Abstract
Due to spatial scaling effects, there is a discrepancy in mineral dissolution rates measured at different spatial scales. Many reasons for this spatial scaling effect can be given. We investigate one such reason, i.e., how porescale spatial heterogeneity in porous media affects overall mineral dissolution rates. Using the bundleoftubes model as an analogy for porous media, we show that the Darcyscale reaction order increases as the statistical similarity between the pore sizes and the effectivesurfacearea ratio of the porous sample decreases. The analytical results quantify mineral spatial heterogeneity using the Darcyscale reaction order and give a mechanistic explanation to the usage of reaction order in Darcyscale modeling. The relation is used as a constitutive relation of reactive transport at the Darcy scale. We test the constitutive relation by simulating flowthrough experiments. The proposed constitutive relation is able to model the solute breakthrough curve of the simulations. Our results imply that we can infer mineral spatial heterogeneity of a porous media using measured solute concentration over time in a flowthrough dissolution experiment.
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1 Introduction
Geochemical reactions such as mineral dissolution play an essential role in determining water chemistry, soil formation, biogeochemical cycling, and global climate (Wen and Li 2017). Mineral reactions can also occur by injecting CO\(_{2}\) in subsurface reservoirs during geothermal energy extraction (Randolph and Saar 2011; Adams et al. 2021; Ezekiel et al. 2022) or calcite mineralization (Pogge von Strandmann et al. 1983). One of the most significant obstacles to understanding the geochemical reactivity of natural subsurface environments stems from the multitude of spatial scales that have to be considered (Noiriel et al. 2012). Due to spatial scaling effects, mineral dissolution rates are known to be 3–6 orders of magnitude lower in the field than when measured in the laboratory (White and Brantley 2003; NavarreSitchler and Brantley 2007; Maher 2010; Moore et al. 2012). The deviation in mineral dissolution rates strongly limits the extrapolation of kinetic dissolution models and parameters characterized at the laboratory to natural systems (Li et al. 2020).
Many factors are responsible for the spatial scaling effects of mineral dissolution rates. This work focuses on how poresize distribution—which contributes to hydraulic heterogeneity—and spatial mineral distribution causes such spatial scaling effects. Li et al. (2007) performed simulations with various settings of spatial mineral distribution and concluded that spatial mineral distribution has a significant scaling effect when the reactive minerals are of small but typical proportions. Experimental studies using a column packed with quartz and magnesite have confirmed the significant role of spatial heterogeneities in subsurface reactive transport and can be used to quantify the effect of spatial mineral distribution on dissolution rates (Salehikhoo et al. 2013; Li et al. 2014; Li and Salehikhoo 2015). Xray microtomography provides observations of the impact of physical and chemical heterogeneity on reaction rates in multimineral porous media (Tutolo et al. 2015; Luhmann et al. 2017; AlKhulaifi et al. 2017, 2018, 2019; Menke et al. 2016, 2018). Fischer et al. (2014) and Fischer and Luttge (2017) studied how mineral surface roughness at the nanometer scale affects surface reaction rates and proposed to upscale the mineral reaction rate using Monte Carlo simulations. Ma et al. (2019, 2021) quantified the accessible surface area of minerals in a sandstone using scanning electron microscopy (SEM) images and Brunauer–Emmett–Teller (BET) surface area measurements.
Regarding the influence of hydraulic heterogeneity, Wen and Li (2017) and Jung and NavarreSitchler (2018a) performed reactive transport simulations on stochastically generated permeability fields and studied how hydrologic heterogeneity affects mineral dissolution rates. Using Monte Carlo simulations, Jung and NavarreSitchler (2018b) further studied the time dependency of mineral dissolution rates, and Wen and Li (2018) developed an upscaled rate law for mineral dissolution in heterogeneous media under variable residencetime and lengthscale conditions. Li et al. (2020) upscaled mineral dissolution rates in a porous medium with a random permeability field using the fluid traveltime distribution function. The works mentioned above used the transition state theory (Lasaga 1998) to model mineral dissolution rates, with a macroscale reaction order of unity \((n=1)\). This is reasonable since there are no physical explanations why this macroscale reaction order should not be one (Lasaga 1998; Brantley and Conrad 2008).
However, there are rate models with a macroscale reaction order of 2 in kinetics of crystal growth (Nancollas 1968; Reddy 1975, 1977). Such secondorder kinetics are used for modeling surface spiral growth (Nielsen 1984). Considering calcite as our mineral of particular interest, fitting experimental data using a reaction order larger than 1 is common, especially when the saturation is close to equilibrium (Plummer and Wigley 1976; Plummer et al. 1978; Palmer 1991; Svensson and Dreybrodt 1992). A higher reaction order is also observed in modeling calcite dissolution in seawater (e.g., Subhas et al. 2015; Naviaux et al. 2019).
