Upscaling geochemical reaction rates using pore-scale ...cap/papers/Li Peters Celia AdWR 2006.pdf · Upscaling geochemical reaction rates using pore-scale ... network modeling as

Advances in Water Resources 29 (2006) 1351–1370

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Upscaling geochemical reaction rates using pore-scalenetwork modeling

Li Li, Catherine A. Peters *, Michael A. Celia

Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA

Received 19 May 2005; received in revised form 14 October 2005; accepted 20 October 2005Available online 6 December 2005

Abstract

Geochemical reaction rate laws are often measured using crushed minerals in well-mixed laboratory systems that are designed toeliminate mass transport limitations. Such rate laws are often used directly in reactive transport models to predict the reaction andtransport of chemical species in consolidated porous media found in subsurface environments. Due to the inherent heterogeneities ofporous media, such use of lab-measured rate laws may introduce errors, leading to a need to develop methods for upscaling reactionrates. In this work, we present a methodology for using pore-scale network modeling to investigate scaling effects in geochemicalreaction rates. The reactive transport processes are simulated at the pore scale, accounting for heterogeneities of both physicaland mineral properties. Mass balance principles are then used to calculate reaction rates at the continuum scale. To examine thescaling behavior of reaction kinetics, these continuum-scale rates from the network model are compared to the rates calculatedby directly using laboratory-measured reaction rate laws and ignoring pore-scale heterogeneities. In this work, this methodologyis demonstrated by upscaling anorthite and kaolinite reaction rates under simulation conditions relevant to geological CO2 seques-tration. Simulation results show that under conditions with CO2 present at high concentrations, pore-scale concentrations of reac-tive species and reaction rates vary spatially by orders of magnitude, and the scaling effect is significant. With a much smaller CO2

concentration, the scaling effect is relatively small. These results indicate that the increased acidity associated with geological seques-tration can generate conditions for which proper scaling tools are yet to be developed. This work demonstrates the use of pore-scalenetwork modeling as a valuable research tool for examining upscaling of geochemical kinetics. The pore-scale model allows theeffects of pore-scale heterogeneities to be integrated into system behavior at multiple scales, thereby identifying important factorsthat contribute to the scaling effect.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Pore-scale network modeling; Reactive transport modeling; Upscaling; Geochemical kinetics; Geological CO2 sequestration

1. Introduction

Geochemical modeling of reactive transport of chem-ical species in subsurface systems has been widelyapplied to improve our understanding of the complexprocesses involved in fluid–rock interactions [1–3], andto solve environmental problems related to groundwaterand subsurface contamination [4–7]. Reactive transport

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* Corresponding author. Tel.: +1 609 2585645; fax: +1 609 2582799.E-mail address: [email protected] (C.A. Peters).

modeling is generally classified into forward modelingand inverse modeling [8]. Forward modeling predictsthe transport and reaction of chemical species givenspecified geochemical reactions and reaction rate param-eters. Inverse modeling utilizes measured chemical com-positions to infer geochemical reactions that haveoccurred along flow paths. These two types of modelingapproaches serve as powerful tools to understand, pre-dict, and interpret reactive transport of chemical speciesin subsurface environments, and provide insights intopossible regulation and management strategies.

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1352 L. Li et al. / Advances in Water Resources 29 (2006) 1351–1370

Because reactive transport modeling is typicallyapplied at large scales, it necessarily ignores spatial het-erogeneities at scales smaller than the size of model dis-cretization. For mineral dissolution and precipitation,there is a mismatch between the scale at which rate dataare needed (the scale of model discretization) and thescale for which rate data are available. Mineral dissolu-tion and precipitation is typically modeled using geo-chemical reaction kinetics measured in laboratorysystems that are very different from natural systems. Inthe laboratory, geochemical reaction rates are typicallymeasured in well-mixed systems using crushed mineralsuspensions [9]. These experiments are intentionallydesigned to homogenize mineral samples and to elimi-nate aqueous-phase mass transport limitations. Thisway the aqueous concentrations reflect reaction progressand can be used to determine the forms and parametersof reaction rate laws. Such rate laws accurately deter-mine the so-called surface-controlled reaction kinetics,but they provide limited information about reactionrates in consolidated porous media.

Natural porous media, such as sedimentary rocks andother geological media encountered in subsurface for-mations, are inherently heterogeneous [10]. They aremixtures of solid matrix and void space, with the latterconsisting of interconnected pores of different shapes,volumes, and surface areas. Furthermore, the mineralcomposition of surfaces may vary from pore to pore.These pore-to-pore heterogeneities, hereafter referredto as pore-scale heterogeneities, can produce variabili-ties in the transport and reaction processes that takeplace within a porous medium, and can result in spatialvariations in aqueous concentrations and reaction rates.In such heterogeneous systems, the laboratory-measuredgeochemical reaction kinetics may not be applicable atthe scale of intact porous media. Consequently, spatialheterogeneities at scales smaller than the scale of discret-ization may be important in determining mineral disso-lution and precipitation in reactive transport modeling.

Such scaling effects have been termed ‘‘non-classical’’behavior as they cannot be predicted by existing theoriesinferred from laboratory systems [11]. They may beamong the potential causes of the oft-cited and dis-cussed field-lab discrepancy, namely, the observedorder-of-magnitude differences between lab-measuredreaction rates and those derived from field measure-ments [12–16]. In particular, the scaling effect of geo-chemical reaction kinetics has been observed in severalstudies. For example, Szecsody et al. [17] observed thatfor reactions of CoII/IIIEDTA with Fe oxide-containingsediments, the reaction parameters measured in batchexperiments could not be applied directly to describethe reactive transport processes in column experiments,due to the pore-scale heterogeneity in adsorption sites.They found that stochastic modeling efforts that incor-porated pore-scale heterogeneity did reproduce the

observations. Lichtner and Tartakovsky [18] used vol-ume-averaging methods to upscale the reaction rate con-stants for quartz dissolution, incorporating theheterogeneity in mineral grain size. They found that,contrary to the usual modeling assumptions, theupscaled effective reaction rate constants varied as afunction of time, which is attributed to heterogeneousdistribution of grain sizes.

In this work, we present a methodology to exploreupscaling of geochemical reaction kinetics using pore-scale network modeling. A pore-scale network modelis one that discretizes the void space of a porous mediuminto a collection of pores within an interconnected net-work. Such models have been widely used to explore avariety of phenomena related to multiphase flow in por-ous media [19–21], including, for example, drying pro-cesses [22–25], moisture transport [26], and bubbletransport [27]. In recent years, pore-scale network mod-els have also been applied to examine problems concern-ing chemical and biological processes, such asdissolution of organic liquids [28–32], biomass growth[33–38], adsorption [39,40], and the formation of worm-holes in porous media [41]. However, this approach hasnever been used to address scaling questions associatedwith geochemical reaction kinetics. As such, this workpresents a novel application of pore-scale networkmodeling.

Within the pore network constructed for this work,the individual pores are considered to be homogeneoussystems. Reactive transport processes are simulated atthe pore scale, incorporating variations in chemicaland physical properties from pore to pore. Using massbalance principles, the continuum-scale reaction ratesare obtained from the network model and are comparedto continuum-scale rates calculated by ignoring pore-scale details and using laboratory-measured reactionkinetics directly. In addition, we use volume-averagingprinciples to quantify the errors that may occur whenfield measured concentrations are used in inverse model-ing. The differences between these continuum-scale ratesquantify the scaling effect of geochemical reactionkinetics.

This work focuses on the reaction kinetics of mineralsthat are considered to be important in the context ofgeological sequestration of CO2. In geological sequestra-tion, CO2 is injected into deep subsurface formationsand trapped by a variety of mechanisms [42–44]. Studieshave found that, for both geological sequestration andnatural long-term regulation of atmospheric CO2 levels,Ca- and Mg-rich feldspars provide the primary sinks forthe net consumption of CO2 [45–51]. In this work, wefocus on a Ca-rich feldspar, anorthite, as the model min-eral, with kaolinite as the coexisting reactive mineral.The reactions of interest are those that would occurunder the acidic conditions created when CO2 dissolvesin formation waters under high pressures. The reaction

L. Li et al. / Advances in Water Resources 29 (2006) 1351–1370 1353

system is an idealized case from the geochemical point ofview, but one which is illustrative in that it makes use ofthe typical nonlinear rates and equilibrium expressions.Application of the pore-scale network model is demon-strated by upscaling reaction kinetics of anorthite andkaolinite under simulation scenarios that are relevantto geological CO2 sequestration.

In the remainder of this paper, we first present thelaboratory-measured reaction rate laws for anorthiteand kaolinite. The methodology to simulate reactivetransport processes at the pore scale is then described,followed by construction of the network model. Afterthat we discuss the methodology to calculate the contin-uum-scale reaction rates. Two simulation scenarios rele-vant to CO2 sequestration are then described: oneinvolving high acidity and one involving moderate acid-ity. Finally, we present the simulation results, explainthe reasons for the observed scaling effects, and discusstheir implications for reactive transport modeling.

