Permeability evolution due to dissolution and precipitation of carbonates using reactive transport modeling in pore networks



[1] A reactive transport model was developed to simulate reaction of carbonates within a pore network for the high-pressure CO2-acidified conditions relevant to geological carbon sequestration. The pore network was based on a synthetic oolithic dolostone. Simulation results produced insights that can inform continuum-scale models regarding reaction-induced changes in permeability and porosity. As expected, permeability increased extensively with dissolution caused by high concentrations of carbonic acid, but neither pH nor calcite saturation state alone was a good predictor of the effects, as may sometimes be the case. Complex temporal evolutions of interstitial brine chemistry and network structure led to the counterintuitive finding that a far-from-equilibrium solution produced less permeability change than a nearer-to-equilibrium solution at the same pH. This was explained by the pH buffering that increased carbonate ion concentration and inhibited further reaction. Simulations of different flow conditions produced a nonunique set of permeability-porosity relationships. Diffusive-dominated systems caused dissolution to be localized near the inlet, leading to substantial porosity change but relatively small permeability change. For the same extent of porosity change caused from advective transport, the domain changed uniformly, leading to a large permeability change. Regarding precipitation, permeability changes happen much slower compared to dissolution-induced changes and small amounts of precipitation, even if located only near the inlet, can lead to large changes in permeability. Exponent values for a power law that relates changes in permeability and porosity ranged from 2 to 10, but a value of 6 held constant when conditions led to uniform changes throughout the domain.

1. Introduction

[2] Over the last decade, numerous studies have been undertaken to understand the physics that govern the transport of supercritical CO2 in the subsurface and its subsequent trapping or leakage in the context of geological carbon sequestration [Bachu and Adams, 2003; Birkholzer et al., 2009; Brosse et al., 2010; Burton et al., 2008; Celia et al., 2011; Doughty and Pruess, 2004; Grimstad et al., 2009; Nogues et al., 2012; Pawar et al., 2009; Pruess et al., 2003]. All of these studies have considered static material properties, such as permeability and porosity, for geologic formations and leakage pathways. The evolution of permeability and porosity due to geochemical reactions has yet to be considered in models that consider leakage risks. To do so, transport models must incorporate methods for predicting dynamic material properties either by numerically upscaling constitutive relationships, developing simplified empirical rules or by using multiscale/multiphysics methods (as the one developed by Flemisch et al. [2011]). All of these methods would require an understanding of the reactions that can substantially alter the rock matrix, and the different ways the rock matrix evolves under different conditions.

[3] The relevant geochemical reactions that can substantially alter the rock matrix are those between CO2-acidified brines and carbonate minerals because these reactions are sufficiently fast and potentially lead to changes in the permeability and porosity in short time scales [Assayag et al., 2009; Ellis et al., 2013, 2011; Luquot and Gouze, 2009; Noiriel et al., 2004; Noiriel et al., 2005; Smith et al., 2012]. In a geological sequestration operation, carbon dioxide dissolves into the resident brine forming carbonic acid. The resulting thermodynamic state of disequilibrium drives dissolution of the mineral matrix. This then has the effect of releasing cations, such as Ca2+, Mg2+, or Fe3+ from the mineral matrix, which may buffer the acid to some extent. When these reactions occur in caprock sealing formations that overlay CO2 injection formations, permeability and porosity changes may jeopardize the trapping mechanisms considered for reliable CO2 storage. Conversely, there are scenarios in which mixing of fluids creates thermodynamic disequilibrium that drives precipitation of carbonate minerals, thereby reducing porosity [Emmanuel and Berkowitz, 2005; Xu et al., 2005; Zhang et al., 2010].

[4] Several modeling studies have examined the role of geochemical reactions and their effects on permeability and porosity within the context of geological carbon sequestration. For instance, Gherardi et al. [2007] simulated changes in porosity in carbonate-rich shales due to the geochemical reactions between calcium-rich waters and resident brine, using the TOUGH-REACT simulator [Xu et al., 2006]. Through numerical experiments, they found conditions that caused the porosity at the boundary between formation rock and caprock to decrease from 15% to 0% porosity due to precipitation of calcite. Andre et al. [2007] used the TOUGH-REACT simulator as well, to study CO2 storage in the carbonate-rich Dogger aquifer in the Paris Basin, France. They saw that when CO2-saturated water was injected, the porosity near the injection well increased significantly due to dissolution of the porous material. Finally, Liu et al. [2011] used TOUGH-REACT to model CO2 injection in the Mt. Simon sandstone formation, USA. Like Andre et al. [2007], they saw a dissolution front caused by the acidified brine. In general, most of these studies have shown that the geochemical reactions between CO2-rich brines and carbonate rocks have effects on the permeability and porosity due to the dissolution or precipitation of carbonate minerals.

[5] All of the reactive transport models used in the previous studies described the interplay of geochemical reactions and transport at the continuum or Darcy scale. Permeability is a continuum-scale property that manifests from the collective conductivities of flow between many pores, and thus is sensitive to reaction-induced alterations at the pore scale. Furthermore, it can be argued that geochemical reactions are best described at the pore scale because they are driven by small-scale chemical gradients and because heterogeneities in porous rocks create variability in local chemical conditions [Kim et al., 2011; Li et al., 2006; Steefel et al., 2005]. Reactive-transport modeling in pore networks is an approach in which geochemical reactions and transport of species is modeled at the pore scale enabling simulation of the effects of the small-scale heterogeneities, and simulation of permeability and porosity for the entire domain.

[6] At the pore scale, relevant studies have found that (i) dissolution and precipitation of carbonate minerals can occur simultaneously [Kang et al., 2010]; (ii) at the scale of a single pore the reduction in flow area is proportional to the reactive and convective processes [Li et al., 2010]; and (iii) mixing of disparate waters can lead to precipitation and reduction in porosity [Tartakovsky et al., 2007]. Szymczak and Ladd [2009] performed pore-scale simulations in 2-D and showed that the creation and evolution of wormholes had a close dependence on the relationships between reaction and flow rates. None of these pore-scale studies linked the changes in mineral volume fraction (or porosity) to changes in permeability. Moreover, their results mainly focused on regular or periodic networks in one or two dimensions or systems, where pore-to-pore heterogeneity was not taken into consideration. Algive et al. [2012] developed a methodology to use a reactive pore network model to extract upscaling factors to tie the pore-scale effects of reactive transport to the core-scale values of permeability and porosity. Their work simplified the geochemistry and transport by incorporating most geochemical dynamics and transport into dimensionless variables, and in doing so prevented a full description of the different physical and chemical parameters. Mehmani et al. [2012] developed a novel approach that coupled several pore-scale models using mortar coupling domain decomposition to study the evolution of precipitation-induced cementation of calcite. They were able to study large changes in permeability and porosity by coupling of 64 pore-scale models (1 mm × 1 mm × 1 mm each) but with a limited description of the chemistry represented by two main parameters, the Damkohler number (Da) and an “alpha” parameter which described the deviation from equilibrium of the precipitation reaction. Mehmani et al. [2012] showed that a low Da and “alpha” number created precipitation at the inlet of the network and that a high Da and “alpha” combination created more uniform precipitation. More recently, the work by Yoon et al. [2013] was able to simulate CaCO3 precipitation due to transverse mixing in a 2-D microfluidic pore network. They captured the essential physiochemical dynamics of precipitation that was characterized by a fast initial precipitation rate that leveled off as the mixing was hindered.