In this work, we use analytical techniques to develop a constitutive relation of mineral dissolution kinetics in porous media based on models with a reaction order \({n>1}\). We characterize hydraulic heterogeneity and mineral spatial heterogeneity by the longitudinal dispersivity and the reaction order. We use the bundleoftubes analogy to show how the reaction order relates to both hydraulic and spatial mineral heterogeneity in porous media. Furthermore, we simulate experimental scenarios involving advective and dispersive transport using such a constitutive relation. Our results show how concentration breakthrough curves of the reactive species reveal information of both the hydraulic and chemical heterogeneity of porous media.
2 Materials and Methods
This section first introduces reactive transport models at the pore scale and the Darcy scale. Then, we lay out statistical distributions of pore sizes and effectivesurfacearea ratios. Such distributions can thus define the volumeaveraged concentration. Constitutive relations based on the Darcyscale reaction order is established using Taylor series expansions. We check the applicability of the constitutive relations by comparing the modeled concentration and the volumeaveraged concentration using a goodnessoffit measure, the Jensen–Shannon divergence. Finally, we explain how one can apply the proposed constitutive relation using a flowthrough experiment.
2.1 Reactive Transport at the Pore Scale
At the pore scale, we model the transport of a mineralforming solute by
where \(C^{*}\) is the solute concentration in the fluid (mol m\(^{3}\)), \(\vec {u}\) is the fluid velocity (m s\(^{1}\)), D is the molecular diffusivity (m\(^{2}\) s\(^{1}\)). In Eq. (1), we consider \(C^{*}\) as the unknown variable, and the fluid velocity, \(\vec {u}\), is given. The heterogeneous reaction is defined as
where \({\hat{{\varvec{n}}}}\) is the unit normal vector pointing outwards from the fluid to the solid surface, and \(R_{\mathrm{het}}\) is the rate of the heterogeneous reaction per surface area (mol m\(^{2}\) s\(^{1}\)) as a function of \(C^{*}\). Since we consider a bundle of capillary tubes as a model of a porous medium, we introduce Eq. (1) described in cylindrical coordinates:
and the boundary condition, Eq. (2),
where r is the radius of the capillary tube (m). We assume fully developed axisymmetric fluid flow, \(u_{r}=u_\theta =0\), constant molecular diffusivity, and axisymmetric solute concentration. Furthermore, we assume the heterogeneous reaction causes negligible change to the tube radius and the mineral surface area. Regarding dissolution and precipitation reactions leading to changes in the pore geometry of porous media and fractures, we refer the reader to the works of Sallés et al. (1993), Békri et al. (1995, 1997) for further discussion. Following Arce et al. (2005), the areaaveraged concentration of the capillary tube is
To develop an areaaveraged or upscaled equation, we perform areaaveraging of Eq. (3) to obtain
Evaluate the integral and rearrange:
Following Paine et al. (1983), we further expand the convective term using Gray (1975)’s representation:
where \({\tilde{u}}_{z}\) and \({\widetilde{C}}^{*}\) are the spatial deviation terms of the fluid velocity and solute concentration, respectively. Considering only transport, Paine et al. (1983) showed that the dispersive transport term can be represented by the Taylor–Aris theory of dispersion:
with the constraints \(\langle u_{z} \rangle r/D \gg 1\) and \(D t^{*} /r^{2} \gg 1\). Ananthakrishnan et al. (1965) delineated dispersion effects for \(\langle u_{z} \rangle r/D = O(1)\) and introduced a timedependent dispersion coefficient (m\(^{2}\) s\(^{1}\)) to model dispersion in the regime not covered by the Taylor–Aris dispersion. We utilize the timedependent dispersion coefficient obtained by the method of moments (Barton 1983) and asymptotic techniques (Vrentas and Vrentas 1988)
where \(j_{n}\) is the nth root of Bessel function of the first kind of order 1 (Meng and Yang 2017). The timedependent term in \({\mathscr{D}}_{{\mathrm{L}}}\) serves as a correction term for Taylor–Aris dispersion. Such a definition of dispersive transport is suitable for simple initial conditions, e.g., injecting a pulse of solute. Considering general initial conditions under steady flow conditions, one has to include source terms to address the complexity of dispersive transport in a cylindrical tube (Taghizadeh et al. 2020). Since the averaged convection term can be described by averaged quantities, we divert our focus to the source term on the righthand side of Eq. (7):
