7
Solute transport in a single fracture with time-dependent aperture due to chemically medicated changes Zhihong Zhao a,n , Longcheng Liu b , Ivars Neretnieks b , Lanru Jing c a Department of Geological Sciences, Bolin Centre for Climate Research, Stockholm University, Stockholm SE-106 91, Sweden b Department of Chemical Engineering and Technology, Royal Institute of Technology, Stockholm, Sweden c Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden article info Article history: Received 20 September 2012 Received in revised form 27 August 2013 Accepted 24 December 2013 Available online 25 January 2014 Keywords: Solute transport Pressure solution Rock fracture Time-dependent aperture abstract In addition to mechanical loading, the transport properties of rock fractures are also affected by chemically mediated changes, such as pressure solution, stress corrosion and free-face dissolution, among others. Based on a time-dependent model of fracture closure under constant normal stresses, the transport behavior of contaminants in a slowly closing fracture is studied using a nite difference scheme. The results show that the contaminant penetrating along the fracture plane gradually becomes slow or almost negligible during the process of fracture closure induced by chemical processes, whereas the matrix diffusion of contaminants is active all the time. This nding indicates that diffusion into the rock matrix perpendicular to the advective ow direction always plays an important role in determining the fate of contaminant in rock fractures. The smaller the uid ow due to fracture closure and the larger impact the matrix diffusion can further delay the solute transport. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Solute transport in fractured geological media has been a major research topic in Earth sciences, including geological radioactive waste repositories, naturally fractured petroleum elds, geother- mal reservoirs and deep CO 2 sequestration projects [14]. The fracture networks in crystalline rocks (e.g. granites) provide the major pathways for uid ow and solute transport, because the permeability of rock matrix is usually negligible compared with that of fractures. Mass migration through a rock fracture is a complex natural phenomenon, which involves different mechan- isms, such as advection, hydrodynamic dispersion, matrix diffu- sion, sorption and chemical reactions [1,5]. It is well known that rock fractures can close, open or propagate due to mechanical loading induced by processes of excavation, thermal expansion, tectonic movements or glaciation [6]. In addition, experiments show that chemically mediated changes also play an important role in the evolution of mechan- ical, hydraulic and transport properties of fractured rocks [79]. Compared with mechanically mediated changes that commonly occur transiently after stress condition changes, such as normal closure and shear-induced dilation, chemically mediated changes are usually time-dependent and need much longer time to reach a steady state [10]. A relatively small change of fracture aperture can signicantly inuence transport properties of rock fractures, which was shown in recent developments to investigate the mechanical effects on solute transport in a single rock fracture [11,12]. Also mechanical effects on solute migration in complex fracture systems have been addressed [1315], but the feedback of chemically mediated changes on contaminant transport in frac- tured rocks has not yet been attempted. In this study, we aim to gain some basic insights into how the time-dependent aperture induced by chemical processes can inuence the transport behaviors of contaminants (e.g. radioactive nuclides or mineral solute species) that are carried by the owing uid in a single fracture in crystalline rocks where matrix diffusion is active. Although aperture closure and changing conducting channels over time in stressed natural fractures was experimen- tally observed [16,17], the main causes have still been under discussions. The predictive models presented in the literature are still under development and there are several mechanisms that still need to be understood in detail. In spite of this, it is certain that chemically induced changes of the fracture aperture may also affect contaminant transport, by changing the ow rate and the diffusion between water and porous rock matrix. These coupled effects have not been addressed for fracture ow yet, and there- fore serves as the motivation of this paper. As a rst step to gain some insights, we simply use the constitutive model developed by Yasuhara et al. [8] to account for the fracture aperture changes mediated by chemical processes, without comparing with other existing models of time-dependent fracture aperture that is beyond the scope of the present study. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijrmms International Journal of Rock Mechanics & Mining Sciences 1365-1609/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2013.12.004 n Corresponding author. Tel.: þ468 164 765. E-mail address: [email protected] (Z. Zhao). International Journal of Rock Mechanics & Mining Sciences 66 (2014) 6975