[7] In this study, we present a new reactive transport model that simulates carbonic acid-driven reactions in a 3-D network of pores in order to predict the changes in permeability and porosity at the continuum scale. Through a series of simulations, the main questions that are addressed are: (1) How do the degree of acidity and the calcite saturation state, controlled by the amount of CO2 and calcium in inflowing waters, affect the changes in permeability and porosity? (2) What are the effects, on the network permeability and porosity, of different flow conditions (i.e., pressure gradients)? (3) How do different patterns of mixing reactive boundary waters promote precipitation and reduction in permeability and porosity? and (4) How do single parameter power law equations perform when predicting the evolution of permeability due to porosity changes? Ultimately, these four questions help tackle the more general question of what are the most plausible changes that can occur in a CO2 sequestration operation.

[8] The pore network used in this study was developed with the aid of the algorithm proposed by Raoof and Hassanizadeh [2010]. The pore network structure was based on a statistical characterization of a synthetic microcomputed-tomography (μ-CT) image of an oolithic dolostone from Biswal et al. [2009]. To focus on the effects of physical heterogeneity, i.e., pore volumes and pore throats, the network was spatially uniform with respect to mineralology. The reactive transport model developed for this work accounts for reactions of acidic fluids with both carbonates and aluminosilicates. The model predicts evolution of pore sizes and pore-to-pore conductivities. In this paper, we present numerous simulation results focused on the permeability-altering effects of different chemical, flow and mixing conditions. The findings are examined in the context of subcontinuum-scale variations in geochemical conditions and reaction rates.

2. Pore Network Creation

[9] The random pore network used in this study relies on the algorithm presented by Raoof and Hassanizadeh [2010] where each pore body can have a maximum of 26 connections along 13 different directions within a regular cubic lattice. To accurately represent a carbonate rock we used statistics derived from a synthetic 3-D µ-CT image of a dolomitized oolithic grainstone (Figure 1a) developed by Biswal et al. [2009], which follows a stochastic geometry model known as a germ-grain model [see Biswal et al., 2007]. The statistics were extracted using the 3DMA-Rock software package [Lindquist et al., 1996; Shin et al., 2005]. The statistical distributions of pore volumes, pore-to-pore connections, pore surface areas, and pore center coordinates are shown in Figure 1. The extracted network consisted of 20,128 pores with an average coordination number of 5, and an average pore volume size of 4.0 × 10−13 m3. The coordination number refers to the number of connection a pore has, for instance, a coordination number of 4 means that a pore is connected to four other pores.

Figure 1.

Synthetic dolomitized oolithic grainstone from Biswal et al. [2009] and the corresponding statistics derived using 3DMA-Rock: (a) Three-dimensional µ-CT image, (b) histogram of pore coordination numbers, (c) histogram of pore body surface areas, (d) histogram of pore body volumes, and (e) pore surface area versus pore volume.

[10] For this study, a smaller network was created in order to have a faster computational model. The network is a cube that is 1.87 mm in length on each axis and is composed of 1728 pores. The porosity (ϕ) is defined as,

display math(1)

where Vi (L3) corresponds to the volume of pore i and VT (L3) is the constant network volume. Table 1 outlines the properties of the network. A representation of the network in a stick-and-ball format is shown in Figure 2, along with histograms of pore volumes and coordination numbers. The pore volume histogram in Figure 2b matches with the original pore volume histogram from the 3-D synthetic µ-CT image (Figure 1d). For our model, we chose to have more tortuosity in the network, therefore the coordination number distribution does not match exactly the ones from the 3-D synthetic µ-CT image, though the values come from the same population.

Table 1. System Properties of the Pore Network Along with the Mineral Abundances Within the Network
System Properties Mineral% by Volume
Initial porosity0.133Albite5
Initial permeability (m2)8.16E-15Anorthite5
Side of network cube (mm)1.87Calcite10
System volume (m3)6.53E-09Dolomite60
Total number of pores1,728Kaolinite5
Figure 2.

(a) Ball and stick representation of the network with a cross-sectional slice of the pore-to-pore connectivity, (b) histogram of network pore volumes, and (c) histogram of network coordination numbers.

[11] For the model in this study, we have assumed that all pore throats are cylindrical in shape and have a characteristic diameter, which is used to calculate the conductivity across the pores. Moreover, these connecting throats are used only in the calculation of the conductivity value that feed into the pressure equation; they do not hold any volume within the system. The conductivity, Cij (L4TM−1) between pores i and j is defined by the Hagen-Poisueille equation [Sutera, 1993] for an incompressible fluid,

display math(2)

where dij is the diameter (L) of the “effective” cylinder connecting pores i and j; lij is the distance (L) between the centroids of adjacent pores; and math formula is the viscosity (MT−1L−1) of the flowing fluid. The initial pore-to-pore conductivities of the network were assigned using the methods developed by Li et al. [2006]. Conductivities were sampled from a lognormal distribution in which there is correlation with the total volume of adjacent pores. In this work, the geometric mean of Cij was chosen such that the intrinsic permeability of the system would be around 10−14 m2 (10 mD). The geometric standard deviation of Cij was equal to unity.

[12] The solid matrix comprises of five reactive minerals plus quartz in the relative amounts shown in Table 1. The carbonate minerals that dominate carbonate formations are calcite and dolomite [Al-Jaroudi et al., 2007; Mukherji and Young, 1973; Stehli and Hower, 1961]. A dolostone is in its majority dolomite (50–90%), with 3–10% clays [Usenmez et al., 1988]. For this study, the mineralogy assigned to the entire network was distributed uniformly. Each pore in the system was allotted mineral volumes according to the relative amounts in Table 1.