The heterogeneous reaction term depends on the solute concentration at the solid–fluid boundary in the radial direction. For reaction–diffusion systems in a cylindrical tube with a linear irreversible heterogeneous reaction, \({R_{\mathrm{het}}=k C^{*} _{r^{*}=r}}\), Arce et al. (2005) performed an orderofmagnitude estimate and claimed that when \(kr/D \ll 1\), the solute concentration on the boundary can be approximated by
where k is a rate constant (m s\(^{1}\)). The constraint \(kr/D \ll 1\) gives us a good range of k, since the pore radius is usually some tens of micrometers. Combining Eqs. (7)–(12), the areaaveraged solute transport equation in a cylindrical pore reads
This is an ad hoc approach of developing the averaged model. Though not perfect, such a onedimensional (1D) expression does represent reactive transport in a pore throat in many porenetwork models (Algive et al. 2010; Raoof et al. 2012, 2013; Varloteaux et al. 2013a, b; Qin and Hassanizadeh 2015; Bekri et al. 2015; Gostick et al. 2016; Xiong et al. 2016; Esteves et al. 2020). Rigorous upscaling of reactive flow in thin geometries has been performed with the following considerations: general mass action kinetics (van Duijn and Pop 2004), dominant Péclet and Damköhler numbers (Mikelić et al. 2006; van Duijn et al. 2008), changes in porescale geometry (van Noorden 2009b; Kumar et al. 2011), changes in porescale geometry with nonisothermal effects (Bringedal et al. 2015, 2016), changes in porescale geometry with twophase flow (von Wolff and Pop 2021), and coexisting homogeneous reactions (Boso and Battiato 2013). For perforated porous media, rigorous upscaling of reactive flow involving dissolution or precipitation processes has been performed (Kumar et al. 2016), with considerations of changing porescale geometry (van Noorden 2009a) as well as multiphase reaction–diffusion systems (Redeker et al. 2016).
The dimensionless number, kr/D, is often identified as the porescale Thiele modulus squared \(\phi ^{2}\) or the Damköhler number Da. Balakotaiah et al. (1995) used the invariant manifold expansion for advection and heterogeneous reaction in a cylindrical tube and showed that when \(\phi ^{2} \ll 1\), dispersion effects can be modeled by the Taylor–Aris theory. For reaction–diffusion systems in porous media, ValdésParada et al. (2017) showed that the constraint \(\phi ^{2} \ll 1\) can be loosened to \(\phi ^{2} \le 1\) by modifying the effective diffusivity and the effective reaction rate constant at the macro scale, and Bourbatache et al. (2020) recovered classical homogenized diffusive equations for small values of Damköhler numbers (defined by the length scale of the representative elementary volume).
2.1.1 SecondOrder Dissolution Kinetics
Throughout this work, the heterogeneous reaction of interest is mineral dissolution. A common mineral dissolution model is of second order:
where \(k_{\mathrm{d}}\) is the dissolution rate constant (mol m\(^{2}\) s\(^{1}\)) and \(\smash [b]{k_{\mathrm{p}}^{\mathrm{II}}}\) is the precipitation rate constant of secondorder kinetics (mol\(^{1}\) m\(^{4}\) s\(^{1}\)). We assume the solution is dilute, such that the activity coefficient of the solute is unity. We relate the dissolution model with a more prevalent formulation involving the solubility product constant, \(K_{\mathrm{sp}}\), and the ion activity product, IAP:
One can switch between the two forms by stating \(\mathrm{IAP}=C^{*2}\) and \(K_{\mathrm{sp}}=k_{\mathrm{d}}/k_{\mathrm{p}}^{\mathrm{II}}\). Using calcite as an example, when pH\(>5.5\), the dominant dissolving species at the diffusion boundary layer are Ca\(^{2+}\) and CO\(_{3}^{2}\) (Sjöberg and Rickard 1984). Thus, at intermediate pH values, \(\mathrm{IAP} = C^{*}_{{\mathrm{Ca}}^{2+}} C^{*}_{{\mathrm{CO}}_{3}^{2}}\). The assumption of electroneutrality close to the mineral surface yields \(\mathrm{IAP} = (C^{*}_{{\mathrm{Ca}}^{2+}})^{2}\), which results in secondorder dissolution kinetics (Ebigbo et al. 2012; Levenson and Emmanuel 2013). For magnesite dissolution at neutral to alkaline pH regimes, such secondorder kinetics is also suitable (Salehikhoo et al. 2013; Wen and Li 2017). Secondorder dissolution/precipitation kinetics also appears in upscaling reactive transport processes in porous media with attention to moving solid–fluid interfaces (Ray et al. 2019; Bringedal et al. 2020; Gärttner et al. 2020). Combining Eqs. (13) and (14) yields