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Page 1: Solute transport in a single fracture with time-dependent aperture due to chemically medicated changes

Solute transport in a single fracture with time-dependent aperturedue to chemically medicated changes

Zhihong Zhao a,n, Longcheng Liu b, Ivars Neretnieks b, Lanru Jing c

a Department of Geological Sciences, Bolin Centre for Climate Research, Stockholm University, Stockholm SE-106 91, Swedenb Department of Chemical Engineering and Technology, Royal Institute of Technology, Stockholm, Swedenc Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 20 September 2012Received in revised form27 August 2013Accepted 24 December 2013Available online 25 January 2014

Keywords:Solute transportPressure solutionRock fractureTime-dependent aperture

a b s t r a c t

In addition to mechanical loading, the transport properties of rock fractures are also affected bychemically mediated changes, such as pressure solution, stress corrosion and free-face dissolution,among others. Based on a time-dependent model of fracture closure under constant normal stresses, thetransport behavior of contaminants in a slowly closing fracture is studied using a finite differencescheme. The results show that the contaminant penetrating along the fracture plane gradually becomesslow or almost negligible during the process of fracture closure induced by chemical processes, whereasthe matrix diffusion of contaminants is active all the time. This finding indicates that diffusion into therock matrix perpendicular to the advective flow direction always plays an important role in determiningthe fate of contaminant in rock fractures. The smaller the fluid flow due to fracture closure and the largerimpact the matrix diffusion can further delay the solute transport.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Solute transport in fractured geological media has been a majorresearch topic in Earth sciences, including geological radioactivewaste repositories, naturally fractured petroleum fields, geother-mal reservoirs and deep CO2 sequestration projects [1–4]. Thefracture networks in crystalline rocks (e.g. granites) provide themajor pathways for fluid flow and solute transport, because thepermeability of rock matrix is usually negligible compared withthat of fractures. Mass migration through a rock fracture is acomplex natural phenomenon, which involves different mechan-isms, such as advection, hydrodynamic dispersion, matrix diffu-sion, sorption and chemical reactions [1,5].

It is well known that rock fractures can close, open orpropagate due to mechanical loading induced by processes ofexcavation, thermal expansion, tectonic movements or glaciation[6]. In addition, experiments show that chemically mediatedchanges also play an important role in the evolution of mechan-ical, hydraulic and transport properties of fractured rocks [7–9].Compared with mechanically mediated changes that commonlyoccur transiently after stress condition changes, such as normalclosure and shear-induced dilation, chemically mediated changesare usually time-dependent and need much longer time to reach asteady state [10]. A relatively small change of fracture aperture can

significantly influence transport properties of rock fractures,which was shown in recent developments to investigate themechanical effects on solute transport in a single rock fracture[11,12]. Also mechanical effects on solute migration in complexfracture systems have been addressed [13–15], but the feedback ofchemically mediated changes on contaminant transport in frac-tured rocks has not yet been attempted.

In this study, we aim to gain some basic insights into how thetime-dependent aperture induced by chemical processes caninfluence the transport behaviors of contaminants (e.g. radioactivenuclides or mineral solute species) that are carried by the flowingfluid in a single fracture in crystalline rocks where matrix diffusionis active. Although aperture closure and changing conductingchannels over time in stressed natural fractures was experimen-tally observed [16,17], the main causes have still been underdiscussions. The predictive models presented in the literature arestill under development and there are several mechanisms thatstill need to be understood in detail. In spite of this, it is certainthat chemically induced changes of the fracture aperture may alsoaffect contaminant transport, by changing the flow rate and thediffusion between water and porous rock matrix. These coupledeffects have not been addressed for fracture flow yet, and there-fore serves as the motivation of this paper. As a first step to gainsome insights, we simply use the constitutive model developed byYasuhara et al. [8] to account for the fracture aperture changesmediated by chemical processes, without comparing with otherexisting models of time-dependent fracture aperture that isbeyond the scope of the present study.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijrmms

International Journal ofRock Mechanics & Mining Sciences

1365-1609/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijrmms.2013.12.004

n Corresponding author. Tel.: þ468 164 765.E-mail address: [email protected] (Z. Zhao).