[13] In the work of Peters [2009], it was shown that the surface area coverage by authigenic clay minerals in consolidated grainstones can be substantially larger than the relative abundance of these clay minerals on a volume basis. For this work, we selected surface area allocations that are consistent with those trends. We allotted 50% of the surface area to kaolinite, while dolomite received 10% and both anorthite and albite received 5% of the surface area each. Since calcite is considered to be at equilibrium, surface area was not needed to calculate reaction rate.

3. Reactive Transport Modeling

[14] The reactive transport model developed for this work has as its main purpose the modeling of permeability and porosity evolution due to carbonic acid-driven precipitation and dissolutions in a pore network system. The model development follows methods previously presented [Li et al., 2006; Kim et al., 2011]; this work is distinct in the focus on carbonates, which are present in large amounts and which are relatively highly reactive, and with the added capability to account for changes to the pore volumes induced by precipitation and dissolution of mineral phases. The reactive transport architecture follows that of the STOMP ECKEChem model [White and McGrail, 2005] and proposed by Fang et al. [2003]. The two processes of transport and geochemical reaction/speciation are separated and solved sequentially in a noniterative manner, a method presented by Steefel and MacQuarrie [1996] under the name of Sequential Non-Iterative Approach (SNIA). A sensitivity analysis was done to choose the adequate time step of the model and justify the sequential approach—the time step was fixed at a value at which the system porosity and permeability evolution did not change significantly from previous time steps. The model first solves the pressure equation, which provides the pore-to-pore fluid flow velocities, under the assumption of an incompressible and constant-density fluid. Then, the solute transport equations are solved for the selected basis components. After the new component concentrations are determined, a geochemical batch reactor module is applied to each pore element. Finally, the model accounts for the changes in volume and pore-to-pore conductivity due to precipitation or dissolution of minerals. The model processes involved are outlined below, with details given in Nogues [2012].

3.1. Reactive Species

[15] The model considers a total of 18 aqueous species and 5 mineral species. Through the Tableau method the basis components were determined to be the total carbonate-bearing species, CT, the total calcium-bearing species, CaT, the total magnesium-bearing species, MgT, the total silica-bearing species, SiT, the total aluminum-bearing species, AlT, the total chloride-bearing species, ClT, and the total sodium-bearing species, NaT. In our model, we replace the total aqueous protons concentration, HT, by the charge balance equation in order to reduce the number of equations we solve when considering transport. In addition to the eight mole and charge balance equations, there were 14 independent reaction equations, corresponding to the reactions listed in Table 2.

Table 2. Chemical Reactions and Equilibrium Constants at 50°Ca
 Reactionslog Keq
  1. a

    The equilibrium values are from the EQ3/6 database [Wolery et al., 1990].

Equilibrium math formula−13.3
math formula−6.15
math formula−10.3
math formula−9.83
math formula−12.6
math formula8.76
math formula18.9
math formula27.3
math formula33.2
math formula−8.66
Kinetic math formula21.7
math formula3.80
math formula−1.67
math formula1.63

[16] Calcite was considered to be at equilibrium, based on its fast reaction rate [Plummer et al., 1978] and based on preliminary batch reaction simulations we did which showed that the reaction rates of this mineral were much higher than the others. We saw that the time for calcite to reach equilibrium, after being perturbed, was much faster (in the order of tenths of seconds) in comparison to the other minerals (hundreds to thousands of seconds). Treating the reaction of calcite as an equilibrium reaction thereby avoided numerical issues of stiffness.

[17] The complete system (equilibrium and kinetic) of nonlinear geochemical equations is solved iteratively, on a pore-by-pore basis, using a Newton method.

3.2. Transport and Pressure Equations

[18] In the model, the pressure field is determined by solving a system of flow equations in which the net flow into and out of each pore is balanced:

display math(3)

where Qij is the flow rate (L3T−1) from pore i to pore j, nc is the number of pores connected to pore i and Pi and Pj are the fluid pressures (ML−1T−2) in pore i and j, respectively. The permeability of the network, k (m2), is defined from the Darcy equation as

display math(4)

where Q is the total flow out of the network (L3T−1), μ is the dynamic viscosity (MTL−1), L is the length across the network (L), A is the outflow area (L2), and ΔP is the pressure difference across the system (MT2L−1).

[19] The basis components are transported from pore-to-pore by advection and diffusion. The transport equations are modeled at the pore scale as follows,

display math(5)

where math formula represents the concentration (ML−3) of a specific component in pore i, math formula is the effective molecular diffusion coefficient (L2T−1) for that component, aij is the cross-sectional area (L2) between pores i and j and is equal to math formula, and math formula is the mass rate (MT−1L−3) of change due to kinetic or equilibrium reactions with mineral phases. Through operator splitting, the last term is not solved in conjunction with the transport of the solutes.

3.3. Equilibrium Reactions

[20] The equilibrium reactions are represented by the mass action equations [Steefel and MacQuarrie, 1996],

display math(6)

where math formula and math formula are the molar concentrations (ML−3) of the secondary and primary component species, respectively, and γs and γp are their respective activity coefficients. The equilibrium constant is Ks, the number of components is Nc, and math formula is the stoichiometric reaction coefficient for a specific component species. The ionic strength was modeled as a constant because variations in the concentrations of reactive species would negligibly affect it. The activities were calculated using the Davies equation presented by Butler [1982], with a constant ionic strength of 0.45 M, which represents the limit of applicability of the Davies equation. Use of this ionic strength rather than the actual value of 1.2 M contributes negligible error.

3.4. Kinetic Rate Laws

[21] The four kinetically controlled minerals in this study are albite, anorthite, dolomite, and kaolinite. The rate of mass change due to reaction for each is written as,

display math(7)

where Mg is the mass of mineral g, normalized by the pore volume (ML−3). The kinetic rate is expressed by rg (ML−2T−1) and Ar,g is the specific reactive area (L2L−3) of mineral g, which is used for both dissolution and precipitation. The kinetic rate rg captures the rate of both dissolution and precipitation and it is given by transition state theory as shown by Lasaga [1998] and takes the following nonlinear form,

display math(8)

where kOH, kH2O, and kH are the temperature-dependent reaction rate constants (ML−2T−1), {OH}, {H2O}, and {H+} are the activities of the chemical species that have a catalytic or prohibitory effect, which are raised by constant exponents nOH, nH2O, and nH. Ω is the ionic activity product (IAP) over the equilibrium constant Keq, which gets modified by the empirical parameter m. The reaction rate constants and the empirical parameters are shown in Table 3.