2. The reaction rate laws

2.1. Theory of reaction rate laws

Reaction rate laws are functions that relate mineraldissolution and precipitation rates to the concentrationsof aqueous species involved in the reactions (althoughsome rate laws have been described as functions of con-centrations of sorbed species [52,53]). Based on transi-tion state theory and the principle of detailedbalancing, reaction rate laws take the following generalnonlinear form [9]:

r ¼ kY

s

fsgns f ðDGÞ; ð1Þ

where r is the reaction rate (mol cm�2 s�1); k is the reac-tion rate constant; s represents a chemical species thathas catalytic or prohibitory effects on the reaction; {s}denotes the activity of species s, the product of the activ-ity coefficient and the aqueous concentration [s]; and ns

is a constant that describes the degree of rate depen-dence on {s}. The saturation state function f(DG) cap-tures the dependence on the activities of chemicalspecies that directly participate in the reaction.

For the expression of f(DG), the following practicalform is used in this work [9]:

f ðDGÞ ¼ 1� Xm; ð2Þwhere X is the saturation state, and m is a constant thatdescribes the dependence of rate on saturation state. Themagnitude of the saturation state X is an indicator ofhow far away the reaction is from equilibrium. Theoret-ically, it is equal to the ratio of the ionic activity prod-uct, IAP, to the equilibrium constant, Keq. As thevalue of X usually covers orders of magnitude, the satu-

ration index, SI = logX, is often used to quantify themagnitude of the saturation state. If X < 1, thenSI < 0, and the net reaction is dissolution; otherwise,the net reaction is precipitation. If X = 1 and SI = 0,the reaction is at equilibrium.

Mineral dissolution and precipitation reactions aregenerally driven by several different parallel mecha-nisms, with each catalyzed by a different chemical spe-cies, for instance, H+, H2O, and OH�. The rate lawfor each mechanism takes the same general form as inEq. (1):

rH ¼ kHfHþgnHð1� XmÞ; ð3ÞrH2O ¼ kH2Oð1� XmÞ; ð4ÞrOH ¼ kOHfOH�gnOHð1� XmÞ. ð5Þ

The total reaction rate is the sum of these three:

r ¼ rH þ rH2O þ rOH. ð6ÞThe value of rH dominates under acidic conditions, andrOH dominates under alkaline conditions. There may ormay not be neutral conditions where the rate is domi-nated by rH2O, depending on the relative magnitude ofthese rates. The full reaction rate law requires the fol-lowing kinetic parameters: the reaction rate constants,kH, kH2O and kOH; the degrees of rate dependence onpH, nH and nOH; and the degree of dependence on satu-ration state, m.

2.2. Anorthite and kaolinite reactions

In this section, we review the literature on anorthiteand kaolinite reaction rates and establish the rate lawsused in the present work. The anorthite reaction is asfollows:

CaAl2Si2O8(s) + 8Hþ�Ca2þ+ 2Al3þ+ 2H4SiO4.

It is well known that under conditions far from equilib-rium (i.e., X� 1), the anorthite dissolution rate dependson proton activity in acidic conditions [54]. Examinationof published experimental data of anorthite dissolutionkinetics suggests that the degree of dependence, nH, var-ies from 1 to 3 [55–57]. In this work, we use kineticparameters measured by Oelkers and Schott [57], whoobserved that at 25 �C the reaction order of anorthitedissolution with respect to {H+} is 1.5 under acidic con-ditions. For neutral conditions, we use the reported va-lue for kH2O of 10�15.6 mol cm�2 s�1 [55]; this value isnearly identical to that reported by Helgeson et al.[58]. For alkaline conditions, we use data from Bradyand Walther [56], who observed that the exponentdescribing rate dependence on {OH�} is 0.33. As anor-thite dissolution is thermodynamically favorable in mostenvironments and almost never reaches equilibrium,there are no data on the form of its saturation statefunction. For completeness we include a saturation state

Table 1Reaction kinetic parameters at 25 �C

logk (mol cm�2 s�1) nH nOH m Ea (kJ mol�1)

logkH log kH2O logkOH

Anorthite �7.32a �15.6b �17.5c 1.5a 0.33c 1.0 �18.4a

Kaolinite �14.8d – �19.7e 0.4d 0.3e 0.9f �29.3d

a Data from Oelkers and Schott [57], based on measurements for pH 2.4–3.2.b Data from Amrhein and Suarez [55] and Helgeson et al. [58], based on measurements for pH 5–9.c Data from Brady and Walther [56], based on compiled data from various measurements for pH 7–12.d Data from Ganor et al. [59], based on measurements for pH 2–4.2.e Data from Carrol and Walther [60], based on measurements for pH 7–12.f Data from Nagy et al. [61], based on measurements for pH around 3.

0 2 4 6 8 10 12 14

–18

–16

–14

–12

–10

–8

pH

log

10 r

(m

ol c

m–2

s–1

)

Anorthite

Kaolinite

Fig. 1. Dissolution rates of anorthite and kaolinite as a function of pHunder conditions far from equilibrium using the parameters summa-rized in Table 1.


function, and use Eq. (2), assuming m = 1. Combiningall these parameters (summarized in Table 1), the anor-thite reaction rate law takes the following form:

rA ¼ ðkHfHþg1:5 þ kH2O þ kOHfOH�g0:33Þð1� XAÞ; ð7Þwith the expression for the saturation state for the anor-thite reaction as follows:

XA ¼fCa2þgfAl3þg2fH4SiO4g2

fHþg8Keq;A

; ð8Þ

where Keq,A is the equilibrium constant of anorthitereaction.

The kaolinite reaction is

Al2Si2O5(OH)4(s) + 6Hþ� 2Al3þ+ 2H4SiO4 + H2O.

Examination of kinetics for kaolinite dissolution farfrom equilibrium reveals that the kaolinite reaction ratedoes not have an obvious pH-independent region [60].Under acidic conditions, the kaolinite dissolution rateis reported to be proportional to {H+} raised to thepower of 0.35–0.40 [59,60]. In this work, a value of 0.4is used, based on data from Ganor et al. [59]. The liter-ature on the kaolinite dissolution rate under alkalineconditions suggests that the rate dependence on{OH�} is approximately 0.3 [60], which is the value usedin this work. Using the saturation state function fromNagy et al. [61] and assuming that the principle of de-tailed balancing applies, the overall reaction rate lawof kaolinite is as follows:

rK ¼ ðkHfHþg0:4 þ kOHfOH�g0:3Þð1� X0:9K Þ; ð9Þ

with the following expression for the saturation state forkaolinite reaction:

XK ¼fAl3þg2fH4SiO4g2

fHþg6Keq;K

; ð10Þ

where Keq,K is the kaolinite dissolution reaction equilib-rium constant. The values of all the kinetic parametersare listed in Table 1. The thermodynamic parametersare discussed in Section 5.

The dissolution rates of both minerals as a functionof pH at conditions far from equilibrium are shown in

Fig. 1. Of these two reactions, anorthite dissolutionhas the larger magnitude, leading to the fast release ofCa2+, Al3+, and H4SiO4. As a result, in a system wherethese two minerals coexist, the net reaction of kaoliniteis mainly precipitation, consuming the Al3+ and H4SiO4

released by anorthite dissolution.

3. Pore-scale network modeling

In pore-scale network modeling, the processes ofinterest are simulated in individual pores, and informa-tion at the pore scale is summed up and averagedover the network to obtain descriptions of processes atthe continuum scale. In this way a pore-scale networkmodel serves as a computational tool to upscale pro-cesses from the pore scale to the continuum scale. Thissection presents (i) the equations for reaction and trans-port at the pore scale, (ii) the construction of the porenetwork, including parameter specification for porousmedium properties, and (iii) the method for flow fieldcalculation.


3.1. Reactive transport processes at the pore scale

Within each pore, the mass of each species in theaqueous phase changes due to transport (advectionand diffusion) and reaction processes. The relevant reac-tions include aqueous reactions, which are considered tobe instantaneous, and mineral dissolution and precipita-tion reactions, which are described kinetically. Examina-tion of different aqueous reactions have shown thatthere are nine important instantaneous reactions,including the water dissociation reaction, and the speci-ation reactions of carbon-bearing species, silica-bearingspecies, and aluminum-bearing species. The two kineticreactions include the dissolution and precipitation ofanorthite and kaolinite. All reactions and their equilib-rium constants are listed in Table 2. In total 14 aqueousspecies are involved in these reactions: Ca2+; H+; OH�;the carbon-bearing species H2CO�3, HCO�3 , CO2�

3 ; thesilica-bearing species H4SiO4, H3SiO�4 , H2SiO2�

4 ; andthe aluminum-containing species Al3+, Al(OH)2+,AlðOHÞþ2 , Al(OH)3, AlðOHÞ�4 .