International Journal of Rock Mechanics & Mining Sciences 66 (2014) 69–75

Page 2: Solute transport in a single fracture with time-dependent aperture due to chemically medicated changes

2. Method

2.1. Chemical controls on fracture aperture

In fractured rocks, the two rough surfaces of a fracture are incontact with each other at some asperities but not in others. Underapplied normal loading, the local stresses at the contactingasperities are higher than the apparent average normal stress ofthe whole fracture due to stress concentrations. Mechanicalcrushing (damage) of the mineral grains that form the asperitiesmay occur at locations where the localized stresses are sufficientlyhigher than the grain strength.

When rocks are exposed to a corrosive environment (fluid),chemical stress corrosion may also occur at contacting asperitieswhere local stresses are smaller than rock strength, and contributeto crushing or micro-spalling [10]. In addition, at the contactingasperities of highly concentrated localized stresses the solubility ofthe minerals is higher than where there is no contact (fracturevoids), so the minerals will tend to dissolve there and precipitatewhere the stresses are lower (Fig. 1). This process is called pressuresolution, which, given time, may also contribute to a significantclosure of the fracture. A further mechanism of mass relocation isdissolution of small crystals (mineral particles), which have largersolubility than large crystals (mineral particles) even if withoutstresses applied, which is so called Ostwald ripening [18]. Incontrast, under-saturation of the fluid in fracture voids maypromote free-face dissolution and result in fracture opening [19].

The situation becomes increasingly complicated when there isadvective flow through the fracture voids that can carry awaydissolved species. Yasuhara et al. [8] showed that the flow rate ofwater through a fracture can affect the rate of fracture closure.Overall, the complex evolution of fracture aperture is generallycontrolled by the above processes.

A few of flow-through experiments were conducted to examinethe evolution of fracture permeability for different types of rocksunder various stress and temperature conditions (Table 1). Theirresults showed that the fracture aperture and permeability basi-cally decrease with time under various external confinements(stresses) and temperatures for most rocks [16,20–25], but thefracture permeability could increase for limestone or KH2PO4

crystals during the flow of highly reactive fluids due to thedomination of free-face dissolution [25–29].

Based on these experimental results, a few constitutive rela-tionships were developed to account for the fracture aperturechanges mediated by chemical processes [8,9,19,25]. Among them,Yasuhara et al. [8] presented a time-dependent model of fractureclosure where the main mechanisms were pressure solution atmineral grain contacts, diffusion of the dissolved species out to thefracture voids and re-precipitation on the faces of crystals. Theenhanced concentration of dissolved silica in a thin slit betweenthe mineral grains in contact, the concentration at the mouth ofthe slit and the equilibrium solubility at the un-stressed surface ofgrains were used to obtain the driving force for dissolution at thecontact spots. Rates of dissolution and precipitation as functions of

σn

σn

Pressure solution at asperties(1) Dissolution (2) Diffusion (3) Precipitation

Δ b(1 ) (2)

(3)z

x0

Fig. 1. Schematic views of fracture aperture reduction due to pressure solution under normal stress.

Table 1Summary of fracture permeability change during flow-through experiments.

