculation. The numerical simulation the canister and bentonite buffer within the deposition hole 1 For the purposes of this verification exercise, the thermal conductivity of the buffer is assumed to be constant rather than a function of saturation (i.e.

description for the base-case model.

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Unsaturated Heat Transport

case model1of the Sensitivity Analysis Task of the Äspö Engineered Barrier is investigated. A single deposition hole is modelled in a 2D axisymmetric

model, including the bentonite buffer and the host rock. The canister is not explicitly representedheat source term is applied on a canister buffer interface. Emphasis of the analysis is othe temperature and saturation with time along fixed positions on the bentonite buffer.

ole is based on the KBS-3V [ 8 ] specifications. Dimensions of the computational domain and its features are schematically illustrated in Figure 2-24.

of the computational domain (dark grey is the host rock, light grey is the bentonite yellow is the area of canister. This figure is reproduced from [8]

The modelling process is subdivided into two phases (i.e. Phase 1 and Phase 2). Phase 1 considers an open deposition hole (i.e. with the canister and bentonite

(BCs) for the pressure are specified on the surface of the depositi(i.e. atmospheric pressure) and on the upper and lower boundaries of the model. Ais applied along the axis of symmetry and at the other outer boundaries. Hydrostatic pressure is set as an

throughout the rest of the domain and a transient simulation is performed untilprovides the steady state pressure field for Phase 2, as an IC. Phas

calculation. The numerical simulation begins on the date of emplacement of the canister and bentonite buffer within the deposition hole, and a transient calculation is performed for

For the purposes of this verification exercise, the thermal conductivity of the buffer is assumed to be

constant rather than a function of saturation (i.e. ( ) ll SS ⋅+−⋅ 3.117.0 W/(m·K)) prescribed in the Task

case model.

Page 20

Engineered Barrier single deposition hole is modelled in a 2D axisymmetric

model, including the bentonite buffer and the host rock. The canister is not explicitly represented; instead a heat source term is applied on a canister buffer interface. Emphasis of the analysis is on the variation of the temperature and saturation with time along fixed positions on the bentonite buffer.

specifications. Dimensions of the computational domain

of the computational domain (dark grey is the host rock, light grey is the bentonite

The modelling process is subdivided into two phases (i.e. Phase 1 and Phase 2). Phase 1 is a purely considers an open deposition hole (i.e. with the canister and bentonite absent).

the surface of the deposition hole A Neumann ‘no-flow’ BC,

Hydrostatic pressure is set as an simulation is performed until steady

as an IC. Phase 2 is a on the date of emplacement of

a transient calculation is performed for

For the purposes of this verification exercise, the thermal conductivity of the buffer is assumed to be

) prescribed in the Task


100 years after the installation. The at the outer edge of the canister –heat generated by nuclear fuel wasteThe initial power output ��is 1700

�� ∑ �� exp ��

��

The ICs and BCs employed in the modelling ofThe physical parameters used in the modelling

Figure 2-25 Summary of the ICs and BCs

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100 years after the installation. The BCs applied in Phase 1 are used, with the addit– bentonite interface. A heat source applied at the interface

nuclear fuel waste. The power decay used in the modelling is given by Equatiis 1700 W, whilst parameters it and ia are listed in Table

the modelling of both phases are schematically summarisedused in the modelling are presented in Table 2-9 and Table

Summary of the ICs and BCs for Phase 1 and Phase 2.

Page 21

with the addition of a Neumann BC e applied at the interface emulates the

given by Equation (1). Table 2-8.

�1�

ses are schematically summarised in Figure 2-25. Table 2-10.


The standard Van Genuchten model (

2-10 for the host rock), is fully implemented bentonite buffer, a modified Van Genuchten functionTable 2-10). In addition, a cubic law relating the relative permeability and the saturation is also prescribed. This functionality is easily implemented in ConnectFlow with the aid of two user defined external routinesuspcap.f and uskrel.f. These versatile

In total, the grid consists of approximately 2500 bentonite buffer, the canister – bentonite interface, as well as the points used for the sampling of temperature and saturation are presented.

I

1

2

3

4

5

6

7

Table 2-8 Constants and coefficients for the power output from the canister

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The standard Van Genuchten model (i.e. the functional forms ( )capl

PS , )(lrel

Sk

fully implemented in ConnectFlow. The task specifications require thata modified Van Genuchten function, is to be used (with an extra term i

a cubic law relating the relative permeability and the saturation is also prescribed. mplemented in ConnectFlow with the aid of two user defined external routinesse versatile routines enable the user to freely specify the relationships used

the grid consists of approximately 2500 finite elements. In Figure 2-26, the discretisation along the bentonite interface, as well as the points used for the sampling of

temperature and saturation are presented.

ti (years) �� 20 0.0601

50 0.7050

200 -0.0547

500 0.249

2000 0.0254

5000 -0.0094

20000 0.023

and coefficients for the power output from the canister, as described in Equation (1)

Page 22

), as presented in Table

specifications require that, for the extra term in (��,�� ) in

a cubic law relating the relative permeability and the saturation is also prescribed. mplemented in ConnectFlow with the aid of two user defined external routines,

the relationships used.

, the discretisation along the bentonite interface, as well as the points used for the sampling of

0.0601

0.7050

0.0547

0.2498

0.0254

0.0094

0.0239

as described in Equation (1)


Parameter

Fluid Properties

Density ( lρ )

Viscosity ( lµ )

Host Rock Properties

Density (Rρ )

Porosity (Rn )

Initial Liquid Saturation ( 0,lS )

Capillary Pressure ( capRP , )

Relative permeability (liquid) ( relRk ,

Tortuosity�!"� Bentonite Properties

Density (Bρ )

Porosity (Bn )

Initial Liquid Saturation ( 0,lS )

Capillary Pressure ( capBP , )

Tortuosity (Bτ )

Relative permeability (liquid) ( relBk ,

Table 2-9 Thermodynamic, transport and hydraulic properties, are presented for the fluidand the bentonite buffer

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Value

1000 kg / m3

1.0 x 10-3 Pa·s

2700 kg / m3

0.003

1.0

) ( ) cap

caplP

PPS

−+=

0

1

P0 = 1.74 MPa, m=0.60

rel )

1

11)(

−−=

m

m

lllrelSSSk

m=0.6

1.0

2780 kg / m3

0.438

0.61

( )1

1

0

,

,1

mcapB

capBlP

PPS

−+=

−

−

P0 = 5.523 MPa, m=0.16, P1 = 950 MPa, m1=1.6.

1.0

lrel, ) 3lS

Thermodynamic, transport and hydraulic properties, are presented for the fluidand the bentonite buffer

Page 23

m

m

−

−

1

1

,

=0.60

2

,

1

1

,1

m

capB

m

P

P

+

=1.6.

Thermodynamic, transport and hydraulic properties, are presented for the fluid, the host rock


Parameter

Thermal conductivity ( lλ )

Specific heat capacity ( lc )

Host

Thermal conductivity ( Rλ )

Specific heat capacity ( Rc )

Bentonite Properties

Thermal conductivity (Bλ )

Specific heat capacity ( Bc

Table 2-10 Thermal properties of the

2.5.3 Results

In Figure 2-27, a series of surface plots of the temperature distribution (Phase 2 calculation)different times are displayed. The temperature variation is shown for the whole extent of the comdomain. The Task Description, prescribes six sampling pointcollection of temperature and saturation measurements.are directly compared to ones obtained by(Figure 2-28) and saturation profiles (ConnectFlow is found to be smaller by few degreeson the slope of saturation rate, full saturation is achieved for all points at approximately the same timeConnectFlow predicting a fully saturated state at

Deviations are to be expected. Even though the same grnumerical solution scheme is adoptedis opted for ConnectFlow. In particulardifferent. TOUGH2 uses a multiphase calculation[11]. Within this investigation, emphasis was placed simulation, thus an approximation to Richards equation.

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Value

Fluid Properties

0.6 W/(m·K)

) 4.183 kJ/(kg·K)

Host Rock Properties

) 2.4 W/(m·K)

) 770 J/(kg·K)

Bentonite Properties

) 1.2 W/(m·K)

B ) 800 J/(kg·K)

hermal properties of the fluid, the host rock and the buffer.

, a series of surface plots of the temperature distribution (Phase 2 calculation). The temperature variation is shown for the whole extent of the com

The Task Description, prescribes six sampling points (see Figure 2-26) to be used for the collection of temperature and saturation measurements. In this study, results obtained using ConnectFlow

to ones obtained byTOUGH2. A good agreement is shown forration profiles (Figure 2-29). The maximum temperature achieved for

ler by few degrees. However, even though there is a moderate deviation l saturation is achieved for all points at approximately the same time

ConnectFlow predicting a fully saturated state at slightly earlier times.

ven though the same grid is used for both computationserical solution scheme is adopted in TOUGH2 (i.e. Finite Difference), while the Finite Element method

In particular, the set of equations solved by ConnectFlow and TOUGH2 are multiphase calculation, whilst ConnectFlow solves Richards

emphasis was placed on the minimisation of gas effects within, thus an approximation to Richards equation is justified.

Page 24

, a series of surface plots of the temperature distribution (Phase 2 calculation) collected at . The temperature variation is shown for the whole extent of the computational

) to be used for the obtained using ConnectFlow

good agreement is shown for both temperature achieved for all points by

there is a moderate deviation l saturation is achieved for all points at approximately the same time, with

id is used for both computations, a different while the Finite Element method

by ConnectFlow and TOUGH2 are s Richards equation [9], [10] &

minimisation of gas effects within the TOUGH2


Figure 2-26 Part of the mesh used discretisation of the buffer. Temperature and saturation profiles are obtained for the six points in the buffer and presented in layer (red elements) between the canister and the buffer is shown.

.

Figure 2-27 The temperature field in the

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used for the computations is given in the figure on the leftdiscretisation of the buffer. Temperature and saturation profiles are obtained for the six

and presented in Figure 2-28 & Figure 2-29. On the right, the interfacelayer (red elements) between the canister and the buffer is shown.

ture field in the model is plotted at different times.

Page 25

n the figure on the left, showing the discretisation of the buffer. Temperature and saturation profiles are obtained for the six

. On the right, the interface


Figure 2-28 Variation of Temperature with time, simulations.

Figure 2-29 Variation of Saturation with time, simulations.

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iation of Temperature with time, calculated by ConnectFlow (�) and TOUGH2(

Variation of Saturation with time, calculated by ConnectFlow (�) and TOUGH2(

Page 26

) and TOUGH2(- -)

) and TOUGH2(- -)


2.6 1D Transient Unsaturated Flow

2.6.1 Overview

This case models transient flow in a 50 m horizontal section of clay, where the initiaand the ends are maintained at a different pressure.

For small variations in pressure the problem has a semi


Figure 2-30 Schematic of problem

The region was meshed using a line of 40 hexahedral elements.

The unsaturated behaviour for relative permeability and capillary pressure were modelled using Van Genuchten functions.

Symbol

K

φ

Ss

ρ

µ

Van Genuchten

n

Pr

Slr

Table 2-11 Input parameter values

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1D Transient Unsaturated Flow

This case models transient flow in a 50 m horizontal section of clay, where the initiaand the ends are maintained at a different pressure.

For small variations in pressure the problem has a semi-analytical solution.


The region was meshed using a line of 40 hexahedral elements.

The unsaturated behaviour for relative permeability and capillary pressure were modelled using Van

Parameter Value

Hydraulic Conductivity 5.0E

Porosity 0.18

Specific storage coefficient 2.0E

Density 1000 kg/m

Viscosity 1.0E

Exponent

Entry pressure 8.0E6 Pa

Residual Saturation 0.01

Input parameter values

Page 27

This case models transient flow in a 50 m horizontal section of clay, where the initial pressure is constant

The unsaturated behaviour for relative permeability and capillary pressure were modelled using Van

Value

5.0E-14 m/s

0.18

2.0E-6 1/m

1000 kg/m3

1.0E-3 Pa.s

1.5

8.0E6 Pa

0.01


As the variation in saturation across the model is small (<3%), we can approximate the equations for transient unsaturated flow as a diffusion

2

2

x

P

t

P

∂

∂=

∂

∂α

where

α is the diffusivity [m

x is distance [m];

P is pressure;

t is time.

and α is given by

P

SgSS

Kk

s

r

∂

∂+

=

φρ

α

where S is the liquid saturation and

For the case considered here, the values in the above equation are averaged between the initial conditions and the boundary conditions to provide an approximate constant diffusivity.

Given this approximate constant diffusivity, a semi

( )( )∑

∞

= +−+=

0

101 exp12

4

n nPPPP

π

Where Po is the uniform initial pressure field domain length of 50 m.

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As the variation in saturation across the model is small (<3%), we can approximate the equations for transient unsaturated flow as a diffusion equation, so that, in 1D

is the diffusivity [m2s-1];

is distance [m];

where S is the liquid saturation and rk is the relative permeability and g is gravity.

For the case considered here, the values in the above equation are averaged between the initial conditions and the boundary conditions to provide an approximate constant diffusivity.

constant diffusivity, a semi-analytical solution can be obtained from

( ) ( )

+

+−

2

2212

sin12

expL

xn

L

tn παπ

is the uniform initial pressure field -1.1E7, P1 the boundary pressure of -1.0E7, and

Page 28

As the variation in saturation across the model is small (<3%), we can approximate the equations for

For the case considered here, the values in the above equation are averaged between the initial conditions

analytical solution can be obtained from

1.0E7, and L the


2.6.3 Variations

2.6.3.1 Crank Nicholson

The transient behaviour is modelled using Crank Nicholson with 150 time steps of size 1.0E11 seconds.

2.6.3.2 Gears method

The transient behaviour is modelled using Gears predictortime step of 1.0E11 seconds. A total of

2.6.4 Results

The results show good agreement with the semisemi-analytical model assuming constant saturation with time.

Figure 2-31 Crank Nicholson - Transient unsaturated pressures

-1.08E+07

-1.07E+07

-1.06E+07

-1.05E+07

-1.04E+07

-1.03E+07

-1.02E+07

-1.01E+07

-1.00E+07

0 10 20

Distance [m]

Resid

ual

Pre

ssu

re [

Pa]

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transient behaviour is modelled using Crank Nicholson with 150 time steps of size 1.0E11 seconds.

The transient behaviour is modelled using Gears predictor-corrector time stepping method with an initial time step of 1.0E11 seconds. A total of 23 time steps were used.

The results show good agreement with the semi-analytical solution. The small differences are due to the analytical model assuming constant saturation with time.

Transient unsaturated pressures

30 40 50

Distance [m]

ConnectFlow 5.0E12 seconds

Semi analytical 5.0E12 seconds





Page 29

transient behaviour is modelled using Crank Nicholson with 150 time steps of size 1.0E11 seconds.

corrector time stepping method with an initial

analytical solution. The small differences are due to the








Figure 2-32 Gears method - Transient unsaturated pressures

An additional comparison was made with Tough2v2 software [

Calculation method

ConnectFlow – Crank Nicholson

ConnectFlow – Gears Method

Semi analytical

Tough2

Table 2-12 Comparison with Tough2v2

Both ConnectFlow and Tough2 show the same behaviour relative to the semi analytical solution.

-1.08E+07

-1.07E+07

-1.06E+07

-1.05E+07

-1.04E+07

-1.03E+07

-1.02E+07

-1.01E+07

-1.00E+07

0 10 20

Distance [m]

Resid

ual

Pre

ssu

re [

Pa]

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Transient unsaturated pressures

An additional comparison was made with Tough2v2 software [12].

Pressure at x=25, t=1.0E13

Crank Nicholson -1.0426E7

Gears Method -1.0421E7

-1.0438E7

-1.0423E7

Comparison with Tough2v2


30 40 50

Distance [m]







Page 30

Pressure at x=25, t=1.0E13









2.7 Seepage Face

2.7.1 Overview

A seepage face is a boundary between the saturated flow field and the atmosphere, along which groundwater discharges. A seepage face iDirichlet boundary condition (BC) head#(set to be equal to the elevation i.e. $ � 0). The BC along the ground surface above the upper limit (a point for a 2D models or a line in 3D models) of the seepage face, is usually a Neumann BC (i.e. no flow). The challenge in modelling a seepage face is that its upper extent is unknown during the problem formulation and its determination forms a part of the solution. This verification case considers the steady state radial flow to a well, for which the Dupuit –Forchheimer well discharge formula holds (analytical solutioassumptions) [13].


A schematic diagram of the problem (segment of a cylinder) is presented inpolar coordinate system �&, '�, the well axis is aligned with the imposed at the bottom and the top of the model. A far distance himposed on the right of the model, setting up the initial height of the water table (i.e. head BC (# � #(), is imposed at the bottom of the well (blue line) . Along the side of the well (red line)seepage face boundary condition is applied. The resulting flow is radially symmetric between the circular equipotential boundaries located at

Figure 2-33 Schematic illustration of seepage face.

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A seepage face is a boundary between the saturated flow field and the atmosphere, along which groundwater discharges. A seepage face is homologous to a free surface boundary condition, so a

suffices for its mathematical description, either in terms of hydraulic (set to be equal to the elevation ' i.e. # � '), or in terms of total pressure $

). The BC along the ground surface above the upper limit (a point for a 2D models or a line in 3D models) of the seepage face, is usually a Neumann BC (i.e. no flow). The challenge in modelling a

extent is unknown during the problem formulation and its determination This verification case considers the steady state radial flow to a well, for which

Forchheimer well discharge formula holds (analytical solution based on the Dupuit

A schematic diagram of the problem (segment of a cylinder) is presented in Figure , the well axis is aligned with the ' axis. A no-flow boundary condition is

imposed at the bottom and the top of the model. A far distance head boundary condition (green line) is imposed on the right of the model, setting up the initial height of the water table (i.e.

), is imposed at the bottom of the well (blue line) . Along the side of the well (red line)seepage face boundary condition is applied. The resulting flow is radially symmetric between the circular equipotential boundaries located at & � ) and & � &(.

Schematic illustration of the conceptual model of a steady state radial flow to a well with a

Page 31

A seepage face is a boundary between the saturated flow field and the atmosphere, along which s homologous to a free surface boundary condition, so a

suffices for its mathematical description, either in terms of hydraulic $ (set to atmospheric

). The BC along the ground surface above the upper limit (a point for a 2D models or a line in 3D models) of the seepage face, is usually a Neumann BC (i.e. no flow). The challenge in modelling a

extent is unknown during the problem formulation and its determination This verification case considers the steady state radial flow to a well, for which

n based on the Dupuit

Figure 2-33. In a cylindrical flow boundary condition is

ead boundary condition (green line) is imposed on the right of the model, setting up the initial height of the water table (i.e. # � #*), whilst a fixed

), is imposed at the bottom of the well (blue line) . Along the side of the well (red line), a seepage face boundary condition is applied. The resulting flow is radially symmetric between the circular

the conceptual model of a steady state radial flow to a well with a


For the above configuration and based on the Dupuit assumptionssteady volumetric discharge into the well can be derived. This expression known as the Dupuit Forchheimer well discharge formula, is given as:

+, � -. /012 /,2345" 6,7 8 .

Further, the phreatic surface elevation

/ � 9:�6�, where :�6� �

Symbol

.

ρ

µ

/,

/*

6,

"

Table 2-13 Input parameters.

.

2.7.3 Variations

2.7.3.1 Volumetric discharge

The first verification test involves the direct comparison of the volumetic flow rates to the well, as obtained by ConnectFlow, to the ones suggested by the Dupuitapproach incorporates a thin wedge shapedapproximation to the conceptual axissymetric model. Three flow domains are investigated. For each case, only the radius of the domain ) is altered (i.e. computational domain, the radius of the well, the head at the elevation of the well and the initial head of the aquifer are kept fixed as, ; �seepage face BC as implemented within Conin Figure 2-33). The next task, takes into account the performance of the implemented BC, by investigatihow closely the head approximates the elevation along the seepage face, as well as the recovery of the length of the seepage face.

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For the above configuration and based on the Dupuit assumptions [13] , an analytical expression for the steady volumetric discharge into the well can be derived. This expression known as the Dupuit Forchheimer well discharge formula, is given as:

8 .

Further, the phreatic surface elevation # � #�&� is given as:

� � � /,2 < �/∞2 � /,2 � =>5" 67 8

=>5" 6,7 8.

Parameter Value

Hydraulic conductivity 9.81E-7 m/s

Density 1000.0 kg/m3


Hydraulic Head at the well 100 m

Initial water table 450 m

Radius of the well 50 m

Radius of influence i)1000 m ii)2000 m iii)3000 m

The first verification test involves the direct comparison of the volumetic flow rates to the well, as obtained by ConnectFlow, to the ones suggested by the Dupuit- Forchheimer well discharge formula. The modelling approach incorporates a thin wedge shaped computational domain in ConnectFlow, which serves as an approximation to the conceptual axissymetric model. Three flow domains are investigated. For each case,

is altered (i.e. ) � 1000, 2000, 3000ABC� ,whilst the heightcomputational domain, the radius of the well, the head at the elevation of the well and the initial head of

800ABC, &( � 50ABC, #( � 100ABC and #* � 450seepage face BC as implemented within ConnectFlow, is applied along the periphery of the well (red line

). The next task, takes into account the performance of the implemented BC, by investigatihow closely the head approximates the elevation along the seepage face, as well as the recovery of the

Page 32

lytical expression for the steady volumetric discharge into the well can be derived. This expression known as the Dupuit –

( 1 )

( 2 )

Value

i)1000 m ii)2000 m iii)3000

The first verification test involves the direct comparison of the volumetic flow rates to the well, as obtained Forchheimer well discharge formula. The modelling

computational domain in ConnectFlow, which serves as an approximation to the conceptual axissymetric model. Three flow domains are investigated. For each case,

,whilst the height of the computational domain, the radius of the well, the head at the elevation of the well and the initial head of

450ABC, respectively. The nectFlow, is applied along the periphery of the well (red line

). The next task, takes into account the performance of the implemented BC, by investigating how closely the head approximates the elevation along the seepage face, as well as the recovery of the


2.7.4 Results

The results in terms of volumetric flow rates (solution (1) are in a good agreement. In particular it can be observed that an increase in the aspect ratio defined as the radius of influence

Radius of the domain

Aspect Ratio" 6,7

" � GHHHAIC 20

" � 2HHHAIC 40

" � JHHHAIC 60

Table 2-14 Volumetic flow rates to the well, calculated using the Dupuit ConnecFlow, for different radius of influence.

Considering the actual performance of the seepage face BC, it can be observed from along the length of the seepage face (selength of the seepage face is shown to be sensitive to the radius of influence, which appears to decrease with an increasing radius of influence. boundary of the well, and further supplements the above observation. Clearly the upper extent of the seepage face (position where the flowincreasing radius of influence. In Table presented, where it is also accompanied by the relative error of the computed head with respect to the actual elevation. Evidently, in all three cases the error is below #~') and a satisfactory performance for

Figure 2-34 On the left graph the Head is plotted as function of the elevation, whilst the figure on the right displays the fraction of the elevation over the head, as function of the elevation.

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olumetric flow rates (Table 2-14), as obtained by ConnectFlow and the analytical solution (1) are in a good agreement. In particular it can be observed that an increase in the aspect ratio defined as the radius of influence over the radius of the well, results in more accurate prediction

Aspect Ratio

7Dupuit-Forchheimer

IJL7

ConnectFlow IJ

L7

0.1980 0.1672

0.1608 0.1472

0.1448 0.1378

Volumetic flow rates to the well, calculated using the Dupuit –Forchheimer formula and ConnecFlow, for different radius of influence.

Considering the actual performance of the seepage face BC, it can be observed from along the length of the seepage face (see Table 2-15), the head closely approximates the elevation. The length of the seepage face is shown to be sensitive to the radius of influence, which appears to decrease

ius of influence. Figure 2-35, portrays the behaviour of the saturation along the boundary of the well, and further supplements the above observation. Clearly the upper extent of the seepage face (position where the flow is fully saturated i.e. � � 1), is found to be decreasing with an

Table 2-15, the position of the upper extent of the seepage facepresented, where it is also accompanied by the relative error of the computed head with respect to the actual elevation. Evidently, in all three cases the error is below 1%, suggesting a good approximation (i.e.

) and a satisfactory performance for the BC.

On the left graph the Head is plotted as function of the elevation, whilst the figure on the right displays the fraction of the elevation over the head, as function of the elevation.

Page 33

), as obtained by ConnectFlow and the analytical solution (1) are in a good agreement. In particular it can be observed that an increase in the aspect ratio

over the radius of the well, results in more accurate prediction.