The aqueous concentrations are assumed to be spa-tially uniform on the scale of an individual pore. Thisassumption is justified by the calculation of a Damkoh-ler number, a dimensionless number which compares therelative rates of reaction and transport [62,1]. In thiscontext, the Damkohler number is defined as the ratioof the mineral surface reaction rate to the molecular dif-fusion rate in an average pore. The fastest reaction ofconcern is anorthite dissolution under the most acidiccondition. The reaction rate is calculated as the productof the area-normalized reaction rate and the averageanorthite surface area (3.0 · 10�2 cm2). At pH = 2.9the area-normalized anorthite dissolution rate usingEq. (7) is 10�11.6 mol cm�2 s�1 so the average anorthitedissolution rate is on the order of 10�13.3 mol s�1 perpore. The diffusion rate is estimated using the expressionD� Ac � DC

l , with D being the diffusion coefficient(approximately 10�4 cm�2 s�1 for hydrogen ion [9]), l

being the length of the pore (0.017 cm), Ac being the

Table 2Reactions of interest, and their equilibrium constants adjusted to current sim

Reactions

Kinetic reactions CaAl2Si2O8(s) + 8H+� C

Al2Si2O5(OH)4(s) + 6H+�

Instantaneous reactions H2O�H+ + OH�

H2CO�3�HCO�3 þHþ

HCO�3�CO2�3 þHþ

H4SiO4�H3SiO�4 þHþ

H3SiO�4�H2SiO2�4 þHþ

Al3+ + OH�� Al(OH)2+

AlðOHÞ2þ þOH��AlðOHAlðOHÞþ2 þOH��AlðOHAlðOHÞ3 þOH��AlðOH

a The method of adjustment is described in Section 5.

characteristic cross-sectional area for diffusion in a pore,conservatively estimated as l2, and DC (mol cm�3) beingthe difference in concentrations at the center of the poreand at locations close to the mineral surface. For hydro-gen ion, DC is approximately 10�5.9 mol cm�3 at pH of2.9, which leads to the diffusion rate of 10�11.6 mol s�1.This rate is approximately 1.7 orders of magnitude lar-ger than the reaction rate. This relatively fast diffusionrate, combined with the much faster advection rate, issufficient to eliminate the concentration gradient withinthe fluid in individual pores.

The mass balance equations describing the masschange of chemical species within individual pores weredeveloped following Lichtner�s systematic formulationfor multicomponent systems that involve both kineti-cally controlled and instantaneous reactions [63,64].We chose to develop the mass balance equations for fivekey components: Ca2+, CT, SiT, AlT, and HT, with CT

collectively representing all carbon-bearing species, SiTcollectively representing all silica-bearing species, AlTcollectively representing all aluminum-bearing species,and HT representing the total aqueous proton. The con-centration of HT is computed as ½HT� ¼ �½AlðOHÞ2þ��2½AlðOHÞþ2 � � 3½AlðOHÞ3� � 4½AlðOHÞ�4 � þ ½H4SiO4��½H2SiO2�

4 � þ ½H2CO�3� � ½CO2�3 � þ ½Hþ� � ½OH��. For this

system, five mass balance equations, together with thenine mass action laws corresponding to the nine instan-taneous reactions, are sufficient to describe the masschange in the aqueous phase.

The mass balance equation (expressed in molar units)for pore i takes the following general form:

V id½��idt¼XQij>0

Qij½��j þXQij<0

Qij½��i

þXnc

j¼1

D�ij;�aij

ð½��j � ½��iÞl

þ Si;�; ð11Þ

where [Æ] represents the molar concentration of one ofthe five key components; Vi is the volume of pore i; j

is the index for a pore that is connected to pore i; nc

ulation conditions (50 �C and 107 Pa)

logKeqa

a2+ + 2Al3+ + 2H4SiO4 21.72Al3+ + 2H4SiO4 + H2O 3.80

�13.2�6.15�10.0�9.2�12.4

8.76Þþ2 18.9Þ3 27.3Þ�4 33.2


is the total number of neighbor pores; Qij symbolizes theflow rate from pore j to i; D�ij;� is the effective diffusioncoefficient; aij is the cross-sectional area between poresi and j; l is the distance between the centroids of adjacentpores and is a constant for the regular-latticed network;and Si,Æ denotes the mass change rate due to kinetic reac-tions. On the right-hand side of the above equations, thefirst two terms take into account advective mass inflowand outflow rates, and the third term is the mass changerate due to diffusion between adjacent pores.

For each of the four components that is a collectiverepresentation of several species, its ‘‘effective’’ diffusioncoefficient D�ij;� is derived by writing the reactive trans-port equations for each constitutive species. Summationof these transport equations leads to a derived effectivediffusion coefficient that is essentially a weighted averageof the diffusion coefficients of involved species weightedby their concentration differences between pore i andpore j. Such a formulation is consistent with what isdone in reactive transport modeling [64]. For example,for CT,

D�ij;CT¼

DH2CO�3ð½H2CO�3�j � ½H2CO�3�iÞ þ DHCO�3

ð½HCO�3 �j � ½HCO�3 �iÞ þ DCO2�3ð½CO2�

3 �j � ½CO2�3 �iÞ

ð½H2CO�3�j � ½H2CO�3�iÞ þ ð½HCO�3 �j � ½HCO�3 �iÞ þ ð½CO2�3 �j � ½CO2�

3 �iÞ. ð12Þ

The diffusion coefficients for individual species aretaken from the literature [65,9]. Note that this formula-tion of mass balance equations leads to continuum-scaledispersion effects by taking into account fluctuations inlocal flow velocities. Hence, dispersion is not modeledexplicitly and the diffusion process is restricted topore-to-pore diffusion within the network.

The last term in Eq. (11), Si,Æ, represents the masschange rate due to kinetic reactions. It takes differentforms for different components, depending on the reac-tions involved (expressed in molar units):

Si;Ca2þ ¼ rA;iAA;i; ð13Þ

Si;CT¼ 0; ð14Þ

Si;SiT ¼ 2rA;iAA;i þ 2rK;iAK;i; ð15Þ

Si;AlT ¼ 2rA;iAA;i þ 2rK;iAK;i; ð16Þ

Si;HT¼ �6rA;iAA;i � 4rK;iAK;i; ð17Þ

where AA;i and AK;i are, respectively, the surface areasof anorthite and kaolinite in pore i. The two reactionrates, rA,i and rK,i, are calculated using the respectivereaction rate laws (Eqs. (7) and (9)) and the local con-centrations in pore i. The mass change rates due to ki-netic reactions depend on the amount of the reactivemineral in each pore. In this pore network model, twotypes of pores are identified with regard to the mineral-ogy of the solid phases: reactive pores, with their solid

phases containing both anorthite and kaolinite, andnon-reactive pores, with their solid phases being quartz.For non-reactive pores, as the reactive minerals are notpresent, the surface areas of anorthite and kaolinite arezero and the mass change rates due to reaction are equalto zero.

Such a representation of reactive pores is not a prob-lem for modeling dissolution, but could pose a problemfor modeling precipitation. Precipitation normally pro-ceeds in two steps: nucleation and crystal growth[9,66]. For many minerals, nucleation is the rate-limitingstep [43], because it has to overcome a surface freeenergy barrier and requires some degree of supersatura-tion. In the absence of a solid mineral phase, precipita-tion of that mineral requires both steps, while in thepresence of a solid mineral phase, precipitation of thatmineral can start as crystal growth. The large differencein these rates is the basis for our modeling assumptionthat precipitation can proceed only in so-called reactivepores.

3.2. Network construction

A 3D regular-lattice network model was constructedto represent a consolidated sandstone with reactive min-erals occupying approximately 10% of the solid phase,which is within the typical range of the percentage offeldspars and clay minerals in sandstones. The porosityof the network is 0.14, and the permeability is7.9 · 10�10 cm2, which falls in the typical range of con-solidated sandstone properties [67,68]. The size of thenetwork is 5.0 · 1.7 · 5.0 mm3, and the network con-tains 9000 pores (30 · 10 · 30 in x-, y-, z-directions,respectively). The bulk flow direction is in the z-direc-tion. This network has several characteristics that distin-guish it from network models that have beenconstructed for multiphase flow. First, pores do nothave prescribed geometries; they are defined only interms of surface area and volume. Second, pore throatsare not represented explicitly except to assign an effec-tive resistance to the flow and an effective cross-sectionalarea for diffusion. All void volume is contained in porebodies.

Solution of the pore-scale mass balance equationsrequires the following values for each pore i: volume;conductances at pore junctions (for the calculation offlow rates); effective cross-sectional areas at connectionsof adjacent pores (for the calculation of diffusion terms);and surface areas. These parameters characterize eachpore, while across the network they form statistical dis-


tributions. The parameters for each pore are randomlyassigned from their respective statistical distributions,as discussed in this section.

3.2.1. Pore volume

Based on literature data for a Fontainebleau sand-stone [69], pore volume is assumed to have a log-normaldistribution. The mean for the log-transformed porevolume (mm3) is �3.42, with the standard deviation ofthe log-transformed pore volume being 0.51. This corre-sponds to an arithmetic mean of pore volume being6.6 · 10�4 mm3.