We relate the dimensional and nondimensional quantities by
where t is the nondimensional time, z is the nondimensional space, C is the nondimensional solute concentration, and \(L_{z}\) is the length of the cylindrical pore (m). Then, we nondimensionalize Eq. (16) by the following scaling of time and concentration
where [t] is the diffusive time scale (s). The variables in square brackets remove the physical dimension of the starred variables and refer to characteristic quantities. Thus we have a nondimensional equation of reactive transport
where \(\epsilon \) is the aspect ratio, Pe is the Péclet number, and Da is the Damköhler number. In Eq. (20), we further relate Da to \(\epsilon \) and the porescale Thiele modulus (of secondorder reactions), denoted as \(r\sqrt{k_{\mathrm{d}} \smash [b]{k^{\mathrm{II}}_{\mathrm{p}}}}/D\), to show that for such a definition of the Damköhler number, \({\text{Da}} > 1\) does not necessarily break the assumption of a small Thiele modulus. For a slender cylindrical pore, \(\epsilon \ll 1\), and a controlled flow rate, \({\text{Pe}}\, \epsilon \ll 1\), we neglect the effect of dispersion. The \(\exp \left( {j_{n}^{2}\epsilon ^{2}\, t}\right) \) term in the dispersion correction term also indicates such a correction term vanishes rapidly over time. Hence, the nondimensional reactive transport equation reads
2.1.2 FirstOrder Dissolution Kinetics
Firstorderkinetic models are uncommon, since chemical reactions often involve two reagents (Cussler 2009). If one wanted to make use of firstorder kinetics, one would have to assume the concentration of a mineralforming ion is in excess or constant (Meile and Tuncay 2006) or limit the usage of firstorder kinetics to low solute concentrations (Kaufmann and Dreybrodt 2007). Nonetheless, we introduce the model of firstorder mineral dissolution
where \(k_{\mathrm{p}}^{\mathrm{I}}\) is the precipitationrate constant of firstorder kinetics (m s\(^{1}\)). Firstorder dissolution kinetics is utilized to model evolution of karst aquifers (Gabrovšek and Dreybrodt 2001). They apply to dissolution rates of various minerals such as gypsum, rocksalt, calcium carbonate, and quartz (Jeschke and Dreybrodt 2002). Considering a general surface reaction, firstorder kinetic models resemble adsorption and desorption kinetics studied by, e.g., Zhang et al. (2017), as well as the Noyes–Whitney model of drug dissolution (Dokoumetzidis and Macheras 2006). Firstorder kinetics is also applied to heterogeneous reactions between living cells and extracellular fluids (SantosSánchez et al. 2016). Combining Eqs. (13) and (22) and ignoring dispersion effects, we scale time and concentration by
such that the Péclet and the Damköhler numbers are
Notice the porescale Thiele modulus squared, \(\phi ^{2} = kr/D\), appears in the form of the precipitationrate constant, \(k_ {\mathrm{p}}^{\mathrm{I}}\). Since \(\phi ^{2}\) is proportional to \({\text{Da}} \, \epsilon ^{2}\), the assumption of \(\phi ^{2} \ll 1\) is fulfilled, similar to the relation between the Damköhler number and the porescale Thiele modulus of secondorder kinetics. The nondimensional reactive transport equation of firstorder kinetics is therefore
As for other possible models of heterogeneous kinetics, Qiu et al. (2017) provided a comprehensive review of upscaling reactive transport processes in porous media, e.g., Michaelis–Mententype kinetics (Wood et al. 2007; Dadvar and Sahimi 2007), Monodtype kinetics (Heße et al. 2009), nonlinear kinetics that reduce to firstorder kinetics when the reaction order is 1 (Guo et al. 2015).
We define the average velocity in a cylindrical pore using the Hagen–Poiseuille equation
where \(\eta \) is the dynamic viscosity of the fluid (Pa s) and \(\Delta P\) is the pressure difference between the inlet and the outlet (Pa).