Abosulute Error %

3.0804

1.3551

0.7035

Forchheimer formula and

Considering the actual performance of the seepage face BC, it can be observed from Figure 2-34 that ), the head closely approximates the elevation. The

length of the seepage face is shown to be sensitive to the radius of influence, which appears to decrease , portrays the behaviour of the saturation along the

boundary of the well, and further supplements the above observation. Clearly the upper extent of the ), is found to be decreasing with an

, the position of the upper extent of the seepage face is presented, where it is also accompanied by the relative error of the computed head with respect to the

, suggesting a good approximation (i.e.

On the left graph the Head is plotted as function of the elevation, whilst the figure on the right displays the fraction of the elevation over the head, as function of the elevation.


Radius of the

domain

Aspect Ratio

" 6,7

" � GHHHAIC 20

" � 2HHHAIC 40

" � JHHHAIC 60

Table 2-15 The upper extent of the seepage face, its actual length and the mean value of the relative error along the whole extent of the seepage face, measuring the performance of the seepage face BC.

Figure 2-35 Saturation N, along the well boundary is plotted as function of the elevation.

The computed phreatic surfaces obtained for the three flow domains, are grouped and presented, along with the ones obtained after applying the DupuitFigure 2-36. It must be noted that within the Dupuit assumptions the seepage face is neglected, and that the analytical approximation is only valid far from the well. As can be seen, the computed phreatic surface closely resembles the one obtained from Dupuit of depression is found to be decreasing as the radius of influence decreases.

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Aspect Ratio

7

Upper extent of the

seepage face

AIC

Seepage face

length

AIC

340 90

320 70

310 60

The upper extent of the seepage face, its actual length and the mean value of the relative error along the whole extent of the seepage face, measuring the performance of the seepage

, along the well boundary is plotted as function of the elevation.

The computed phreatic surfaces obtained for the three flow domains, are grouped and presented, along with the ones obtained after applying the Dupuit-Forchheimer model (see equation 2 ) are presented in

. It must be noted that within the Dupuit assumptions the seepage face is neglected, and that the analytical approximation is only valid far from the well. As can be seen, the computed phreatic surface

y resembles the one obtained from Dupuit – Forchheimer for increasing values of of depression is found to be decreasing as the radius of influence decreases.

Page 34

Mean Relative

Error

�/ � OO �%

0.3348

0.2888

0.2641

The upper extent of the seepage face, its actual length and the mean value of the relative error along the whole extent of the seepage face, measuring the performance of the seepage

, along the well boundary is plotted as function of the elevation.

The computed phreatic surfaces obtained for the three flow domains, are grouped and presented, along Forchheimer model (see equation 2 ) are presented in

. It must be noted that within the Dupuit assumptions the seepage face is neglected, and that the analytical approximation is only valid far from the well. As can be seen, the computed phreatic surface

Forchheimer for increasing values of &. Finally, the cone


Figure 2-36 Water table position, based on the computations (lines) and the Dupuit Forchheimer model (squares).

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osition, based on the computations (lines) and the Dupuit Forchheimer model

Page 35

osition, based on the computations (lines) and the Dupuit Forchheimer model



2.8.1 Overview

This case considers salt water intrusion into a vertical slice of an isotropic homogeneous confined aquifer.

The variant modelled is a modified version of the original Henry‘s test case, as recommended in [modified case halves the fresh water inflow rate, which increases the sensitivity of the solution to the variation in density.


A schematic of the test case is shown in

Figure 2-37 Henry's Problem

A uniform hexahedral grid was used of size 80x40 elements.

Symbol

K Hydraulic conductivity of rock

D

Q

0ρ

maxρ

Lα

Tα

φ

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led is a modified version of the original Henry‘s test case, as recommended in [modified case halves the fresh water inflow rate, which increases the sensitivity of the solution to the

st case is shown in Figure 2-37 and the input parameters are given in

A uniform hexahedral grid was used of size 80x40 elements.

Parameter Value


Coefficient of molecular diffusion

1.886E-5 m

Freshwater inflow per unit depth

3.3E-5 m

Reference density 998 kg/m

Saltwater density 1023 kg/m

Longitudinal dispersivity 0 m

Transverse dispersivity 0 m

Porosity 0.35

Page 36


led is a modified version of the original Henry‘s test case, as recommended in [14]. The modified case halves the fresh water inflow rate, which increases the sensitivity of the solution to the

and the input parameters are given in Table 2-16.

Value

2 m/s

5 m2/s

5 m2/s

998 kg/m3

1023 kg/m3

0 m

0 m

0.35



The model makes use of the ConnectFlow “reference waters” capability which allows fluids with different properties to be defined. The steady solution is obtained by running a transient solstate is reached.

2.8.3 Results

The test case has an analytical solution, represented by an infinite double Fourier series. A truncated form of the series gives a non-linear system which can then be iteratively solved. Results from this preported in [14] and these are used for the comparisons in location is less than 3%.

Figure 2-38 Non-dimensional concentration of salinity

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.20 0.40 0.60

y [

m]

ConnectFlow 25%

Semi analytical 25%

ConnectFlow 50%

Semi analytical 50%

ConnectFlow 75%

Semi analytical 75%

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The model makes use of the ConnectFlow “reference waters” capability which allows fluids with different properties to be defined. The steady solution is obtained by running a transient solution until the steady

The test case has an analytical solution, represented by an infinite double Fourier series. A truncated form linear system which can then be iteratively solved. Results from this p

] and these are used for the comparisons in Figure 2-38. The maximum error in contour

dimensional concentration of salinity

0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

X [m]

ConnectFlow 25%

Semi analytical 25%

ConnectFlow 50%

Semi analytical 50%

ConnectFlow 75%

Semi analytical 75%

Page 37

The model makes use of the ConnectFlow “reference waters” capability which allows fluids with different ution until the steady

The test case has an analytical solution, represented by an infinite double Fourier series. A truncated form linear system which can then be iteratively solved. Results from this process are

. The maximum error in contour

2.00


2.9 1D Rock Matrix Diffusion (RMD)

2.9.1 Overview

This case considers the transient one diffusion between the fractured rock and the adjacent rock matrix is modelled. This test case assumes a constant density.

The test case is taken from the SKB R


Figure 2-39 depicts the modelled 1D domain.

Figure 2-39 Schematic of problem

Symbol

K

D

ρ

maxρ

Lα

Tα

φ

µ

τ

mφ

iD

σ Fracture surface area per unit


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1D Rock Matrix Diffusion (RMD)

This case considers the transient one dimensional transport of salinity through fractured rock where diffusion between the fractured rock and the adjacent rock matrix is modelled. This test case assumes a

The test case is taken from the SKB R-04-78 report [15] and has a semi-analytical solution.

depicts the modelled 1D domain.


Parameter Value

Rock permeability 1.0E-11 m

Salt diffusion coefficient 1.0E-9 m

Density 998.3 kg/m

Saltwater density 998.3 kg/m



Fracture porosity 0.01


Tortuosity 1.0

Matrix porosity 0.1

Intrinsic diffusion coefficient 5.0E-11 m

Fracture surface area per unit volume.

2 m-1

Page 38

dimensional transport of salinity through fractured rock where diffusion between the fractured rock and the adjacent rock matrix is modelled. This test case assumes a

nalytical solution.

Value

11 m2

9 m2/s

998.3 kg/m3

998.3 kg/m3

100 m

10 m

0.01

3 Pa.s

11 m2/s

1


In this case the rock matrix diffusion slows down the increaseproportion of the salinity is diffusing into the surrounding rock matrix. The time taken for the midrelative concentration to reach 0.99 is increased by more than a factor 10 due to RMD.

2.9.3 Variations


In this variation the non-linear system is solved using Newton iteration on each time step for all variables as a single group. Automatic time stepping is used to gradually increase the time step size from its initial value of 1.0E6 seconds.

2.9.3.2 Sequential Inner Iteration

Sequential inner iteration solves the nonto a user-specified sequence of Newton iterations on subgroups of the full system of equations. In this test the pressure field is time independent, so just the salt concentration is solved for.

This setup also uses nodal quadrature in order to allow an optimized fast equation assembly process. Both the fast assembly and normal assembly were tested.

Automatic time stepping is used to gradually increase the time step size from its initial value of 1.0E6 seconds.

2.9.3.3 Increased Salt Diffusion Coefficient

The effective dispersion of concentration is given by

)( vD

Lατ

ϕρ +

Where v is the pore water velocity in the fractured rock, which for this case has a constant value of

4.981E-5 m/s. This means that τ

D

This variation sets the dispersion length

results in a modified test with the same effective diffusion and hence the same semi

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In this case the rock matrix diffusion slows down the increase in salinity in the fractured rock, as a proportion of the salinity is diffusing into the surrounding rock matrix. The time taken for the midrelative concentration to reach 0.99 is increased by more than a factor 10 due to RMD.

linear system is solved using Newton iteration on each time step for all variables as a single group. Automatic time stepping is used to gradually increase the time step size from its initial

tial Inner Iteration

Sequential inner iteration solves the non-linear system of equations that arise at each time step according specified sequence of Newton iterations on subgroups of the full system of equations. In this test

is time independent, so just the salt concentration is solved for.

This setup also uses nodal quadrature in order to allow an optimized fast equation assembly process. Both the fast assembly and normal assembly were tested.

sed to gradually increase the time step size from its initial value of 1.0E6

Increased Salt Diffusion Coefficient

The effective dispersion of concentration is given by

is the pore water velocity in the fractured rock, which for this case has a constant value of

D = 1.0E-9 and vLα = 4.981E-3.

This variation sets the dispersion length Lα to zero, and the salt diffusion coefficient D to 4.981E

results in a modified test with the same effective diffusion and hence the same semi

Page 39

in salinity in the fractured rock, as a proportion of the salinity is diffusing into the surrounding rock matrix. The time taken for the mid-point relative concentration to reach 0.99 is increased by more than a factor 10 due to RMD.

linear system is solved using Newton iteration on each time step for all variables as a single group. Automatic time stepping is used to gradually increase the time step size from its initial

linear system of equations that arise at each time step according specified sequence of Newton iterations on subgroups of the full system of equations. In this test

This setup also uses nodal quadrature in order to allow an optimized fast equation assembly process. Both

sed to gradually increase the time step size from its initial value of 1.0E6

is the pore water velocity in the fractured rock, which for this case has a constant value of

to zero, and the salt diffusion coefficient D to 4.981E-3. This

results in a modified test with the same effective diffusion and hence the same semi-analytical solution.


2.9.3.4 Finite Volume RMD

This variant uses the alternative finite volume RMD method indiscretisation of the rock matrix into a 1D column of cells at each solution node in the model. The discretisation is specified by the user, e.g. reasonable accuracy for a given level of refinement.compatible with reactive transport. The model used for verification is thmatrix porosity of 0.3.

Figure 2-40 Schematic of the distribution of cell sizes in the rock matrix

2.9.4 Results

The semi-analytical solution is derived using Laplace transforms, which are then inverted numerically. The details of this process are covered in [

The results show good agreement with the swithin 1% of the semi-analytical solution for all variations. The concentration profiles for three time values are shown in Figure 2-41.

Fracture

Original distribution (x = D/5)

Distribution 2 (x = D/15)




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This variant uses the alternative finite volume RMD method in ConnectFlow that provides an explicit discretisation of the rock matrix into a 1D column of cells at each solution node in the model. The discretisation is specified by the user, e.g. Figure 2-40. This method provides computational efficiency with reasonable accuracy for a given level of refinement. Unlike the original RMD method, this method is compatible with reactive transport. The model used for verification is the same as the base case, but with a

Schematic of the distribution of cell sizes in the rock matrix.

derived using Laplace transforms, which are then inverted numerically. The details of this process are covered in [15].

The results show good agreement with the semi-analytical solution, the ConnectFlow concentrations being analytical solution for all variations. The concentration profiles for three time values

Rock matrix Fracture

x 2x 3x 4x 5x

x 2x 4x 6x 8x

x 2x 4x 8x 16

5x 4x 3x 2x x

x x x x x

D

Page 40

ConnectFlow that provides an explicit discretisation of the rock matrix into a 1D column of cells at each solution node in the model. The

This method provides computational efficiency with Unlike the original RMD method, this method is

e same as the base case, but with a

derived using Laplace transforms, which are then inverted numerically. The

analytical solution, the ConnectFlow concentrations being analytical solution for all variations. The concentration profiles for three time values


The results for the finite volume RMD method are shown in agreement with the analytical solution, but accuracy increases as the cell sizes closest to the fracture decrease in size, either due to an increase in the number of cells or by using a distribution tin cell size with distance from the fracture.

Figure 2-41 Crank Nicholson - Relative concentration

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

9.00E-01

1.00E+00

0 2000 4000

Rela

tive C

on

cen

trati

on

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00E+00

Re

lativ

e c

on

cent

ratio

n

FV RMD 5

FV RMD 5 (distribution 2)

FV RMD 5 (distribution 3)

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The results for the finite volume RMD method are shown in Figure 2-42 to Figure 2agreement with the analytical solution, but accuracy increases as the cell sizes closest to the fracture decrease in size, either due to an increase in the number of cells or by using a distribution tin cell size with distance from the fracture.

Relative concentration

4000 6000 8000 10000

Distance [m]






Semi analytical 5E9 seconds

2.00E+03

Distance (m)

FV RMD 5 (distribution 4) FV RMD 20

FV RMD 5 (distribution 5) Analytical RMD

FV RMD 10

Page 41

2-44. There is good agreement with the analytical solution, but accuracy increases as the cell sizes closest to the fracture decrease in size, either due to an increase in the number of cells or by using a distribution that increases






Semi analytical 5E9 seconds

4.00E+03

Analytical RMD


Figure 2-42 Relative concentration along the column at time t=1.0matrix diffusion (RMD) with varying number of matrix cells and different cell sizes compared to an analytical solution.

Figure 2-43 Relative fracture water concentration midway along the column against time, using finite volume rock matrix diffusion with varying number of rock matrix cells compared to an analytical solution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00E+00 2.00E+09

Rel

ativ

e co

ncen

trat

ion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00E+00 2.00E+09

Rel

ativ

e co

ncen

trat

ion

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Relative concentration along the column at time t=1.0·108 s, using finite volume (FV) rock matrix diffusion (RMD) with varying number of matrix cells and different cell sizes compared to an

Relative fracture water concentration midway along the column against time, using finite volume rock matrix diffusion with varying number of rock matrix cells compared to an analytical solution.

4.00E+09 6.00E+09 8.00E+09

Time (s)

Finite volume RMD (5 cells)



Analytical RMD

4.00E+09 6.00E+09 8.00E+09

Time (s)




Analytical RMD

Page 42

108 s, using finite volume (FV) rock matrix diffusion (RMD) with varying number of matrix cells and different cell sizes compared to an


8.00E+09




8.00E+09





Figure 2-44 Relative fracture water concentration midway along the column against time, using finite volume rock matrix diffusion with varying number of rock matrix cells compared to an analytical solution.

2.10 1D Nuclide Transport with S

2.10.1 Overview

This case considers transient onesorption and decay are considered.

The test case is taken from [16] and has an analytical solution.


Figure 2-39 depicts the modelled 1D domain.

Figure 2-45 Schematic illustration of the problem definition

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1D Nuclide Transport with Sorption and Decay

This case considers transient one-dimensional nuclide transport due to advection and diffusion. Both sorption and decay are considered.

] and has an analytical solution.

depicts the modelled 1D domain.

Schematic illustration of the problem definition

Page 43


dimensional nuclide transport due to advection and diffusion. Both


Symbol

k

U

D

Lα

Tα

Rn1

Rn2

1nλ

2nλ

τ

ρ

µ

φ


Constant pressure boundary conditions are applied to generate the desired Darcy velocity. The modelled is 20 days which also corresponds to the half life of the parent nuclide N1.

The mesh consists of 100 CB08 elements along the length of the flow domain, and the time step is taken to be 1E4 seconds.

2.10.3 Variations

2.10.3.1 Fast Linear Transport

Four runs are carried out to illustrate the effect of sorption and decay on the solution.

► No decay or sorption

► Sorption only

► Decay only

► Sorption and decay

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Parameter Value


Darcy velocity 3.486E-7 m/s

Nuclide diffusion coefficient 1.162E-7



Retardation factor nuclide 1 2.0

Retardation factor nuclide 2 1.0

Decay constant nuclide 1 4.011E-7

Decay constant nuclide 2 1.0E-20 1/s

Tortuousity 1.0

Density 1000.0 kg/m


Porosity 0.3

Constant pressure boundary conditions are applied to generate the desired Darcy velocity. The modelled is 20 days which also corresponds to the half life of the parent nuclide N1.

The mesh consists of 100 CB08 elements along the length of the flow domain, and the time step is taken

runs are carried out to illustrate the effect of sorption and decay on the solution.

Page 44

Value

9 m2

7 m/s

7m2/s

0 m

0 m

7 1/s

20 1/s

1000.0 kg/m3

3 Pa.s

Constant pressure boundary conditions are applied to generate the desired Darcy velocity. The timescale modelled is 20 days which also corresponds to the half life of the parent nuclide N1.

The mesh consists of 100 CB08 elements along the length of the flow domain, and the time step is taken

runs are carried out to illustrate the effect of sorption and decay on the solution.



This variation repeats the sorption and decay case using the Crank Nicholson transient s

2.10.4 Results

The results show excellent agreement with the analytical solution [nuclide N1 is compared with the analytical solution for each of the four fast linear transport runs in 2-46, Figure 2-47, Figure 2-48 and

The Crank Nicholson results were very close to the Fast Transient results, with the peak daughter nuclide concentration being 1% higher and the parent nuclide concentration being identical.

Figure 2-50 shows the ConnectFlow daughter nuclide N2 concentration for the sorption and decay case.

Figure 2-46 N1 concentration with no sorption or decay

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 1

Co

ncen

trati

on

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This variation repeats the sorption and decay case using the Crank Nicholson transient s

The results show excellent agreement with the analytical solution [17]. The concentration of the parent nuclide N1 is compared with the analytical solution for each of the four fast linear transport runs in

and Figure 2-49.

The Crank Nicholson results were very close to the Fast Transient results, with the peak daughter nuclide her and the parent nuclide concentration being identical.

shows the ConnectFlow daughter nuclide N2 concentration for the sorption and decay case.

N1 concentration with no sorption or decay

2 3 4

Distance [m]

ConnectFlow

Analytical Solution

Page 45

This variation repeats the sorption and decay case using the Crank Nicholson transient solver.

]. The concentration of the parent nuclide N1 is compared with the analytical solution for each of the four fast linear transport runs in Figure

The Crank Nicholson results were very close to the Fast Transient results, with the peak daughter nuclide her and the parent nuclide concentration being identical.

shows the ConnectFlow daughter nuclide N2 concentration for the sorption and decay case.

ConnectFlow

Analytical Solution


Figure 2-47 N1 concentration with sorption only

Figure 2-48 N1 concentration with decay only

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 1

Co

ncen

trati

on

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 1

Co

ncen

trati

on

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N1 concentration with sorption only

N1 concentration with decay only

2 3 4

Distance [m]

ConnectFlow

Analytical Solution

2 3 4

Distance [m]

ConnectFlow

Analytical Solution

Page 46

ConnectFlow

Analytical Solution

ConnectFlow

Analytical Solution


Figure 2-49 N1 concentration with both sorption and decay

Figure 2-50 Parent and daughter concentrations for the

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 1

Co

ncen

trati

on

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 1

Co

ncen

trati

on

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N1 concentration with both sorption and decay

Parent and daughter concentrations for the sorption and decay case

2 3 4

Distance [m]

ConnectFlow

Analytical Solution

2 3 4

Distance [m]

Page 47

ConnectFlow

Analytical Solution

Parent - N1

Daughter - N2


2.11 Mass Flux Calculations

2.11.1 Overview

The mass conserving algorithm was originally implemented in ConnectFlow for particle tracking calculations. The fluxes generated during particle tracking are useful in their own right for calculaflux passing through a particular region of a CPM model. This verification case compares a flux calculation using the mass conserving algorithm with an analytic solution to the same problem.


Consider the following 3D CPM model

► A regular mesh of 4 layers of 16 x 16 cuboidal (CB08) finite elements, oriented parallel to the xplane. The dimensions of each finite element are 100 m x 100 m x 100 m giving a model with dimensions 1600 m in the x and y directions and 400 m in the

► The finite elements consist of rock 1 with permeability 1x10finite elements (4 layers of 4 x 4) which consist of rock 2 with permeability 1x102-51).

► Pressure (Dirichlet) boundary conditions are applied to the top (z=400 m) and bottom (z=0model and no-flow boundary conditions are applied to the other sides of the mpressure gradient along the z-

The symmetry of the problem allows an analytic solution to the flow through each finite element using the following expression

ρµ

Adz

dpkF =

where

A is the area = 10,000 m2.

k is the permeability

µ is the viscosity = 0.001 Ns/m

ρ is the density = 1000 kg/m

dp/dz is the pressure gradient = 250 N/m.

F is the mass flux through the element.

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Mass Flux Calculations

The mass conserving algorithm was originally implemented in ConnectFlow for particle tracking calculations. The fluxes generated during particle tracking are useful in their own right for calculaflux passing through a particular region of a CPM model. This verification case compares a flux calculation using the mass conserving algorithm with an analytic solution to the same problem.

Consider the following 3D CPM model,

A regular mesh of 4 layers of 16 x 16 cuboidal (CB08) finite elements, oriented parallel to the xplane. The dimensions of each finite element are 100 m x 100 m x 100 m giving a model with dimensions 1600 m in the x and y directions and 400 m in the z direction.

The finite elements consist of rock 1 with permeability 1x10-14 m2 except for a vertical column of 64 finite elements (4 layers of 4 x 4) which consist of rock 2 with permeability 1x10

Pressure (Dirichlet) boundary conditions are applied to the top (z=400 m) and bottom (z=0flow boundary conditions are applied to the other sides of the model. These provide a

-axis of 250 N/m.

The symmetry of the problem allows an analytic solution to the flow through each finite element using the

.

is the viscosity = 0.001 Ns/m2

is the density = 1000 kg/m3

dp/dz is the pressure gradient = 250 N/m.

F is the mass flux through the element.

Page 48

The mass conserving algorithm was originally implemented in ConnectFlow for particle tracking calculations. The fluxes generated during particle tracking are useful in their own right for calculating the flux passing through a particular region of a CPM model. This verification case compares a flux calculation using the mass conserving algorithm with an analytic solution to the same problem.

A regular mesh of 4 layers of 16 x 16 cuboidal (CB08) finite elements, oriented parallel to the x-y plane. The dimensions of each finite element are 100 m x 100 m x 100 m giving a model with

except for a vertical column of 64 finite elements (4 layers of 4 x 4) which consist of rock 2 with permeability 1x10-12 m2 (see Figure

Pressure (Dirichlet) boundary conditions are applied to the top (z=400 m) and bottom (z=0 m) of the odel. These provide a

The symmetry of the problem allows an analytic solution to the flow through each finite element using the


Thus the vertical flux through the finite elements of rock type 2 is 2.5 kg/s aneach of the other finite elements is 0.025 kg/s. The total flux through all of the elements in each layer is then 43.6 kg/s.

2.11.3 Results

Using the command, >> CALCULATE CONSERVED MASS FLUXsolution to this problem exactly.

Figure 2-51 The rock types used for the mass flux verification test. Red represents rock type 2 and blue denotes rock type 1. Note that the rock types are constant along the z

2.12 Reactive Transport

2.12.1 Overview

This case considers transient onechemical reactions [18]. The results are compared with equivalent calculations carried out by Phast [and contrasted with ConnectFlow calculations without reactions included.


For each case, a horizontal column, 10 m in length and 1 m in width and height, of grid cells is considered, as shown in Figure 2-52. The grid is discretised into cells that are cells in total. The properties used are given in used by Phast leads to numerical dispersion equal to half the cell size and therefore it is necessary to take this into account when setting the dispersion lengths, although it is a small effect relative to the diffusion in these cases. The dispersion length in held constant and is uniform across the model

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Thus the vertical flux through the finite elements of rock type 2 is 2.5 kg/s and the vertical flux through each of the other finite elements is 0.025 kg/s. The total flux through all of the elements in each layer is

>> CALCULATE CONSERVED MASS FLUX, ConnectFlow recreates the analytic

The rock types used for the mass flux verification test. Red represents rock type 2 and blue denotes rock type 1. Note that the rock types are constant along the z-axis.

Reactive Transport

This case considers transient one-dimensional, multi-component reactive transport due to advection and ]. The results are compared with equivalent calculations carried out by Phast [

and contrasted with ConnectFlow calculations without reactions included.

e, a horizontal column, 10 m in length and 1 m in width and height, of grid cells is considered, . The grid is discretised into cells that are 0.05 m in each dimension, giving 80,000

cells in total. The properties used are given in Table 2-19. Note that the spatial discretisation schemes o numerical dispersion equal to half the cell size and therefore it is necessary to take

this into account when setting the dispersion lengths, although it is a small effect relative to the diffusion in these cases. The dispersion length in Table 2-19 includes this numerical dispersion. The fluid density is held constant and is uniform across the model.