3.2.2. Pore hydraulic conductanceFor the distribution of pore conductances, we have

made two assumptions: (1) the distribution follows alog-normal distribution; and (2) the conductancebetween pores is correlated to the sum of the logarithmof their pore volumes. Regarding the first assumption,although data on distributions of pore-scale conduc-tances generally do not exist, in the literature there isindirect evidence supporting the idea that the distribu-tion of the conductances should be related to the distri-bution of grain or pore size, which is usually log-normally distributed [70]. The second assumption isbased on observations for sandstones from experimentalstudies [71,72].

We define Cij (cm4 g Pa�1) as the conductancebetween two adjacent pores i and j. The log-transformedconductance Xij = lnCij has a normal distribution, withmean lX, and variance r2

X . The summation of the log-transformed volumes of adjacent pores, Yij = ln-Vi + lnVj, also takes a normal distribution, with meanlY = 2llnV and variance r2

Y ¼ 2r2ln V . Assuming Xij and

Yij are jointly Gaussian distributed, the mean and vari-ance of the log-transformed conductance for a pair ofconnected pores i and j, (lX)ij and ðr2

X Þij, are calculatedas follows [73]:

ðlX Þij ¼ lX þ qrX

rYðY ij � lY Þ; ð18Þ

ðr2X Þij ¼ r2

X ð1:0� q2Þ; ð19Þ

where q is the correlation coefficient. For the connectionbetween pores i and j, the conductance is sampled fromthe log-normal distribution with mean (lX)ij and vari-ance ðr2

X Þij, which depend on the log-volume summationYij. In this work, the value of lX and r2

X were chosen tobe �6.1 and 1.0 to be consistent with the permeabilityvalue (7.9 · 10�10 cm2), as described in the next section.The value of q is assumed to be 0.9 as we assume strongcorrelation between conductance and adjacent porevolumes.

3.2.3. Pore-to-pore cross-sectional area

The cross-sectional area between connected pores i

and j, aij, is required for the calculation of the mass

transport due to diffusion. Considering an ‘‘effective’’cylinder between pore i and j, with conductance Cij,and length l, the diameter of the cylinder dij can be cal-culated by rearranging Poiseuille�s law:

dij ¼128ml� Cij

p

� �14

; ð20Þ

with m being the viscosity of the flowing fluid. Corre-spondingly, the ‘‘effective’’ cross-sectional area aij ¼14pd2

ij. The arithmetic average of the cross-sectional areafor this network is about 1.02 · 10�4 mm2.

3.2.4. Surface area

Because there is no reliable literature data on thepore-scale distribution of surface area for consolidatedsandstone, we use the following procedure. The surfacearea of pore i is randomly sampled from an exponentialdistribution, with the restriction of not being less thanthe minimum surface area, which is calculated fromthe pore volume assuming a spherical pore. The valuesof surface areas of consolidated sandstones measuredby Brunauer–Emmet–Teller (BET) method, oftenreported as the surface-to-volume ratio, cover threeorders of magnitude, ranging from 103 to 105 cm�1

[74–76]. In this work, the mean of the pore surfacearea was chosen to be 6.0 mm2, which results in a sur-face-to-volume ratio for the entire network of 1.3 ·104 cm�1.

As described earlier, the solid surface for an individ-ual pore is modeled as either containing reactive miner-als or not. Within 9000 pores in total, 900 of themcontain reactive minerals, which is consistent with theassumption that 10% of the solid phase is reactive min-eral. We have chosen to randomly distribute the reactiveminerals in clusters of size 2 · 2 · 15 pores (in the x-, y-,z-directions, respectively), which is within the range ofthe spatial scale of feldspars distribution in sandstone[77]. In pores within the reactive clusters, 50% of the sur-face area is anorthite and 50% is kaolinite. The resultingdistributions of values of AA;i and AK;i for the networkinclude a large number of zeros for the non-reactivepores and a truncated exponential distribution for thesurface areas in reactive pores.

3.3. Determination of the steady-state flow field

To determine the steady-state flow field of the net-work, fixed fluid pressures are imposed at the bound-aries of the network in the z-direction, with no-flowboundary conditions applied in the other two direc-tions, resulting in macroscopically one-dimensionalflow. The flow rate between connected pores i and j

is proportional to the pressure difference through theconductance:

Qij ¼ CijðP i � P jÞ; ð21Þ


where Pi and Pj are fluid pressures in pore i and j,respectively. The inflows and outflows must balancefor each pore i:

Xnc

j¼1

Qij ¼ 0. ð22Þ

Combination of Eqs. (21) and (22) results in a linearalgebraic system for the unknown pressure field. Giventhe boundary pressures, this system is solved for thepressure in each pore, from which the steady-state flowfield is calculated.

At the continuum scale, the total flow rate followsDarcy�s law:

QT ¼jmK

DPL; ð23Þ

where QT (cm3 s�1) denotes the total flow rate throughthe network in the main flow direction, z; j is the intrin-sic permeability of the network (cm2); K represents thecross-sectional area of the network in the directionorthogonal to the main flow (cm2); DP symbolizes thepressure difference between the two boundaries in z-direction; and L is the length of the network in z-direc-tion (cm). By rearrangement of the above equation, thepermeability of the network can be calculated. The per-meability is thus a function of the mean and variance ofthe conductance field, lX and r2

X , and the correlationwith local volumes. As mentioned previously, theseparameters were chosen so that the permeability corre-sponds to a typical value for sandstone. An iterative ap-proach was used whereby the statistical parameters werevaried until a flow field was generated that resulted inthe desired permeability.

4. Continuum-scale reaction rates

4.1. The rates from the network model: RN

For a porous medium, the ‘‘true’’ reaction rate at thecontinuum scale is the total mass change rate due toreactions at every reactive surface, normalized by thetotal amount of surface area of reactive minerals. Thenetwork model allows us to estimate such a ‘‘true’’ reac-tion rate at the continuum scale. The pore-scale massbalance equations are used to simulate changes in aque-ous concentration over time in each pore. At each timestep, the sum of the mass change rates due to mineralreactions for the entire network, divided by the total sur-face area of reactive minerals, gives the ‘‘true’’ reactionrate for the network. For example, for anorthite, theeffective continuum-scale reaction rate for the networkmodel is as follows:

RN;A ¼Pn

i¼1AA;irA;iPni¼1AA;i

; ð24Þ

where n is the total number of pores. Expressed inmol cm�2 s�1, this rate is equivalent to the rate calcu-lated by summing the mass change rate of Ca2+ due toanorthite reaction across the network, and then dividingby the total anorthite surface area. An equation analo-gous to Eq. (24) is used to calculate the continuum-scalerate for kaolinite. The continuum-scale rates, RN,A andRN,K, depend on local reaction rates (rA,i and rK,i,respectively), as well as the coupling between the localrates and local surface area of reactive minerals. Fornon-reactive pores, as values of AA;i are zero, the termAA;irA;i is not counted in the calculation of RN. As such,concentrations in non-reactive pores are irrelevant to thereaction rates, although they enter indirectly via thetransport equations. The continuum-scale reaction ratesare dependent only on the concentrations in reactivepores.

4.2. The rates from the continuum model: RC

This section presents the methodology for calculat-ing continuum-scale reaction rates using a continuummodel, which is the fundamental building block of tradi-tional forward modeling. Forward modeling generallyemploys a continuum approach [63,64,78], in which por-ous media are characterized by spatially-averaged prop-erties at the scale of the grid block, thereby ignoringdetails at the sub-grid scale. It is common practice toapply reaction rate laws determined from well-mixedlaboratory systems to calculate the reaction rates forthe porous medium in an entire grid block. Further-more,the mathematical forms and parameters of theserate laws are assumed to hold at the grid block scaledespite possible nonlinearities in upscaling.

In a similar manner, the continuum model uses spa-tially-averaged values to represent system properties,assumes uniform concentrations in the aqueous phase,and applies lab-measured reaction rates directly at thecontinuum scale. In so doing, it ignores pore-scale heter-ogeneities a priori, and assumes no mass transport lim-itation from pore to pore. In the calculation thatfollows, the continuum model is constrained to havethe same hydrodynamic properties, total surface areaof reactive minerals, initial conditions and boundaryconditions as the network model.