2.2 Reactive Transport at the Darcy Scale
We use the bundleoftubes analogy to model reactive transport at the Darcy scale (Kozeny 1927). We define the specific mineral surface area (m\(^{1}\)) of the porous medium as
where A is the mineral surface area, V is the bulk volume of the porous media (m\(^{3}\)), \(\omega \) is the ratio between mineral surface area and the total surface area of a pore, and N is the total number of pores. We assume that the pores have the same length, \(L_{z}\), as the porous medium, such that the tortuosity is 1. The porosity of the porous medium is
where \(V_{\mathrm{f}}\) is the fluid volume. Dividing Eq. (27) by Eq. (28), we obtain
Recall the Damköhler number for firstorder kinetics, Eq. (24), the Damköhler number at the Darcy scale is therefore
where the characteristic length L is chosen to be the length of the porous medium, \(L_{z}\). The Darcyscale Damköhler number for secondorder kinetics is
Since we consider the porous medium as a bundle of tubes, the seepage velocity of the porous medium can be defined using a volumeaveraged velocity,
Hence, the Péclet number at the Darcy scale is
Dispersion effects arise when the pore sizes are not uniform (Carbonell 1979; Arriaza and Ghezzehei 2013; Meng and Yang 2017). Therefore, we introduce a longitudinal dispersion coefficient (m\(^{2}\) s\(^{1}\)),
where \(\alpha _{\mathrm{L}}\) is the longitudinal dispersivity with the unit of length (m). We derive the longitudinal dispersion coefficient using the spatial moments of the averaged solute concentration of the bundle of tubes model
The shape factor f(r) abbreviates the expression of pore sizes in Eq. (35). Detailed derivations are presented in “Appendix 1”. The longitudinal dispersivity is
where \(\bar{z}\) is the center of mass of an instantaneous source injected in the porous domain. The spatial moment analysis showed that the longitudinal dispersivity increases linearly as the distance traveled of an instantaneous source, which is reasonable in an infinite domain. In recent studies based on porenetwork modeling, Mahmoodlu et al. (2020) observed the longitudinal dispersivity increases as travel distance increases.
By scaling the longitudinal dispersion coefficient by the molecular diffusivity yields the nondimensional reactive transport model at the Darcy scale,
where \(q_{\mathrm{d}}\) is the reaction term that is yet to be defined. Such a Darcyscale model resembles the upscaled formulation of heterogeneous reaction systems including convective transport in porous media, which incorporates the effect of pore geometry in the dispersivity tensor and the effective reactionrate constant (Wood et al. 2007; Battiato et al. 2009; Battiato and Tartakovsky 2011; ValdésParada et al. 2011; Guo et al. 2015; Qiu et al. 2017). Our proposed model assumes the reaction term does not contribute to the overall dispersion coefficient. The flowthrough experiment introduced in Sect. 2.7 circumvents such effects by eliminating spatial variation of solute concentration. Note that when all pores have the same radius and effectivesurfacearea ratio, Eq. (37) reduces to the equation for a single pore. In the next section, we discuss statistical distributions of pore sizes and effectivesurfacearea ratios.
2.3 Statistical Distributions of Pore Sizes and EffectiveSurfaceArea Ratios
The pore sizes of porous media usually follow a lognormal distribution (Shi et al. 1991; Hefny et al. 2020). The probability density function of lognormally distributed pore sizes, r, is
where \(\mu \) and \(\sigma \) are the mean and standard deviation of the variable’s natural logarithm, respectively. The sum of pore sizes raised to the power of m can be described by a raw statistical moment,
If the poresize distribution is known, the intrinsic permeability, \(k_{\mathrm{I}}\), of the bundle can be calculated as
Using the moments of the lognormal distribution, we can now clarify the shape factor,
Since we are interested in not only the poresize distribution but also the effective surface area of the reactive minerals, we utilize the effectivesurfacearea ratio, \(\omega \), which has a value between zero and one. A convenient choice for modeling the distribution of a variable bounded by zero and one is the beta distribution. The probability density function of the beta distribution is
where \(\Gamma (\cdot )\) is the gamma function. Variables a and b shape the beta distribution.
The specific surface area over porosity, \(S/\psi \), which is a part of the Darcyscale Damköhler number, Eqs. (30) and (31), can be defined as
Such a definition of \(S/\psi \) depends on the pore geometry. Inferring from Hussaini and Dvorkin (2021)’s compilation of specific surface area versus porosity using digital images of natural rocks, \(S/\psi \) can range from 0.15 (Fountainebleau sandstone) to 0.7 (Kentucky sandstone) when E\((\omega ) = 1\).
2.4 A Constitutive Relation for Reaction at the Darcy Scale
A constitutive relation is an additional equation that specifies properties of a material. For example, the longitudinal dispersivity describes the dispersion of solute introduced by variations in fluid velocities in a porous material. In this section, we develop a constitutive relation that models the average solute concentration of a porous medium with varying reaction rates in pores.