Authors Rock type Temperature (1C) Normal stress (MPa) Permeabity change

Lin and Daily [20] Topopah Spring welded tuff 60–160 5 DecreaseMoore et al. [21] Westerly granite 300–500 50 (effective) DecreaseMorrow et al. [22] Westerly granite 150–500 50 (effective) DecreaseDurham et al. [23] Carrara marble Room temperature 0.2 DecreasePolak et al. [16] Arkansas novaculite 20, 80, 120, 150 3.5 (effective) DecreaseYasuhara et al. [17,27] Arkansas novaculite 20–120 1.4 Initially decrease but ultimately increasePolak et al. [26] Limestone Room temperature 3.5 (effective) Initially decrease but ultimately increaseDetwiler et al. [28] KH2PO4 crystals 27.5 0 IncreaseDetwiler [29] KH2PO4 crystals 24.5 0.21 IncreaseYasuhara et al. [24] Mizunami granite 20–90 5 and 10 DecreaseMcGuire et al. [25] Capitan massive limestone 20 2.5–10 Decrease (pH 7 fluid)

Increase (pH 6 and 5 fluids)

Note: ‘effective’ represents the effective normal stress (¼normal stress—fluid pressure).

Z. Zhao et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 69–7570

Page 3: Solute transport in a single fracture with time-dependent aperture due to chemically medicated changes

the driving forces were also investigated. The model also includedadvective flow in the fracture, which could carry away dissolvedsilica.

2.2. Chemically mediated fracture closure model

In this study, the mechanically mediated changes in fractureapertures are assumed to have finished already, so that thefracture is in a mechanically static state. The chemically mediatedchanges then start and have continuous and long lasting effects onfracture aperture evolution.

Both pressure solution at contacting asperities and free-facedissolution at fracture voids contribute to fracture sealing oropening, respectively [19], but we focus on the first mechanismin this study, with the pressure solution consisting of three serialprocesses: dissolution at asperity contacts, diffusion along theinterfacial water film and precipitation at the fracture walls. Themodel and experimental data in Yasuhara et al. [8] are used asa basis for the present study. The rock tested was novaculite witha measured density of 2650 kg m�3 and a small porosity of o1%.

Yasuhara et al. [8] found that the hydraulically measuredaperture reduction rates varied in the range 2.5�10�13–

2.5�10�11 m s�1 with time. In addition they justified that, if theaperture reduction is assumed to be mainly dependent on thedissolution at contacting asperities, the aperture reduction rate(db=dt) can be related to the dissolution rate at contactingasperities in the form of,

dbdt

¼ �dMdiss

dt1

2Acρð1Þ

with the dissolution rate at the contacting asperities given by

dMdiss

dt¼

3V2mðsa �sc ÞkAcρ

RT saZsc

0 saosc

(ð2Þ

where Ac is the total contacting area; ρ is the grain mineraldensity; Vm is molar volume of the solid; sa and sc are themicroscopic effective stress applied at contact area and criticalstress that defines the stress state where pressure solution ceases,respectively; k is the dissolution rate constant of grain; R is the gasconstant; and T is the temperature of the system.

In writing Eq. (2), the microscopic effective stress has beendefined as,

sa ¼sef f

Rcð3Þ

where seff is the effective stress defined as the macroscopicnormal stress applied on the fracture minus the water pressure;Rc is the contact ratio that in reality changes over time in acomplicate manner.

As a result, substituting Eqs. (2) and (3) into Eq. (1) gives,

dbdt

¼�3V2

mðsef f =Rc �scÞk2RT

sef f

RcZsc

0 sef f

Rcosc

8<: ð4Þ

where the contact ratio Rc can be calculated, based on theregression model between contact area and aperture [8], as

Rc ¼ �γ3 ln2b�γ1γ2

� �þRc0 ð5Þ

where Rc0 represents the initial contact ratio when mechanicalmediation has finished and given an initial hydraulic aperture ofb0; b is the hydraulic aperture in the unit of mm; γ1 γ2 and γ3 areexperimental constants that can be determined from curve fittingof the test data, and the conditions 2b0�γ1 ¼ γ2 and 2b4γ1 arerequired for Eq. (5).