Because the reactions and the species involved arethe same as in the network model, we follow thesame systematic formulation to develop mass balanceequations for the five components: Ca2+, CT, SiT, AlT,and HT. For each key component, the mass balanceequations take the following general form, appliedover the entire volume of the pore network definedearlier:

V T

d½��dt¼ QT½��in � QT½��C þ SC;�; ð25Þ


where VT is the total pore volume across the network,[Æ]in corresponds to the concentration at the inflowboundary, and [Æ]C is the uniform concentration in thecontinuum model. On the right side of the equation,the first two terms account for the mass change ratesdue to advection, and the last term, SC,Æ, represents themass change rate due to reaction, which takes differentforms for different components:

SC;Ca2þ ¼ RC;AAA;T; ð26ÞSC;CT

¼ 0; ð27ÞSC;SiT ¼ 2RC;AAA;T þ 2RC;KAK;T; ð28ÞSC;AlT ¼ 2RC;AAA;T þ 2RC;KAK;T; ð29ÞSC;HT

¼ �6RC;AAA;T � 4RC;KAK;T; ð30Þ

where RC,A and RC,K are the reaction rates of anorthiteand kaolinite, respectively, and AA;T and AK;T are thetotal surface areas of anorthite and kaolinite. Thesemass balance equations look similar to those writtenfor an individual pore in the network model, except thatat the continuum scale there is no diffusion term, and theparameters characterizing pore i are replaced by theparameters that describe the continuum-scale proper-ties. The reaction rates from the continuum model,RC,A and RC,K, are calculated using the uniform concen-trations (converted to activities), and by applying thereaction rate laws directly. For anorthite,

RC;A ¼ ðkHfHþg1:5C þ kH2O þ kOHfOH�g0:33

C Þð1� XC;AÞ;ð31Þ

where XC,A is calculated using uniform concentrationsas well. For kaolinite, the rate expression is similar, fol-lowing the reaction rate law shown in Eqs. (9) and (10).These rates represent the rates calculated in one gridblock in forward modeling, where the grid block is takento correspond to the entire network used in the networkmodel.

4.3. The rates from volume-averaged

concentrations: R0N

This section presents the methodology for calculatingcontinuum-scale rates, R0N, using volume-averaged con-centrations in reaction rate laws. The volume-averagedconcentrations are analogous to those measured infield sampling, where groundwater is pumped from awell, and water from a large number of pores is mixed.Concentrations measured in the sample represent spa-tially-averaged values. A typical application in inversemodeling is to use these measured concentrations in con-junction with laboratory-measured reaction rate laws orequilibrium relationships to infer which minerals are dis-solving, and which are precipitating [79].

The purpose of this calculation is not to develop amethod of volume-averaging, but to use the network

model to calculate the rates that resemble those esti-mated from field sampling. This allows us to examinethe extent to which volume-averaged concentrations,½��, provide accurate predictions of continuum-scalereaction rates. From the network model simulationresults, we compute the volume-averaged concentra-tions, such as that of hydrogen ion, as follows:

½Hþ� ¼Pn

i¼1V i½Hþ�iPni¼1V i

. ð32Þ

Since activity coefficients are constant throughout thenetwork (see Section 5), the value of the volume-aver-aged activity, fHþg, is the product of the activity coeffi-cient and the volume-averaged concentration. Then, forthe anorthite reaction, we use the averaged activities tocompute

R0N;A ¼ ðkHfHþg1:5þ kH2O þ kOHfOH�g0:33Þð1� X0N;AÞ;

ð33Þwhere

X0N;A ¼fCa2þgfAl3þg

2

fH4SiO4g2

fHþg8Keq;A

. ð34Þ

A similar rate law applies for kaolinite. In so doing, therate calculated from volume-averaged concentrationsignores pore-scale heterogeneities a posteriori. Thesereaction rates represent the rates that would be inferredfrom the measured concentrations via inverse modeling.

4.4. Comparison between continuum-scale reactionrates

We have defined three continuum-scale reactionrates: (1) RN, derived from explicit mass balance calcu-lation from the network model; (2) RC, derived fromuniform concentrations in the continuum model; and(3) R0N, derived from averaged concentrations from thenetwork model. Of these three upscaled reaction rates,we take RN to represent the ‘‘true’’ reaction rate, as itmost closely takes into account the effects of pore-scaleheterogeneities, and is consistent with the mass balanceof the network system. Both RC and R0N are based onspatially-averaged properties (either a priori or a poste-riori) and on reaction rate laws that were determined inthe laboratory. Comparisons of RC and R0N to RN pro-vide insights into how pore-scale heterogeneity influ-ences continuum-scale reaction rates, and how well theuse of average properties represents these heterogeneoussystems.

For the comparison of RC to RN, we define the ratio:

g � RC

RN

. ð35Þ

If g is close to unity, the differences between these tworates are not significant, which means that the effects


of pore-scale heterogeneities are not significant, and thatthe scaling effect is negligible. In such cases, the applica-tion of the lab-measured reaction kinetics at the con-tinuum scale is appropriate in forward modeling.Otherwise, pore-scale heterogeneities are importantand conventional forward modeling using continuummodeling may introduce significant errors.

For the comparison of R0N and RN, we define theratio:

b � R0NRN

. ð36Þ

If b is close to unity, these two rates are similar, whichmeans that volume-averaged concentrations are suffi-cient to describe the reaction rates and the scaling effectis negligible. Otherwise, it is not appropriate to calculatethe reaction rates using volume-averaged concentra-tions, such as those determined from field samplingand used in inverse modeling.

The deviation of g or b values from unity provides ameasure of the scaling effect of the reaction kinetics. Ifeither RC or R0N has a different sign from RN, then RC

or R0N predicts a different reaction direction as comparedto RN; that is, one is predicting dissolution while theother is predicting precipitation, or other wise.

5. Simulation conditions and numerical computation

5.1. Simulation conditions and parameter estimation

The application of pore-scale network modeling toupscale geochemical reaction rates is illustrated withtwo simulation scenarios relevant to geological CO2

sequestration. One scenario is that in which the networksystem is near the CO2 injection site so that the bound-ary aqueous phase is in equilibrium with CO2 of a typ-ical injection pressure, 107 Pa, and is highly acidic(referred to hereafter as case A). The other scenario isthat in which the network system is relatively far awayfrom the injection site so that the boundary aqueousphase has a much lower CO2 concentration. Here, wehave chosen to represent this as an aqueous phase inequilibrium with CO2 of a relatively low pressure,105 Pa (referred to hereafter as case B). The differencein CO2 concentrations generates different acidities inthe aqueous phases that flow into the network system.Except for this difference, the simulation conditionsare the same for the two cases.

In geological sequestration, CO2 would typically beinjected into subsurface formations at a depth of 1 kmor greater. To be consistent with this depth, the simula-tions were run at a temperature of 323 K and a pressureof 107 Pa. The brine is represented by saline water withtotal dissolved salinity (TDS = 0.45 mol L�1) that hastaken into account the contribution of all important

aqueous species in typical brine. Initially the brine isassumed to be in equilibrium with quartz and kaolinite,at pH 7.5. The electronic condition is satisfied withthe addition of Ca2+, which produces an initial concen-tration that is very small (7.9 · 10�6 mol L�1). Thereare no carbon-bearing species present in the brine. Thebrine at the inflowing boundary is assumed to have thesame [Ca2+] as in the initial aqueous phase withinthe network. The total carbon concentration and thusthe pH in the boundary brine is determined using thesemi-empirical relationship from Duan and Sun [80],based on equilibrium with separate phase carbon diox-ide at a given temperature, pressure, and salinity. In caseA, due to the large amount of CO2 dissolved in theboundary brine, the boundary pH is 2.9. In case B, theboundary pH is 4.1. We chose to run the simulationsfor a nominal value using the flow velocity, q = 5.8 ·10�3 cm s�1 [67].

The salinity is high enough that the activity coeffi-cients of the reactive species deviate considerably fromunity. These values were calculated using the Daviesequation [81]. Because the brine TDS is high, it isassumed that the dissolution and precipitation of reac-tive species does not change the TDS significantly, andthe activity coefficient of each species is considered con-stant both in time and space.

Equilibrium constants were taken from Morel andHering [81] and adjusted to the system temperatureand pressure using the software SUPCRT92 [82], aslisted in Table 2. The reaction rate constants wereadjusted to the given temperature using the Arrheniusequation [9], with the activation energies listed in Table1. The diffusion coefficients were obtained and adjustedto the system temperature and salinity [9,65]. The effectsof pressure on reaction rate constants and diffusion coef-ficients were considered insignificant, because the pres-sure effect on these parameters is relatively smallcompared to the effect of temperature [9].

Simulations were run within a limited time framesuch that the total amount of minerals reacted producedonly small porosity changes. Using empirical relation-ships between permeability and porosity [83–85], thepredicted porosity changes result in changes of perme-ability less than 5%. Therefore, the porosity and perme-ability of the porous medium are considered constantduring the simulation process. The system eventuallyreaches steady state, after which the concentrations ofthe aqueous species and the reaction rates do not varyas a function of time.

5.2. Numerical computations

For the network model, given initial and boundaryconditions, as well as the predetermined flow field, weuse the adaptive Runge–Kutta algorithm to simulta-neously solve the mass balance equations for the con-

5

10

15

20

25

30

x


centrations of the five components in each pore at eachtime step [86]. The values of these five concentrations,coupled with the nine laws of mass action correspondingto the nine instantaneous reactions, are then used to cal-culate concentrations of all 14 species iteratively using ahybrid algorithm combining both the Newton–Raphsonand bisection method. In this way we obtain the concen-trations of all 14 aqueous species in each pore at eachtime step. The concentrations of the aqueous species inthe continuum model are solved in the same way.