Page 49

d the vertical flux through each of the other finite elements is 0.025 kg/s. The total flux through all of the elements in each layer is

, ConnectFlow recreates the analytic

The rock types used for the mass flux verification test. Red represents rock type 2 and blue

omponent reactive transport due to advection and ]. The results are compared with equivalent calculations carried out by Phast [19]

e, a horizontal column, 10 m in length and 1 m in width and height, of grid cells is considered, 0.05 m in each dimension, giving 80,000

. Note that the spatial discretisation schemes o numerical dispersion equal to half the cell size and therefore it is necessary to take

this into account when setting the dispersion lengths, although it is a small effect relative to the diffusion in includes this numerical dispersion. The fluid density is


The column is initially filled with a water (aqueous solution) in Then a second water with a different composition is introduced at the upstream end of the column and allowed to flow advectively into it. The flow is specified as a flux boundary condition at the inflow end of tmodel and a zero pressure boundary condition at the outflow end, which also has an outflow (zero dispersive flux) boundary condition to allow the solutes to flow advectively from the model. The advective transport velocity is 0.1 m/y and the simulation the incoming water to be advectively transported along the full length of the column, although dispersion and diffusion processes will cause some spreading out of the front. All concentrations armass fractions in kilograms per kilogram of solution (kg/kg

Figure 2-52 Column model used for verification of reactive transport, coloured by head

Symbol

k

φ

D

Lα Longitudinal dispersion length

Tα Transverse

T

µ

ρ

U

dt

nt

Table 2-19 Input parameters for reactive transport verification

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The column is initially filled with a water (aqueous solution) in equilibrium with one or more mineral phases. a second water with a different composition is introduced at the upstream end of the column and

allowed to flow advectively into it. The flow is specified as a flux boundary condition at the inflow end of tmodel and a zero pressure boundary condition at the outflow end, which also has an outflow (zero dispersive flux) boundary condition to allow the solutes to flow advectively from the model. The advective transport velocity is 0.1 m/y and the simulation is run for 120 years, allowing more than sufficient time for the incoming water to be advectively transported along the full length of the column, although dispersion and diffusion processes will cause some spreading out of the front. All concentrations armass fractions in kilograms per kilogram of solution (kg/kgs).

olumn model used for verification of reactive transport, coloured by head

Parameter Value


Porosity 1.0E-

Solute diffusion coefficient 1.0E-9 m

Longitudinal dispersion length 0.05 m

Transverse dispersion length 0.05 m

Temperature 25 °C

Fluid viscosity 1.0E-3 Pa.s

Fluid density 1.0E3 kg/m

Darcy velocity 1.0E-5 m/y

Time step size 1.0 y

Number of time steps 120

reactive transport verification.

Page 50

ne or more mineral phases. a second water with a different composition is introduced at the upstream end of the column and

allowed to flow advectively into it. The flow is specified as a flux boundary condition at the inflow end of the model and a zero pressure boundary condition at the outflow end, which also has an outflow (zero dispersive flux) boundary condition to allow the solutes to flow advectively from the model. The advective

is run for 120 years, allowing more than sufficient time for the incoming water to be advectively transported along the full length of the column, although dispersion and diffusion processes will cause some spreading out of the front. All concentrations are represented as

olumn model used for verification of reactive transport, coloured by head.

Value

17 m2

-4

9 m2/s

0.05 m

0.05 m

25 °C

3 Pa.s

1.0E3 kg/m3

5 m/y

1.0 y


2.12.3 Variations

2.12.3.1 Mineral Equilibration

This variant considers equilibration of a mixture of pure water and an oxygenated saline water with calcite and pyrite. It provides an example of a redox reaction. Thedissolution/precipitation and the oxidation of iron (II) and sulphide in pyrite to iron(III) and sulphate:

4PQ

4PQ

Equilibration with CO2(g) has not been considered. with a sufficient initial quantity (10.0 mol/waters are given in Table 2-20. The column is initially filled with pure water calcite and pyrite and the saline water is introduced to the upby adjusting the chloride mass fraction and prephreeqc.dat thermodynamic database is used.

pH

pe

Component mass fractions (kg/kgs)

C

Ca

Cl

Fe

Na

S

Table 2-20 Water composition for calcite and pyrite equilibration

2.12.3.2 Ion Exchange

This variant considers the reaction of a mixture of two aqueous solutions and an ion exchanger that is able to exchange cations. The reactions are defined in the standard phreeqc.dat thermodynamic databaseusing the example exchanger ‘X’. The model initially containssmall amount of potassium, and a the column. The composition of the waters is given in exchanger X with an initial quantity of 0.011 moles per kilogram of water (mol/initialised by equilibrium with the NaCl water.

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considers equilibration of a mixture of pure water and an oxygenated saline water with calcite It provides an example of a redox reaction. The following reactions are significant for calcite

dissolution/precipitation and the oxidation of iron (II) and sulphide in pyrite to iron(III) and sulphate:

RRST ↔ RVW < RSTVX

;RST ↔ ;W < RSTVX

�V < 14SV < 4;VS ↔ 4PQVW < 8�SYVX < 8;W

VW < SV < 4;W ↔ 4PQTW < 2;VS

(g) has not been considered. Both the calcite and pyrite are present in the model with a sufficient initial quantity (10.0 mol/kgw) such that they will not be depleted. The composition

. The column is initially filled with pure water that has been equilibratedcalcite and pyrite and the saline water is introduced to the upstream end. Both waters are charge balanced by adjusting the chloride mass fraction and pre-equilibrated with calcite and pyrite. The standard

thermodynamic database is used.

Saline water Pure water

7.00 7.00

13.00 4.00

0.00 0.00

0.00 0.00

3.55E-5 0.00

0.00 0.00

2.30E-5 0.00

0.00 0.00

ater composition for calcite and pyrite equilibration.

considers the reaction of a mixture of two aqueous solutions and an ion exchanger that is able to exchange cations. The reactions are defined in the standard phreeqc.dat thermodynamic database

. The model initially contains a sodium chloride (NaCl) small amount of potassium, and a calcium chloride (CaCl2) solution is introduced at the upstream end of the column. The composition of the waters is given in Table 2-21. The column also contains exchanger X with an initial quantity of 0.011 moles per kilogram of water (mol/kgw), whose composition is initialised by equilibrium with the NaCl water.

Page 51

considers equilibration of a mixture of pure water and an oxygenated saline water with calcite following reactions are significant for calcite

dissolution/precipitation and the oxidation of iron (II) and sulphide in pyrite to iron(III) and sulphate:

(2-1)

(2-2)

(2-3)

(2-4)

Both the calcite and pyrite are present in the model ) such that they will not be depleted. The compositions of the

that has been equilibrated with stream end. Both waters are charge balanced

equilibrated with calcite and pyrite. The standard

Pure water

7.00

4.00

0.00

0.00

0.00

0.00

0.00

0.00

considers the reaction of a mixture of two aqueous solutions and an ion exchanger that is able to exchange cations. The reactions are defined in the standard phreeqc.dat thermodynamic database,

(NaCl) solution, with a solution is introduced at the upstream end of

. The column also contains the ion ), whose composition is


pH

pe


Ca

Cl

K

Na

Table 2-21 Water composition for

2.12.3.3 Kinetic dissolution of K-feldspar

This variant considers the dissolution of Kequilibrium under the timescales of interest. Therefore, it is appropriate to model this reaction using a kinetic approach. The reactions are defined in the standard phreeqc.dat thermodynamic databasefollows:

Z[\�]TS^ < 8The model initially contains a salineof the column. The composition of the waters is given in0.02 moles of K-feldspar per kilogram of water in the model2-23. The rate expressions are contained in the thermodynamic database.

pH

pe


Al

Cl

K

Na

Si

Table 2-22 Water composition for

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CaCl2 water NaCl water

7.00 7.00

4.00 4.00

2.40E-4 0.00

4.25E-4 3.90E

0.00 3.91E

0.00 2.30E

ater composition for ion exchange.

feldspar

dissolution of K-feldspar, which is a slow process and so does not tend to reach equilibrium under the timescales of interest. Therefore, it is appropriate to model this reaction using a

ons are defined in the standard phreeqc.dat thermodynamic database

8;VS → ZW < [\�S;�YX < 3;Y�]SY

saline solution and a dilute acidic solution is introduced at the upstream end of the column. The composition of the waters is given in Table 2-22. There is a uniform initial quantity of

feldspar per kilogram of water in the model. The kinetic parameters are given in The rate expressions are contained in the thermodynamic database.

Dilute water Saline water

4.00 7.00

4.00 4.00

0.00 1.17E

3.09E-6 3.08E

0.00 1.69E

2.00E-6 2.00E

0.00 3.64E

ater composition for K-feldspar dissolution.

Page 52

water

7.00

4.00

0.00

-4

-5

-4

feldspar, which is a slow process and so does not tend to reach equilibrium under the timescales of interest. Therefore, it is appropriate to model this reaction using a

ons are defined in the standard phreeqc.dat thermodynamic database as

(2-5)

solution is introduced at the upstream end There is a uniform initial quantity of

The kinetic parameters are given in Table

water

7.00

4.00

-6

-3

-6

-3

-6


Parameter

Time step subdivision

Runge-Kutta subintervals

P1 P2

Tolerance

Table 2-23 Kinetic parameters for

2.12.4 Results

2.12.4.1 Mineral Equilibration

Figure 2-53 shows the mass fraction profiles for iron. Iron is not present in the initial are no plots for the ConnectFlow case without chemistry. However, iron is produced in solutiooxidation of pyrite by the oxygenated inflowing saline water. ConnectFlow and Phast are in excellent agreement in the calculation of the total amount of iron in solution. profiles for sulphate, again showing very good agreement between Phast and ConnectFlow. in good agreement for the calculated 2-55.

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Value

10

3

1.36E4 dm-1

0.1

1.0E-8

for K-feldspar dissolution.

shows the mass fraction profiles for iron. Iron is not present in the initial are no plots for the ConnectFlow case without chemistry. However, iron is produced in solutiooxidation of pyrite by the oxygenated inflowing saline water. ConnectFlow and Phast are in excellent agreement in the calculation of the total amount of iron in solution. Figure 2-54 shows the mass fraction profiles for sulphate, again showing very good agreement between Phast and ConnectFlow.

calculated pe values (a measure of the redox conditions), as shown in

Page 53

shows the mass fraction profiles for iron. Iron is not present in the initial waters, hence there are no plots for the ConnectFlow case without chemistry. However, iron is produced in solution during the oxidation of pyrite by the oxygenated inflowing saline water. ConnectFlow and Phast are in excellent

shows the mass fraction profiles for sulphate, again showing very good agreement between Phast and ConnectFlow. They are also

pe values (a measure of the redox conditions), as shown in Figure


Figure 2-53 Comparison of mass fraction profiles of iron for the calcite and pyrite equilibration case between ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5column.

0.0E+00

2.0E-09

4.0E-09

6.0E-09

8.0E-09

1.0E-08

1.2E-08

1.4E-08

0.0 1.0 2.0

Fe

(k

g/k

gs

)

CF chemistry 10 y

Phast 10 y

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Comparison of mass fraction profiles of iron for the calcite and pyrite equilibration case een ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model

2.0 3.0 4.0 5.0 6.0 7.0 8.0

Distance (m)

CF chemistry 30 y CF chemistry 120 y

Phast 30 y Phast 120 y

Page 54

Comparison of mass fraction profiles of iron for the calcite and pyrite equilibration case m intervals along the model

9.0 10.0

CF chemistry 120 y

Phast 120 y


Figure 2-54 Comparison of mass fraction profiles of sulphate for the calcite and pyrite equilibration case between ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5column.

0.0E+00

1.0E-08

2.0E-08

3.0E-08

4.0E-08

5.0E-08

0.0 1.0 2.0

Su

lph

ate

(k

g/k

gs

)

CF chemistry 10 y

Phast 10 y

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Comparison of mass fraction profiles of sulphate for the calcite and pyrite equilibration case between ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model

2.0 3.0 4.0 5.0 6.0 7.0 8.0

Distance (m)



Page 55

Comparison of mass fraction profiles of sulphate for the calcite and pyrite equilibration case m intervals along the model

9.0 10.0

CF chemistry 120 y

Phast 120 y


Figure 2-55 Comparison of pe profiles for the calcite and pyrite equilibration case between ConnectFlow (CF) and Phast. Values are sampled at 0.5 m intervals along the model column.

2.12.4.2 Ion Exchange

Figure 2-56 shows the mass fraction profiles of chloride for component and there is good agreement between ConnectFlow (with and without chemistry) and Phast, showing that the transport is equivalent in each calculation. for calcium. This is not present in solution initially in thmass fraction rises with time. However, its mass fraction is reduced compared to the ConnectFlow case without chemistry calculations because the calcium ions are taken up by the ion exchanger, which releases sodium and potassium ionsexchanger are shown in Figure 2-and Phast.

-6.9

-6.8

-6.7

-6.6

0.0 1.0 2.0 3.0

pe

CF chemistry 10 y

Phast 10 y

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Comparison of pe profiles for the calcite and pyrite equilibration case between ConnectFlow (CF) and Phast. Values are sampled at 0.5 m intervals along the model column.

shows the mass fraction profiles of chloride for three different times. This is a noncomponent and there is good agreement between ConnectFlow (with and without chemistry) and Phast, showing that the transport is equivalent in each calculation. Figure 2-57 shows the mass fraction profiles for calcium. This is not present in solution initially in the model, but as it is introduced in the CaClmass fraction rises with time. However, its mass fraction is reduced compared to the ConnectFlow case without chemistry calculations because the calcium ions are taken up by the ion exchanger, which releases sodium and potassium ions as a consequence. The corresponding profiles for calcium in the

58. There is excellent agreement between the results for ConnectFlow

3.0 4.0 5.0 6.0 7.0 8.0 9.0

Distance (m)



Page 56

Comparison of pe profiles for the calcite and pyrite equilibration case between ConnectFlow

times. This is a non-reactive component and there is good agreement between ConnectFlow (with and without chemistry) and Phast,

shows the mass fraction profiles e model, but as it is introduced in the CaCl2 water its

mass fraction rises with time. However, its mass fraction is reduced compared to the ConnectFlow case without chemistry calculations because the calcium ions are taken up by the ion exchanger, which

The corresponding profiles for calcium in the There is excellent agreement between the results for ConnectFlow

9.0 10.0

CF chemistry 120 y

Phast 120 y


Figure 2-56 Comparison of mass fraction profiles of chloride for the ion exchange case between ConnectFlow (CF), with and without chemistry, and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

3.9E-04

4.0E-04

4.1E-04

4.2E-04

4.3E-04

0.0 1.0 2.0

Cl

(kg

/kg

s)

CF chemistry 10 y

CF no chemistry 10 y

Phast 10 y

NaCl water

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Comparison of mass fraction profiles of chloride for the ion exchange case between ConnectFlow (CF), with and without chemistry, and Phast. Mass fractions are sampled at 0.5 m intervals

2.0 3.0 4.0 5.0 6.0 7.0 8.0

Distance (m)


CF no chemistry 30 y CF no chemistry 120 y


CaCl2 water

Page 57

Comparison of mass fraction profiles of chloride for the ion exchange case between ConnectFlow (CF), with and without chemistry, and Phast. Mass fractions are sampled at 0.5 m intervals

9.0 10.0

CF chemistry 120 y


Phast 120 y


Figure 2-57 Comparison of mass fraction profiles of calcium for the ion exchange case between ConnectFlow (CF), with and without chemistry, and Phast. Mass fractions are sampled at 0.5 m intalong the model column.

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

0.0 1.0 2.0

Ca

(k

g/k

gs

)

CF chemistry 10 y


Phast 10 y

NaCl water

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Comparison of mass fraction profiles of calcium for the ion exchange case between ConnectFlow (CF), with and without chemistry, and Phast. Mass fractions are sampled at 0.5 m int

3.0 4.0 5.0 6.0 7.0 8.0

Distance (m)


CF no chemistry 30 y CF no chemistry 120 y


CaCl2 water

Page 58

Comparison of mass fraction profiles of calcium for the ion exchange case between ConnectFlow (CF), with and without chemistry, and Phast. Mass fractions are sampled at 0.5 m intervals

9.0 10.0

CF chemistry 120 y


Phast 120 y


Figure 2-58 Comparison of the CaX2 distribution (in moles per kilogram of water) across the model for the ion exchange case between ConnectFlow (CF) and Phast. Quantities are sampled at 0.5 m intervals along the model column.

2.12.4.3 K-feldspar dissolution

Figure 2-59 shows the increase in the the dissolution of K-feldspar. The decrease in the amount of KFigure 2-60. There is excellent agreement between the results for ConnectFlow and Phast.

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

0.0 1.0 2.0

Ca

X2 (

mo

l/k

gw

)

CF chemistry 10 y

Phast 10 y

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Comparison of the CaX2 distribution (in moles per kilogram of water) across the model for the ion exchange case between ConnectFlow (CF) and Phast. Quantities are sampled at 0.5 m intervals along

increase in the mass fraction of aluminium over time at three differentThe decrease in the amount of K-feldspar due to dissolution is shown in

There is excellent agreement between the results for ConnectFlow and Phast.

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Distance (m)



Page 59

Comparison of the CaX2 distribution (in moles per kilogram of water) across the model for the ion exchange case between ConnectFlow (CF) and Phast. Quantities are sampled at 0.5 m intervals along

three different times due to to dissolution is shown in

There is excellent agreement between the results for ConnectFlow and Phast.

9.0 10.0

CF chemistry 120 y

Phast 120 y


Figure 2-59 Comparison of mass fraction profiles of ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

1.1E-06

1.2E-06

1.3E-06

1.4E-06

1.5E-06

1.6E-06

0.0 1.0 2.0

Al

(kg

/kg

s)

CF kinetics 1 year

Phast 1 year

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Comparison of mass fraction profiles of aluminium for the K-feldspar dissolutionConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

2.0 3.0 4.0 5.0 6.0 7.0 8.0Distance (m)

CF kinetics 3 years CF kinetics 12 years

Phast 3 years Phast 12 years

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feldspar dissolution case between ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

8.0 9.0 10.0

CF kinetics 12 years

Phast 12 years


Figure 2-60 Comparison of K-feldspar quantities(CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

9.5E-03

9.6E-03

9.7E-03

9.8E-03

9.9E-03

1.0E-02

0.0 1.0 2.0

K-f

eld

sp

ar

(mo

l/k

gw

)

CF kinetics 1 year

Phast 1 year

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feldspar quantities for the K-feldspar dissolution case between ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

2.0 3.0 4.0 5.0 6.0 7.0Distance (m)

CF kinetics 1 year CF kinetics 3 years CF kinetics 12 years

Phast 3 years Phast 12 years

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case between ConnectFlow (CF) and Phast. Mass fractions are sampled at 0.5 m intervals along the model column.

8.0 9.0 10.0

CF kinetics 12 years

Phast 12 years


3. Discrete Fracture Network Verification

A summary of the DFN test cases is given in

Case Title

3.1 3D Fracture distributions

3.2 3D Fracture connectivity.

3.3 3D Fracture connectivity, power

law distribution.

3.4 Upscaling from DFN


3.5 Three fracture intersections.

3.7 Steady state flow in fractured

rock.

3.8 Henry’s salt transport.

3.9 Salt transport

3.10 Salt upconing

3.11 Grouting of a single fracture.

3.12 Grouting of fractures

intersecting a fracture in specified set.

Table 3-1 DFN verification test

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Discrete Fracture Network Verification

mmary of the DFN test cases is given in Table 3-1.

Overview

3D Fracture distributions

Generation and export of random fractures. The exported fractures are analysed to check the orientation and side length against the expected values.

3D Fracture connectivity. Evaluation of the critical areal density in connecting 1 m square fractures, with a uniform distribution of orientations within a cube.

3D Fracture connectivity, power law distribution.

Evaluation of the critical areal density as a function of domain size for square fractures with a power law distribution exponent of 2.5 for fracture size and uniform distribution for orientation.

Upscaling from DFN to CPM.

Calculation of permeability distribution from a DFN model with and without guard zones. Import of permeability data into CPM model. Comparison of steady state flow distributions.

Radial steady state flow. Steady state groundwater flow with a borehole used to apply the required mass flow boundary condition.

Three fracture intersections. Steady state groundwater flow between three intersecting fractures. Forward and backward particle tracks are generated.

Steady state flow in fractured Steady state groundwater flow through intersecting fractures within a rock matrix. Rock matrix is modelled using an imported IFZ lattice and also via the in built matrix lattice option.

Henry’s salt transport.

Steady state groundwater flow. Density dependent on salinity. Two variants are modelled, the first modelling the transport of salinity and the second using an imported density field.

Salt transport One-dimensional transport of salt due to a pressure gradient

Salt upconing Upconing of salt water into a tunnel in the centre of a thin 3D model. Comparison of CPM and DFN results.

Grouting of a single fracture. Grouting of a fracture – external surface intersection.

Grouting of fractures intersecting a fracture in

specified set.

Grouting of fracture-fracture intersections. One of the fractures is supposed to represent some physical features such as a tunnel.

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Discrete Fracture Network Verification

Generation and export of random fractures. The exported fractures are analysed to check the orientation and side length against the expected

Evaluation of the critical areal density in connecting 1 m square fractures, with a uniform

s within a cube.

Evaluation of the critical areal density as a function square fractures with a power

law distribution exponent of 2.5 for fracture size and uniform distribution for orientation.

Calculation of permeability distribution from a DFN model with and without guard zones. Import of permeability data into CPM model. Comparison of

Steady state groundwater flow with a borehole used to apply the required mass flow boundary

Steady state groundwater flow between three intersecting fractures. Forward and backward

through intersecting fractures within a rock matrix. Rock matrix is modelled using an imported IFZ lattice and also via

Steady state groundwater flow. Density dependent on salinity. Two variants are modelled, the first modelling the transport of salinity and the second

dimensional transport of salt due to a

Upconing of salt water into a tunnel in the centre of a thin 3D model. Comparison of CPM and DFN

external surface

fracture intersections. One of the fractures is supposed to represent some physical features such as a tunnel.


3.1 3D Fracture Distributions

3.1.1 Overview

This case generates sets of random fractures using a number of commonly used distributions for length and orientation. The fractures are exported and analysed to verify tthe expected values.


Random fractures are generated in a 100 m x 100 m x 100 m cube. Six variations are considered with the following combination of properties.

Symbol

Ψ

α

ω

fd

L1

L2

Table 3-2 Input parameters for variations 1

The range of exponents in Table 3where small fractures dominate (exponent 3.5). This is illustrated in

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3D Fracture Distributions

This case generates sets of random fractures using a number of commonly used distributions for length and orientation. The fractures are exported and analysed to verify that the observed distribution matches

Random fractures are generated in a 100 m x 100 m x 100 m cube. Six variations are considered with the following combination of properties.

Parameter Value

Dip angle Uniform distribution 0degrees

Dip direction Uniform distribution 0degrees

Orientation Uniform distribution 0degrees

Fracture density 0.02

Fracture side length 1 Truncated power law distribution

Minimum 1 m

Maximum 150 m

Exponent 2.5, 3.0, 3.5 corresponding to variations 1-3

Fracture side length 2 Truncated power law distribution

Minimum 1 m

Maximum 150 m

Exponent 2.5, 3.0, 3.5 corresponding to variations 1-3

Input parameters for variations 1-3

3-2 covers cases where large fractures dominate (exponent 2.5) and alwhere small fractures dominate (exponent 3.5). This is illustrated in Figure 3-1, Figure

Page 63

This case generates sets of random fractures using a number of commonly used distributions for length hat the observed distribution matches

Random fractures are generated in a 100 m x 100 m x 100 m cube. Six variations are considered with the

Value

Uniform distribution 0-180



Truncated power law

2.5, 3.0, 3.5 corresponding

Truncated power law


covers cases where large fractures dominate (exponent 2.5) and also Figure 3-2 and Figure 3-3.


Figure 3-1 Fractures for exponent of 2.5, connectivity dominated by large fractures

Figure 3-2 Fractures for exponent of 3.0

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Fractures for exponent of 2.5, connectivity dominated by large fractures

Fractures for exponent of 3.0

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Fractures for exponent of 2.5, connectivity dominated by large fractures


Figure 3-3 Fractures for exponent of 3

Symbol

Ψ

α

ω

fd

L

Fracture side length of square

Table 3-3 Input parameters for variations 4

As the fisher parameter increases in variations 4time the standard deviation of the log of the length is increasing in the log norrise to an increased number of larger fractures. This is illustrated in

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Fractures for exponent of 3.5, connectivity dominated by small fractures

Parameter Value

Dip angle

Fisher distribution with dispersion 5.0, 20.0, 100.0 corresponding to variations 4-6

Dip direction Uniform distribution 0degrees

Orientation Uniform distributiondegrees

Fracture density 0.02

Fracture side length of square fractures

Log normal distribution

Mean of log(length) 2

Standard deviation of log(length).