When there are other sinks for the dissolved minerals, e.g.advective flow bringing in water that has lower concentration ofminerals than the equilibrium concentration, the driving force fordissolution increases and dissolution rate increases. This wasaccounted for in the basic Yasuhara–Elsworth model [8]. Anothersink for dissolved minerals that was however not considered is theexchange by diffusion with dissolved minerals in the porous rockmatrix. This can be a sink as well as a source of dissolved minerals.With time the contact area subjected to stresses increases andthus the local stresses decrease, resulting in the decreasingdissolution rate, even if the normal stresses are kept constant.This process is included in the present model.

2.3. Solute transport in a single fracture with time-dependentaperture

The setting for the model development is that the groundwaterflowing through a slowly closing fracture, provided a contaminantsource of constant strength at the origin of the fracture [30]. Thismay represent a system that is initially free of nuclide and then thenuclide concentration is suddenly increased to a value of c0 at theinlet of the fracture [31].

A basic solute transport model in a single fracture considers theadvection along fracture plane and matrix diffusion perpendicularto the flow direction. Considering the mass balance of solute in thefracture and in the porous matrix, respectively, one can obtain thefollowing differential equations for describing the contaminanttransport in a single fracture of time-changing aperture,

∂bcf∂t

þvb∂cf∂x

�De∂cm∂z

����z ¼ 0

¼ 0 ðfor fractureÞ ð6Þ

and

∂cm∂t

�Da∂2cm∂z2

¼ 0 ðfor rock matrixÞ ð7Þ

where cf and cm are the volumetric concentration of solute in thefracture and matrix, respectively; x is the fracture axis; z is the axisperpendicular to the fracture plane; De and Da are the effective andapparent diffusion coefficients, respectively; v is the mean fluidvelocity in the fracture that can be determined, under theassumption of a quasi-steady flow, by the cubic law,

v¼ b2

12μdPdx

ð8Þ

where μ is the dynamic viscosity; dP=dx is the hydraulic pressuregradient.

The initial condition of the system is,

cf ðx;0Þ ¼ cmðx; z;0Þ ¼ 0 ð9Þwhich expresses that both fracture and matrix are free of con-taminant initially.

The boundary conditions for the fracture are,

cf ð0; tÞ ¼ c0 ð10Þand

∂cf ðL; tÞ∂x

¼ 0 ð11Þ

Eq. (10) represents a continuous injection of solute of constantconcentration at the inlet of the fracture [1], and Eq. (11)represents continuous solute concentration at the outlet of thefracture under the condition that L (fracture length) has beenchosen to be so long that the solute barely penetrates that far.

Likewise, the boundary conditions of the rock matrix can bewritten as,

cmðx;0; tÞ ¼ cf ðx; tÞ ð12Þ

Z. Zhao et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 69–75 71

Page 4: Solute transport in a single fracture with time-dependent aperture due to chemically medicated changes

and

∂cmðx; Lm; tÞ∂z

¼ 0 ð13Þ

Eq. (12) expresses identical (continuous) solute concentrationat the fracture surface, because transverse hydrodynamic disper-sion is usually negligible in a thin rock fracture [30]. Eq. (13)illustrates that solute can transport in the rock matrix till apenetration depth of Lm.

It is to be noted that substituting Eq. (8) into Eq. (6) allows oneto write,

∂bcf∂t

þ b3

12μdPdx

∂cf∂x

�De∂cm∂z

����z ¼ 0

¼ 0 ð14Þ

where hydraulic aperture (b) is a function of time, and can becalculated based on the aperture reduction rate (Eq. (4)).

Since it is impossible or difficult to derive analytical solutionsfor Eqs. (14) and (7) under the initial and boundary conditionsEqs. (9)–(13), a numerical solution as the one presented in thefollowing section was employed.