Table 3. Reaction Kinetic Parameters at 50°Ca
 log10k (mol m−2s−1)    
H2OOHH+ math formula math formula math formulam
  1. a

    All parameters altered to 50°C using Arrhenius Law from the STP values shown in the EQ3/6 database.

  2. b

    Same as the values used in the models of Li et al. [2006] and Kim et al. [2011].

  3. c

    Data from Xu et al. [2007].


3.5. Network Evolution

[22] In order to capture the changes caused by mineral precipitation or dissolution, we developed a mathematical construct to modify the pore-to-pore conductivities by relating these changes to changes in pore volumes. The mathematical construct was chosen following the conceptual understanding that pore throat diameters are likely related to the volumes of the connected pores. For a given pore, as its pore volume is reduced or increased, each pore throat diameter associated with that pore body is updated using the following equation,

display math(9)

where math formula and math formula are the new and prior throat diameter (L) connecting pore i and j, respectively. Equation (9) is consistent with the original method of assigning conductivities, which was based on the assumption that there is a correlation between the pore body volumes and the size of the throats connecting them. Limit constraints are in place to prevent precipitated mineral to occupy more space than the original pore volume, and flow in the pore is prevented once 99% of the pore volume has been filled.

[23] The fraction of the pore surface area assigned to each mineral is scaled down or up as minerals dissolve or precipitate. If a mineral does not exist but it is thermodynamically favored to precipitate, the model assigns a minimal fraction of the surface area to it. The evolution of the fraction of surface area follows an equation similar to equation (9) in which the fraction is scaled by the relative mineral volume in that specific pore.

[24] It is important to note that the while the surface area fractions assigned to different minerals do change, the model does not change the total surface area. The choice to model the total surface area as unchanged is one of convenience. This modeling choice has negligible impact on the simulation results because the reaction rate of the most important reactive mineral, calcite, is independent of surface area because it is modeled as being at equilibrium.

4. Simulation Methods

[25] In all the simulations, flow was driven in the x direction (Figure 2a) by imposing pressure boundary conditions at the boundaries, with no-flow conditions on the other four boundaries. In order to prevent boundary effects to affect the evolution of the pore network, diffusion of ions from the outlet boundary into the domain was prohibited. In order to have comparable systems each simulation was run for 10,000 injected pore volumes through the entire network—effectively removing the dependency of time in the analysis. In all the simulations, the initial condition chosen for the chemical composition of the resident brine was close to equilibrium with respect to calcite and dolomite (Table 4).

Table 4. Initial and Boundary Conditions Used in the First Analysis of This Study
 Resident Brine Initial ConcentrationsInflowing Water Concentrations
CT (mol/L)1.80e-34.2e-4 to 1.5
CaT (mol/L)7.53e-41.0e-4 to 3.34e-2
MgT (mol/L)2.26e-43.33e-5 to 1.11e-2
SiT (mol/L)5.2e-55.2e-5
AlT (mol/L)9.06e-59.06e-5
NaT (mol/L)1.21.2
ClT (mol/L)1.21.2
SI Calcite0.067−6.1 to 0.04
SI Dolomite−0.043−12 to −0.09

4.1. Changes in Permeability and Porosity Due to Different CaT and CT Concentrations

[26] The first question investigated in this study pertains to the effects that different inflowing brines have on the network's porosity and intrinsic permeability. Flow-through simulations were run with different combinations of CaT and CT concentrations for the inflowing water, with the amount of magnesium (MgT) kept constant at one third of CaT, thereby limiting the degrees of freedom to two independent variables. The different boundary conditions were chosen from the ranges shown in Table 4. The maximum allowable CT value of 1.5 mol/L corresponds to the upper limit of CO2 solubility in brines as reported by Duan et al. [2006]. Figure 3a shows the pH values that correspond to the different combinations of CT and CaT. Figures 3b and 3c show the saturation index (SI) values of calcite and dolomite for the same conditions as in Figure 3a. Note that conditions with low CaT and high CT concentrations correspond to negative values of SI which indicates that the water is undersaturated with respect to the mineral and thermodynamically favors dissolution. The selected boundary condition values were bounded by an upper limit of pH 6 and by the SI equilibrium line for calcite. That is, all the inflowing fluids were either undersaturated or at equilibrium with respect to dolomite and calcite.

Figure 3.

The concentrations and conditions of the inflowing boundary waters in the first analysis of this study: (a) pH contours with the equilibrium lines of calcite and dolomite superposed and (b and c) saturation index contours of calcite and dolomite, respectively, over the CT and CaT variable space.

4.2. Evolution of Permeability and Porosity Due to Flow Rate Variation

[27] The second analysis was performed to understand how the network's permeability and porosity evolve as a result of variation in the flow rate. We ran comparative simulations by imposing different pressure gradients across the network. The pressure gradients were imposed such that the pore-scale transport spanned the range of diffusive-dominated and advective-dominated regimes. This analysis was conducted with two different pH boundary conditions, 3 and 5. These concentrations are shown in Table 5 along with the range of pressure gradients imposed.

Table 5. Initial and Boundary Conditions for the Second Analysis, as Well as the Range of Pressure Gradients and the SI Values
 Resident Brine Initial ConcentrationspH 3 Boundary Water ConcentrationspH 5 Boundary Water Concentrations
CT (mol/L)1.80e-31.53.16e-1
CaT (mol/L)7.53e-41.0e-41.15e-2
MgT (mol/L)2.26e-43.33e-53.46e-3
SiT (mol/L)5.2e-55.2e-55.2e-5
AlT (mol/L)9.06e-59.06e-59.06e-5
NaT (mol/L)
ClT (mol/L)
ΔPX (kPa/m) range0.26–13.4
SI calcite0.067−6.1−0.686
SI dolomite−0.043−12−1.54

4.3. Evolution of Permeability and Porosity Due to Mixing Scenarios

[28] The third analysis explored the effects of pore-scale mixing on the evolution of the permeability and porosity of the network. We created three inflow boundary scenarios, which involved injection of two inflowing waters that when mixed would be reactive. The conditions that we were trying to replicate in these scenarios were similar to the ones presented in Tartakovsky et al. [2008] and Zhang et al. [2010], where two waters that were oversaturated with either Ca2+ or CO32− flowed into the porous medium in order to create precipitation as they mixed.

[29] Figure 4 shows the three different inflowing boundary patterns that were simulated. Two fluids flowed through the network at the same time and the initial brine was the same as the one used in the previous simulations with a pH of 8.2 (see Tables 4 or 5). As can be seen from Figure 4 there are a total of 100 boundary nodes separated in two sets of 50. The three different inflowing boundaries are referred as “Half,” “Cross,” and “Check” as suggested by their geometrical shape.