0.2, 0.4, 0.6 corresponding to variations 4-6

Input parameters for variations 4-6

As the fisher parameter increases in variations 4-6, the fractures become increasingly parallel. At the same time the standard deviation of the log of the length is increasing in the log normal distribution which gives rise to an increased number of larger fractures. This is illustrated in Figure 3-4, Figure

Page 65

.5, connectivity dominated by small fractures

Value

Fisher distribution with dispersion 5.0, 20.0, 100.0 corresponding to variations



Log normal distribution


6, the fractures become increasingly parallel. At the same mal distribution which gives

Figure 3-5 and Figure 3-6.


Figure 3-4 Fractures for fisher dispersion 5.0

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Fractures for fisher dispersion 5.0

Page 66


Figure 3-5 Reduced range of angles for fisher dispersion 20.0

Figure 3-6 Reduced range of angles for fisher dispersion 100.0 and larger fractures due to increased standard deviation in log normal distribution.

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Reduced range of angles for fisher dispersion 20.0

Reduced range of angles for fisher dispersion 100.0 and larger fractures due to increased standard deviation in log normal distribution.

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Reduced range of angles for fisher dispersion 100.0 and larger fractures due to increased


3.1.3 Results

All results agree to within 5% of the expected values.

3.1.3.1 Variations 1-3

The volume of the region is 1.0E6 mfractures is 20000. The results for variations 1

Variation Expected Value

Variation 1

Variation 2

Variation 3

Table 3-4 Number of randomly generated fractures

In each of these variations the fracture normal is uniformly distributed over all possible directions. This is tested by calculating the average unit normal vectors over all fractures. The magnitude of the resulting vector should approach zero as the numbefractures being parallel). The results for variations 1

Variation Expected Va

Variation 1

Variation 2

Variation 3

Table 3-5 Magnitude of Average Fracture Normal

For a set of n fractures of side length

distribution exponent η is given in [

1

1 min

ln1

−

=

+= ∑

n

i

i

l

lnη

Applying this to variations 1-3, gives the results shown in


Variation 1

Variation 2

Variation 3

Table 3-6 Estimated power law exponent for fracture side length 1

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hin 5% of the expected values.

The volume of the region is 1.0E6 m3, given a point density of 0.02 the expected number of generated fractures is 20000. The results for variations 1-3 are shown in Table 3-4.

Expected Value Measured Value

20000 19743 1.29%

20000 20116 0.58%

20000 20015 0.08%

Number of randomly generated fractures

In each of these variations the fracture normal is uniformly distributed over all possible directions. This is tested by calculating the average unit normal vectors over all fractures. The magnitude of the resulting vector should approach zero as the number of fractures increases (a value of 1 would correspond to all fractures being parallel). The results for variations 1-3 are shown in Table 3-5.


0.0 0.0031 0.31%

0.0 0.0095 0.95%

0.0 0.0065 0.65%

Magnitude of Average Fracture Normal

For a set of n fractures of side length il and minimum side length minl an estimate for the power law

is given in [20] as

3, gives the results shown in Table 3-6 and Table 3-7.


2.5 2.513 0.52%

3.0 3.035 1.16%

3.5 3.510 0.29%

Estimated power law exponent for fracture side length 1

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iven a point density of 0.02 the expected number of generated

Error

1.29%

0.58%

0.08%

In each of these variations the fracture normal is uniformly distributed over all possible directions. This is tested by calculating the average unit normal vectors over all fractures. The magnitude of the resulting

r of fractures increases (a value of 1 would correspond to all

Error

0.31%

0.95%

0.65%

an estimate for the power law

Error

0.52%

1.16%

0.29%



Variation 1

Variation 2

Variation 3

Table 3-7 Estimated power law exponent for fracture side length 2

3.1.3.2 Variations 4-6

The volume of the region is 1.0E6 mfractures is 20000. The results for variations 4


Variation 4

Variation 5

Variation 6

Table 3-8 Number of randomly generated fractures

For a set of n fractures with angle κ is given in [21] as

2

2

1

)3(

R

RR

−

−=κ where ∑

=

=n

inR

1

cos(1

Applying this to variations 4-6, gives the results shown in


Variation 4

Variation 5

Variation 6

Table 3-9 Fisher distribution parameter

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2.5 2.513 0.52%

3.0 3.017 0.57%

3.5 3.514 0.40%

Estimated power law exponent for fracture side length 2

The volume of the region is 1.0E6 m3. Given a point density of 0.02, the expected number of generated fractures is 20000. The results for variations 4-6 are shown in Table 3-8.


20000 19740 1.30%

20000 19896 0.52%

20000 19771 1.15%

Number of randomly generated fractures

For a set of n fractures with angle θ to the average normal, an estimate for the fisher dispersion parameter

)cos(θ

6, gives the results shown in Table 3-9


5.0 5.14 2.90%

20.0 20.11 0.55%

100.0 100.06 0.06%

Fisher distribution parameter

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Error

0.52%

0.57%

0.40%

. Given a point density of 0.02, the expected number of generated

Error

1.30%

0.52%

1.15%

to the average normal, an estimate for the fisher dispersion parameter

Error

2.90%

0.55%

0.06%


For a log normal distribution, the arithmetic mean and standard deviation are given in [

2

2l

l

e

σµ

µ+

= and

2

2=+ l

l

eeσ

σµ

σ

Where lµ and lσ are the mean and standard deviation of the log of the distribution. Applying this to the

fracture length distribution in variations 4


Variation 4

Variation 5

Variation 6

Table 3-10 Average side length for log normal distribution


Variation 4

Variation 5

Variation 6

Table 3-11 Standard deviation of side length for log normal distribution

3.2 3D Fracture Connectivity

3.2.1 Overview

The DFN software has the ability to create networks of randomcase uses percolation theory to check the generation and connectivity of random fractures within a cube.


The modelled domain is a cube of side length 20 m. This is filled with randomly generated square fractures with side length 1 m and with random orientation and position. This is illustrated in

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For a log normal distribution, the arithmetic mean and standard deviation are given in [

12

−lσ

are the mean and standard deviation of the log of the distribution. Applying this to the

fracture length distribution in variations 4-6 gives the results shown in Table 3-10 and


7.53 7.48 0.68%

8.00 7.96 0.62%

8.85 8.73 1.29%

Average side length for log normal distribution


1.52 1.54 1.17%

3.33 3.29 1.20%

5.82 5.59 3.92%

Standard deviation of side length for log normal distribution

3D Fracture Connectivity

The DFN software has the ability to create networks of random fractures with specified properties. This case uses percolation theory to check the generation and connectivity of random fractures within a cube.

The modelled domain is a cube of side length 20 m. This is filled with randomly generated square fractures with side length 1 m and with random orientation and position. This is illustrated in

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For a log normal distribution, the arithmetic mean and standard deviation are given in [22] as

are the mean and standard deviation of the log of the distribution. Applying this to the

and Table 3-11.

Error

0.68%

0.62%

1.29%

Error

1.17%

1.20%

3.92%

fractures with specified properties. This case uses percolation theory to check the generation and connectivity of random fractures within a cube.

The modelled domain is a cube of side length 20 m. This is filled with randomly generated square fractures with side length 1 m and with random orientation and position. This is illustrated in Figure 3-7.


Symbol

Ψ

α

ω

l

L


The critical areal density at which all surfaces of the cube should be connected by the fracture network is given in [23] as 1.23.

3.2.3 Results

Thirty different random numbers seeds were used. For each seed a progression of areal densities were used in order to estimate the critical density. The average resulting critical density was 1.32, 7.3% higher than in [23].

Figure 3-7 Fractures at critical areal density

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Parameter Value

Dip angle Uniform distribution 0

degrees

Dip direction Uniform distribution 0

degrees

Orientation Uniform distribution 0

degrees

Fracture side length 1 m

Cube side length 20 m

The critical areal density at which all surfaces of the cube should be connected by the fracture network is

seeds were used. For each seed a progression of areal densities were used in order to estimate the critical density. The average resulting critical density was 1.32, 7.3% higher

Fractures at critical areal density

Page 71

Value

Uniform distribution 0-90 degrees



1 m

20 m

The critical areal density at which all surfaces of the cube should be connected by the fracture network is

seeds were used. For each seed a progression of areal densities were used in order to estimate the critical density. The average resulting critical density was 1.32, 7.3% higher


3.3 3D Fracture Connectivity with a Power Law Size Distribution

3.3.1 Overview

This case considers the connectivity of random fractures within a cube, where the fracture lengths follow a power law distribution. The solution is taken from [


The modelled domain is a cube for a range of increasing sizes. This is square fractures with side length corresponding to a truncated power law.

Symbol

Ψ

α

ω

a

L


When the exponent is less than 3 the critical fracture density varies with domain size and the connectivity is dominated by the larger fractures. This is illustrated in

For large values of the exponent the connectivity of the smaller fractures dominates and the critical density does not vary with domain size for sufficiently large domains. This is similar

3.3.3 Results

A comparison of the dependence of the critical density on domain size is shown in reference data is taken from [23].

The results from ConnectFlow show good agreement, particularly in terms of thediscussed in [23], varies significantly with the exponent

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This case considers the connectivity of random fractures within a cube, where the fracture lengths follow a power law distribution. The solution is taken from [24].

The modelled domain is a cube for a range of increasing sizes. This is filled with randomly generated square fractures with side length corresponding to a truncated power law.

Parameter Value

Dip angle Uniform distribution 0

degrees

Dip direction Uniform distribution 0

degrees

Orientation Uniform distribution 0

degrees

Exponent in power law 2.5

Cube side length 5 - 300 m

When the exponent is less than 3 the critical fracture density varies with domain size and the connectivity is dominated by the larger fractures. This is illustrated in Figure 3-9.

For large values of the exponent the connectivity of the smaller fractures dominates and the critical density does not vary with domain size for sufficiently large domains. This is similar to case

A comparison of the dependence of the critical density on domain size is shown in

The results from ConnectFlow show good agreement, particularly in terms of the slope which, as ], varies significantly with the exponent a.

Page 72


This case considers the connectivity of random fractures within a cube, where the fracture lengths follow a

filled with randomly generated

Value



distribution 0-90 degrees

300 m

When the exponent is less than 3 the critical fracture density varies with domain size and the connectivity

For large values of the exponent the connectivity of the smaller fractures dominates and the critical density to case 3.1.

A comparison of the dependence of the critical density on domain size is shown in Figure 3-8. The

slope which, as


Figure 3-8 Critical fracture density variation with domain size

1.00E-04 1.00E-03

Fracture Density

Do

main

Sid

e L

en

gth

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ty variation with domain size

1

10

100

1000

1.00E-02 1.00E-01 1.00E+00

Fracture Density

ConnectFlow

Reference Data

Page 73

ConnectFlow

Reference Data


Figure 3-9 Fracture network generated using a power law size distribution (

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Fracture network generated using a power law size distribution (a=2.5)

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3.4 Upscaling from DFN to CPM

3.4.1 Overview

This case is taken from the DECOVALEX Task C [m region of fractured rock, generated from fracture mapping data from Sellafield UK.

The fracture set consists of 7797 fractures, with lengths varying from 0.5 to >30 m, and apertures varying from 1 to 200 microns.

The upscaled permeabilties are then used in a CPM calculation and the mass flows are compared.


The modelled domain consists of a 20 m x 20 m x 1 m region with an applied horizontal pressure gradient of 1.0E4 Pa/m, as illustrated in Figure


The fracture distribution is shown in

Symbol

ρ

µ

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Upscaling from DFN to CPM

This case is taken from the DECOVALEX Task C [25] and calculates the permeability through a 20m region of fractured rock, generated from fracture mapping data from Sellafield UK.

The fracture set consists of 7797 fractures, with lengths varying from 0.5 to >30 m, and apertures varying


The modelled domain consists of a 20 m x 20 m x 1 m region with an applied horizontal pressure gradient Figure 3-10.


The fracture distribution is shown in Figure 3-11 and the input parameters in Table

Parameter Value

Density 1000 kg/m

Viscosity 1.0E-3 Ns/m

Page 75

calculates the permeability through a 20 m x 20 m region of fractured rock, generated from fracture mapping data from Sellafield UK.

The fracture set consists of 7797 fractures, with lengths varying from 0.5 to >30 m, and apertures varying


The modelled domain consists of a 20 m x 20 m x 1 m region with an applied horizontal pressure gradient

Table 3-14.

Value

1000 kg/m3

3 Ns/m2



Figure 3-11 Fractured Rock

The DFN permeabilities are calculated on a regular grid of cells and the CPM calculation uses a regular grid of the same resolution. The finest grid resolution tested was 400x400 cells.

3.4.3 Variations

3.4.3.1 Cellular Model Calculation

The DFN region is overlayed with a

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permeabilities are calculated on a regular grid of cells and the CPM calculation uses a regular grid of the same resolution. The finest grid resolution tested was 400x400 cells.

Cellular Model Calculation

The DFN region is overlayed with a grid of cells of the required resolution.

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permeabilities are calculated on a regular grid of cells and the CPM calculation uses a regular


3.4.3.2 Cellular Model with Guard Zone

Each cell is stretched by a factor 3 in the x and z directions and then the middle section is used to export permeabilities for the area of interest. This helps improve accuracy wherthan one of the cells sides.

3.4.3.3 Regional Model Calculation

The DFN model region is used to define the grid of cells.

3.4.4 Results

The overall flow through the left hand boundary x=from [25] and the upscaled CPM results (with guard zone). The CPM result is 3% larger than the TUL result.

Approach

TUL

ConnectFlow DFN

ConnectFlow CPM (400 x 400 upscaling)

DFN and CPM Results without guard zones agreed to within 6%, whether calculated through region elements or through cellular model calculation.

The distribution of outflow at x = -10 m is compared between DFN and CPM models in range of upscaling resolutions. The data is averaged over 1 m intervals in order to better compare the bulk profile. Flow orthogonal to the pressure gradient at the outflow boundary z =

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Cellular Model with Guard Zone

Each cell is stretched by a factor 3 in the x and z directions and then the middle section is used to export permeabilities for the area of interest. This helps improve accuracy where single fractures intersect more

Regional Model Calculation

The DFN model region is used to define the grid of cells.

The overall flow through the left hand boundary x=-10 is compared against the stress free TUL resu] and the upscaled CPM results (with guard zone). The CPM result is 3% larger than the TUL

Flow X=-10

9.40E-5 m3/s

9.67E-5 m3/s

ConnectFlow CPM (400 x 400 upscaling) 9.71E-5 m3/s

DFN and CPM Results without guard zones agreed to within 6%, whether calculated through region elements or through cellular model calculation.

10 m is compared between DFN and CPM models in range of upscaling resolutions. The data is averaged over 1 m intervals in order to better compare the bulk profile. Flow orthogonal to the pressure gradient at the outflow boundary z = 10 m is shown in

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Each cell is stretched by a factor 3 in the x and z directions and then the middle section is used to export e single fractures intersect more

10 is compared against the stress free TUL results ] and the upscaled CPM results (with guard zone). The CPM result is 3% larger than the TUL

DFN and CPM Results without guard zones agreed to within 6%, whether calculated through region

10 m is compared between DFN and CPM models in Figure 3-12 for a range of upscaling resolutions. The data is averaged over 1 m intervals in order to better compare the bulk

10 m is shown in Figure 3-13.


Figure 3-12 Outflow distribution at left boundary (X=

Figure 3-13 Outflow distribution at top boundary (Z=10 m) for cellular model upscaling with no guard zone

The overall outflow through the 20 m top surface is 5.45E6.01E-6 m3/s as computed by the CPM 400x400 model. This is around 1/20aligned with the pressure gradient.

Figure 3-14 compares the outflow distribution across the three variations in resolution of 400x400. The Cellular and Model Region variationthe guard zone improves the agreement as compared with the DFN solution.

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

1.00E-05

1.20E-05

1.40E-05

-10 -5

Ou

tflo

w m

3/s

-8.00E-06

-6.00E-06

-4.00E-06

-2.00E-06

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

1.00E-05

-10 -5Ou

tflo

w m

3/s

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Outflow distribution at left boundary (X=-10 m) for cellular model upscaling with no guard zone

Outflow distribution at top boundary (Z=10 m) for cellular model upscaling with no guard zone

The overall outflow through the 20 m top surface is 5.45E-6 m3/s as computed by the DFN model and 6 m3/s as computed by the CPM 400x400 model. This is around 1/20th of the outflow in the direction

aligned with the pressure gradient.

compares the outflow distribution across the three variations in 3.4.3 for an upscaling resolution of 400x400. The Cellular and Model Region variations are virtually identical as expected. Use of the guard zone improves the agreement as compared with the DFN solution.

-5 0 5 10

z [m]

0 5 10

z [m]

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aling with no guard zone

Outflow distribution at top boundary (Z=10 m) for cellular model upscaling with no guard zone

uted by the DFN model and of the outflow in the direction

for an upscaling s are virtually identical as expected. Use of

DFN

CPM 400x400

DFN

CPM 400x400


Figure 3-14 Comparison of outflows at left boundary (X=

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

1.00E-05

1.20E-05

1.40E-05

-10 -5

Ou

tflo

w m

3/s

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Comparison of outflows at left boundary (X=-10m) using a CPM resolution of

-5 0 5 10

z [m]

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10m) using a CPM resolution of 400x400

DFN

Cellular

Region

Guard Zone



3.5.1 Overview


The example has a simple analytical


This case is the DFN equivalent of


For a fracture, the aperture, e, is related to the hydraulic conductivity,

g

Ke

ρ

µ12=

The values in Table 3-15 have scaled Q from 1.0E

A 5 degree sector of the disk is modelled and is tessellated using 100 single fractures along its length. As these fractures are not rectangular, ConFigure 3-16.

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The example has a simple analytical solution and can be used to test a range of modelling choices.

This case is the DFN equivalent of Case 2.1.


, is related to the hydraulic conductivity, K by

have scaled Q from 1.0E-7 in Case 2.1 to give an equivalent head distribution.

A 5 degree sector of the disk is modelled and is tessellated using 100 single fractures along its length. As these fractures are not rectangular, ConnectFlow breaks them up into right angle triangles as illustrated in

Page 80


solution and can be used to test a range of modelling choices.

to give an equivalent head distribution.

A 5 degree sector of the disk is modelled and is tessellated using 100 single fractures along its length. As nectFlow breaks them up into right angle triangles as illustrated in


Figure 3-16 Fractures in sector of

The fractures used model the circumference of the sector as a straight line. As a result the accuracy of the simulation increases as the angle of the sector decreases.

The outflow at the axis is modelled using a borehole with a specified flow rate worthogonal to the disk.

Symbol

h

Q

K

R

e

r

ρ

µ

g


3.5.3 Results


)ln(2

)()(r

R

Kd

QRhrh

π−=

The results from ConnectFlow show good agreement with the analytical solution, as illustrated in 3-17.

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disk

The fractures used model the circumference of the sector as a straight line. As a result the accuracy of the angle of the sector decreases.

The outflow at the axis is modelled using a borehole with a specified flow rate which has direction

Parameter Value

Head at disk circumference 0 m


Hydraulic Conductivity 1.0E-8 m/s

Radius of disk 2000 m

Aperture of fracture 1.11E-


Density 1000 kg/m


Gravity 9.8 m/s


The results from ConnectFlow show good agreement with the analytical solution, as illustrated in

Page 81

The fractures used model the circumference of the sector as a straight line. As a result the accuracy of the

hich has direction

Value

0 m

14 m3/s

8 m/s

2000 m

-7 m

2000 m

1000 kg/m3

3 Pa.s

9.8 m/s2

The results from ConnectFlow show good agreement with the analytical solution, as illustrated in Figure


Figure 3-17 Variation of head with distance from the axis with an outflow modelled with a borehole

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Radius [m]

He

ad

[m

]

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Variation of head with distance from the axis with an outflow modelled with a borehole

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

Page 82

Variation of head with distance from the axis with an outflow modelled with a borehole

ConnectFlow

Analytical Solution


3.6 Three Fracture Intersections

3.6.1 Overview

This case models steady groundwater flow and particle tracking in a simple two dimensional fracture network.


The problem is a variant of test C1 from DarcyTools [region as shown in Figure 3-18.

Figure 3-18 Three fracture intersections

A 3x3 system of linear equations is derived by applying intersections. This is used to calculate the heads at a, b and c and hence the flow rates in each fracture segment.

The fracture geometry used for this test is defined in

X Start Y Start

0

0

0

Table 3-16 Fracture locations

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Three Fracture Intersections

This case models steady groundwater flow and particle tracking in a simple two dimensional fracture

The problem is a variant of test C1 from DarcyTools [26]. Three fractures are placed in a rectangular

Three fracture intersections

A 3x3 system of linear equations is derived by applying mass conservation at the three fracture intersections. This is used to calculate the heads at a, b and c and hence the flow rates in each fracture

The fracture geometry used for this test is defined in Table 3-16.

Y Start X End Y End

2 12

3 12

5 12

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This case models steady groundwater flow and particle tracking in a simple two dimensional fracture

]. Three fractures are placed in a rectangular

mass conservation at the three fracture intersections. This is used to calculate the heads at a, b and c and hence the flow rates in each fracture

Y End

5

3

2


3.6.3 Variations

3.6.3.1 Symmetric

For the symmetric case, the fracture analytical flow field where the flow rates along each fracture are almost identical.

3.6.3.2 Non-symmetric

In the non-symmetric case, the fracture apertures are set to 1.0Esection bc, 2.0E-4 m for section ac and 4.0Ethrough ab is removed as shown in

Figure 3-19 Non symmetric fractures

This arrangement means that the flow at b splits, with some of it moving to c and then a, and the rest moving directly towards the left hand boundary.

The particle tracking is tested by releasing 10000 particles at the h=1 surface, with the fraction of particles in each fracture distributed in proportion to the analytical inflow rate.

The proportion of particles leaving the geometry at Y=2, 3 and 5 analytically predicted outflow rates.

Y Analytical inflow distribution

2

3

5

Table 3-17 Analytical inflow/outflow

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For the symmetric case, the fracture aperture is set to 1.0E-4 m for all three fractures. This gives an analytical flow field where the flow rates along each fracture are almost identical.

symmetric case, the fracture apertures are set to 1.0E-4 m for the fracture pa4 m for section ac and 4.0E-4 m for section ab. In addition, the section of fracture passing

through ab is removed as shown in Figure 3-19.

Non symmetric fractures

This arrangement means that the flow at b splits, with some of it moving to c and then a, and the rest moving directly towards the left hand boundary.

The particle tracking is tested by releasing 10000 particles at the h=1 surface, with the fraction of particles in each fracture distributed in proportion to the analytical inflow rate.

The proportion of particles leaving the geometry at Y=2, 3 and 5 should correspond to the ratio of the analytically predicted outflow rates.

Analytical inflow distribution at X=12

Analytical outflow distribution at X=0

7.9% 77.3%

64.9% 10.0%

27.2% 12.7%

Analytical inflow/outflow

Page 84

4 m for all three fractures. This gives an

4 m for the fracture passing through 4 m for section ab. In addition, the section of fracture passing

This arrangement means that the flow at b splits, with some of it moving to c and then a, and the rest

The particle tracking is tested by releasing 10000 particles at the h=1 surface, with the fraction of particles

should correspond to the ratio of the


For this test, the APPROXIMATE PARTICLE TRACKING inflow to outflow and backward tracking to track from outflow to inflow.

3.6.3.3 Symmetric – Exact Particle tracking

This test is identical to the symmetric case, but using the of approximate particle tracking.

The proportion of particles leaving the geometry at Y=2, 3 and 5 should correspond to the ratio of the analytically predicted outflow rates, i.e. it should be the same in all three locations.

3.6.4 Results

3.6.4.1 Symmetric

The predicted results for head are within 1% of the analytical values.

Intersection ConnectFlow

a 0.333

b 0.496

c 0.667

Table 3-18 Heads at fracture intersections

3.6.4.2 Non Symmetric

The ConnectFlow head distribution for this case is shown in and particle flows are within 5% of the analytical values.

Figure 3-20 Head distribution

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APPROXIMATE PARTICLE TRACKING model is used, with forward tracking to track from inflow to outflow and backward tracking to track from outflow to inflow.

Exact Particle tracking

This test is identical to the symmetric case, but using the EXACT PARTICLE TRACKING

The proportion of particles leaving the geometry at Y=2, 3 and 5 should correspond to the ratio of the ed outflow rates, i.e. it should be the same in all three locations.

re within 1% of the analytical values.