2.4. Finite difference solution

A backward difference scheme was used to approximate thetime derivatives in Eqs. (14) and (7) , and the first order spatialderivative ∂cf =∂x in Eq. (14) and the second order spatial derivative∂2cm=∂z2 in Eq. (7) was approximated by central differenceschemes, respectively. For the first order spatial derivative∂cm=∂zjz ¼ b at the fracture walls, an one-sided finite differencetechnique was used. The boundary conditions were introducedinto the governing equations naturally.

A MATLAB code was developed to solve the numerical systemsof transport equations for both fracture and matrix at each timestep. To validate the code, the obtained results were comparedwith the analytical solutions for constant fracture aperture(Eqs. (22) and (23) in Zhao et al. [12]) in Fig. 2, using theparameters listed in Table 2. It shows that the finite differencemethod (FDM) could obtain the nearly identical concentration inthe fracture as that from the closed-form solution. The concentra-tion distribution in the matrix was also in good agreement butnow shown here. Therefore, the developed code was verified andwas used to investigate the solute transport behaviors in singlefractures impacted by the chemically mediated changes.

3. Solute transport impacted by pressure solution inducedfracture closure

Based on literature [8], proper material parameters werechosen and listed in Table 2, which represented a typical fractureclosure due to pressure solution in underground geothermal orhydrothermal situations. Both dissolution rate constant and criticalstress are dependent on temperature, in the form ofk¼ k0expð�Ek=RTÞ and sc ¼ Emð1�T=TmÞ=4Vm, with k0¼1.59 mol m2, Ek¼7.13�104 J mol�1, Em¼8.57�103 J mol�1 andTm¼1883 K [8,32,33]. This yields, in the case of a temperature of150 1C, a dissolution rate constant of 2.51�10�9 mol m2 s�1 and acritical stress of 73.2 MPa. It follows that the fracture closure withtime can be calculated according to the aperture reduction ratemodel (Eq. (4)), and the evolutions of fracture aperture undereffective stresses of 5.0 and 10.0 MPa, respectively, are presentedin Fig. 3. It is shown that the fracture aperture attained a steadystate value after �6 years. Generally, the fracture closure behaviorin this study was in a similar way as the experimental observationsin Polak et al. [26] and Yasuhara et al. [24]. However, exactcomparison between our aperture reduction rate model with otherexperimental results is beyond the scope of this study. The transportequations can be analytically solved when fracture closure reaches asteady state, which is also beyond the interests of the present paperand has been studied by many previous studies e.g. [30].

On the other hand, It is to be noted that molecular diffusioncoefficient (D*) in water is mostly between 1�10�9 and

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Con

cent

ratio

n c/

c 0

Distance along fracture axis (m)

FDM Analytical solution

Fig. 2. Comparison between the developed FDM code and analytical solution interms of the concentration distribution in a fracture of constant aperture.

Table 2Material parameters.

Material parameters Value

Molar volume of quartz Vm (m3 mol�1) 2.27�10�5

Dissolution rate constant k (mol m2 s�1) 2.51�10�9

Effective stress sef f (MPa) 5.0; 10.0Critical stress sc (MPa) 73.2Initial contact ratio Rc0 0.01Gas constant R (J K�1 mol�1) 8.31Temperature T (K) 423.15 (150 1C)Initial aperture b0 (mm) 30Dynamic viscosity μ (Pa s) 0.001Hydraulic pressure gradient dP=dx (Pa m�1) 10Effective diffusion coefficient De (m2 s�1) 1.0�10�12

Apparent diffusion coefficient Da (m2 s�1) 1.0�10�10

Empirical constant γ1, γ2 and γ3 10, 20, 0.05

Fig. 3. Fracture aperture closure with time due to pressure solution based on theaperture reduction rate model (Eq. (4)).