510

3020

100

yz

Fig. 2. Locations of reactive pores for the y = 5 plane and for thex = 18 plane. Reactive pores are shown in red color.

6. Results

In this section, we first present the results on the pore-scale heterogeneities of concentrations and reactionrates for case A, then briefly compare the pore-scale het-erogeneities in case A and case B. After that, the evolu-tion of the continuum-scale reaction rates are described,and the scaling effect is examined in the context ofcontinuum modeling and volume-averaging. Finally,the resulting concentrations in the network model areexamined with respect to network-scale equilibriumrelationships.

6.1. Pore-scale heterogeneities: case A

The locations of reactive pores for two orthogonalplanes in the network are shown in Fig. 2. For the sim-ulation conditions of case A, the steady-state concentra-tion fields of log[Ca2+] and pH for the y = 5 plane areshown in Fig. 3. In general, the Ca2+ concentrationsand pH values are lower in pores near the left boundarydue to the low calcium content and acidic condition ofthe inflowing water. As the inflowing water encounters

Fig. 3. For one layer of pores in the middle of the network (y = 5): steady-staleft (at z = 0).

reactive minerals, anorthite slowly dissolves and releasesCa2+. Correspondingly, there is an increase in pH due tothe increase in alkalinity. The concentrations of Ca2+

and the pH in reactive pores are generally higher thanthose in nearby non-reactive pores, as indicated by thesimilar patterns of high-concentration clusters and lowpH clusters (Fig. 3), to that of reactive pore clusters(Fig. 2). Non-reactive pores may also have elevatedCa2+ concentrations and pH if they are close to reactivepores. For example, near the inflow boundary where x isapproximately 17 or 18, the high [Ca2+] and pH are dueto the proximity of these pores to the reactive clusters inthe y = 3 and y = 4 layer.

The pore-scale concentrations vary considerably overthe network: the values of [Ca2+] cover 2.5 orders of mag-nitude, and hydrogen ion concentrations cover approxi-mately half an order of magnitude. The frequency

te profiles of log[Ca2+] (a) and pH (b). Boundary fluid flows in from the

−5 −4 −30

0.05

0.1

Fre

quen

cy a.

3 3.2 3.4 3.60

0.05

0.1b.0.15↑~~

−5 −4 −30

0.05

0.1

Fre

quen

cy c.

3 3.2 3.4 3.60

0.05

0.1d.0.15↑~~

−5 −4 −30

0.01

0.02

log [Ca 2+ ]i

Fre

quen

cy e.

3 3.2 3.4 3.60

0.01

0.02

pHi

f.

Fig. 4. Steady-state distributions of log[Ca2+] and pH in all pores (aand b); in non-reactive pores (c and d), and in reactive pores (e and f).The dot-dashed lines indicate the uniform concentrations from thecontinuum model. The dotted lines indicate the volume-averagedconcentrations.


distributions of log[Ca2+] and pH for pores in the entirenetwork are shown in Fig. 4a and b. Both distributionsare highly skewed. For Ca2+, approximately 9% of thepores have the minimum concentration values, which isequal to that of the inflowing boundary. Similarly,approximately 15% of the pores have the minimum valueof pH. Calcium concentration has a bimodal distribu-tion, in which a second peak arises at the large end ofthe concentration range.

For comparison, the distributions of log[Ca2+] andpH in non-reactive pores are shown in Fig. 4c and d.Because the non-reactive pores occupy 90% of the pores,their concentration distributions bear great similarity tothe distributions for all pores, except at the right end ofthe concentration range, where concentrations in reac-tive pores play an important role. The distributions ofconcentrations in reactive pores have very differentshapes from those for non-reactive pores, as shown inFig. 4e and f (Note the different scale of frequency axis).

−20 −15 −100

0.02

0.04

0.06

SIA,i

a.

Fre

qu

ency

Fig. 5. For anorthite: steady-state distributions of satura

They are less skewed, and have peaks that fall in thelarge ends of the concentration ranges, indicating therelatively high [Ca2+] and pH in reactive pores.

Also shown in Fig. 4a and b are the logarithm of vol-ume-averaged calcium concentration and the logarithmof volume-averaged hydrogen ion concentration (referredto hereafter as volume-averaged pH), as well as the loga-rithm of the uniform concentrations calculated from thecontinuum model. While both volume-averaged concen-trations lie in the mid-range of the distributions for theentire network, they fall in the tail region at the low endof the concentration range in reactive pores. For pH,more than 90% of the reactive pores have values largerthan the volume-averaged value. The uniform concentra-tions from the continuum model represent concentrationsin reactive pores better, as they are closer to the modes ofthe concentrations in reactive pores.

Due to the dependence of reaction rates on aqueousconcentrations, the pore-scale concentration heterogene-ities lead to spatial variations in saturation states, and inreaction rates. Fig. 5 shows the steady-state distribu-tions of saturation indexes (see Section 2.1) and reactionrates for anorthite. The saturation states cover approxi-mately 10 orders of magnitude. However, they are allorders of magnitude smaller than unity, and thereforeanorthite dissolution remains far from equilibrium inall pores. As such, anorthite dissolution rates are largelydetermined by inherent mineral reactivity and the activ-ities of catalytic species. That is, the value of 1 � XA inEq. (7) is approximately unity. In fact, at steady state, allthe pores in the network have acidic pH conditions suchthat the anorthite dissolution rate in each pore isapproximately:

rA;i ffi kHfHþg1:5i . ð37Þ

Thus, spatial variations in anorthite dissolution rates areentirely due to spatial variation in pH in reactive pores. Asis consistent with the exponent in Eq. (37), the variance inthe logarithm of rA,i is roughly twice that of pH (as ex-pected given that the square of 1.5 is 2.25). The hydrogenion concentrations span roughly 0.65 orders of magni-

−13 −12.5 −12 −11.50

0.02

0.04

0.06

log rA,i

b.

tion indexes (a) and logarithm of reaction rates (b).

–4 –2 0 2 40

0.02

0.04

0.06

0.08

0.1

SIK,i

Fre

qu

ency

Ω < 1 Ω > 1a.

–17 –17 –15 –13 –110

0.02

0.04

0.06

0.08

0.1b.

Dissolution Precipitation

–16log r

K,ilog (–r

K,i )

/ /

/ /

Fig. 6. For kaolinite reaction: steady-state distributions of saturation indexes (a) and logarithm of local reaction rates (b).


tude, and the anorthite dissolution rates cover approxi-mately one order of magnitude, as shown in Fig. 5b.

As shown in Fig. 6, for kaolinite, the saturationindexes include both positive and negative values, indi-cating that dissolution is occurring in some pores andprecipitation is occurring in others. As the values of sat-uration state are either in a similar order of magnitudeas unity, or orders of magnitude larger, the spatial var-iation of local rates depends not only on pH heterogene-ities, but also on concentration heterogeneities of Al3+

and H4SiO4, with the dependence on {H+} raised tothe power of approximately �5.0 (refer to Eqs. (9) and(10)). Such high dependence of rates on concentrationsresults in large spatial variations in kaolinite reactionrates, as illustrated in Fig. 6b. In about 6% of reactivepores, dissolution occurs and the rates span almosttwo orders of magnitude; in more than 90% of reactivepores, precipitation occurs, and the rates span approxi-mately 4.6 orders of magnitude. This is an interestingillustration of the effect of pore-scale heterogeneity.

6.2. Comparison of pore-scale heterogeneities

for case A and B

Case B is different from case A in that its boundaryfluid is much less acidic (with pH of 4.1, compared to

−14 −13.5 −13 −12.50

0.02

0.04

0.06

0.08

0.1

0.12

log rA,i

Fre

quen

cy

a.

Fig. 7. Steady state reaction rate distributions in case B: anorthi

2.9 in case A). Due to the sensitivity of anorthite disso-lution rate on hydrogen ion concentration, the reducedacidity results in smaller anorthite dissolution rates,and smaller differences in concentrations between reac-tive and non-reactive pores. As illustrated by Fig. 7a,the anorthite dissolution rates shift to smaller values,and cover a much narrower range, as compared to caseA (Fig. 5b). For kaolinite reaction (Fig. 7b), not only dothe rates cover a smaller range than in case A (Fig. 6b),but the reaction occurs only in the precipitation direc-tion. Such differences in pore-scale heterogeneitiesbetween the two cases are quantified by the differencesin the range of concentration and reaction rates in Table3.

6.3. Comparison between continuum-scale reaction rates

To illustrate the differences between the rates fromthe continuum model and the continuum-scale ratesfrom the network model, Fig. 8 shows RC and RN as afunction of time for the two reactive minerals in caseA. The anorthite dissolution rate increases with time,due to the gradual influx of highly acidic aqueous phasefrom the boundary. Eventually, the system reachessteady state, where the aqueous concentrations andreaction rates become constants. The release of Ca2+,

−16 −14 −120

0.02

0.04

0.06

0.08

0.1

0.12 b.

log (−rK,i

)

te dissolution rates (a) and kaolinite precipitation rates (b).