ConnectFlow Analytical % Error

0.333 0.333 0.19%

0.496 0.500 0.89%

0.667 0.667 0.10%

Heads at fracture intersections

The ConnectFlow head distribution for this case is shown in Figure 3-20. Predicted results for the head and particle flows are within 5% of the analytical values.

Page 85

model is used, with forward tracking to track from

EXACT PARTICLE TRACKING method instead

The proportion of particles leaving the geometry at Y=2, 3 and 5 should correspond to the ratio of the

% Error

0.19%

0.89%

0.10%

. Predicted results for the head



a 0.062

b 0.937

c 0.599


Y at Exit ConnectFlow

2 7.4%

3 65.6%

5 27.0%

Table 3-20 Particle flow rates (backwards)


2 77.6%

3 9.5%

5 12.9%

Table 3-21 Particle flow rates (forwards)

3.6.4.3 Symmetric – Exact Particle tracking

The predicted proportions of particles leaving the model at each location are within 1% of the analytical values.


2 33.1%

3 33.8%

5 33.1%

Table 3-22 Proportions of particles at each outflow point

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0.061 1.49%

0.968 3.23%

0.600 0.15%


ConnectFlow Analytical Error

7.4% 7.9% 0.5%

65.6% 65.0% 0.6%

27.0% 27.2% 0.2%

Particle flow rates (backwards)

ConnectFlow Analytical Error

77.6% 77.3% 0.3%

9.5% 10.0% 0.5%

12.9% 12.7% 0.2%

Particle flow rates (forwards)

Exact Particle tracking

of particles leaving the model at each location are within 1% of the analytical


33.1% 33.3% 0.2%

33.8% 33.3% 0.5%

33.1% 33.3% 0.2%

Proportions of particles at each outflow point using exact particle tracking

Page 86

% Error

1.49%

3.23%

0.15%

Error

0.5%

0.6%

0.2%

Error

0.3%

0.5%

0.2%

of particles leaving the model at each location are within 1% of the analytical

% Error

0.2%

0.5%

0.2%

using exact particle tracking



3.7.1 Overview

The example is taken from Level 1 of the international HYDROCOIN project for verification of groundwater flow codes [27] and is the same problem as CPM C

It models steady state flow in a twoinclined fractures which intersect one another at depth, and have a higher permeabilsurrounding rock.

The topography has been made simple so that it consists of two valleys located where the fracture zones meet the surface. To simplify the problem definition, the shape of the surface is described by straight lines. Although the surface topography is symmetric, the flow is influenced by the asymmetry of the fracture zones.

This problem is based on an idealized version of the hydrogeological conditions encountered

at a potential site for a deep repository in Swedish bedrock. A

model of this was made in a separate study [


Figure 3-21 depicts the modelled domain.

Figure 3-21 Fractured rock

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The example is taken from Level 1 of the international HYDROCOIN project for verification of groundwater and is the same problem as CPM Case 2.2.

It models steady state flow in a two-dimensional vertical slice of fractured rock. The rock contains two inclined fractures which intersect one another at depth, and have a higher permeabil


the surface topography is symmetric, the flow is influenced by the asymmetry of the fracture


at a potential site for a deep repository in Swedish bedrock. A detailed three-dimensional

model of this was made in a separate study [28].

depicts the modelled domain.

Page 87

The example is taken from Level 1 of the international HYDROCOIN project for verification of groundwater

dimensional vertical slice of fractured rock. The rock contains two inclined fractures which intersect one another at depth, and have a higher permeability than the


the surface topography is symmetric, the flow is influenced by the asymmetry of the fracture


dimensional


Symbol

Kr Hydraulic conductivity of rock

Krf Hydraulic conductivity of CPM

e1

e2


The fracture apertures have been derived to match the transmissivities of the fracture regions in case 2.2.2. For the DFN model, the transmissivity is proportional to aperture cubed and for the CPM model it is proportional to the cross sectional thickness of the region representing the fracture.

The rock is modelled using an arrayperspective to make it easier to visualise.

Figure 3-22 Modelled fractures

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Parameter Value


Hydraulic conductivity of CPM fractured region

1.0E-6 m/s

Thickness of fracture F1 2.053E-

Thickness of fracture F2 2.620E-

The fracture apertures have been derived to match the transmissivities of the fracture regions in case . For the DFN model, the transmissivity is proportional to aperture cubed and for the CPM model it is

proportional to the cross sectional thickness of the region representing the fracture.

The rock is modelled using an array of fractures as shown in Figure 3-22. The figure is drawn in perspective to make it easier to visualise.

Page 88

Value

8 m/s

6 m/s

-4 m

-4 m

The fracture apertures have been derived to match the transmissivities of the fracture regions in case . For the DFN model, the transmissivity is proportional to aperture cubed and for the CPM model it is

proportional to the cross sectional thickness of the region representing the fracture.

. The figure is drawn in


3.7.3 Variations

3.7.3.1 IFZ Rock Matrix

In this variant, the rock matrix is modelled using a two dimensional grid of fractures that are read in as an Implict Fracture Zone (IFZ) file.

3.7.3.2 Matrix Lattice Option

In this variant, the rock matrix is modelled using a three dimensional grid of fractures generated by the command GENERATE MATRIX LATTICE. approximately twice as wide as that set manually in t

3.7.3.3 Current Value

In this variant, a subset of the domain is modelled, x = 100 to 1500 m and y = illustrated in Figure 3-23.The boundary pressure is taken from the CPM solution the >>CURRENT VALUE boundary condition option. The rock matrix is modelled approach as in 3.7.3.1.

Figure 3-23 Current Value

3.7.4 Results

The results presented here comparagreement with the HYDROCOIN study.

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In this variant, the rock matrix is modelled using a two dimensional grid of fractures that are read in as an

In this variant, the rock matrix is modelled using a three dimensional grid of fractures generated by the GENERATE MATRIX LATTICE. In this case, the fracture widths are set automatically and are

approximately twice as wide as that set manually in the IFZ Rock Matrix variant.

In this variant, a subset of the domain is modelled, x = 100 to 1500 m and y = -300 to .The boundary pressure is taken from the CPM solution 2.2.3.1boundary condition option. The rock matrix is modelled using the same IFZ

The results presented here compare the head profile at a height of y=-200 m and show excellent agreement with the HYDROCOIN study.

Page 89

In this variant, the rock matrix is modelled using a two dimensional grid of fractures that are read in as an

In this variant, the rock matrix is modelled using a three dimensional grid of fractures generated by the In this case, the fracture widths are set automatically and are

300 to -100 m. This is 2.2.3.1 and applied using using the same IFZ

200 m and show excellent


Figure 3-24 Head at height -200 m (IFZ Rock Matrix)

Figure 3-25 Head at height -200 m (Matrix Lattice)

100

105

110

115

120

125

0 400

Distance [m]

Head

[m

]

100

105

110

115

120

125

0 400

Distance [m]

Head

[m

]

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200 m (IFZ Rock Matrix)

200 m (Matrix Lattice)

800 1200 1600

Distance [m]

ConnectFlow

Hydrocoin

800 1200 1600

Distance [m]

ConnectFlow

Hydrocoin

Page 90

ConnectFlow

Hydrocoin

ConnectFlow

Hydrocoin


Figure 3-26 Head at height -200 m (Current Value)

In addition, the ConnectFlow results were compared against the Feftra base case results [differences in head between the two codes being less than 0.5% for the IFZ Rock Matrix and Reduced Domain variants and less than 1% for the Matrix Lat

The latter is marginally less accurate due to the increased permeability contribution of the wider rock matrix fractures. This was checked by making the fractures in the IFZ Rock Matrix variant deliberately wider and the same trend was seen.

Figure 3-27 shows the head distribution through the fracture network.

100

105

110

115

120

125

0 400

Distance [m]

Head

[m

]

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200 m (Current Value)

In addition, the ConnectFlow results were compared against the Feftra base case results [differences in head between the two codes being less than 0.5% for the IFZ Rock Matrix and Reduced Domain variants and less than 1% for the Matrix Lattice.

The latter is marginally less accurate due to the increased permeability contribution of the wider rock matrix fractures. This was checked by making the fractures in the IFZ Rock Matrix variant deliberately wider and the same trend was seen.

shows the head distribution through the fracture network.

800 1200 1600

Distance [m]

ConnectFlow

Hydrocoin

Page 91

In addition, the ConnectFlow results were compared against the Feftra base case results [3], with the differences in head between the two codes being less than 0.5% for the IFZ Rock Matrix and Reduced

The latter is marginally less accurate due to the increased permeability contribution of the wider rock matrix fractures. This was checked by making the fractures in the IFZ Rock Matrix variant deliberately

ConnectFlow

Hydrocoin


Figure 3-27 Fractures coloured by head (IFZ Ro

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Fractures coloured by head (IFZ Rock Matrix)

Page 92



3.8.1 Overview


The variant modelled is a modified version of the original Henry‘s test case, as recommended in [modified case halves the fresh water inflow rate, which increases the sensitivity of the solution to the variation in density.

This case is the DFN equivalent of C



Figure 3-28 Schematic illustration of the Henry test case

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The variant modelled is a modified version of the original Henry‘s test case, as recommended in [modified case halves the fresh water inflow rate, which increases the sensitivity of the solution to the

nt of Case 2.8.

A schematic of the test case is shown in Figure 3-28 and the input parameters are given in

Schematic illustration of the Henry test case

Page 93


The variant modelled is a modified version of the original Henry‘s test case, as recommended in [14]. The modified case halves the fresh water inflow rate, which increases the sensitivity of the solution to the



Symbol


D

Q

0ρ

maxρ

Lα

Tα

φ


The domain is represented by a single fracture tessellated into 0.025 m square thickness is set to 1.8516E-4 m in order to generate the desired hydraulic conductivity of 1.0E

The fracture has a porosity of 1.0 rather than 0.35 in [in order to account for this and to get the appropriate velocity field.

3.8.3 Variations

3.8.3.1 Interpolated Density

The density is interpolated from the CPM solution of test Ccalculation.

3.8.3.2 Salt Transport

The salt transport is modelled and the density calculated from the salinity.

3.8.4 Results

3.8.4.1 Interpolated Density

The interpolated density field used is shown in

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Parameter Value



1.886E-5 m


9.43E-5 m





Porosity 1

The domain is represented by a single fracture tessellated into 0.025 m square sub4 m in order to generate the desired hydraulic conductivity of 1.0E

The fracture has a porosity of 1.0 rather than 0.35 in [14]. The inflow rate is increased by a factor of 1/0.35 in order to account for this and to get the appropriate velocity field.

from the CPM solution of test Case 2.8 and kept fixed throughout the

The salt transport is modelled and the density calculated from the salinity.

The interpolated density field used is shown in Figure 3-29.

Page 94

Value

2 m/s

5 m2/s

5 m2/s

998 kg/m3

1023 kg/m3

0 m

0 m

sub-fractures. The fracture 4 m in order to generate the desired hydraulic conductivity of 1.0E-2 m/s.

]. The inflow rate is increased by a factor of 1/0.35

and kept fixed throughout the


Figure 3-29 Interpolated density

The pressure field is compared against the reference CPM solution, the latter having been separately checked against the analytical solution via the salt concentration.

Plots of the two pressure fields are shown in within 5%.

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field is compared against the reference CPM solution, the latter having been separately checked against the analytical solution via the salt concentration.

Plots of the two pressure fields are shown in Figure 3-30 and Figure 3-31. The pressure fields agree to

Page 95

field is compared against the reference CPM solution, the latter having been separately

. The pressure fields agree to


Figure 3-30 DFN residual pressure

Figure 3-31 CPM Residual pressure

3.8.4.2 Salt Transport

This test case has an analytical solution, represented by an infinite double Fourier series. A truncated form of the series gives a non linear system which can then be iteratively solved. Results from this process are

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DFN residual pressure

M Residual pressure


Page 96



reported in [14] and these are used for comparisons in is less than 2%.

Figure 3-32 Non-dimensional concentration of salinity

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.20 0.40 0.60

z [

m]

ConnectFlow 25%

Semi analytical 25%

ConnectFlow 50%

Semi analytical 50%

ConnectFlow 75%

Semi analytical 75%

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] and these are used for comparisons in Figure 3-32. The maximum error in contour location

dimensional concentration of salinity

0.60 0.80 1.00 1.20 1.40 1.60 1.80

X [m]

ConnectFlow 25%

Semi analytical 25%

ConnectFlow 50%

Semi analytical 50%

ConnectFlow 75%

Semi analytical 75%

Page 97

mum error in contour location

2.00


3.9 Salt Transport

3.9.1 Overview

This case considers 1D transport of



Figure 3-33 A schematic of the single fracture test case using an outflow (zero dispersive flux) boundary condition. The single fracture is 2 m wide and 1 m high, tessellated to a length of 0.025

Parameter

Fracture transmissivity


Reference density

Saltwater density

Salinity dispersion


The domain is represented by a single fracture tessellated into 0.025 m square subtransmissivity is set to 5.3E-6 m2/s. The is introduced through its bottom edge. The boundary conditions are zero dispersive flux and a constant pressure on the top surface, and no flow on the sides of the fracture.

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This case considers 1D transport of salt water due to a pressure gradient.


A schematic of the single fracture test case using an outflow (zero dispersive flux) boundary ingle fracture is 2 m wide and 1 m high, tessellated to a length of 0.025

Value

5.3E-6 m2/s

3.49E-8 m3/s

1000 kg/m3

1025 kg/m3

1.886E-5 m2/s

The domain is represented by a single fracture tessellated into 0.025 m square sub/s. The fracture initially contains fresh water throughout, and saline water

is introduced through its bottom edge. The boundary conditions are zero dispersive flux and a constant pressure on the top surface, and no flow on the sides of the fracture.

Page 98


A schematic of the single fracture test case using an outflow (zero dispersive flux) boundary ingle fracture is 2 m wide and 1 m high, tessellated to a length of 0.025 m.

The domain is represented by a single fracture tessellated into 0.025 m square sub-fractures. The fracture fracture initially contains fresh water throughout, and saline water

is introduced through its bottom edge. The boundary conditions are zero dispersive flux and a constant


3.9.3 Results

A steady state salt transport calculation was carried out, the anticipated solution being a constant salinity of 1.0 throughout the model. Results agreed with this solution to within 2%.

3.10 Salt Upconing

3.10.1 Overview

This case considers the upconing of salt water ia 2D case). To verify the case, equivalent CPM and DFN calculations were carried out to ensure that they produced consistent results.



Figure 3-34 Model of a vertical crossand 4 m high.

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teady state salt transport calculation was carried out, the anticipated solution being a constant salinity of 1.0 throughout the model. Results agreed with this solution to within 2%.

This case considers the upconing of salt water into a tunnel in the centre of a thin 3D model (representing a 2D case). To verify the case, equivalent CPM and DFN calculations were carried out to ensure that they


Model of a vertical cross-section, 800 m high and 1200 m wide. The open tunnel is 3

Page 99

teady state salt transport calculation was carried out, the anticipated solution being a constant salinity

nto a tunnel in the centre of a thin 3D model (representing a 2D case). To verify the case, equivalent CPM and DFN calculations were carried out to ensure that they


m wide. The open tunnel is 3 m wide


Symbol

K Hydraulic conductivity (CPM)

φ

D Coefficient of molecular

0ρ

maxρ

Lα Longitudinal

Tα Transverse dispersion length

Table 3-26 Salt transport input parameters

To ensure that the same parameters are used in both cases, the conductivities are transmissivities to be input to the DFN. The tunnel is represented by a void in the model, with a depth range of 400 m to 404 m in the centre of the model.

There is zero dispersive flux at the tunnel and a residual pressure of The specified initial condition is a depthof:

+−

−≥>−=

−

−

)800(10983.11

400010172.5

3

4

z

zzs

The salinity profile used as the boundary condition uses the same values.

In order to achieve convergence, the model requires a parameterstep has D = 1.0E-4 m2/s, and the result is used as the initial condition for a step with D

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Parameter Value

Hydraulic conductivity (CPM) Equivalent hydraulic conductivity (DFN)

=

−

−

−

−

/108.1

/102.1

/106.1

/109.3

11

10

9

8

m

m

m

m

K

Porosity

=

−

−

−

−

1011.6

1015.1

1070.2

1090.7

6

5

5

5

ϕ


1.0E

Reference density 998.217 kg/m


Longitudinal dispersion length

Transverse dispersion length

Salt transport input parameters

To ensure that the same parameters are used in both cases, the conductivities are transmissivities to be input to the DFN. The tunnel is represented by a void in the model, with a depth range of 400 m to 404 m in the centre of the model.

There is zero dispersive flux at the tunnel and a residual pressure of -4.0E6 Pa (atmThe specified initial condition is a depth-dependent salinity, equivalent in both cases. This has a salinity

−≥>− .800400

,400

z

The salinity profile used as the boundary condition uses the same values.

convergence, the model requires a parameter-stepping technique be used: the first /s, and the result is used as the initial condition for a step with D

Page 100

Value

−≥>−

−≥>−

−≥>−

−≥>

.800400/

,400150/

,15050

,500/

zs

zs

zs

zs

−≥>−

−≥>−

−≥>−

−≥>

.800400

,400150

,15050

,500

z

z

z

z

1.0E-6 m2/s

998.217 kg/m3

1042 kg/m3

3 m

1 m

To ensure that the same parameters are used in both cases, the conductivities are converted to transmissivities to be input to the DFN. The tunnel is represented by a void in the model, with a depth

4.0E6 Pa (atmospheric pressure). dependent salinity, equivalent in both cases. This has a salinity

stepping technique be used: the first /s, and the result is used as the initial condition for a step with D = 1.0E-6 m2/s


3.10.3 Results

The results shown in Figure 3-35 (DFN) and upconing of salt into the tunnel as antici

Results for the model are taken in a central line from 405 m to 800 m depth at fivefrom just below the tunnel to the bottom of the model), as plotted in results agreed to within 10%, with the largest differences close to the tunnel.

Figure 3-35 DFN Salinity profile

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(DFN) and Figure 3-36 (CPM) are a good match and demonstrate upconing of salt into the tunnel as anticipated.

Results for the model are taken in a central line from 405 m to 800 m depth at five-from just below the tunnel to the bottom of the model), as plotted in Figure 3-37. The CPM and DFN results agreed to within 10%, with the largest differences close to the tunnel.

Page 101

(CPM) are a good match and demonstrate

-metre intervals (i.e. . The CPM and DFN


Figure 3-36 CPM Salinity profile

Figure 3-37 Difference between CPM and DFN results in the upconing case.

-800

-750

-700

-650

-600

-550

-500

-450

-400

0 0.1 0.2 0.3

De

pth

[m

]

DFN Result

CPM Result

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Difference between CPM and DFN results in the upconing case.

0.3 0.4 0.5 0.6 0.7 0.8

Salinity

Page 102

0.9 1



3.11.1 Overview

The grouting of the intersection of a single fracture and an external surface can be solved analytically and is used for verification of the implementation of a grouting model.


Grouting of fractures can be specified in a variety of manners. In this case a grois used. That is, the flow after applying the grout is reduced by a factor of 10. In terms of hydraulic aperture this means a reduction of aperture by a factor f,

f = (1-e)1/3 = 0.464.

The model is defined as follows.

The region is a simple rectangular region with the coordinates specified in

Top

Bottom

Left

Right

Front

Back

Table 3-27 Region coordinates for fracture

One ‘known’ fracture is added in the yz plane, connecting the top and bottom, with a width (w) of 2.0The height (h) is the distance between

The pressure boundary condition at the top is zero and at the bottom it is

where ρ is the fluid density (1000 kg/mparameter to specify an overpressure.

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Grouting of a fracture – surface intersection

ersection of a single fracture and an external surface can be solved analytically and is used for verification of the implementation of a grouting model.

Grouting of fractures can be specified in a variety of manners. In this case a grouting efficiency (e) of 90% is used. That is, the flow after applying the grout is reduced by a factor of 10. In terms of hydraulic aperture this means a reduction of aperture by a factor f,

egion is a simple rectangular region with the coordinates specified in Table 3-

(m)

0

-10

-5

5

-5

5

Region coordinates for fracture-surface intersection grouting

One ‘known’ fracture is added in the yz plane, connecting the top and bottom, with a width (w) of 2.0The height (h) is the distance between the intersections at the top and bottom surfaces, i.e. 10.0

The pressure boundary condition at the top is zero and at the bottom it is

P = ρ g (h – s),

is the fluid density (1000 kg/m3), g the gravitational acceleration (9.81 kg m/sparameter to specify an overpressure.

Page 103

ersection of a single fracture and an external surface can be solved analytically and

uting efficiency (e) of 90% is used. That is, the flow after applying the grout is reduced by a factor of 10. In terms of hydraulic aperture

-27.

One ‘known’ fracture is added in the yz plane, connecting the top and bottom, with a width (w) of 2.0 m. the intersections at the top and bottom surfaces, i.e. 10.0 m.

), g the gravitational acceleration (9.81 kg m/s2) and s is a


For this simple problem, the Darcy flow equation can be solved explicitly and the solution is:

or, with the pressures shown as above:

Q = T w s / h,

where T is the transmissivity of the fracture. The ungrouted transmissivity is taken to be 10grouted transmissivity will be a factor of 10 smaller.

The overpressure parameter is 2.0specified in terms of residual pressure, so the value at the bottom is 19620.0

3.11.3 Results

The results are presented in Table ungrouted flows out of the top surface .The second column is the result of a ConnectFlow calculation with the fracture untessellated and the last column is the same but with a tessellation length of 1.0leads to it being broken up into 20 tesselthe whole fracture is grouted.

Analytical

Ungrouted 4.000E

Grouted 4.000E

Table 3-28. Grouting results for fracture


3.12.1 Overview

Fracture-fracture intersections will only be grouted if one of the factures belongs trepresenting some physical features, such as a tunnel, shaft or deposition hole.

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Q = -(T w / ρ g) (∆P - ρ g),

or, with the pressures shown as above:

f the fracture. The ungrouted transmissivity is taken to be 10grouted transmissivity will be a factor of 10 smaller.

The overpressure parameter is 2.0 m. Note that the pressure boundary condition in a DFN model is al pressure, so the value at the bottom is 19620.0 Pa.

Table 3-28. The first column has the analytical results for the grouted anungrouted flows out of the top surface .The second column is the result of a ConnectFlow calculation with the fracture untessellated and the last column is the same but with a tessellation length of 1.0leads to it being broken up into 20 tessellates. Note that a large penetration depth is used to assure that

Analytical Untessellated fracture Tessellated fracture

4.000E-5 m3/s 4.000E-5 m3/s 3.997E

4.000E-6 m3/s 4.000E-6 m3/s 3.997E

. Grouting results for fracture-surface intersection

Grouting of a fracture – fracture intersection

fracture intersections will only be grouted if one of the factures belongs to a set of fractures representing some physical features, such as a tunnel, shaft or deposition hole.

Page 104


f the fracture. The ungrouted transmissivity is taken to be 10-4 m2/s. The

m. Note that the pressure boundary condition in a DFN model is

. The first column has the analytical results for the grouted and ungrouted flows out of the top surface .The second column is the result of a ConnectFlow calculation with the fracture untessellated and the last column is the same but with a tessellation length of 1.0 m, which

lates. Note that a large penetration depth is used to assure that

Tessellated fracture

3.997E-5 m3/s

3.997E-6 m3/s

o a set of fractures



The ‘feature’ is the horizontal fracture in the figure below. It is connected by a single fracture to the top surface and another to the bottom surface, as shown in

Figure 3-38. Fractures used for fracture

The model region is a simple rectangular volume with the coordinates specified in

Top

Bottom

Left

Right

Front

Back

Table 3-29. Region coordinates for fracture

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The ‘feature’ is the horizontal fracture in the figure below. It is connected by a single fracture to the top surface and another to the bottom surface, as shown in Figure 3-38.

. Fractures used for fracture-fracture intersection grouting

The model region is a simple rectangular volume with the coordinates specified in Table

(m)

0

-10

-5

5

-5

5

. Region coordinates for fracture-fracture intersection grouting

Page 105

The ‘feature’ is the horizontal fracture in the figure below. It is connected by a single fracture to the top

Table 3-29.


The Darcy flow equation for this system can be solved explicitly. The parameters used are indicated in the vertical slice in Figure 3-39.