2.4.1 FirstOrder Kinetics
Before we approach the full reactive transport problem—Eq. (37)—we start with a single pore involving only mineral dissolution,
where \(C_{0}\) is the initial concentration of the mineralforming solute in the pore. We consider an aspect ratio, \(r/L_{z}\), small enough such that a volumeaveraged concentration is representative. Using Eq. (24), the Damköhler number in a cylindrical pore is
where \({2\omega /r}\) defines the porescale geometry and \({L_{z}^{2} k_{\mathrm{p}}^{\mathrm{I}}/D}\) defines the physics of the problem. In this work, we focus on analyzing how porescale geometry affects solute concentration and reaction rate over time, assuming constant \({L_{z}^{2} k_{\mathrm{p}}^{\mathrm{I}}/D}\). We abbreviate \({L_{z}^{2} k_{\mathrm{p}}^{\mathrm{I}}/D}\) to \({\text{Da}}_{\mathrm{p}}\). The solution for the singlepore reaction problem is
We define the volumeaveraged solution as
The bounds of the integral over pore size, r, should correspond to the bounds of the prescribed poresize distribution. Consider \(C_{0}=0\) and expand the volumeaveraged solution using a Taylor series around \(t=0\):
Let us now describe the Darcyscale reaction in the same manner as we did for a single pore, Eq. (44). In this case, the definition of the Darcyscale Damköhler number,
leads to a firstorder approximation of the volumeaveraged concentration. To better approximate \(C_{\mathrm{avg}}\), we propose a nonlinear reactionrate model as a constitutive relation,
where n is the reaction order (Lasaga 1998). Many researchers have attempted to explain values of the reaction order in terms of dissolution or precipitation processes (Blum and Lasaga 1987; Teng et al. 2000). However, attributions of a process on the basis of this exponent are generally not defensible without further observations (Brantley 2008). For further discussion, see Brantley (2003, 2008).
The solution to the nonlinear reactionrate model, Eq. (54), is
Its Taylor series expansion around \(t=0\) while \(C_{0} = 0\) is
We observe that the Darcyscale Damköhler number, Eq. (53), still matches the firstorder term of the volumeaveraged solution, Eq. (52). If we define
then the nonlinear reactionrate model approximates the volumeaveraged concentration to the second order with respect to time. By Cauchy–Schwarz inequality, we infer \(n \ge 1\), which agrees with experimental observations. The inverse square root of this definition of the reaction order, n, is also known as Tucker’s congruence coefficient, which assesses similarity between two variables (LorenzoSeva and ten Berge 2006).
Figure 1 shows scatter plots of pore sizes and effectivesurfacearea ratios. Each point represents an observation of the pore size and the effectivesurfacearea ratio in a porous sample. Tucker’s congruence coefficient, \(r_{\mathrm{c}}\), measures the similarity between pore size and effectivesurfacearea ratio. From the leftmost figure to the rightmost figure, the congruence coefficient decreases as the observations become less similar, or more heterogeneous. Since the reaction order, n, is the squared inverse of the congruence coefficient, the reaction order increases as the heterogeneity increases. Such a definition of the reaction order is a function of the geometric variables \(\omega \) and r. Thus we can use the reaction order to infer porescale spatial heterogeneity of minerals.
2.4.2 SecondOrder Kinetics
We model the singlepore problem with secondorder kinetics by
where the solution is
Using Eq. (20), the Damköhler number in a cylindrical pore is
where \({L_{z}^{2} \sqrt{k_{\mathrm{d}} \smash [b]{k_{\mathrm{p}}^{\mathrm{II}}}}/D}\), which we abbreviate to \({\text{Da}}_{\mathrm{p}}\), defines the physics of the problem for secondorder kinetics. The volumeaveraged concentration is defined using Eq. (47), and we apply Taylor series expansion to the volumeaveraged concentration,
where \(B_{2k}\) is the Bernoulli number (Oldham et al. 2009). The Taylor series expansion of the hyperbolic tangent function, Eq. (60), converges for \({{\text{Da}}\, t < \pi /2}\), which is not of concern since we utilize only the derivatives of \({C_{\mathrm{avg}}(t=0)}\).
We propose the following constitutive relation that describes the solute concentration at the Darcy scale,
We do not attempt a solution for \({\overline{C}}\). However, we can still expand \({\overline{C}}\) around \(t=0\) with Taylor series:
See “Appendix 2” for detailed derivations. Comparing the thirdorder term of Eq. (64) with that of Eq. (66) yields
One can utilize the quadratic formula to explicitly determine n,
where we consider only the larger value of n as a solution. Note that the reaction orders for first and secondorder kinetics are nondimensional and, most importantly, independent of the length scale and the reaction rate constants.