Figure 4.

Flow patterns for inflowing waters over the y-z space at the network boundary.

[30] The compositions of the two brines were chosen by trial and error to find pairs for which only the mixing could create conditions of dissolution and precipitation. In other words, if each sample of water were to flow alone through the network there would be no significant dissolution or precipitation. The selected pairs of brines are presented in Table 6. The brines chosen for the dissolution case were close to equilibrium with respect to calcite and dolomite. The brines chosen for the precipitation case were oversaturated with respect to calcite and dolomite.

Table 6. Boundary Conditions of the Inflowing Brines for the Analysis of Mixing Patterns
 Dissolution CasePrecipitation Case
Conc. IConc. IIConc. IConc. II
CT (mol/L)1.101.5e-20.977.35e-4
CaT (mol/L)0.11.6e-24.18e-57.65e-2
MgT (mol/L)3.01e-23.86e-31.17–52.93e-2
SiT (mol/L)1.95e-73.65e-72.63e-74.94e-7
AlT (mol/L)3.82e-71.97e-77.67e-77.83e-7
NaT (mol/L)1.410.611.490.19
ClT (mol/L)1.620.640.470.39
SI Calcite0.0740.141.812.15
SI Dolomite−0.027−0.00013.414.22

[31] The pressure boundary condition was the same for each of the three different mixing simulations. In the dissolution case, the pressure gradient imposed across the system was 0.26 kPa/m. For the precipitation case, two pressure gradients were tested, one set of simulations with a 0.26 kPa/m gradient and another with a 13.4 kPa/m. Both sets of simulations were run until 10,000 pore volumes flowed through the network or a simulation time of 300 days had passed (whichever was reached first).

5. Results and Discussion

[32] While the modeling of pore-scale transport and geochemical reactions allows for detailed views of how each individual pore evolves regarding its chemistry, flow rates, volume, and surface area, we focus our results on the integrated, upscaled values of permeability and porosity.

5.1. Changes in Permeability and Porosity Due to Different CaT and CT Concentrations

[33] The first analysis looked at how changes in the network permeability and porosity depended on the chemistry and thermodynamic potentials of the inflowing waters, while keeping pressure differences across the network constant. Figure 5 shows the resulting changes in porosity (Δϕ) and relative changes in permeability (k/ko), after 10,000 injected pore volumes, over the variable space of CaT and CT in the inflowing water. The highest changes seen for both permeability and porosity correspond to the smallest concentration of CaT and the highest concentrations of CT, where the pH equals 3, with a difference in porosity of 0.37 and a change in permeability of 2000 times the initial value. Conversely, the smallest changes in permeability and porosity are seen for conditions nearest calcite saturation, as expected. For example, when the CaT concentration is 10−2.9 mol/L, CT is 10−1.4 mol/L, and pH is 6, the difference in porosity is 0.03 with a corresponding change in permeability of two times the initial value.

Figure 5.

Contours showing (a) the changes from initial porosity and (b) the relative change in permeability with respect to the initial permeability (ko) over the CT and CaT variable space, after 10,000 injected pore volumes. The two dots labeled “A” and “B” show a case where boundary condition with a higher SI value produces a larger change in the permeability and porosity.

[34] Final values of permeability and porosity are dependent in a complex way on the amounts of CT and CaT in the inflowing waters. First, we can see that no single geochemical variable (CT, CaT, pH, saturation index, nor alkalinity, which is essentially proportional to CaT) alone is a good indicator of evolution of porosity and permeability. That is, the contours in Figure 5 do not mimic the contours of any of the possible independent variables. It is a minimum of two independent geochemical variables that are needed for predictive purposes. While this is not surprising from a scientific point of view, it bears practical relevance because it means there is not a simple relationship that can defensibly be developed for reduced-order models, for the range of conditions examined here. The only simplification that can be made is that for values of CaT less than 10−3 mol/L, the changes in permeability and porosity are dependent only on CT. Only at CaT concentrations greater than 10−2.5 mol/L does dissolved calcium begin to hinder changes in porosity and permeability.

[35] Furthermore, the amounts of CT and CaT dictate the paths of evolution of the permeability, porosity, pH, and SI values, and may lead to counter-intuitive results. For instance, in Figure 5 two dots are marked “A” and “B” for two different inflowing boundary conditions with the same pH value but case A has a calcite SI = −3 and case B has calcite SI = −5.5. Their resulting changes in permeability and porosity are interesting. For the case that is closer to equilibrium (case A) there is a larger change in permeability and porosity than for the smaller SI value (case B). This at first might seem counter-intuitive but the fact is that the boundary condition with a higher SI value has a higher concentration of CT, and therefore requires the addition of more calcium out of the mineral matrix in order to reach an SI value of “0.” These results indicate that it is both the relative and absolute amounts of CT and CaT that determines the ultimate evolution of the system.

[36] To better understand this, we conducted a simple numerical simulation of calcite dissolution in a single pore with an initial condition corresponding to the boundary water condition of interest. For initial conditions corresponding to cases A and B, Figure 6a shows the evolution of calcite dissolution (on the x axis) as the pore reaches calcite saturation. Also shown in Figure 6 are the evolution paths of pH, carbonate ion, and calcium ion. As the equilibration of calcite is done instantaneously in our model, this numerical simulation is solely for illustrative purposes. In case A, the amount of calcite dissolution is greater than in case B. In case A, there is only a slight pH increase, from 3.8 to 4.6, compared to case B in which the pH is buffered from 3.8 to 5.6 because of the significant buffering that comes from even the small amount of calcite dissolution. This large pH shift means that the speciation of system “B” results in a higher concentration of math formula, which brings the system close to calcite saturation faster. The Saturation Index of calcite is given by,

display math(10)

where math formula and {Ca2+} represent the activities of carbonate and calcium ions, respectively, and Ksp is the solubility product of calcite in solution. This comparative evolution of the two cases with the same pH but different SI conditions is seen regardless of the flow regime, diffusion, or advective dominated. In conclusion, it is the evolutionary path that matters, and a system that is far from calcite saturation because of a low CT concentration can move quickly toward calcite saturation thus slowing ongoing reaction and inhibiting further change in porosity and permeability.

Figure 6.

Evolution of geochemical parameters inside a single pore in two different systems with the same boundary pH and different SI values: (a) Saturation index of calcite, (b) pH of the pore water, (c) concentration of carbonate ions, and (d) concentration of calcium ions.