Figure 3-39. Parameters used for fracturethe fractures

In terms of these z-coordinates (z), lengths betwe(T) the solution for the outflow at the top is:

Q = (P1 - P

where w is the common width of all fractures. With the following choices:

this reduces to

z1, pressure P

La, Ta

Tb, L

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The Darcy flow equation for this system can be solved explicitly. The parameters used are indicated in the

. Parameters used for fracture-fracture intersection grouting shown on a vertical slice through

coordinates (z), lengths between intersections (L), pressures (P) and transmissivities (T) the solution for the outflow at the top is:

P2 + ρ g (La + Lc)) / ((ρ g / w) (La/Ta + Lb/Tb + Lc/Tc)),

where w is the common width of all fractures. With the following choices:

La + Lb = Lc,

Ta = Tb = T,

Tc = f T (f = 1.0 or 0.1),

P2 = P1 + h ρ g,

P1 = 0.0,

, pressure P1

zx

z2, pressure P2

Lc, Tc

, Lb

Page 106

The Darcy flow equation for this system can be solved explicitly. The parameters used are indicated in the

fracture intersection grouting shown on a vertical slice through

en intersections (L), pressures (P) and transmissivities

)),


Note that h is a parameter, not the distance between top and bottom. A value of han overpressure of 2.0 ρ g. Note that the pressure boundary condition in the DFN model is specified in terms of residual pressure, so a value 19620.0 Pa is used at the bottom.

3.12.3 Results

The results are presented in Table the values:

A large penetration depth is assumed to ensure that the whole of each fracture is grouted. The second column presents the results for a ConnectFlow run with these parameters. The third column represents the same problem but with the external surface split up by planes parallel to the subdivisions in the region specification. This leads to the intersection on the top surface being broken up into 4 intersection segments (not all the same size).

Analytical

Ungrouted 1.667E

Grouted 3.030E

Table 3-30. Grouting results for fracture

3.13 Transient Salt Diffusion

3.13.1 Overview

This case considers diffusion of salinity in a fracture (representing a 1D case) with no flow. The case was verified by comparison with an analytic solution to the 1D diffusion equation in a region with reflecting boundaries, given by:

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Q = w T (La + Lc – h) / ((1+1/f) Lc).

Note that h is a parameter, not the distance between top and bottom. A value of h = g. Note that the pressure boundary condition in the DFN model is specified in

terms of residual pressure, so a value 19620.0 Pa is used at the bottom.

Table 3-30. The first column contains the analytical results corresponding to

w = 1.0 m

T = 1.0 10-5 m2/s

La = 4.0 m

Lc = 6.0 m

f = 1.0 (ungrouted) or 0.1 (grouted)

assumed to ensure that the whole of each fracture is grouted. The second column presents the results for a ConnectFlow run with these parameters. The third column represents the same problem but with the external surface split up by planes parallel to the xz plane, using 29 subdivisions in the region specification. This leads to the intersection on the top surface being broken up into 4 intersection segments (not all the same size).

Analytical Simple region Subdivided region

1.667E-6 m3/s 1.667E-6 m3/s 1.647E

3.030E-7 m3/s 3.101E-7 m3/s 3.000E

. Grouting results for fracture-fracture intersection

Transient Salt Diffusion

of salinity in a fracture (representing a 1D case) with no flow. The case was verified by comparison with an analytic solution to the 1D diffusion equation in a region with reflecting

Page 107

= 12.0 m is used to give g. Note that the pressure boundary condition in the DFN model is specified in

. The first column contains the analytical results corresponding to

assumed to ensure that the whole of each fracture is grouted. The second column presents the results for a ConnectFlow run with these parameters. The third column represents the

xz plane, using 29 subdivisions in the region specification. This leads to the intersection on the top surface being broken up

Subdivided region

1.647E-6 m3/s

3.000E-7 m3/s

of salinity in a fracture (representing a 1D case) with no flow. The case was verified by comparison with an analytic solution to the 1D diffusion equation in a region with reflecting


∑∞

−∞=

−

+=

nittD

CtxC exp

)(4),( 0

π

where C(x,t) is the salinity at position x and time t, D is the diffusion coefficient, L is the domain length, xis the position of the peak, and ti is the initial time position of the peak. Csalinity is in the region - essentially a scale factor fti allow us to tune the initial profile such the initial salinity peak has a sensible value and shape. The sum over n represents reflections at the boundary. We only take this sum to ±3 because the become vanishingly small as n increases, and to save on calculation. For the initial salinity profile only the n=0 component is used.

This case tests the diffusion part of transient salt transport calculations.



Figure 3-40 Model of domain, 2 m long and 0.05

The domain consisted of a single fracture tessellated into 300 subfractures.

Zero dispersive flux for salinity

Zero residual Specified initial saline concentration

Gaussian

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+

+−−

ittD

nLxx

)(4

)( 2

0

salinity at position x and time t, D is the diffusion coefficient, L is the domain length, xis the initial time position of the peak. C0 is a measure of how much

essentially a scale factor for the area under the curve, with units of metres. Callow us to tune the initial profile such the initial salinity peak has a sensible value and shape. The sum

over n represents reflections at the boundary. We only take this sum to ±3 because the become vanishingly small as n increases, and to save on calculation. For the initial salinity profile only the

This case tests the diffusion part of transient salt transport calculations.

hematic of the test case is shown in Figure 3-40 and the input parameters are given in

m long and 0.05 m wide.

The domain consisted of a single fracture tessellated into 300 subfractures.

Single tessellated fracture

Specified initial saline concentration

Gaussian peak centred in fracture centre

Zero dispersive flux

Page 108

salinity at position x and time t, D is the diffusion coefficient, L is the domain length, x0 is a measure of how much

or the area under the curve, with units of metres. C0 and allow us to tune the initial profile such the initial salinity peak has a sensible value and shape. The sum

over n represents reflections at the boundary. We only take this sum to ±3 because the higher-n the terms become vanishingly small as n increases, and to save on calculation. For the initial salinity profile only the

and the input parameters are given in Table 3-31


Zero residual


Symbol

T

D

0ρ

maxρ

Lα Longitudinal


C0

ti

x0

L


There is no flow in the model (obtained by specifying the residual pressure to be 0 at both ends of the model) and zero flux boundary conditions for salinity at each end of the model. The specified initial condition is a Gaussian distribution for the salinity given by the n=0 component of the above equation.

−

+=

4

(exp

)(4),( 0

iD

x

ttD

CtxC

π

A transient salt transport calculation was undertaken with timesteps of ∆t=0.1 seconds from t=0.1s to t=1

∆t=100s from t=100 to t=1000s.

3.13.3 Results

The results shown in Figure 3-41 and with the analytic solution given above. The results agree to within 5.5 %, with the largest difference being at the centre of the plume at time t=100 s.

Results for the model are taken on a line in x, through the centre of the fracture in the y and z dimensions.

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Parameter Value

Transmissivity 1.00E-4 m


1.00E-3 m





Initial salinity 0.12 m

Initial time 1.2s

Peak position 1 m

Domain length 1.998 m


There is no flow in the model (obtained by specifying the residual pressure to be 0 at both ends of the model) and zero flux boundary conditions for salinity at each end of the model. The specified initial

Gaussian distribution for the salinity given by the n=0 component of the above equation.

+

−

)(

) 2

0

ittD

xx

A transient salt transport calculation was undertaken with timesteps of ∆t=0.01s fromt=1s, ∆t=1s from t=1 to t=10s, ∆t=10 seconds from t=10s

and Figure 3-42 show that the calculated results are in good agreement with the analytic solution given above. The results agree to within 5.5 %, with the largest difference being

e of the plume at time t=100 s.


Page 109

Value

4 m2/s

3 m2/s

1000 kg/m3

1025 kg/m3

0.05 m

0.02 m

0.12 m

1.2s

1 m

1.998 m

There is no flow in the model (obtained by specifying the residual pressure to be 0 at both ends of the model) and zero flux boundary conditions for salinity at each end of the model. The specified initial

Gaussian distribution for the salinity given by the n=0 component of the above equation.

from t=0 to t=0.1s, and t=10s to t=100s, and

show that the calculated results are in good agreement with the analytic solution given above. The results agree to within 5.5 %, with the largest difference being



Figure 3-41 Comparison of calculated and analytic profiles at

Figure 3-42 Comparison of calculated and analytic profiles at time t=1000

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.00 0.20 0.40 0.60

C

DFN result Analytical

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00 0.20 0.40

C

DFN result Analytical

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Comparison of calculated and analytic profiles at time t=100

Comparison of calculated and analytic profiles at time t=1000

t=100

0.60 0.80 1.00 1.20 1.40 1.60

x(m)

t=1000

0.60 0.80 1.00 1.20 1.40 1.60

x(m)

Page 110

1.80 2.00

1.80 2.00


3.14 1D Advection of Salinity

3.14.1 Overview

This case considers advection of salinity in a fracture (representing a 1D case). comparison with a simple analytic solution which predicts the time taken for a saltwater/freshwater interface to flush out of the model. For a fracture which is initially full of saltwater with freshwater flowing in from one end, the time taken for the interface to flush is given by:

fFvt /=

where v is the volume of the fracture, and (m3/s).



Figure 3-43 Model of domain 2 m long and 0.05

Zero salinity

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1D Advection of Salinity

This case considers advection of salinity in a fracture (representing a 1D case). The case was verified by comparison with a simple analytic solution which predicts the time taken for a saltwater/freshwater interface to flush out of the model. For a fracture which is initially full of saltwater with freshwater flowing in

he time taken for the interface to flush is given by:

is the volume of the fracture, and Ff is the volumetric flow rate of freshwater into the fracture


Model of domain 2 m long and 0.05 m wide.

Single tessellated fracture

Zero dispersive flux

Specified initial saline concentration

100% everywhere

Page 111

The case was verified by comparison with a simple analytic solution which predicts the time taken for a saltwater/freshwater interface to flush out of the model. For a fracture which is initially full of saltwater with freshwater flowing in

is the volumetric flow rate of freshwater into the fracture



Zero residual


Symbol

T

a Equivalent aperture

D Coefficient of molecular

0ρ Reference density

maxρ Saltwater density

Lα Longitudinal dispersion

Tα Transverse dispersion

Js Flux value at inflow model

A Model inflow surface area

L Fracture length

W


The domain consisted of a single fracture tessellated into 300 fractures.

There is a specified volumetric flux of 1.00Eresidual pressure of 0.00 Pa at the rightend of the model and a zero dispersive flux boundary condition at the rightfor the salinity is that C=1.0 everywhere.

With the parameters described above, the analytical estimate for the time taken for the int

from the fracture can be calculated. With

A transient calculation was carried out with timesteps offrom t=2.4s to t=2.55s.

3.14.3 Results

The results shown in Figure 3-44 and with the analytic solution given above. The results agree to within 1%.

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Parameter Value

Transmissivity 1.00E-4 m2

Equivalent aperture 4.97E-4 m


1E-9 m2/s



Longitudinal dispersion length

0.0001 m

Transverse dispersion length

0.001 m

Flux value at inflow model surface

1.00E-3 m3/m

Model inflow surface area 2.00E-2 m

Fracture length 1.998 m

Fracture width 0.05 m


of a single fracture tessellated into 300 fractures.

There is a specified volumetric flux of 1.00E-3 m3/m2/s at the left-hand end of the model, and a specified residual pressure of 0.00 Pa at the right-hand end. There is a specified salinity of C=0.0 at tend of the model and a zero dispersive flux boundary condition at the right-hand end. The initial condition for the salinity is that C=1.0 everywhere.

With the parameters described above, the analytical estimate for the time taken for the int

from the fracture can be calculated. With LWav = and AJF sf = , the time taken

A transient calculation was carried out with timesteps of ∆t=0.1s from t=0 to t=2.4s, and

and Figure 3-45 show that the calculated results are in good agreement en above. The results agree to within 1%.

Page 112

2/s

4 m

/s

1000 kg/m3

1025 kg/m3

0.0001 m

0.001 m

/m2/s

2 m2

1.998 m

hand end of the model, and a specified hand end. There is a specified salinity of C=0.0 at the left-hand

hand end. The initial condition

With the parameters described above, the analytical estimate for the time taken for the interface to flush

, the time taken st 48.2= .

, and ∆t=0.01 seconds

show that the calculated results are in good agreement


Due to the non-zero diffusion and dispersion lengths, there is a spreading of the saltwater/freshwater interface as time progresses. The interface location is taken to be the point at which salinity C = 0.5.This position is tracked by visualising the fracture with two colour bands, blue representing concentrations below 50% and red concentrations above 50%.

Figure 3-44 Salinity profile at time t=1.3s.

Figure 3-45 Salinity profile at time t=2.48 s

3.15 Transient Salt Upconing

3.15.1 Overview

This case considers transient upconing of salt water into a tunnel in the centre of a fracture (representing a 2D case). The results of the test case were verified by comparison with an analytic solution given by Dagan and Bear [29], which predicts (for small displacement) the shape of the upconed salt plume and the peak displacement of the saltwater/freshwater interface at a given time. A fverifying that the results at long time tended to the steady



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zero diffusion and dispersion lengths, there is a spreading of the saltwater/freshwater interface as time progresses. The interface location is taken to be the point at which salinity C = 0.5.This position is tracked by visualising the fracture with two colour bands, blue representing concentrations below 50% and red concentrations above 50%.

Salinity profile at time t=1.3s.

Salinity profile at time t=2.48 s

Transient Salt Upconing

This case considers transient upconing of salt water into a tunnel in the centre of a fracture (representing a he test case were verified by comparison with an analytic solution given by

], which predicts (for small displacement) the shape of the upconed salt plume and the peak displacement of the saltwater/freshwater interface at a given time. A further check is made by verifying that the results at long time tended to the steady-state DFN calculation.


Page 113

zero diffusion and dispersion lengths, there is a spreading of the saltwater/freshwater interface as time progresses. The interface location is taken to be the point at which salinity C = 0.5.This position is tracked by visualising the fracture with two colour bands, blue representing concentrations

This case considers transient upconing of salt water into a tunnel in the centre of a fracture (representing a he test case were verified by comparison with an analytic solution given by

], which predicts (for small displacement) the shape of the upconed salt plume and the urther check is made by



Figure 3-46 Model of a vertical crossand 4 m high.

Symbol

K Effective hydraulic conductivity of model (for comparison with

a

D

0ρ

maxρ

Lα Longitudinal dispersion length


F

A


The tunnel is represented by a void in the model, with a depth range of model.

Open tunnelSalinity profile

Specified pressure

Specified initial salinity

Zero flow

Specified pressure

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Model of a vertical cross-section, 800 m high and 1200 m wide. The open tunnel is 3

Parameter Value

Effective hydraulic conductivity of model (for comparison with

analytic solution) 2.012E-5 m/s

Fracture aperture 4.969E-


1.0E-9 m

Reference density 988.217 kg/m




Mass flux at the tunnel 6.32 E-6 kg/m

Surface area of tunnel 1.4 m


The tunnel is represented by a void in the model, with a depth range of -400m to -404m in the centre of the

Open tunnel Salinity profile

Specified pressure

Specified initial salinity

Zero flow

Specified pressure

Page 114

m wide. The open tunnel is 3 m wide

Value

5 m/s

-6 m

9 m2/s

988.217 kg/m3

1042 kg/m3

2.0 m

0.5 m

6 kg/m2/s

1.4 m2

404m in the centre of the


The boundary conditions for residual pressure are set to current value on boundaries, with an initial pressure condition set such that there is a small crossflow from left to right. The boundary conditions for salinity are set to current value on the leftboundaries, with an initial condition set such that concentration is 0 between z=0 and z=between z=-600m and z=-800m (the bottom quarter of the fracture). At the tunnel the boundary condition for salinity is set to zero dispersive flux. There is a corepresent an outflow.

A transient calculation was undertaken with a timestep beginning with s, and with ∆t=8.0E7 s between t=2E8 s and t=6E8 s.

3.15.3 Results

The results shown in Figure 3-47, simulation and the predictions of the analytic solutions [analytic solution. The literature suggests that the limit of applicability of the analysis occurs for peak displacements somewhere between d/4 and d/2, where d is the distance between the original interface and the tunnel. This supports the observation of discrepancies in peak height the DFN model that are visible in the later times of (with d = 200 m). Comparing the gradie

6

6

08.1

25.1

−

−

=

=

Edt

dz

Edt

dz

analytical

napsac

where z is peak height. Comparing the area under the salinity interface for time t=3.2E7 (find an error of 13.6%. This is likely due to the fairly coarse discretisation in the z coordinate leading to a poorly resolved peak for small displacements.

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The boundary conditions for residual pressure are set to current value on the left-hand, rightboundaries, with an initial pressure condition set such that there is a small crossflow from left to right. The boundary conditions for salinity are set to current value on the left-hand, right-hand, top and bottom

ries, with an initial condition set such that concentration is 0 between z=0 and z=800m (the bottom quarter of the fracture). At the tunnel the boundary condition

for salinity is set to zero dispersive flux. There is a constant mass flux boundary condition at the tunnel to

A transient calculation was undertaken with a timestep beginning with ∆t=4.0E7 s between t=0 and t=2E8 t=8.0E7 s between t=2E8 s and t=6E8 s.

, Figure 3-48 demonstrate the comparisons between DFN transient salt simulation and the predictions of the analytic solutions [29]. The rate of upconing is in agreement with the

ture suggests that the limit of applicability of the analysis occurs for peak displacements somewhere between d/4 and d/2, where d is the distance between the original interface and the tunnel. This supports the observation of discrepancies in peak height between the analysis and the DFN model that are visible in the later times of Figure 3-47, once the peak displacement passes 50m (with d = 200 m). Comparing the gradients of both sets of data up to the d/4, we find that:

where z is peak height. Comparing the area under the salinity interface for time t=3.2E7 (find an error of 13.6%. This is likely due to the fairly coarse discretisation in the z coordinate leading to a poorly resolved peak for small displacements.

Page 115

hand, right-hand and top boundaries, with an initial pressure condition set such that there is a small crossflow from left to right. The

hand, top and bottom ries, with an initial condition set such that concentration is 0 between z=0 and z=-600m, and1

800m (the bottom quarter of the fracture). At the tunnel the boundary condition nstant mass flux boundary condition at the tunnel to

t=4.0E7 s between t=0 and t=2E8

ween DFN transient salt ]. The rate of upconing is in agreement with the

ture suggests that the limit of applicability of the analysis occurs for peak displacements somewhere between d/4 and d/2, where d is the distance between the original interface

between the analysis and , once the peak displacement passes 50m

nts of both sets of data up to the d/4, we find that:

where z is peak height. Comparing the area under the salinity interface for time t=3.2E7 (Figure 3-48), we find an error of 13.6%. This is likely due to the fairly coarse discretisation in the z coordinate leading to a


Figure 3-47 Peak interface height varying with time. Comparison between analytical result and transient DFN simulation.

Figure 3-48 Interface height varying in x. Comparison between different times (cf Figure

0

10

20

30

40

50

60

70

80

90

100

0.E+00 1.E+07 2.E+07 3.E+07

peak in

terf

ace h

eig

ht

(m)

Analysis Napsac

0

10

20

30

40

50

60

-600 -400

Inte

rfa

ce

he

igh

t (m

)

t=4.4E7s analytical t=3.2E7s Analytical

t=4.4E7s Napsac t=3.2E7s Napsac

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Peak interface height varying with time. Comparison between analytical result and transient

Interface height varying in x. Comparison between analytical result and DFN simulation at two Figure 3-47)

3.E+07 4.E+07 5.E+07 6.E+07 7.E+07 8.E+07

Time (s)

-200 0 200 400

x(m)

t=3.2E7s Analytical

t=3.2E7s Napsac

Page 116

Peak interface height varying with time. Comparison between analytical result and transient

analytical result and DFN simulation at two

9.E+07 1.E+08

600


Figure 3-49 and Figure 3-50 demonstrate upconing of salt into the tunnel as anticipated, with the longtransient result showing good agreement with the steady squalitatively similar to the case presented by Werner et al. [is yet to be made.

Figure 3-49 Salinity profile at time t=6.0E8 seconds.

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demonstrate upconing of salt into the tunnel as anticipated, with the longtransient result showing good agreement with the steady state solution. The transient behaviour is also qualitatively similar to the case presented by Werner et al. [30], though a rigorous quantitative comparison

Salinity profile at time t=6.0E8 seconds.

Page 117

demonstrate upconing of salt into the tunnel as anticipated, with the long-term tate solution. The transient behaviour is also

], though a rigorous quantitative comparison


Figure 3-50 Steady state salinity profile

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Steady state salinity profile

Page 118


4. Combined DFN/CPM Verification

A summary of the DFN/CPM test cases is given in

Case Title


4.2 2D steady state flow

4.3 2D steady state flow with

particle tracks.

4.4 2D variable density flow with

particle tracks

4.5 Henry’s salt transport

Table 4-1 DFN/CPM verification tests

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Combined DFN/CPM Verification

A summary of the DFN/CPM test cases is given in Table 4-1

Overview

Radial steady state flow.

Steady state groundwater flow with a borehole used to apply the required mass flow boundary condition. The DFN/CPM interface is modelled

using the Mass Lumped and Linearly Distributed options. Truncated variations of the domain are

used to test current value boundary conditions in the DFN and CPM part of the model.

2D steady state flow Steady state groundwater flow from a CPM region

with anisotropic permeability to a DFN reg

2D steady state flow with particle tracks.

Particle tracking through steady state groundwater flow with varying permeability. Variations include mass-conserving and regular particle tracking.

2D variable density flow with particle tracks

Steady state groundwater flow with an imported spatially varying density field. Particle tracking

using forward, backward, conservative and regultracking options.

Henry’s salt transport Steady state groundwater flow. Density dependent

on imported density field from CPM calculation.

DFN/CPM verification tests

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with a borehole used to apply the required mass flow boundary condition. The DFN/CPM interface is modelled

using the Mass Lumped and Linearly Distributed options. Truncated variations of the domain are

used to test current value boundary conditions in DFN and CPM part of the model.

Steady state groundwater flow from a CPM region with anisotropic permeability to a DFN region.

Particle tracking through steady state groundwater flow with varying permeability. Variations include

conserving and regular particle tracking.

Steady state groundwater flow with an imported spatially varying density field. Particle tracking

using forward, backward, conservative and regular

Steady state groundwater flow. Density dependent on imported density field from CPM calculation.



4.1.1 Overview

This case models steady groundwater flow in a 2D disk where water is removed from the centre at a constant rate and the outer disk bounda


The problem definition and solution is taken from Feftra [The model domain is split up into two regions, fractured rock in the centre of the disk surrounded by porous medium rock, as is illustrated in rock are selected such that the solution is the same as in case


Symbol

h

Q

Kf Hydraulic Conductivity Fracture

Kr Hydraulic Conductivity Rock

R

e

d

r

ρ

µ

g


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The problem definition and solution is taken from Feftra [3] and is the DFN/CPM equivalent of case The model domain is split up into two regions, fractured rock in the centre of the disk surrounded by porous medium rock, as is illustrated in Figure 4-1. The properties of the fractures and porous medium rock are selected such that the solution is the same as in case 2.1.


Parameter Value

Head at r = R 0 m


Hydraulic Conductivity Fracture 8.17E-3

Hydraulic Conductivity Rock 8.17E-7 m/s

Radius of rock 2000 m

Aperture of fracture 1.0E-4 m

Thickness of disk 1 m


Density 1000 kg/m


Gravity 9.8 m/s

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] and is the DFN/CPM equivalent of case 2.1. The model domain is split up into two regions, fractured rock in the centre of the disk surrounded by

. The properties of the fractures and porous medium

Value

0 m

6 m3/s

3 m/s

7 m/s

2000 m

4 m

1 m

2000 m

1000 kg/m3

Pa.s

9.8 m/s2


The product of the hydraulic conductivities and thickness are chosen to be constant across the domain, with the hydraulic conductivity of the fracture

µ

ρ

12

2ge

K f =

The outflow rate is then set to make the solution the same as in

4.1.3 Variations

The modelled domain consists of a 15 degree sector. The region r<1000 m is modelled as a planar tessellated fracture using the DFN model. The region 1000<r<2000 m is modelled as model. The fractures and mesh are shown in

Figure 4-2 Fractures and CPM Mesh

4.1.3.1 Mass lumped ConnectFlow Fluxes

The interface between the two regions is modelled using the interface options, FLUX and MASS LUMP NAPSAC FLUXES

4.1.3.2 Current Value in CPM Region

In this variant, the modelled domain is truncated at a radius of 1500the CPM case 2.1.3.1 is applied as a pressure boundary condition. The interface option is used.

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The product of the hydraulic conductivities and thickness are chosen to be constant across the domain, with the hydraulic conductivity of the fracture being derived from the fracture aperture as follows.

The outflow rate is then set to make the solution the same as in 2.1.5.

The modelled domain consists of a 15 degree sector. The region r<1000 m is modelled as a planar tessellated fracture using the DFN model. The region 1000<r<2000 m is modelled as model. The fractures and mesh are shown in Figure 4-2.

Fractures and CPM Mesh

lumped ConnectFlow Fluxes

The interface between the two regions is modelled using the interface options, CONSERVE FRACTURE MASS LUMP NAPSAC FLUXES.