2.5 Goodness of Fit Between the PoreScale and the DarcyScale Concentrations
There exists a variety of goodnessoffit measures between models and experimental observations. For example, the coefficient of determination, \(R^{2}\), is often used to determine the kinetic rate law when applying the integral method (Brantley and Conrad 2008; Zhao and Skelton 2014). The meansquared error is also a goodnessoffit measure, and the leastsquares approach tends to minimize such a metric. We use another goodnessoffit measure, the Jensen–Shannon divergence, which is based on the Kullback–Leibler divergence (Kullback and Leibler 1951). The Kullback–Leibler divergence between some unknown distribution, p(x), and an approximating distribution, q(x), is:
The Kullback–Leibler divergence satisfies \({\text{KL}}(p\Vert q) \ge 0\) with equality if, and only if, \(p(x)=q(x)\) (Bishop 2006). Although \({\text{KL}}(p\Vert q) \ge 0\), it may diverge to infinity depending on the underlying densities (Nielsen 2020). Thus we use the Jensen–Shannon divergence,
which is bounded between 0 and 1 when using base2 logarithms (Lin 1991). Throughout this work, we use the Jensen–Shannon distance, which is defined as the square root of the Jensen–Shannon divergence.
Such a metric measures the distance between probability distributions (Endres and Schindelin 2003; Österreicher and Vajda 2003; Levene and Kononovicius 2019). The following describes how we apply this measure to solute concentration over time, \(C_{\mathrm{avg}}(t)\) and \({\overline{C}}(t)\). Suppose we regard solute concentration over time as cumulative distribution functions. In that case, we measure the Jensen–Shannon divergence of their derivatives, which can be seen as the probability density functions or the reaction rates over time.
When the observed solute concentration is not monotonically increasing over time like a cumulative distribution function, we simply use the root mean square error (RMSE) as a quality measure of the constitutive relation,
2.6 Determination of the DarcyScale Damköhler Number, \({\text{Da}}_{\mathrm{d}}\), and the Reaction Order, n, using power series
We test the constitutive relation using the volumeaveraged concentration \(C_{\mathrm{avg}}(t)\), which is an analogy of solute concentration measurement from a dissolution experiment. Assume the solute concentration can be described by the constitutive relation within a certain error. Then we can use the Taylor series expansions, Eqs. (57) and (66), to obtain the Darcyscale Damköhler number,
which is the initial rate of reaction. When the kinetics is of first order, the reaction order is obtained by differentiating Eq. (57) twice,
Differentiating Eq. (66) thrice and rearranging yields the reaction order for secondorder kinetics,
This method of determining \({\text{Da}}_{\mathrm{d}}\) and n utilizes powerseries expansion and requires only the derivatives of concentration at \(t=0\), given \(C_{\mathrm{avg}}(t=0)=0\).
We consider three sets of lognormally distributed pore sizes, \(R_{1}\), \(R_{2}\), and \(R_{3}\), which have \(S/\psi \approx 0.6\) but different variances. The pore sizes are chosen such that they range from 10 to 80 \(\upmu \)m (Gong et al. 2020). Likewise, we assume the effective surface area follows the beta distribution, where \(\Omega _{1}\) considers most pores are fully reactive, \(\Omega _{2}\) assumes a larger variance of mineral surface area in the pores, and \(\Omega _{3}\) implies that the reactive mineral constitutes a small portion of the porous sample. Figure 2 shows the details of the aforementioned probability distributions.
The products of the random variables R and \(\Omega \) form nine scenarios of the bundleoftubes model, which can be used as benchmarks for our powerseries approach to obtain the Darcyscale Damköhler number and the reaction order. We compare this method with a goodnessoffit minimization using both \({\text{Da}}_{d}\) and n as unknowns, similar to the ideas of nonlinear leastsquares model fitting (Fogler 2016). Initially, the pores are filled with dissolving fluid with no solute concentration, \({C=0}\). Then the mineral starts to dissolve into the fluid, such that the solute concentration increases. We assume we can observe the average concentration, \(C_{\mathrm{avg}}\), without transport effects. To capture the full reaction behavior, the simulation ends when concentration \(C_{\mathrm{avg}}\) is larger than 0.99. The physicsrelated parameters, \({\text{Da}}_{\mathrm{p}}\), is set as 50, such that the Darcyscale Damköhler numbers of the scenarios are at a similar scale.