[37] It should also be noted that for the simulations that produced smaller increases in permeability and porosity, it took a longer time to reach 10,000 pore volumes because the conductivities throughout the network did not increase as fast as those near the inlet. For the system as a whole, this did not result in an increase in flow rate. The differences in how these systems respond as functions of pore volumes injected can be seen in Figure 7, which shows a set of curves corresponding to different boundary CaT and CT values. Figure 7a shows the evolution of the porosity with respect to the injected pore volumes and Figure 7b shows the evolution of the network permeability with respect to injected pore volumes. In these figures, we can see two distinct regions of evolution: a linear increase in permeability and porosity at early times followed by a semilogarithmic increase at later times. The linear increase is mainly due to the dissolution of calcite, which is an instantaneous reaction in our model. As calcite is depleted, dolomite dissolution, which is kinetically controlled, determines the change in pore volumes and therefore governs the long-term permeability evolution of the network.

Figure 7.

Evolution curves of the (a) porosity and (b) permeability (in log base 10) of the system with respect to injected pore volumes for 50 different inflowing combinations of CaT and CT values. The labels “SI Ca” and “Si Do” stand for the Saturation Index values with respect to calcite and dolomite, respectively. (c) Permeability (in log base 10) versus porosity curves for 50 different inflowing combinations of CaT and CT values.

[38] In Figure 7c, which shows the permeability (in log space) of the network with respect to porosity. This figure shows that a rather singular relationship exists for the variation of permeability with porosity despite the wide range of pH and SI conditions tested. Thus, while there is complexity in how the extent of the change depends on boundary chemistry, there is a tight, unique relationship between permeability and porosity.

[39] The variation that does exist in Figure 7c can be explained by the slight effect of kaolinite precipitation. For example, for the case of pH 6.0 (line labeled “Si Ca = 0.0, Si Do = −0.1” in Figure 7a) kaolinite's original volume increased by 20%, while dolomite and calcite decreased its original volume by 8% and 0%, respectively. The overall effect is dissolution due to the large amount of dolomite to start with (60% by volume) compared to the small amount of kaolinite (5% by volume). In this example, of all the volume added by precipitation and subtracted by dissolution, dolomite dissolution contributed 83% of it. In systems with less calcite and dolomite, the effect of anorthite dissolution and kaolinite precipitation would be more substantial.

[40] It can be argued that the permeability versus porosity curve, regardless of the pH, SI values, and alkalinity of the inflowing waters, is for practical purposes unique—given a homogenous mineral distribution with constant flow conditions. We can also see that if any simplification were to be made to the geochemical modeling it would be appropriate to not include kaolinite, anorthite, and albite in cases where carbonate minerals are abundant.

5.2. Changes in Permeability and Porosity Due to Flow Conditions

[41] Carbonate reactions are transport limited [Bernabé et al., 2003] and the reaction extent is sensitive to how quickly reactive water is distributed throughout the network. Singurindy and Berkowitz [2003] found in column experiments of calcareous sandstones that because dissolution of calcium carbonate is mass transfer limited, higher flow rates cause a more rapid dissolution of the porous medium. In Figure 8, we show the evolution of porosity and permeability for three different pressure gradients (which serves to illustrate the effect of flow rate) and two different pH conditions as functions of injected pore volumes. Here we see that in a high-pressure gradient system (i.e., ΔP = 13.4 kPa/m), in which water flows faster through the network, there is less permeability and porosity change for the same number of pore volumes compared with a slow flowing system (i.e., ΔP = 0.26 kPa/m). This is because the residence time of the slow moving reactive fluid is longer than that of the fast flowing system so there is more time for reaction and a greater extent of change in the slow-flowing system for a given number of pore volumes. If the abscissa were represented as time the curves in these graphs would be switched and faster flowing waters would reach higher permeability and porosity changes in a smaller amount of time compared to a slow flowing system.

Figure 8.

Evolution of the (a) porosity and (b) permeability with respect to injected pore volumes for three different flow regimes. Solid lines correspond to an inflowing pH of 3 and dashed lines correspond to an inflowing pH of 5. (c) The log10(k) versus porosity relationship for three different flow regimes.

[42] The resulting subdomain structure changes are explored in Figure 9, where the average volume changes along the x direction (i.e., the flow direction) are shown when the overall network porosities reach 0.2, 0.4, and 0.6. In a low-pressure gradient regimes, the reaction is first concentrated at the inlet boundary which dissolves first since the inflowing water does not advance that easily through the network, making the dissolution front progress slowly with time. The changes for the smallest pressure gradient are concentrated close to the inflow boundary while for the largest pressure gradient the changes are uniformly distributed across the network. This figure is characteristic of how transport-limited reactions change the permeability and porosity of the network as a function of pressure gradients.

Figure 9.

Average change of volumes of pores across the network for three different final porosities: (a) 0.2, (b) 0.4, and (c) 0.6.

[43] As shown in Figure 8c, there is a distinct difference in the evolution of the permeability and porosity between the two different pH waters flowing through the network. For a pH of 3, the system dissolves all the calcite and dolomite and does not precipitate any kaolinite. For a pH of 5, however, kaolinite precipitates continuously. The highest values of permeability and porosity reached with a pH of 3 are not reached with a pH of 5. After all the dolomite and calcite have dissolved away in the pH 5 scenario, kaolinite keeps precipitating and takes over the evolution of the network. This continual precipitation of kaolinite makes the permeability and porosity relationship take a turn at the end.

[44] In Figure 8c, we see that the simulations with pressure gradients of 1.6 and 13.4 kPa/m follow the same path regardless of their pH, while the simulation with a pressure gradient of 0.26 kPa/m takes a different path. Therefore, slow flow conditions create subdomain-scale structure in porosity evolution, which leads to nonuniqueness in how permeability evolves with porosity. Much like the work presented by Luquot and Gouze [2009] here we see that the dissolution regime—and the subsequent nonuniqueness—can be related back to reactivity and flow rate.