Current Value in CPM Region

In this variant, the modelled domain is truncated at a radius of 1500 m, and the interpolated solution from is applied as a pressure boundary condition. The MASS LUMP NAPSAC FLUXES

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The product of the hydraulic conductivities and thickness are chosen to be constant across the domain, being derived from the fracture aperture as follows.

The modelled domain consists of a 15 degree sector. The region r<1000 m is modelled as a planar tessellated fracture using the DFN model. The region 1000<r<2000 m is modelled as rock using the CPM

CONSERVE FRACTURE

m, and the interpolated solution from MASS LUMP NAPSAC FLUXES


4.1.3.3 Current Value in DFN Region

In this variant, the modelled domain is truncated at a radius of 250 m, and the interpolated solution from the CPM case 2.1.3.1 is applied as a pressure boundary condition. The interface option is used.

4.1.3.4 Linearly Distributed ConnectFlow Fluxes

The interface between the two regions is modelled using the default interface options, FRACTURE FLUX and LINEARLY DISTRIBUTE NAPSAC FLUXES

4.1.4 Results


)ln(2

)()(r

R

Kd

QRhrh

π−=

The results from DFN/CPM calculation show good overall agreement with the analytical solution, as illustrated in Figure 4-3 to Figure 4linear distributed fluxes method gives a smoother and more accurate solution.

Figure 4-3 Variation of head with distance using a mass lumped DFN/CPM interface

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Head

[m

]

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Current Value in DFN Region

In this variant, the modelled domain is truncated at a radius of 250 m, and the interpolated solution from is applied as a pressure boundary condition. The MASS LUMP NAPSAC FLUXES

Linearly Distributed ConnectFlow Fluxes

The interface between the two regions is modelled using the default interface options, LINEARLY DISTRIBUTE NAPSAC FLUXES.


The results from DFN/CPM calculation show good overall agreement with the analytical solution, as 4-6. The mass lumping method is more computationally efficient, but the

ted fluxes method gives a smoother and more accurate solution.

Variation of head with distance using a mass lumped DFN/CPM interface

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

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In this variant, the modelled domain is truncated at a radius of 250 m, and the interpolated solution from MASS LUMP NAPSAC FLUXES

The interface between the two regions is modelled using the default interface options, CONSERVE

The results from DFN/CPM calculation show good overall agreement with the analytical solution, as . The mass lumping method is more computationally efficient, but the

Variation of head with distance using a mass lumped DFN/CPM interface

ConnectFlow

Analytical Solution


Figure 4-4 Variation of head with distance using a current value boundary in the CPM region and a mass lumped DFN/CPM interface

Figure 4-5 Variation of head with distance using a current value boundary in the DFN reglumped DFN/CPM interface

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Head

[m

]

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Head

[m

]

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Variation of head with distance using a current value boundary in the CPM region and a mass lumped DFN/CPM interface

Variation of head with distance using a current value boundary in the DFN reglumped DFN/CPM interface

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

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Variation of head with distance using a current value boundary in the CPM region and a mass

Variation of head with distance using a current value boundary in the DFN region and a mass

ConnectFlow

Analytical Solution

ConnectFlow

Analytical Solution


Figure 4-6 Variation of head with distance using a linearly distributed DFN/CPM interface

4.2 Flow to Fracture Network

4.2.1 Overview

This case models steady state groundwaterflow enters the network through porous medium rock with anisotropic permeability.


The problem is a variant of test 3.6


The fracture properties are the same as in Section

The CPM properties are defined in such a way as to ensure the following conditions at the interface.

-8

-7

-6

-5

-4

-3

-2

-1

0

0 500

Head

[m

]

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Variation of head with distance using a linearly distributed DFN/CPM interface

Flow to Fracture Network

This case models steady state groundwater flow in a simple two-dimensional fracture network where the flow enters the network through porous medium rock with anisotropic permeability.

3.6, as illustrated in Figure 4-7.


The fracture properties are the same as in Section 3.6.3.2.


1000 1500 2000

Radius [m]

ConnectFlow

Analytical Solution

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Variation of head with distance using a linearly distributed DFN/CPM interface

dimensional fracture network where the flow enters the network through porous medium rock with anisotropic permeability.


ConnectFlow

Analytical Solution


1== CPMDFN hh m

hKAQQ CPMDFN ∇==

Where K is the hydraulic conductivity

The first equation defines the head gradient in the CPM region, and the second is then used in conjunction with the DFN analytical value of Q to obtain K.

As the flow needs to enter the DFN region throucomponent to the flow in the CPM region. In order to avoid this impacting the head at the fracture inflows the vertical component or permeability is set to be 1000 times higher than the horizontal compo

4.2.3 Results

The calculated heads are all within 4% of the analytical solution.


a 0.062

b 0.936

c 0.598


4.3 2D Steady State Flow with Particle Tracks

4.3.1 Overview

This case is taken from Level 3 of the international HYDROCOIN project for verification of groundwater flow codes [31]. It models steady state flow in a tworegion of higher permeability.

The case has a non-uniform analytical solution and is used in the HYDROCOIN study to test particle tracking.

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Where K is the hydraulic conductivity and A the cross sectional area of the CPM region.

The first equation defines the head gradient in the CPM region, and the second is then used in conjunction with the DFN analytical value of Q to obtain K.

As the flow needs to enter the DFN region through the three fractures, there will also be a vertical component to the flow in the CPM region. In order to avoid this impacting the head at the fracture inflows the vertical component or permeability is set to be 1000 times higher than the horizontal compo

The calculated heads are all within 4% of the analytical solution.


0.061 1.42%

0.968 3.36%

0.598 0.600 0.19%


2D Steady State Flow with Particle Tracks

This case is taken from Level 3 of the international HYDROCOIN project for verification of groundwater ]. It models steady state flow in a two-dimensional vertical slice of rock, containing a circular


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and A the cross sectional area of the CPM region.

The first equation defines the head gradient in the CPM region, and the second is then used in conjunction

gh the three fractures, there will also be a vertical component to the flow in the CPM region. In order to avoid this impacting the head at the fracture inflows the vertical component or permeability is set to be 1000 times higher than the horizontal component.

% Error

1.42%

3.36%

0.19%

This case is taken from Level 3 of the international HYDROCOIN project for verification of groundwater tical slice of rock, containing a circular



This is the equivalent DFN/CPM test of C



The DFN region is represented by a single tessellated fracture. In orconsistent with [31], the following relations must hold.

Balancing the flow rate Q:

Fracture permeability: k

Thickness available for flow:

Taking φ from [31], selecting d = 1 m and scaling the pressure gradient gives the values in

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the equivalent DFN/CPM test of Case 2.3.


The DFN region is represented by a single tessellated fracture. In order to ensure that the solution is ], the following relations must hold.

ekdk fi =

12

2e

k f =

φ/ed =

], selecting d = 1 m and scaling the pressure gradient gives the values in

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der to ensure that the solution is

], selecting d = 1 m and scaling the pressure gradient gives the values in Table 4-4.


Symbol

Lx

Ly Vertical outer region distance

a

P1

P0

ko

ki

kf

e

d

φ

fφ

ρ

µ


Eight particle tracks are released 50 m upstream of the disk centre and at Y values of 10, 12, 14, 16, 18, 20, 22 and 24 m.

The analytical solution for the pathlines is given in the HYDROC

)1/()(

)(2

2

oi

oi

kk

kk

r

aoyy

+

−+= for r > a

o

oi

k

kk

oyy2

)( += for r < a

Where r is the distance from the centre of the disk and ya long distance away from the origin.

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Parameter Value

Upstream and downstream distances

250 m

Vertical outer region distance 240 m

Radius of inner disk 10 m

Upstream pressure 3.0E-4 Pa

Downstream pressure -3.0E-4 Pa

Permeability of outer region 8.333E-

Permeability of inner region 8.333E-

Fracture permeability 8.333E-

Fracture thickness 0.1 m

CPM thickness 1 m

Porosity 0.1

Fracture Porosity 1.0

Density 1000 kg/m



The analytical solution for the pathlines is given in the HYDROCOIN report [7] as

Where r is the distance from the centre of the disk and y0 is a constant representing the height of the track a long distance away from the origin.

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Value

250 m

240 m

10 m

4 Pa

4 Pa

-7 m2

-5 m2

-4 m2

0.1 m

1 m

0.1

1.0

1000 kg/m3

3 Pa.s


esenting the height of the track


4.3.3 Variations

4.3.3.1 Regular Particle Tracking

Regular particle tracking was used within the CPM region. The mesh around the DFN region is shown in Figure 4-9.

Figure 4-9 CPM wrapped mesh

The tessellated fractures within the inner region are shown in

Figure 4-10 Tessellated fractures (larger scale image).

4.3.3.2 Mass Conserving Particle Tracking

Mass conserving particle tracking was used withishown in Figure 4-11.

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Regular Particle Tracking

Regular particle tracking was used within the CPM region. The mesh around the DFN region is shown in

The tessellated fractures within the inner region are shown in Figure 4-10.

Tessellated fractures (larger scale image).

Mass Conserving Particle Tracking

Mass conserving particle tracking was used within the CPM region. The mesh around the DFN region is

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Regular particle tracking was used within the CPM region. The mesh around the DFN region is shown in

n the CPM region. The mesh around the DFN region is


Figure 4-11 CPM regular mesh

The tessellated fractures within the inner region are shown in

Figure 4-12 Tessellated fractures (larger scale im

4.3.4 Results

The particle tracks within the DFN part of the model are treated in a stochastic manner, allowing different tracks to take different paths. Ten particles were released from each of the starting positions and the results for regular particle tracking are shown in

For the tracks that do not pass through the DFN region, the results are within 1% ofor the regular particle tracking, and within 2% of the analytical solution for the mass conserving particle tracking.

The stochastic tracks have more variation as expected, however the average tracks starting from each position also have a travel time and final location within 1% of the analytical solution for the regular particle tracks and within 2% of the analytical solution for the mass conserving particle tracking.

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ated fractures within the inner region are shown in Figure 4-12.

Tessellated fractures (larger scale image).

The particle tracks within the DFN part of the model are treated in a stochastic manner, allowing different tracks to take different paths. Ten particles were released from each of the starting positions and the

tracking are shown in Figure 4-13 and Figure 4-14.

For the tracks that do not pass through the DFN region, the results are within 1% ofor the regular particle tracking, and within 2% of the analytical solution for the mass conserving particle

The stochastic tracks have more variation as expected, however the average tracks starting from each also have a travel time and final location within 1% of the analytical solution for the regular particle

tracks and within 2% of the analytical solution for the mass conserving particle tracking.

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The particle tracks within the DFN part of the model are treated in a stochastic manner, allowing different tracks to take different paths. Ten particles were released from each of the starting positions and the

For the tracks that do not pass through the DFN region, the results are within 1% of the analytical solution for the regular particle tracking, and within 2% of the analytical solution for the mass conserving particle

The stochastic tracks have more variation as expected, however the average tracks starting from each also have a travel time and final location within 1% of the analytical solution for the regular particle

tracks and within 2% of the analytical solution for the mass conserving particle tracking.


Figure 4-13 Regular particle tracks

Figure 4-14 Stochastic behaviour of particle tracks

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Regular particle tracks

Stochastic behaviour of particle tracks

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4.4 2D Variable Density Flow with Particle Tracks

4.4.1 Overview

This case uses a predefined density field, corresponding to an analvelocity field is used to test regular and mass conserving particle tracking.


The modelled domain is shown in

boundary condition on all four sides of the region and the density field


The DFN region is represented by a single tessellated fracture. In order to ensure that the solution is consistent with the CPM region, the following relations must hold.

Balancing the flow rate Q:

Fracture permeability: k

Thickness available for flow:

Taking e = 1.0E-5 and d = 1.0E-3 then gives the following values in

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2D Variable Density Flow with Particle Tracks

This case uses a predefined density field, corresponding to an analytical flow solution. The resulting velocity field is used to test regular and mass conserving particle tracking.

The modelled domain is shown in Figure 4-15. The analytical pressure field 2

xy −boundary condition on all four sides of the region and the density field xy is applied to the interior.


The DFN region is represented by a single tessellated fracture. In order to ensure that the solution is consistent with the CPM region, the following relations must hold.

ekkd f=

12

2e

k f =

φ/ed =

3 then gives the following values in Table 4-5.

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ytical flow solution. The resulting

2x is applied as a

is applied to the interior.

The DFN region is represented by a single tessellated fracture. In order to ensure that the solution is


Symbol

k

kf

e

d

φ

µ


4.4.3 Variations

4.4.3.1 Forward Particle Tracking

This variant tracks 10 particles starting from the upper left hand corner of the The tracks subsequently pass through the DFN region where they behave in a stochastic manner, allowing different tracks to take different paths.

4.4.3.2 Backward Particle Tracking

This variant performs backward tracking on 10 particlwhich corresponds to the same analytical path as the forward track in variation

4.4.3.3 Mass Conserving Particl

This variant tracks ten particles starting from x=1.5, y = 9.5 and uses the mass conserving particle tracking option. The tracks subsequently pass through the DFN region where they behave in a stochastic manner, allowing different tracks to take different paths.

4.4.4 Results

4.4.4.1 Analytical solution

Substituting the Darcy flow equation into the continuity equation gives

0))(.( =∇∇ Pk

µρϕ

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Parameter Value

Permeability of CPM region 8.333E-

Fracture permeability 8.333E-

Fracture thickness 1.0E-5 m

CPM thickness 1.0E-3 m

Porosity 0.01


Forward Particle Tracking

This variant tracks 10 particles starting from the upper left hand corner of the CPM region at x=1.5, y = 9.5. The tracks subsequently pass through the DFN region where they behave in a stochastic manner, allowing different tracks to take different paths.

Backward Particle Tracking

This variant performs backward tracking on 10 particles ending at x=9.07, y=1.57 within the DFN region, which corresponds to the same analytical path as the forward track in variation 4.4.3.1


This variant tracks ten particles starting from x=1.5, y = 9.5 and uses the mass conserving particle tracking option. The tracks subsequently pass through the DFN region where they behave in a stochastic manner,

different paths.

Substituting the Darcy flow equation into the continuity equation gives

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Value

-14 m2

-12 m2

5 m

3 m

0.01

3 Pa.s

CPM region at x=1.5, y = 9.5. The tracks subsequently pass through the DFN region where they behave in a stochastic manner, allowing

es ending at x=9.07, y=1.57 within the DFN region, 4.4.3.1.

This variant tracks ten particles starting from x=1.5, y = 9.5 and uses the mass conserving particle tracking option. The tracks subsequently pass through the DFN region where they behave in a stochastic manner,


Gravity is omitted as 0=∂

∂

z

ρ.

Substituting in the proposed solution

0))((.( 22 =−∇∇ xyxy

which reduces to

4)2()2( 22 −=∂

∂+−

∂

∂xyxy

yyx

x

This confirms that xy=ρ and P

From the pressure field, the particle tracks can then be determined as

represents the starting location of the particle track. The corresponding travel time being

4.4.4.2 Pressure and Density Fields

Figure 4-16 and Figure 4-17 show the pressure and density fields. The former being calculated and the latter being prescribed.

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Substituting in the proposed solution xy=ρ and 22

xyP −= then gives

04 =+ xyxy

22xyP −= is a solution of the Darcy flow equations with variable density.

From the pressure field, the particle tracks can then be determined as x

yxy 00= where

represents the starting location of the particle track. The corresponding travel time being

sity Fields

show the pressure and density fields. The former being calculated and the

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is a solution of the Darcy flow equations with variable density.

where 00, yx

represents the starting location of the particle track. The corresponding travel time being )ln(2 ox

x

kt

µφ= .

show the pressure and density fields. The former being calculated and the


Figure 4-16 Calculated residual pressure

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Calculated residual pressures.

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Figure 4-17 Applied densities.

4.4.4.3 Forward particle tracking

Figure 4-18 shows 10 particle tracks released from the DFN region the tracks behave in a stochastic manner. The background of the plot is coloured by residual pressure.

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Forward particle tracking

shows 10 particle tracks released from the top left corner of the CPM region. On entering the DFN region the tracks behave in a stochastic manner. The background of the plot is coloured by residual

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top left corner of the CPM region. On entering the DFN region the tracks behave in a stochastic manner. The background of the plot is coloured by residual


Figure 4-18 Forward particle tracks

An averaged particle track plot is compared against the analytical solution in agreeing to within 2% of the analytical solution for position and in the DFN region, it is necessary to set the model reference density to a value representative of that of the fluid which the track passes through.

Figure 4-19 Averaged forward tracks

0

1

2

3

4

5

6

7

8

9

10

0 2 4

X [m]

Y [

m]

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Forward particle tracks

veraged particle track plot is compared against the analytical solution in Figure agreeing to within 2% of the analytical solution for position and travel time. In order to predict travel times in the DFN region, it is necessary to set the model reference density to a value representative of that of the fluid which the track passes through.

Averaged forward tracks

6 8 10

X [m]

ConnectFlow

Analytical Solution

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Figure 4-19, the results travel time. In order to predict travel times

in the DFN region, it is necessary to set the model reference density to a value representative of that of the

Analytical Solution


4.4.4.4 Backward Particle Tracking

Figure 4-20 shows 10 particle tracks tracked backwards from the lower right hand corner of the DFN region. Within the DFN region the tracks behave in a stochastic manner. The background of the plot is coloured by residual pressure.

Figure 4-20 Backward particle tracks

An averaged particle track plot is compared against the analytical solution in agreeing to within 2% of the analytical solution for position

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Backward Particle Tracking

shows 10 particle tracks tracked backwards from the lower right hand corner of the DFN in the DFN region the tracks behave in a stochastic manner. The background of the plot is

Backward particle tracks

An averaged particle track plot is compared against the analytical solution in Figure agreeing to within 2% of the analytical solution for position and travel time.

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shows 10 particle tracks tracked backwards from the lower right hand corner of the DFN in the DFN region the tracks behave in a stochastic manner. The background of the plot is

Figure 4-21, the results


Figure 4-21 Averaged backward particle tracks

4.4.4.5 Mass Conserving Particle Tracking

The mass conserving tracks behave in a similar manner to the nonresults agreeing to within 2% of the analytical solution for position and travel time

Figure 4-22 Averaged mass conserving particle tracks

0

1

2

3

4

5

6

7

8

9

0 2 4

X [m]

Y [

m]

0

1

2

3

4

5

6

7

8

9

10

0 2 4

X [m]

Y [

m]

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Averaged backward particle tracks


The mass conserving tracks behave in a similar manner to the non-mass conserving equivalents, with the ts agreeing to within 2% of the analytical solution for position and travel time

mass conserving particle tracks

6 8 10

X [m]

ConnectFlow

Analytical Solution

6 8 10

X [m]

ConnectFlow

Analytical Solution

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mass conserving equivalents, with the

Analytical Solution

Analytical Solution


4.5 Henry’s Salt Transport Using Interpolated Density

4.5.1 Overview

This case considers the groundwater pressures arising from a spatially varying density field.

The geometry, boundary conditions and density field are taken from the modified Henry CPM



Figure 4-23 Henry’s Problem

Symbol


Q

ρ

φ

e


The DFN domain is represented by a single fracturefracture thickness is set to 1.8516Em/s. The DFN region covers the left 75% of the domain, which has been selected so that the interfbetween the CPM and DFN region is in a location where the density and pressure fields are rapidly changing.

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Henry’s Salt Transport Using Interpolated Density

considers the groundwater pressures arising from a spatially varying density field.

The geometry, boundary conditions and density field are taken from the modified Henry CPM


Parameter Value



9.43E-5 m

Density Interpolated

Porosity 0.35

Fracture thickness 1.8516E-

The DFN domain is represented by a single fracture tessellated into 0.025 m square subfracture thickness is set to 1.8516E-4 m in order to generate the desired hydraulic conductivity of 1.0Em/s. The DFN region covers the left 75% of the domain, which has been selected so that the interfbetween the CPM and DFN region is in a location where the density and pressure fields are rapidly

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considers the groundwater pressures arising from a spatially varying density field.

The geometry, boundary conditions and density field are taken from the modified Henry CPM Case 2.8.


Value

2 m/s

5 m2/s

Interpolated

0.35

-4 m

tessellated into 0.025 m square sub-fractures. The 4 m in order to generate the desired hydraulic conductivity of 1.0E-2

m/s. The DFN region covers the left 75% of the domain, which has been selected so that the interface between the CPM and DFN region is in a location where the density and pressure fields are rapidly


The CPM region has thickness /e

The CPM region is meshed with the same spatial resolution as the sub

The interpolated density field used is shown in

Figure 4-24 Interpolated density field

4.5.3 Results

The residual pressure field of the DFN/CPM calculation is compared against the CPM pressure field from case 2.8, the latter having been separately checked against the analytical solution via the salt concentration solution.

Plots of the two pressure fields are shown in vary in shape near the inflow, in absolute terms the pressure fields agree to within 5%.

The influence of the density variation in this test case was checked by redensity instead of the interpolated density. In this case the difference in compared with the CPM solution.

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φ/ so that the cross sectional area available to flow remains constant.

The CPM region is meshed with the same spatial resolution as the sub-fractures.

The interpolated density field used is shown in Figure 4-24.

Interpolated density field

The residual pressure field of the DFN/CPM calculation is compared against the CPM pressure field from , the latter having been separately checked against the analytical solution via the salt

Plots of the two pressure fields are shown in Figure 4-25 and Figure 4-26. Although the pressure contours r the inflow, in absolute terms the pressure fields agree to within 5%.

The influence of the density variation in this test case was checked by re-running with a constant average density instead of the interpolated density. In this case the difference in the pressure fields was 30% when

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so that the cross sectional area available to flow remains constant.

The residual pressure field of the DFN/CPM calculation is compared against the CPM pressure field from , the latter having been separately checked against the analytical solution via the salt

. Although the pressure contours r the inflow, in absolute terms the pressure fields agree to within 5%.

running with a constant average the pressure fields was 30% when


Figure 4-25 Residual pressure in the DFN/CPM model

Figure 4-26 Residual pressure in the CPM model

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Residual pressure in the DFN/CPM model

Residual pressure in the CPM model

Page 141


5. Further Testing

This section references further testing that has taken place on the ConnectFlow suite of software, which complements and extends the testing covered in the earlier sections.

5.1 Continuum Porous Media

5.1.1 Verification

5.1.1.1 SKB – Implicit Representation of Fractures

Section 3.3 of SKB report R-01-49 [rock matrix by modifying CPM permeabilties. The approach is tested on a simple 3D case with a semianalytical solution. The results are all within 20% of the expected solution, with the accuracy improving for larger features. The algorithm allows small features to be modelled more accurately but with increased computational cost.

5.1.1.2 SKB – Heat Transport

Appendix F of SKB report R-06-98 [The first case models 1D heat conduction with a timecase models 1D heat conduction with a specifieHYDROCOIN Level 1 Test Case 5 for transient thermal convection in a saturated permeable medium containing a heat source with decaying power output. The first two cases have an analytical solution anthe third a semi-analytical solution. In all cases ConnectFlow shows excellent agreement with the expected solutions.

5.1.1.3 SKB R-09-24

Appendix G of SKB report R-09-24 [modelling the inflow of fresh water into a repository under open repository conditions. The ConnectFlow simulation modeled the repository region as a combined DFN/CPM, with boundary conditions being applied from a separate DFN site scale model. fine grid refinement around the tunnelsthe repository agreed to within 20%.

amecfw.com

Further Testing

This section references further testing that has taken place on the ConnectFlow suite of software, which complements and extends the testing covered in the earlier sections.

Continuum Porous Media

Implicit Representation of Fractures

49 [32] describes the modelling of a set of fractures within a background rock matrix by modifying CPM permeabilties. The approach is tested on a simple 3D case with a semianalytical solution. The results are all within 20% of the expected solution, with the accuracy improving for larger features. The algorithm allows small features to be modelled more accurately but with increased

98 [33] describes three verification tests on heat transport in ConnectFlow. D heat conduction with a time-varying temperature boundary condition. The second

case models 1D heat conduction with a specified heat flux boundary condition. The final case models the HYDROCOIN Level 1 Test Case 5 for transient thermal convection in a saturated permeable medium containing a heat source with decaying power output. The first two cases have an analytical solution an

analytical solution. In all cases ConnectFlow shows excellent agreement with the expected

24 [34] describes a comparison of DarcyTools and ConnectFlow in water into a repository under open repository conditions. The ConnectFlow

simulation modeled the repository region as a combined DFN/CPM, with boundary conditions being applied from a separate DFN site scale model. DarcyTools used a single regional-scale Efine grid refinement around the tunnels. Both models used the same Hydro-DFN data. The overall inflow to the repository agreed to within 20%.