Table 3Comparison of pore-scale heterogeneities between the two simulationscenarios

[Ca2+] [H+] rA,i rK,i

Case A 2.6 0.65 1.1 4.6Case B 1.3 0.25 0.5 2.2

Values shown for each variable are the ranges in order of magnitude.


Al3+, and H4SiO4 from anorthite dissolution leads to theoversaturation of the aqueous phase with respect tokaolinite. As such, kaolinite precipitates, and theincrease in its precipitation rate parallels the increasein anorthite dissolution rate.

For each reactive mineral, the two models predictvery different evolutions of the reaction rates. The con-tinuum model overestimates the anorthite dissolutionrate for the duration of the simulation. At steady state,

0 100 200 300 4000

0.5

1

1.5x 10–12

time (s)

Rat

e (m

ol c

m–2

s–1

)

RC

RN

a.

Fig. 8. For case A: comparison of RC and RN ev

0 100 200 300 4000

0.5

1

1.5x 10–12

time (s)

Rat

e (m

ol c

m–2

s–1

)

R ′N

RN

a.

Fig. 9. For case A: comparison of R0N and RN ev

it overestimates the rate by 30%. For kaolinite, the con-tinuum model initially predicts its dissolution (although,these slightly positive values of RC are difficult to see inFig. 8b), which contradicts the prediction from the net-work model. Over time, the continuum model predictskaolinite precipitation, but with a much smaller rate.At steady state, it underestimates the kaolinite precipita-tion rate by almost two orders of magnitude (with theratio of RC over RN being 0.05). Overestimation of theanorthite dissolution rate and underestimation of kaoli-nite precipitation rate by the continuum model lead toan overestimation of the net dissolving rate of the solidphase by approximately a factor of 1.5 at steady state.

Fig. 9 shows a comparison of the time evolution ofthe rate calculated using volume-averaged concentra-tion, R0N, and the continuum-scale rate from the networkmodel, RN, for each mineral. The values of R0N;A under-estimate those of RN,A at the very start of the simula-

0 100 200 300 400–8

–7

–6

–5

–4

–3

–2

–1

0

1x 10–14

time (s)

b.

olution for anorthite (a) and kaolinite (b).

0 100 200 300 400–8

–7

–6

–5

–4

–3

–2

–1

0

1x 10–14

time (s)

b.

olution for anorthite (a) and kaolinite (b).


tion, and then significantly overestimate RN,A. At steadystate, the ratio of R0N;A over RN,A is 2.3. For kaolinite,R0N;K predicts its dissolution over the entire time period(although R0N;K is close to zero, it is actually positive),which is in contrast with the precipitation predicted bythe network model. At steady state the ratio of R0N;K overRN,K is �1.8 · 10�3. As such, values of R0N;K not onlypredict the wrong reaction rate, but also the wrong reac-tion direction.

Compared to those in case A, the differences betweencontinuum-scale rates are much smaller in case B. Table4 presents the differences between the continuum-scalerates for the two simulation scenarios, in terms of theratios between continuum-scale reaction rates. For caseB, these ratios are much closer to unity than in case A,which means that the scaling effect is much smaller. Forexample, the value of bK is positive in case B, indicatingthe correct prediction of reaction direction by using vol-ume-averaged concentrations. This smaller scaling effectis attributed to the less significant pore-scale variationsin concentrations and reaction rates.

6.4. Equilibrium relationships at the continuum scale

As a result of pore-scale concentration heterogene-ities, the equilibrium relationships for instantaneousreactions that are applicable at the pore scale may nothold at the continuum scale. For example, in aqueous

0 100 200 300 400–14

–13

–12

–11

–10

–9

time (s)

log

IAP

a.

Fig. 10. Evolution of logIAP for case A: (a) water dissociation reaction H2OThe dashed lines indicate the value of logKeq for each reaction.

Table 4Comparison of the ratios of continuum-scale reaction rates for the twosimulation scenarios

gA gK bA bK

Case A 1.3 0.046 2.3 �0.0018Case B 1.1 0.56 1.2 0.089

solutions under the temperature, pressure, and salinityconditions of these simulations, the product of the activ-ities of hydrogen ion and hydroxide ion equals the equi-librium constant 6.6 · 10�14. However, the ionic activityproduct (IAP) values calculated from volume-averagedactivities are not equal to their equilibrium constants.To illustrate this we show the temporal evolution oflogIAP in case A for the water dissociation reactionH2O� H+ + OH�, and Al3+ speciation reaction Al3þ þ4OH��AlðOHÞ�4 in Fig. 10. For both reactions, thedeviations of logIAP from logKeq are very large at anearly time period, where the IAP values are orders ofmagnitude different from the values of the equilibriumconstants. At steady state, the deviations reach constantvalues, IAP values 1.1 and 3.6 times their respective val-ues of equilibrium constants for water and aluminum. Incase B, the deviations of IAP values from equilibriumconstants are much smaller. For example, at steadystate, the IAP values are 1.0 and 1.2 times their res-pective values of equilibrium constants for water andaluminum.

7. Discussion

Simulation results have shown that under highlyacidic conditions relevant to geological CO2 sequestra-tion, heterogeneities in physical and mineral propertiescan lead to pore-scale heterogeneities in concentrationsand reaction rates, which results in scaling effects of reac-tion kinetics. With different boundary pH values, case Aand case B differ significantly in the extent of the scalingeffect. In both cases, the time scale of advection, calcu-lated as the ratio of the network length to average flowvelocity (L/q), is approximately 100 s. The diffusion timescale, calculated as the ratio of the square of the network

0 100 200 300 40022

24

26

28

30

32

34

36

time (s)

b.

= H+ + OH�; (b) Al3+ speciation reaction Al3þ þ 4OH� ¼ AlðOHÞ�4 .


length to the diffusion coefficient (L2/D), is about 103–104 s. The time scale of reaction, however, differs in caseA and B due to their different boundary pH values. Forpurpose of comparison, we estimate the time it wouldtake for reaction alone to effect a change in concentra-tion from the initial to the steady state condition. In caseA, the steady-state anorthite dissolution rate is in theorder of 10�10.6 mol s�1 (almost three orders of magni-tude larger than the average rate in individual poresdue to the much larger surface area of reactive mineralsat the network scale). Initially the anorthite dissolutionrate is orders of magnitude smaller because of the highinitial pH values. Therefore, on average, the anorthitedissolution rate is about 10�10.9 mol s�1. In case A, theconcentrations of calcium varied from the initial7.9 · 10�6 to 1.58 · 10�4 mol L�1 at steady state, whichcorresponds to a mass change of 8.9 · 10�10 mol. Assuch, the corresponding reaction time scale is 70 s,slightly smaller than the advection time scale. This meansthat the rates of advection and reaction are comparable,with none of them dominating. The comparable rates ofreaction and advection lead to large spatial variations inconcentrations, which results in relatively large scalingeffect. In case B, due to the much larger pH value, theaverage anorthite dissolution rate is in the order of10�12.7 mol s�1, corresponding to a reaction time scaleon the order of 103 s that is much larger than the advec-tion time scale. Compared to case A, the extent of spatialvariations in concentrations is much smaller, and thescaling effect is smaller.

With the same extent of spatial variations in concen-trations, spatial variations in reaction rates can vary fordifferent reactions, due to the difference in rate depen-dence on concentrations. In general, for a given rangeof spatial variation in concentration, the larger the ratedependence on concentration, the larger the spatial var-iation in local reaction rates. For the anorthite reaction,the very large range of calcium concentrations observedin case A has little effect on the spatial variation in rates,because in a system in which anorthite dissolution ishighly thermodynamically favorable and under acidicconditions, the anorthite dissolution rate is dependentonly on hydrogen ion concentration. Although the var-iation of hydrogen ion concentration in case A spansless than one order of magnitude, due to the nonlineardependence of the anorthite dissolution rate on thehydrogen ion activities, the values of rA,i cover morethan an order of magnitude. For case B, the nonlineardependence of reaction rate on {H+} also producesgreater variability in the reaction rate, but the effect ismuch less important than in case A. For kaolinite pre-cipitation, due to the high dependence of rates on hydro-gen ion activity (raised to the power of �5.0), the pHvariations lead to significant spatial variation in rates,spanning approximately five orders of magnitude incase A and two orders of magnitude in case B. Nonlin-

ear dependence on activities of chemical species, espe-cially on {H+}, is commonly observed for rate lawsfor mineral dissolution and precipitation reactions[9,87]. As such, we expect to see larger or smaller spatialvariation in local reaction rates than those in local con-centrations, depending on the mathematical form of therate laws.