5.3. Changes in Permeability and Porosity Due to Mixing

[45] The changes seen for the different mixing scenarios for the dissolution case are shown in Figure 10. Each pattern of mixing produces a different evolution of the permeability and porosity curve. When each boundary brine flows through the network by itself (curves labeled pH 4.5 and 6.2), the changes in permeability and porosity are minimal. Conversely, the scenario with the “Check” pattern produces the highest amount of change in porosity, while the mixing pattern that separates the boundary in two halves produces the least amount of change. This indicates that more mixing near the boundary leads to more rapid changes in porosity. However, this change does not correspond to a proportional change in permeability as can be asserted by Figures 10a and 10b. The “Cross” and “Half” patterns reach the same permeability values but with much less change in porosity. This indicates that as before the diffusion-dominated regime concentrates dissolution at the inlet and therefore the outlet sees minimal change in porosity. In Figure 10c, we also see the comparative results, which produce a nonunique evolution of the network's permeability and porosity. We can see that the mixing alone produces very different permeability evolutions. From Figure 10c, we see that for the “Half” mixing pattern there is a larger change in permeability for the same change in porosity, compared to the “Check” pattern. This shows the importance of localized dissolution and its effect on the overall conductivity of the system. The simulation of the “Half”-pattern mixing caused dissolution in particular areas of the network where the changes in the pore-to-pore conductivities lead to a change in continuum-scale permeability that was larger than the “Check”-pattern produced for the same amount of volume change. These results indicated the importance of localized dissolution, which creates pathways across the network that increases the continuum-scale conductivity.

Figure 10.

Evolution of the (a) porosity and (b) permeability of the system with respect to injected pore volumes for the analysis of dissolution for different mixing patterns at the boundary. (c) The log10(k) versus porosity relationship for these simulations.

[46] The fact that the increase in permeability and porosity was significant for all three scenarios indicates that the mixing was not only limited to the pores next to the boundary but that the entire network had enough mixing of reactive fluids to create dissolution along the whole length of the network. Figure 11 shows the average pore volume change across the network corresponding to an initial porosity of 0.3 for all three mixing scenarios. No clear pattern of dissolution is evident; it seems that the entire network saw some form of dissolution. However, due to the low-pressure gradient imposed for these simulations, most of the dissolution happened near the inlet. The “Check” pattern produces somewhat of a uniform dissolution front at the boundary because it is the one that promotes the most amount of mixing across the faces of the pore network. This observation indicates that mixing lengths are very small compared to the overall size of the volume. That is, enough mixing is promoted in the first pores of the network to create uniform dissolution.

Figure 11.

Set of plots showing the average volume change (averaged in the y direction, transverse to flow) along the x direction and z direction for different mixing scenarios. Black lines correspond to the “Check” pattern, dashed magenta lines correspond to the “Cross” pattern, and the dash and dotted blue lines correspond to the “Half” pattern.

[47] The results of porosity and permeability evolution for the precipitation-induced simulations, in which we used two solutions that when mixed are oversaturated, are shown in Figure 12. As expected, the introduction of only one fluid at a time (i.e., pH 8.48 or 8.69) produces minimal change. However, when mixtures of these fluids are injected, the changes are more substantial. The decrease in permeability and porosity is largest when three fluids are mixing; that is, when the resident brine (pH 8.2) and the two inflowing waters are mixed. Once the initial resident brine is completely flushed from the system there is a more gradual decrease in the permeability and porosity due to the smaller amounts of calcium and carbonate ions in the two inflowing waters. This shows that the mixing of the three waters creates a more effective scenario for reducing the permeability than when only the two boundary waters mix. Emmanuel and Berkowitz [2005] found similar results when they modeled a continuum scale 2-D-channel with increase precipitation and porosity reduction in regions where significant mixing occurred.

Figure 12.

Evolution of the (a) porosity and (b) permeability with respect to injected pore volumes for three different mixing patterns at the boundary for the case of over-saturated mixtures. (c) The log10(k) versus Porosity relationship for three mixing scenarios.

[48] The main mineral precipitated throughout the evolution of the network was calcite. In Figure 12c, we see that, just as in the dissolution case, there is a difference in the relationships between permeability and porosity depending on how the waters mix, implying nonuniqueness in the evolution of the network. The “Half” mixing scenario sees a larger decrease in porosity but the same change in permeability compared to the “Check” pattern, indicating the presence of localized precipitation, which does not reduce the boundary-to-boundary conductance. More importantly, decrease in permeability due to precipitation is a much slower mechanism than an increase in permeability due to dissolution. While dissolution opens up the network creating more space for unreacted water to flow in, precipitation hinders the movement of the unreactive water and therefore for the same amount of time the increase in permeability and porosity is greater in the dissolution scenario.

[49] Much like in the dissolution case, due to the low-pressure gradient imposed on the network, most of the reaction was concentrated near the inflow boundary. There is, however, a fundamental difference in the precipitation scenario because the precipitation front does not move across the network with time. Once calcite precipitates, the extra ions are depleted from solution reducing the thermodynamic drive for further precipitation, and downstream transport is inhibited due to reduction in pore-to-pore conductivity. This has a direct effect on the permeability of the system, which decreases a significant amount without much change in porosity. In general, it can be said that in a diffusive-dominated scenario small changes in porosity would lead to large change in permeability. The only way to get a significant reduction in permeability and porosity due to precipitation is by changing to an advective-dominated system, where transport of ions across the network are on a time scale that is faster relative to the reaction time scale. In Figure 13, we show this occurrence by plotting the average volume changes across the network for two flow scenarios, the original 0.26 kPa/m (Figure 13a) scenario and a highly advective scenario with an imposed pressure gradient of 13.4 kPa/m (Figure 13b). We plot the average percent volume change as a function of distance away from the boundary after 400 injected pore volumes. The resulting porosity changes in each flow scenario, regardless of the mixing condition, are about the same. However, the higher advection scenario produces more precipitation throughout the network due to its ability to transport ions farther from the inflow boundary. Also, in the highly advective scenario the mixing patterns create different evolutions across the network. The “Check” pattern has precipitated mostly near the inflow boundary because it is quickly mixed, while the “Half” and “Cross” scenarios take longer to mix and therefore spread the precipitation across the network. These simulation scenarios show that precipitation of carbonate minerals would happen in a very small region unless there is a high advective drive that distributes the precipitation.

Figure 13.

For the analysis of precipitation due to over-saturated mixtures, percent volume change averaged over the y-z space for imposed pressure gradients of (a) 0.26 kPa/m and (b) 13.4 kPa/m after 400 injected pore volumes.