Z. Zhao et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 69–7572

Page 5: Solute transport in a single fracture with time-dependent aperture due to chemically medicated changes

2�10�9 m2 s�1 for small solutes and ions. A value of 1.6�10�9 m2 s�1 was chosen so the effective diffusion coefficient isabout 1.0�10�12 m2 s�1 for a rock matrix of porosity (θ) of 0.01,based on Archie0s law (De ¼Dnθ1:6, [34]). Consequently, an appar-ent diffusion coefficient (Da ¼De=θ) of 1.0�10�10 m2 s�1 can beset for the rock matrix. In this way, the time history of soluteconcentration distributions in fracture and rock matrix can beobtained by numerically solving Eqs. (4), (7) and (14). L¼10 m andLm¼5 m were chosen to avoid boundary effects.

Two cases were considered in the following analysis. For theconservative tracers (without matrix diffusion), the water resi-dence time for a fracture of a length of 2 mwere about 0.085 years,when no effective stress was applied (for a constant aperture of

30 mm). The water residence time increased to 0.088 and 0.092years under effective stress of 5 and 10 MPa, respectively. In otherwords, the water residence time under effective stress of 5 and10 MPa would increase by 4 and 8%, respectively, compared withthat for a fracture of constant aperture. However, for a muchlonger fracture of a length of 20 m, the water residence timewould increase from 0.85 years to 1.41 (under an effective stress of5 MPa) and 1.61 years (under an effective stress of 10 MPa). Thisindicates that fracture closure induced by pressure solution is arelatively slow process, and its influence on advection (waterresidence time) became more significant in a long fracture (orafter sufficient time).

When matrix diffusion was included, the solute migrationprocess became more complex. Figs. 4 and 5 show the soluteconcentration distributions in the fracture and the rock matrixwith time, respectively. As indicated in a previous study [30], thesolute concentration in both fracture and matrix increased withtime. The solute concentration in the fracture increased drasticallyduring the initial period (0–1 years), but the increase of soluteconcentration became gentle thereafter (after 1 years) (Fig. 6).Under an effective stress of 5 MPa, the concentration in thefracture still increased with time, but the concentration was muchsmaller than that in a constant aperture (Figs. 4 and 6). Similarly,the contaminant continuously penetrated to further distance inthe matrix during the process of fracture aperture closure. Underan effective stress of 10 MPa, this larger effective stress inducedfracture closure more quickly and significantly (Fig. 3). The con-taminant concentration exhibited a trend of first increase and thendecrease with time (Fig. 4). This trend also was indicated from thebreakthrough curve in the fracture (Fig. 6). The penetration ofcontaminant along the fracture became negligible after about2 years under an effective stress of 10 MPa, but the contaminantdiffused into the matrix during the whole process, but more slowlythan the diffusion under effective stress of 5.0 MPa (Fig. 5).

Fig. 4. Concentration distributions in a fracture subjected to pressure dissolutionunder different effective stress (upper three lines: sef f ¼ 0 MPa; middle three lines:sef f ¼ 5 MPa; lower three lines: sef f ¼ 10 MPa) at different times.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

0

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

00.10.20.30.40.50.60.70.80.91

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

Dis

tanc

e in

to m

atrix

(m)

00.010.10.20.30.40.50.60.70.80.91

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance along fracture axis (m)

0

0.2

0.4

0.6

0.8

110 years 10 years 10 years

2 years 2 years 2 years

6 years6 years6 years

Fig. 5. Concentration distributions in the rock matrix with time under different effective stresses. (a) sef f ¼ 0 MPa (b) sef f ¼ 5 MPa (c) sef f ¼ 10 MPa.

Z. Zhao et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 69–75 73

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These simulations clearly demonstrated that, due to the frac-ture aperture closure under chemically mediated changes, theaverage fluid velocity became smaller with time. The rate ofsolute mass entering into the fracture void became slower, andthe concentration in the fracture increased slowly or evendecreased after some time (Fig. 6). However, the matrix diffusion(into the rock matrix) was active all the time and played animportant role in solute transport in single fractures [31], evenconsidering the effects of aperture decreasing with time. Consid-ering the 1% penetration length (cf =c0 ¼ 0:01) in the fracture, itremained about 0.93 m at time scales of 2, 6 and 10 years,respectively, under an effective stress of 10.0 MPa. This negligiblepenetration indicates that, over long time period, the solute migra-tion along the fracture may become less important, but matrixdiffusion is more significant.