The very large spatial variation in local concentra-tions has significant implications for the error producedby reaction rates estimated using volume-averaged con-centrations, as is analogous to what is done when inter-preting field measurements using inverse modeling. Theconsumption of hydrogen ions from anorthite dissolu-tion raises the pH such that the pH in reactive pores ishigher than the pH in the non-reactive pores. However,volume averaging is a non-discriminating calculationthat includes concentrations in both reactive and non-reactive pores. Because the reactive pores occupy only10% of the network, the volume-averaged pH is morerepresentative of pH in non-reactive pores, which under-estimates the pH in reactive pores. Because chemicalreaction rates depend only on concentrations in reactivepores, the concentrations in non-reactive pores play norole in driving or inhibiting the reaction rates in the por-ous medium (except indirectly via mass transport mech-anisms). Therefore, although volume averaging producesconcentrations that are true representations of the totalmass of a species per total fluid volume, these valuesdo not always serve as good indicators of the concentra-tions that drive the chemical reactions. Our work hasshown that when volume-averaged concentrations areused in the prediction of continuum-scale reaction rates,significant errors can result and the scaling effect is large,especially in case A where there exist significant differ-ences between concentrations in reactive and non-reac-tive pores. These errors led to the overestimation of theanorthite dissolution rate and the incorrect predictionof kaolinite reaction direction. In case B, with smallerdiscrepancies between concentrations in reactive andnon-reactive pores, the scaling effect is much smaller.

A discrepancy between the reaction rate from thecontinuum model and that from the network modelwas also observed, especially in case A. This differencewas less significant than the error produced from vol-ume-averaging. The difference between RC and RN

may appear to be small, however, this ‘‘small’’ differencein rates can lead to significant differences in predictionsof the dynamics of reactive transport processes for alonger time scale. Overestimation of mineral dissolutionrates by a factor of 2 will nominally lead to underestima-tion of reaction time scales by a factor of half. Further-more, in models that predict long-term reactionprogress, the overestimation of reaction rates will even-tually overestimate the rates of porosity change. Thestrong dependence of permeability on porosity, oftenraised to the power of 3, can magnify the effects of the


initial ‘‘small’’ difference in reaction rates to produce aconsiderable error in the predicted flow field. The over-estimated flow rates will further produce even largerreaction rates. Therefore, the error in rates can amplifyto a significant amount, and the continuum model mayultimately predict highly erroneous reactive transportdynamics relative to the true system.

The explanation for the discrepancy between RC andRN is slightly different than the explanation for the dif-ference between R0N and RN. Pore-scale heterogeneitiescreate concentration gradients that enhance the extentto which pore-to-pore mass transport of reactive speciesgoverns reaction kinetics. A continuum model, such as isused in forward modeling, is designed to ignore pore-scale heterogeneities, and mathematically representsthe entire system as if it is well mixed. At a given pointin time, the incoming fluid instantaneously comes incontact with all the reactive minerals in the system.Thus, the continuum model implicitly assumes no masstransport limitation at the pore scale. As such, it fails tocapture the extent of spatial variations in concentrationthat arises from the slow mass transport of reactive spe-cies from pore to pore, especially in case A. In case B,with relatively small spatial variations in concentrations,the scaling effect is much smaller.

The results presented here are consistent with theobservation of the lab-field discrepancy, that is, the directuse of reaction kinetics at scales larger than lab measure-ments often leads to the overestimation of reaction ratesunder natural conditions [12]. Although many factors,such as reaction duration, surface area and coating, havebeen considered as causes of such discrepancy [15], thiswork casts light on another potential factor: the spatialvariations in physical and mineral properties of naturalporous media. This work shows that spatial heterogene-ities in physical and chemical properties do affect large-scale reaction rates under natural conditions, and theextrapolation of lab-measured reaction kinetics may needto incorporate the effect of such spatial heterogeneities.

The fact that the equilibrium relationships that applyat the pore scale may not hold at the continuum scalehas potentially important implications for both forwardand inverse modeling. In forward modeling, equilibriumrelationships for instantaneous reactions are coupledwith reaction rate laws for kinetic reactions to describethe reactive processes. In inverse modeling, equilibriumrelationships are applied using measured aqueous con-centrations in groundwater samples to infer the mineralsthat are dissolving, and the minerals that are precipitat-ing. Both types of model can impose equilibrium rela-tionships that do not really exist at the continuum scale.

Equilibrium relationships are theoretically estab-lished to describe chemical speciation in spatially uni-form systems. If the pore-to-pore variations inconcentrations are sufficiently large, as in case A, equi-librium relationships may not hold for the spatially-

averaged concentrations used to describe continuum-scale phenomena in porous media. Such discrepancyhas been encountered in many reaction and transportphenomena in subsurface systems (see [11], and refer-ences therein), and calls for efforts incorporating effectsof porous medium heterogeneities in these relationships.This remains an open research question.

8. Summary and conclusions

This work establishes a novel methodology to usepore-scale network modeling to examine scaling effectsin geochemical reaction rates. Pore-scale network mod-els are constructed to represent the detailed physical andmineral properties of porous media. Reactive transportprocesses are simulated at the pore scale, and the contin-uum-scale reaction rate RN is calculated for the entirenetwork and is taken to be a measure of the ‘‘true’’ reac-tion rate. To explore the scaling effects, we also calculatetwo additional continuum-scale rates that ignore pore-scale heterogeneities and use the lab-measured reactionrates directly: RC, the rate obtained through continuummodeling, and R0N, the rate obtained using volume-aver-aged concentrations. Values of RC are compared to RN

to examine the extent to which the continuum approachused in forward modeling would lead to erroneous pre-diction of reaction rates in porous media. Values of R0Nare compared with RN to examine the extent to whichfield-measured concentrations, such as those used ininverse modeling, accurately predict the continuum-scale reaction rates.

The application of this method was illustrated with apore-scale network model that represents a consolidatedsandstone, with reactive minerals being anorthite andkaolinite occupying 10% of the pore space and gatheringin pore clusters of size 2 · 2 · 15. The network contains9000 pores, corresponding to the spatial scale of severalmillimeters in x-, y-, z-direction. Simulations were rununder conditions relevant to CO2 sequestration: onewith the boundary fluid in equilibrium with a high pres-sure of CO2 (case A), and the other with a less amountof CO2 (case B).

Simulation results have shown that in case A, the rel-atively large anorthite reaction rates arising from thehighly acidic boundary fluid leads to order-of-magnitudespatial variations in concentrations and reaction rates,which can be poorly represented by uniform or averagedconcentrations from models that ignore pore-scale heter-ogeneities. As such, the rates calculated from these sin-gle-valued concentrations in lab-measured reaction ratelaws would introduce large errors and the scaling effectsare significant. Under such conditions, the equilibriumrelationships that apply at the pore scale may not holdat the continuum scale, which suggests that commonpractice of reactive transport simulations may impose


equilibrium relationships that do not really exist at thecontinuum scale. In case B, with reduced acidity ofboundary fluid, the magnitude of the pore-scale hetero-geneities is much smaller than in case A, which eventu-ally leads to smaller scaling effects. The differences inthe scaling effects in the two cases suggest that elevatedCO2 concentrations can produce conditions under whichscaling effects are significant.

These findings have potentially significant implica-tions for reactive transport modeling. In general, the sig-nificance of pore-scale heterogeneity will depend on avariety of factors, including, for example, hydrodynamicconditions, and the total amount and spatial distribu-tions of reactive minerals. Although the observed signif-icance of the pore-scale heterogeneities and scaling effectare specific to our current network model and simula-tion conditions, the results imply that elevated CO2 con-centrations in deep aquifer systems can generateconditions under which scaling becomes important forreactive transport modeling. Under conditions wherespatial heterogeneities exist and scaling becomes impor-tant, geochemical modeling must be approached care-fully, and that general scaling laws incorporating theheterogeneity effect need to be developed.

Overall, pore-scale network modeling has proven tobe an effective research tool to upscale reactive transportprocesses from the pore scale to the continuum scale. Itallows for the investigation of details at the pore scalewhich cannot be done with traditional experimentalstudies, and thereby allows for identification of impor-tant pore-scale details that are essential to describe thereactive transport processes at the continuum scale. Thisnovel application of pore-scale network modeling maybe useful to study scaling issues associated with othergeochemical reactions in heterogeneous porous media,and therefore is likely to provide new perspectives forresolving the long-standing lab-field rate discrepancy.

Disclaimer

This report was prepared as an account of worksponsored by an agency of the United States Govern-ment. Neither the United States Government nor anyagency thereof, nor any of their employees, make anywarranty, express or implied, or assumes any legal liabil-ity or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would notinfringe privately owned rights. Reference herein toany specific commercial product, process, or service bytrade name, trademark, manufacturer, or otherwise doesnot necessarily constitute or imply its endorsement, rec-ommendation, or favoring by the United States Govern-ment or any agency thereof. The views and opinions ofauthors expressed herein do not necessarily state or

reflect those of the United States Government or anyagency thereof.

Acknowledgements

This material is based upon work supported by theUS, Department of Energy under Award No. DE-FG02-05ER15636. The authors also acknowledge BPand Ford Motor Co. for their support of the CarbonMitigation Initiative at Princeton University.

Appendix A. Supplementary data

Supplementary data associated with this article canbe found, in the online version at doi:10.1016/j.advwatres.2005.10.011.

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