5.4. Implications for Power Law Approximations

[50] Usually, continuum-scale models of permeability use a simple constitutive relationship that relates porosity to permeability such as:

display math(11)

where ko and ϕo refer to the initial permeability and porosity of the system, respectively, and α is the power law parameter. These relationships are derived theoretically or by fitting parameters to laboratory experiments and usually take a cubic law form. Other relationships include changes in reactive surface area, hydraulic radius, and effective and critical porosity [Bernabé et al., 2003; Gouze and Luquot, 2011; Martys et al., 1994]. In this work, it has been shown that there is not a unique relationship even when considering the same porous material. The evolution of the network in time and in space dictates the relationship between permeability and porosity. Figure 14 explores the relationship between α as shown in equation (11) and the porosity evolution for some of the simulations in this work. The three different sets of α versus porosity relationships in Figure 14 relate to three analyses in this work. For at least the first two scenarios (Figures 14a and 14b), there is a region where the cubic law (α ∼ 3) applies, between the porosity values of 0.15 and 0.25, which coincidentally is within the range of reported porosities in laboratory experiments [Bernabé et al., 1982; Bourbie and Zinszner, 1985; Pape et al., 1999]. However, as porosity increases, higher values of α are needed to relate porosity to permeability. This is important because usually laboratory experiments do not report permeability and porosity relationships for large values of porosity (i.e., greater than 30% porosity). The implication of these results is that a cubic law might be applicable only for a small region of porosity values and as the porosity increases the α value should also increase.

Figure 14.

Values of power law parameter α as a function of network porosities, for (a) different inflowing chemistries; for (b) different pressure gradient conditions, where solid lines correspond to an inflowing pH = 3 and dashed lines for an inflowing pH = 5; and for (c) different mixing scenario for the dissolution simulations with pressure gradient of 0.26 kPa/m.

[51] In Figure 14b, it can also be seen that in high-pressure gradient conditions (1.6 and 13.4 kPa/m) the α value holds constant at 6 between porosities of 0.25 and 0.5 (compared to the other curves). This fairly constant value of α could be explained by realizing that all the pores across the network are dissolving uniformly. The increase in porosity can be thought of being constant and therefore a constant power relationship holds for a longer span of porosity values. This finding was also presented by Martys et al. [1994] in a different form. They showed using a random packing of spheres model that as they decreased the porosity of the system (by increasing sphere size) a single value of α was able to capture the evolution of the permeability with respect to porosity. On the other hand, the evolution of α for the curves for low-pressure gradient simulations (i.e., curves on Figure 14a) reflects the slower movement of the dissolution front across the system, making changes in porosity occur in different section of the network with time, starting from the pores near the boundary and eventually dissolving the nodes at the outlet of the network.

[52] In all the cases, as porosities reach their final value the power law parameter increases almost in a linear fashion as a function of porosity. This increase can be explained by the fact that porosity might change by a factor between 2 and 6 in relation to its original value throughout the evolution of the network, but the permeability might change between 102 and 104, in relation to its original value, and therefore it follows from equation (13) that α should increase as porosity increases. The initial large values of α were explained by Bernabé et al. [2003] by stating that selective enlargement of the well-conducting pores create conditions for a large change in permeability with little change in porosity.

[53] In the third analysis (Figure 14c), the evolution of α at early times is controlled by dissolution near the inlet, and the resulting localized increases in pore-to-pore conductivity. However, as porosity increase becomes more uniform across the network, the evolution paths of α in these mixing scenarios resemble those of the other analyses presented in Figures 14a and 14b.

[54] These results show the wide range of α values that can be expected, even for a relatively simple system, and that in order to accurately capture the evolution of permeability at the continuum scale a single value of α cannot be used, and the use of a value of 3 would seriously underestimate the change in permeability. For this specific study, we see that the value of α can take values from as low 2.5 up to values of 10. In general, the values of α start high then quickly drop down to its lowest value (∼2 or 3) and then slowly move up to a maximum value (∼5–9). The approach taken to represent the evolution of permeability at the continuum scale can be achieved either by carrying more information from the pore scale in order to know a priori the right α values to choose (i.e., pH/SI regime, mixing scenario, boundary conditions, etc.) or making scaling arguments that would eliminate the least likely scenarios. In the examples here, the high-pressure gradient versus the low-pressure gradient curves can be used as bounding curves of the possible α values. The results shown coincide with the laboratory experiments by Gouze and Luquot [2011], which found a range of values to fit a power law relationship that linked porosity to permeability. They found much like we have that this parameter was dependent on the type of rock, the flow conditions, and the reactivity of the system.

6. Conclusions

[55] This study investigated the evolution of continuum-scale parameters, permeability and porosity, induced by transport and reactions at the pore scale. The methodology developed for this study consists of modeling the transport of CO2-rich waters in a pore network model, the subsequent chemical reactions that occur with primary or secondary minerals and the change in pore volumes that affect continuum-scale parameters. This study showed how the permeability and porosity of a pore network evolves depending on the chemical reactivity of the inflowing water, the transport processes that dominate the system (i.e., advective versus diffusive), and the mixing patterns that might occur within a network.

[56] Overall there are six major conclusions that can be made from this study: (1) In general, the higher the solution concentration of carbonic acid in the inflowing water, the faster the system dissolves, with the caveat that the more calcium ions in solution, the less dissolution occurs. Neither the calcite saturation index nor any other single geochemical variable, alone, is sufficient to predict the extent of porosity change. (2) Despite the complexity in the effect of solution chemistry on the extent of change in porosity and permeability, the relationship of the permeability to the porosity of the network can be said to be unique, for a given flow condition and a similar porous structure as the one modeled in this work with homogenous mineral distribution. (3) In a CO2 sequestration operation, it is more likely for dissolution to occur than precipitation due to the anticipated high concentration of CT and the low values of SI and pH that occur. (4) Slow flow conditions create nonunique porosity-permeability relationships, whereas fast flow conditions produce a unique porosity-permeability relationship because these conditions produce uniform changes across the domain. (5) Mixing waters that are by themselves unreactive can create conditions of precipitation or dissolution. The larger the spatial extent of mixing of these waters within the network, the larger the change in permeability and porosity. (6) Using a power law relationship with a single exponent parameter (e.g., cubic law) to equate a porosity change to a permeability change may not capture the correct evolution of the network permeability. The constant values that this parameter takes are between 2 and 10 and depend on the flow rate and geochemistry of the system, but a value of 6 may be representative of high flow rate conditions.

[57] These insights illuminate the complex phenomena involved in the modification of material properties of a porous medium when CO2-rich waters react with carbonate rocks. The insights offered are to be considered when, in the future, continuum-scale models try to account for dynamic material properties that might affect the trapping capabilities of carbonate formations within a geological sequestration of CO2 framework.


[58] This project was supported through funding from the U.S. Department of Energy under award DE-FE0000749 and DE-FG02–09ER64748. The authors also wish to acknowledge Rudolf Hilfer from the University of Stuttgart for providing the synthetic µ-CT images of the carbonate rock, and Brent Lindquist from Stony Brook University for processing the µ-CT image using 3DMA-Rock.