Various effective stresses changed the fracture closure ratesignificantly through changing the dissolution rate (Fig. 3), and thecontaminant transport behavior was also affected as a result.Therefore, it is important that the time dependent fractureaperture reduction model and the involving parameters must becalibrated in practice.

4. Discussions and conclusion

According to a time-dependent fracture closure rate model [8],a FDM scheme was used to study the effects of chemicallymediated changes (e.g. pressure solution) on solute transport ina single fracture located in a porous rock matrix. For the casewhere fracture aperture continuously decreased during the pro-cess of pressure solution, the solute penetration into the fracturebecame slower or relatively negligible (depending on the aperturereduction rate), while the effects of matrix diffusion became moresignificant. This is mainly due to the decreasing fluid velocity andrate of solute mass entering into the fracture. For any spallingfractures (due to combined effects of uneven horizontal stressesand heating by the canister in a nuclear waste repository) in thegranite around a deposition hole subjected to swelling pressure ofbentonite, the possible fracture closure induced by pressuresolution can play a positive role in avoiding the spreading of theleaked nuclides.

This study is a basic attempt to investigate the feedback ofchemically mediated changes on solute migration, but it can beextended further to consider solute transport in single fractureswith time-dependent aperture under various conditions. For

example, during a glaciation period, the movement of ice sheets,including glacial build-up and retread, could not only change thehydraulic conditions, but also induce variations in rock porosity,hydraulic conductivity, pore pressure and fracture aperture evolu-tion by varying loading during the glaciation and deglaciationcycles [35]. The present method can therefore be used to evaluatethe underground nuclear waste repository performance for peri-ods of glaciation and deglaciation. In addition, our model can alsobe used to assess mineral migration processes in geothermalreservoirs or deep CO2 sequestrations.

Some outstanding issues requiring further investigation areaddressed below.

(1) Fracture aperture reduction rate modelIn the present model of fracture closure, pressure solution atasperity contacts was incorporated in the evaluation of frac-ture aperture closure. Under other circumstances, the free-facedissolution (or wormholing) of fracture walls may increase thefracture permeability that is not considered in this study. Thedissolved mass of mineral grains may also precipitate on thefree surfaces of fracture walls, which also induce a decrease offracture aperture. On the other hand the enhanced concentra-tion due to pressure solution is higher than that in the matrixpores where larger mature crystals are present. Matrix diffu-sion can then act as a sink and increase the rate of crystaldissolution. More accurate and generic models of fractureaperture change are expected to be developed in future,including more mechanisms and especially to account forlonger time effects. The present model is general and generic,and can be further developed to accommodate more complexmodels of time-dependent aperture as they become available.

(2) Transport modelIn the present study, only advection in a fracture and diffusionin rock matrix perpendicular to the fracture plane wereconsidered. Other transport mechanisms such as adsorptionon the fracture walls and dissolved grain particles may alsohave impact on solute transport behavior. It is expected thatthe decreasing area of fracture voids may significantly influ-ence the contaminant adsorption. As a result, a ‘real’ fracturegeometry model (rather than parallel plate model) should beapplied.

(3) Contact ratio of rock fractureThe contact area and contact ratio of rough rock fractures arecomplex mechanical and geometric issues that, however, maybe important for transport because of its influences on specificarea (affecting wall surface sorption) and asperity distribution(therefore pressure solution). This effect may be more impor-tant for crystalline rocks.

Acknowledgements

ZZ would like to acknowledge the financial support from theBolin Centre for Climate Research at Stockholm University.

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