Page 142

This section references further testing that has taken place on the ConnectFlow suite of software, which

] describes the modelling of a set of fractures within a background rock matrix by modifying CPM permeabilties. The approach is tested on a simple 3D case with a semi-analytical solution. The results are all within 20% of the expected solution, with the accuracy improving for larger features. The algorithm allows small features to be modelled more accurately but with increased

] describes three verification tests on heat transport in ConnectFlow. varying temperature boundary condition. The second

d heat flux boundary condition. The final case models the HYDROCOIN Level 1 Test Case 5 for transient thermal convection in a saturated permeable medium containing a heat source with decaying power output. The first two cases have an analytical solution and

analytical solution. In all cases ConnectFlow shows excellent agreement with the expected

] describes a comparison of DarcyTools and ConnectFlow in water into a repository under open repository conditions. The ConnectFlow

simulation modeled the repository region as a combined DFN/CPM, with boundary conditions being scale ECPM model with

DFN data. The overall inflow to


5.1.1.4 ANDRA

A verification exercise for ConnectFlow has been performed for ANDRA [groundwater flow and radionuclide transport problems were investigated. The exercise included cases that were diffusion dominated, cases that were advection and dispersion dominated, cases in twodimensions, and cases with permeability contrasts between regions of the models such that the dominant flow and transport processes differed between the regions. With appropriate choices of the grid, the initial and boundary conditions, and the timethe analytical solutions given for all cases considered.

5.1.1.5 KBS-3

ConnectFlow has also been used in two reviews of repository assessments. These reviews compared the results obtained using different programs for the sameKBS-3 study [36], the groundwater heads obtained using ConnectFlow were compared with those obtained using the program GWHRT, for several different cases. In every case, the results agreed to at least six significant figures (the number of figures listed for the output from GWHRT). This gives great confidence that both programs were coded correctly.

5.1.1.6 Gewähr

Results obtained using ConnectFlow were also compared with results obtained using the FEM301 program for the Swiss Project Gewähr [slightly from those obtained using FEM301. These differences were traced to discrepancies between the FEM301 program and its documentation, and differences between treatment of highly distorted elements. When an appropriate temporary modification was made to ConnectFlow to enable it to mimic the behaviour of FEM301, the results obtained agreed to within five significant figures with those obtained from FEM301. It should be stressed that the initial differences were not due to any problems with ConnectFlow.

5.1.1.7 Nirex

The results of a Monte-Carlo study of dispersion in a heterogeneous porous medium [and quite stringent test of the groundwater flow and particle transport algorithms used in ConnectFlow. The fact that good agreement could be obtained between the analytical and numerical results for the dispersion of the particles indicates that ConnectFlow provided an accuratflow in a heterogeneous permeability field. This case therefore also builds confidence in the correctness of ConnectFlow.

amecfw.com

A verification exercise for ConnectFlow has been performed for ANDRA [35]. Cases covering a range of groundwater flow and radionuclide transport problems were investigated. The exercise included cases that were diffusion dominated, cases that were advection and dispersion dominated, cases in two

s with permeability contrasts between regions of the models such that the dominant flow and transport processes differed between the regions. With appropriate choices of the grid, the initial and boundary conditions, and the time-stepping scheme, ConnectFlow was found to accurately reproduce the analytical solutions given for all cases considered.

ConnectFlow has also been used in two reviews of repository assessments. These reviews compared the results obtained using different programs for the same finite-element model. In a review of the Swedish

], the groundwater heads obtained using ConnectFlow were compared with those obtained using the program GWHRT, for several different cases. In every case, the results agreed to at

gnificant figures (the number of figures listed for the output from GWHRT). This gives great confidence that both programs were coded correctly.

Results obtained using ConnectFlow were also compared with results obtained using the FEM301 or the Swiss Project Gewähr [37]. Initially, the results obtained using ConnectFlow differed

slightly from those obtained using FEM301. These differences were traced to discrepancies between the FEM301 program and its documentation, and differences between ConnectFlow and FEM301 in the treatment of highly distorted elements. When an appropriate temporary modification was made to ConnectFlow to enable it to mimic the behaviour of FEM301, the results obtained agreed to within five

e obtained from FEM301. It should be stressed that the initial differences were not due to any problems with ConnectFlow.

Carlo study of dispersion in a heterogeneous porous medium [st of the groundwater flow and particle transport algorithms used in ConnectFlow.

The fact that good agreement could be obtained between the analytical and numerical results for the dispersion of the particles indicates that ConnectFlow provided an accurate solution for the groundwater flow in a heterogeneous permeability field. This case therefore also builds confidence in the correctness of

Page 143

]. Cases covering a range of groundwater flow and radionuclide transport problems were investigated. The exercise included cases that were diffusion dominated, cases that were advection and dispersion dominated, cases in two- and three-

s with permeability contrasts between regions of the models such that the dominant flow and transport processes differed between the regions. With appropriate choices of the grid, the initial

ow was found to accurately reproduce

ConnectFlow has also been used in two reviews of repository assessments. These reviews compared the element model. In a review of the Swedish

], the groundwater heads obtained using ConnectFlow were compared with those obtained using the program GWHRT, for several different cases. In every case, the results agreed to at

gnificant figures (the number of figures listed for the output from GWHRT). This gives great

Results obtained using ConnectFlow were also compared with results obtained using the FEM301 ]. Initially, the results obtained using ConnectFlow differed

slightly from those obtained using FEM301. These differences were traced to discrepancies between the ConnectFlow and FEM301 in the

treatment of highly distorted elements. When an appropriate temporary modification was made to ConnectFlow to enable it to mimic the behaviour of FEM301, the results obtained agreed to within five

e obtained from FEM301. It should be stressed that the initial differences were

Carlo study of dispersion in a heterogeneous porous medium [38] provides a useful st of the groundwater flow and particle transport algorithms used in ConnectFlow.

The fact that good agreement could be obtained between the analytical and numerical results for the e solution for the groundwater

flow in a heterogeneous permeability field. This case therefore also builds confidence in the correctness of


5.1.1.8 Olkiluoto Site

A site scale comparison of FEFTRA and ConnectFlow was carried out as part of the FEFTR[3]. The study showed good agreement with regard to borehole pressures and flow rates near the repository. Flow path positions near the repository wwere shorter in FEFTRA as compared with ConnectFlow.

A further study [39] has compared modelling of pressures and salinity distributions in 15 deep core drilled boreholes between the codes and measuremengroundwater flow and solute transport over the last 8000 years.

5.2 Discrete Fracture Network

5.2.1 Verification

5.2.1.1 Stripa

This testing exercise formed part of an OECD/NEA [radioactive waste in crystalline rock, that involved a detailed characterisation of the abandoned iron ore mine at Stripa in central Sweden.

The testing compared fracture generatiocodes ConnectFlow, FracMan/MAFIC and FMG.

Eight test cases were considered and tsolutions and the results of FracMan/MAFIC and

5.2.1.2 Nirex

This study [41] compared transient flow for the fracture flow codes ConnectFlow and FracMan/MAFIC for six test cases. ConnectFlow and FracMan/MAFIC gave qualitatively similar results. However,quantitative differences were observed even for quite simple test networks. These are thought to be due to inadequate mesh discretisation of the source zone in FracMan/MAFIC. The ConnectFlow results for the early time behaviour compared well wsource zone responses that were inaccurate. The asymptotic ConnectFlow results were shown to tend to the steady state solutions.

5.3 Automated Testing

The test cases presented in this documentanalytical and benchmark data.

amecfw.com

A site scale comparison of FEFTRA and ConnectFlow was carried out as part of the FEFTR]. The study showed good agreement with regard to borehole pressures and flow rates near the

repository. Flow path positions near the repository were also in good agreement although the travel times were shorter in FEFTRA as compared with ConnectFlow.

] has compared modelling of pressures and salinity distributions in 15 deep core drilled boreholes between the codes and measurements based on simulations of the evolution of coupled groundwater flow and solute transport over the last 8000 years.

Discrete Fracture Network

This testing exercise formed part of an OECD/NEA [40] project relating to the final disposal of highly radioactive waste in crystalline rock, that involved a detailed characterisation of the abandoned iron ore mine at Stripa in central Sweden.

The testing compared fracture generation, steady state flow and particle tracking for the fracture flow , FracMan/MAFIC and FMG.

Eight test cases were considered and the ConnectFlow results compared well with available analytical solutions and the results of FracMan/MAFIC and FMG (when a formulation of the test case was possible).

] compared transient flow for the fracture flow codes ConnectFlow and FracMan/MAFIC for six test cases. ConnectFlow and FracMan/MAFIC gave qualitatively similar results. However,quantitative differences were observed even for quite simple test networks. These are thought to be due to inadequate mesh discretisation of the source zone in FracMan/MAFIC. The ConnectFlow results for the early time behaviour compared well with the available analytical solutions, whereas FracMan/MAFIC gave source zone responses that were inaccurate. The asymptotic ConnectFlow results were shown to tend to

Automated Testing

The test cases presented in this document can be automatically re-run to check the results against the

Page 144

A site scale comparison of FEFTRA and ConnectFlow was carried out as part of the FEFTRA verification ]. The study showed good agreement with regard to borehole pressures and flow rates near the

ere also in good agreement although the travel times

] has compared modelling of pressures and salinity distributions in 15 deep core drilled ts based on simulations of the evolution of coupled

] project relating to the final disposal of highly radioactive waste in crystalline rock, that involved a detailed characterisation of the granite formation in an

n, steady state flow and particle tracking for the fracture flow

he ConnectFlow results compared well with available analytical FMG (when a formulation of the test case was possible).

] compared transient flow for the fracture flow codes ConnectFlow and FracMan/MAFIC for six test cases. ConnectFlow and FracMan/MAFIC gave qualitatively similar results. However, significant quantitative differences were observed even for quite simple test networks. These are thought to be due to inadequate mesh discretisation of the source zone in FracMan/MAFIC. The ConnectFlow results for the

ith the available analytical solutions, whereas FracMan/MAFIC gave source zone responses that were inaccurate. The asymptotic ConnectFlow results were shown to tend to

run to check the results against the


This complements the regular ConnectFlow regression test suite, which consists of more than 200 tests and is used to comprehensively check results from new releaversions.

5.4 Peer Review

ConnectFlow is used by a wide variety There are representatives of both the regulatory bodies and the nuclear utilities. Many of thein the iConnect club and provide valuable feedback that helps enhance the software quality.

amecfw.com

This complements the regular ConnectFlow regression test suite, which consists of more than 200 tests and is used to comprehensively check results from new releases of the software against previous

ConnectFlow is used by a wide variety of organisations with an interest in radioactive waste disposal. There are representatives of both the regulatory bodies and the nuclear utilities. Many of thein the iConnect club and provide valuable feedback that helps enhance the software quality.

Page 145

This complements the regular ConnectFlow regression test suite, which consists of more than 200 tests ses of the software against previous

of organisations with an interest in radioactive waste disposal. There are representatives of both the regulatory bodies and the nuclear utilities. Many of these participate in the iConnect club and provide valuable feedback that helps enhance the software quality.


6. References

1 NAMMU Technical Summary, SA/ENV/CONNECTFLOW/8 May 2010

2 NAPSAC Technical Summary, SA/ENV/CONNECTFLOW/12 , May 2010

3 FEFTRA Verification, VTT Research Notes 2385, Espoo 2007.

4 de Marsily, G., 1986. Quantitative Hydrogeology Press INC, Orlando.

5 NEA/SKI, The International HYDROCOIN Project, Level 1: code verification,OECD Publications, 1988.

6 R. Atkinson, T.P. Cherill, A.W. Herbert, D.P. Hodgkinson, C.P. Jackson, J. Rae

and P.C. Robinson, Review of the grUKAEA Report AERE-R. 11140, 1984

7 NEA/SKI, The International HYDROCOIN Project, Level 3: uncertainty and sensitivity analysis ,OECD Publications, 1991.

8 A. Schäfers. Task Force on Engineered

Description. 2014.

9 Richards, L.A., 1931. Capillary conduction of liquids through porous mediums.Physics, 1(5), pp.318-333.

10 Zarba, R.L., Bouloutas, E.T. and Celia, M., 1990. unsaturated flow equation. Water Resources Research WRERAQ

11 Van Genuchten, M.T., 1980. A closedunsaturated soils. Soil science society of America journal

12 K. Pruess, TOUGH2 – A General Purpose Numerical Simulator for Multiphase Fluid and Heat Flow, Lawrence Berkeley Laboratory Report LBL

13 Bear Jacob, Hydraulics of Groundwater,1979, McGraw

14 Improving the worthiness of the Henry problem as a benchmark for density dependent ground water flows. Water Resources Research Vol 40, 2004.

15 A R Hoch, C P Jackson Rock-matrix diffusion in transport of salinity. SKB Report R

16 C. M. Oldenburg and K. Pruess. EOS7R: Radionuclide Transport for Tough2, Lawrence Berkley Laboratory Report LBL-34868, Berkley, CA, November 1995.

17 Javandel, I., C. Doughty, and C.American Geophys. Union, pp. 14

18 Joyce S, Applegate D, Appleyard P, Gordon A, Heath T, Hunter F, Hoek J, Jackson P, Swan D, Woollard H, 2014. Groundwater flow Svensk Kärnbränslehantering AB.

19 Parkhurst D L, Kipp K L, Charlton S R, 2010. PHAST Version 2: a program for simulatingflow, solute transport, and multicomponent geochemical reactiGeological Survey, Denver, Colorado.

amecfw.com

NAMMU Technical Summary, SA/ENV/CONNECTFLOW/8 May 2010

NAPSAC Technical Summary, SA/ENV/CONNECTFLOW/12 , May 2010

FEFTRA Verification, VTT Research Notes 2385, Espoo 2007.

de Marsily, G., 1986. Quantitative Hydrogeology – Groundwater Hydrology for Engineers. Academic

NEA/SKI, The International HYDROCOIN Project, Level 1: code verification,OECD Publications, 1988.

R. Atkinson, T.P. Cherill, A.W. Herbert, D.P. Hodgkinson, C.P. Jackson, J. Rae

and P.C. Robinson, Review of the groundwater flow and radionuclide transport modelling in KBSR. 11140, 1984

NEA/SKI, The International HYDROCOIN Project, Level 3: uncertainty and sensitivity analysis ,OECD

Task Force on Engineered Barrier Systems, “Code Validation”. Code Validation Task

Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Journal of Applied

Zarba, R.L., Bouloutas, E.T. and Celia, M., 1990. General mass-conservative numerical solution for the Water Resources Research WRERAQ, 26(7), pp.1483-

Van Genuchten, M.T., 1980. A closed-form equation for predicting the hydraulic conductivity of cience society of America journal, 44(5), pp.892-898.

A General Purpose Numerical Simulator for Multiphase Fluid and Heat Flow, Lawrence Berkeley Laboratory Report LBL–29400, 1991.

Bear Jacob, Hydraulics of Groundwater,1979, McGraw-Hill Inc.

Improving the worthiness of the Henry problem as a benchmark for density dependent ground water flows. Water Resources Research Vol 40, 2004.

matrix diffusion in transport of salinity. SKB Report R

C. M. Oldenburg and K. Pruess. EOS7R: Radionuclide Transport for Tough2, Lawrence Berkley 34868, Berkley, CA, November 1995.

C. Doughty, and C.-F. Tsang, Groundwater Transport, Handbook of Mathematical Models, American Geophys. Union, pp. 14-19, 1984.

Joyce S, Applegate D, Appleyard P, Gordon A, Heath T, Hunter F, Hoek J, Jackson P, Swan D, Woollard H, 2014. Groundwater flow and reactive transport modelling in ConnectFlow. SKB RSvensk Kärnbränslehantering AB.

Parkhurst D L, Kipp K L, Charlton S R, 2010. PHAST Version 2: a program for simulatingflow, solute transport, and multicomponent geochemical reactions. Techniques and Methods 6Geological Survey, Denver, Colorado.

Page 146

Engineers. Academic


oundwater flow and radionuclide transport modelling in KBS-3,


. Code Validation Task

Journal of Applied

conservative numerical solution for the -1496.

form equation for predicting the hydraulic conductivity of

A General Purpose Numerical Simulator for Multiphase Fluid and Heat Flow,

Improving the worthiness of the Henry problem as a benchmark for density dependent ground water

matrix diffusion in transport of salinity. SKB Report R-04-78.

C. M. Oldenburg and K. Pruess. EOS7R: Radionuclide Transport for Tough2, Lawrence Berkley

F. Tsang, Groundwater Transport, Handbook of Mathematical Models,

Joyce S, Applegate D, Appleyard P, Gordon A, Heath T, Hunter F, Hoek J, Jackson P, Swan D, and reactive transport modelling in ConnectFlow. SKB R-14-19,

Parkhurst D L, Kipp K L, Charlton S R, 2010. PHAST Version 2: a program for simulating groundwater ons. Techniques and Methods 6-A35, U.S.


20 Hall, P. (1982). "On Some Simple Estimates of an Exponent of Regular Variation". Statistical Society, Series B (Methodological) 44 (1): 37

21 A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra. Clustering on the Unit Hypersphere using von MisesFisher Distributions. JMLR, 6:1345

22 Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.

23 Robinson P.C. 1984, Numerical calculations of critical densities for lines and planes. J.Phys. A:Math. Gen. 17, 2823-2830.

24 Bour and Davey 1998, On the connectivity of three dimensional fault networks, Water Resources Research, Vol 34

25 DECOVALEX Task C Stage 1 Project Progress Report, Lanru Jing and John Hudson, 2009

26 DarcyTools, Version 2.1 Verification and validation R

Computer-aided Fluid Engineering AB, Sweden. March 2004

27 NEA/SKI, The International HYDROCOIN Project, Level 1: cod

28 R. Atkinson, T.P. Cherill, A.W. Herbert, D.P. Hodgkinson, C.P. Jackson, J. Rae

and P.C. Robinson, Review of the groundwater flow and radionuclide transport modelling in KBSUKAEA Report AERE-R. 11140, 1984

29 G. Dagan & J. Bear (1968): Solving the problem of local interface upconing in a coastal aquifer by the method of small perturbations, Journal of Hydraulic Research, 6:1. 15 30 A.D. Werner, D. Jakovovic, C.T. Simmons (2009): Experimental observations of salJournal of Hydrology, 337, 230-241 31 NEA/SKI, The International HYDROCOIN Project, Level 3: uncertainty and sensitivity analysis ,OECD Publications, 1991.

32 Lee Hartley, Peter Jackson, Mike Poole Development of hydrogeological modelling tools based on NAMMU SKB Report R-01-49

33 Lee Hartley, Andrew Hoch, Peter Jackson, Steve Joyce, Rachel McCarthy, William Rodwell, Ben Swift, 2006. Groundwater flow and transpoSKB R-06-98, Svensk Kärnbränslehantering AB

34 Joyce S, Simson T, Hartley L, Applegate D, Hoek J, Jackson P, Roberts D, Swan D, Gylling B, Marsic N, Rhén I, 2010. Groundwater flow modellinSKB R-09-24, Svensk Kärnbränslehantering AB

35 S P Watson, M J Poole and A R Hoch, NAMMU Implementation of Test Cases for Flow and Transport in Porous Media, AEA Technology Report 14302/00/1 Issue 1

36 R Atkinson, T P Cherrill, A W Herbert, D P Hodgkinson, C P Jackson, J Rae and P C Robinson, Review of the Groundwater Flow and Radionuclide Transport Modelling in KBS1984.

37 P C Robinson, C P Jackson, A W Herberof the Swiss Project Gewähr, UKAEA Report AERE R.11929, 1986

38 S T Morris, J D Porter and C P Jackson, An Investigation of the Accuracy of the Spectral Turning Bands Random Field Generator, Nirex

amecfw.com

Hall, P. (1982). "On Some Simple Estimates of an Exponent of Regular Variation".

, Series B (Methodological) 44 (1): 37–42.

A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra. Clustering on the Unit Hypersphere using von MisesFisher Distributions. JMLR, 6:1345–1382, Sep 2005.

Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.

nson P.C. 1984, Numerical calculations of critical densities for lines and planes. J.Phys. A:Math.

Bour and Davey 1998, On the connectivity of three dimensional fault networks, Water Resources

1 Project Progress Report, Lanru Jing and John Hudson, 2009

DarcyTools, Version 2.1 Verification and validation R-04-21. Urban Svensson,

aided Fluid Engineering AB, Sweden. March 2004


R. Atkinson, T.P. Cherill, A.W. Herbert, D.P. Hodgkinson, C.P. Jackson, J. Rae

and P.C. Robinson, Review of the groundwater flow and radionuclide transport modelling in KBSR. 11140, 1984

Dagan & J. Bear (1968): Solving the problem of local interface upconing in a coastal aquifer by the method of small perturbations, Journal of Hydraulic Research, 6:1. 15-44

A.D. Werner, D. Jakovovic, C.T. Simmons (2009): Experimental observations of sal241


Lee Hartley, Peter Jackson, Mike Poole Development of hydrogeological modelling tools based on

Lee Hartley, Andrew Hoch, Peter Jackson, Steve Joyce, Rachel McCarthy, William Rodwell, Ben Swift, 2006. Groundwater flow and transport modelling during the temperate period for the SR

, Svensk Kärnbränslehantering AB.

Joyce S, Simson T, Hartley L, Applegate D, Hoek J, Jackson P, Roberts D, Swan D, Gylling B, Marsic N, Rhén I, 2010. Groundwater flow modelling of periods with temperate climate conditions

24, Svensk Kärnbränslehantering AB.

S P Watson, M J Poole and A R Hoch, NAMMU Implementation of Test Cases for Flow and Transport in Porous Media, AEA Technology Report 14302/00/1 Issue 1, 2000.

R Atkinson, T P Cherrill, A W Herbert, D P Hodgkinson, C P Jackson, J Rae and P C Robinson, Review of the Groundwater Flow and Radionuclide Transport Modelling in KBS-3, UKAEA Report AERE R.11140,

P C Robinson, C P Jackson, A W Herbert and R Atkinson, Review of the Groundwater Flow Modelling of the Swiss Project Gewähr, UKAEA Report AERE R.11929, 1986

S T Morris, J D Porter and C P Jackson, An Investigation of the Accuracy of the Spectral Turning Bands Random Field Generator, Nirex Report NSS/R320, 1997.

Page 147

Hall, P. (1982). "On Some Simple Estimates of an Exponent of Regular Variation". Journal of the Royal

A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra. Clustering on the Unit Hypersphere using von Mises-

Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.

nson P.C. 1984, Numerical calculations of critical densities for lines and planes. J.Phys. A:Math.

Bour and Davey 1998, On the connectivity of three dimensional fault networks, Water Resources

1 Project Progress Report, Lanru Jing and John Hudson, 2009

e verification,OECD Publications, 1988.

and P.C. Robinson, Review of the groundwater flow and radionuclide transport modelling in KBS-3,

Dagan & J. Bear (1968): Solving the problem of local interface upconing in a coastal aquifer by the

A.D. Werner, D. Jakovovic, C.T. Simmons (2009): Experimental observations of saltwater up-coning,


Lee Hartley, Peter Jackson, Mike Poole Development of hydrogeological modelling tools based on

Lee Hartley, Andrew Hoch, Peter Jackson, Steve Joyce, Rachel McCarthy, William Rodwell, Ben Swift, rt modelling during the temperate period for the SR-Can assessment,

Joyce S, Simson T, Hartley L, Applegate D, Hoek J, Jackson P, Roberts D, Swan D, Gylling B, Marsic g of periods with temperate climate conditions – Laxemar.

S P Watson, M J Poole and A R Hoch, NAMMU Implementation of Test Cases for Flow and Transport

R Atkinson, T P Cherrill, A W Herbert, D P Hodgkinson, C P Jackson, J Rae and P C Robinson, Review 3, UKAEA Report AERE R.11140,

t and R Atkinson, Review of the Groundwater Flow Modelling

S T Morris, J D Porter and C P Jackson, An Investigation of the Accuracy of the Spectral Turning


39 Lofman J and Meszaros F. Modelling of Groundwater Flow and Solute Transport in Olkiluoto 2011, Posiva Working Report 2011

40 F. Schwartz and G. Lee. CrossTechnical Report 91-29, Stripa Project, 1991

41. Golders Associates -Transient solver cross code comparison between NAPSAC and FRACMAN/MAFIC . 94524312/21 April 1995.

amecfw.com

Lofman J and Meszaros F. Modelling of Groundwater Flow and Solute Transport in Olkiluoto

2011, Posiva Working Report 2011-YY, in prep

F. Schwartz and G. Lee. Cross-verification Testing of Fracture Flow and Mass Transp29, Stripa Project, 1991

Transient solver cross code comparison between NAPSAC and FRACMAN/MAFIC . 94524312/21 April 1995.

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verification Testing of Fracture Flow and Mass Transport Codes.

Transient solver cross code comparison between NAPSAC and

Documents

Release 11.4 July 2016 - Foster Wheeler · ConnectFlow is the suite of Amec Foster Wheeler’s groundwater modelling ... ConnectFlow Command Reference Manual; ... design and evaluation