2.7 FlowThrough Experiment
In the previous section, we test the constitutive relation considering only mineral reaction. To measure the solute concentration of the fluid in a porous sample, one has to push the fluid out of the porous sample. We now discuss the full reactive transport problem. Consider a flowthrough experiment, i.e., injecting fluid that dissolves the mineral in a porous sample. We collect the fluid from the outlet and measure the solute concentration over time. Before the experiment, the porous sample should be saturated by the dissolving fluid, which will result in an initial concentration of the solute in the fluid. Then we start the flowthrough experiment by injecting fluid without solute under high Péclet number. This process aims to flush out all residual concentrations, such that the initial condition of the concentration corresponds to the upscaling approach considering only reaction. We take fluid samples from the outlet and measure the solute concentration. Since we are injecting fluid with zero solute concentration, we expect the measured solute concentration at the outlet to decrease over time. When the outlet concentration is close to zero, we reduce the Péclet number to 0.1–1% of the original Péclet number in order to observe an increase of the outlet solute concentration. The flowthrough experiment creates a Vshaped curve of outlet concentration over time.
We use FEniCS (Alnaes et al. 2015) to solve the transient initial boundary value problem of each pore and apply Eq. (47) to obtain the average concentration. We discuss two types of porous samples, namely the combinations of the poresize distribution and the effectivesurfacearea ratio, \(R_{1} \Omega _{3}\) and \(R_{3} \Omega _{3}\). We test the porous samples for the low and high Damköhlernumber cases, described in the following subsections. In all cases, we consider the molecular diffusivity, D, as \(5\times 10^{9}\) m\(^{2}\) s\(^{1}\).
2.7.1 The Low DamköhlerNumber Case, \({\text{Da}}_{\mathrm{d}} = 0.1\)
The Darcyscale Damköhler number, Eq. (53), is proportional to the length scale squared. This case is suitable when the mineral has low reaction rates or when the porous domain is short (small length scale), e.g., a 5 cm rock sample in a laboratory. We consider this as our “small lengthscale scenario” for which the outlet solute concentration can be measured. Since the Damköhler number is low, the solute concentration during injection of the dissolving fluid should be far from chemical equilibrium. Therefore, we assume the reaction is of first order.
Following the procedure of the flowthrough experiment, we set the initial Péclet number as 10, and the Darcyscale Damköhler number is 0.1. We reduce the Péclet number to 0.01 at 0.15 nondimensional time. The simulation ends at 1.5 nondimensional time, which for the 5 cm rock sample mentioned above would correspond to about 8.7 days.
The solute concentration over time at the outlet of the porous sample is collected from the simulation data. Knowing the Péclet number and that the mineral reaction is of first order, we fit the observed concentration over time with the reactive transport model,
The shape factor, f(r), is defined using Eq. (35). Utilizing the optimization procedures in SciPy (Virtanen et al. 2020), we find \({\text{Da}}_{\mathrm{d}}\) and n by minimizing the RMSE between the observation and the model.
Another method of fitting \({\text{Da}}_{\mathrm{d}}\) and n is to utilize a part of \({C_{\mathrm{avg}}(t)}\), where the diffusion effects are dominant enough (\({{\text{Pe}}_{\mathrm{d}} \ll 1}\)) such that we can treat the concentration as constant over space. Owing to the divergence theorem and the boundary condition \({C(0,t)=0}\) results in
Then, we can perform the leastsquares fitting on the lefthand side to determine \({\text{Da}}_{\mathrm{d}}\) and n. Figure 3 shows the outlet concentration of the flowthrough experiment and highlights the region in which we consider diffusion to be dominant. Such an approach relies heavily on the strongdiffusion assumption and is therefore not suitable for the high Damköhlernumber case, discussed in the next section.
2.7.2 The High DamköhlerNumber Case, \({\text{Da}}_{\mathrm{d}}={4\times 10^{4}}\)
Consider a field experiment in an aquifer, where the length scale is on the order of meters. Here, a fluid is injected in one well and produced at another well. In this case, it may not be possible to observe a concentration breakthrough at the outlet. Thus, we perform simulations of a flowthrough experiment and measure the concentration at the inlet over time. The Darcyscale Damköhler number is \(4\times 10^{4}\). We assume a secondorder kinetics model for the mineral reaction, and our reactive transport model is
We consider three cases of initial Péclet numbers, \(4\times 10^{3}\), \({8\times 10^{3}}\), and \({8\times 10^{4}}\). The injection stops at \({2\times 10^{6}}\) dimensionless time, and the simulation ends at \({1\times 10^{5}}\) dimensionless time, which is roughly 231.5 days considering a 100 m simulation domain.
Though it is practically not possible to observe the inlet concentration during the injection phase, we perform the fitting of \({\text{Da}}_{\mathrm{d}}\), n, and f(r) using all information of \(C_{\mathrm{avg}}\) at the inlet. In the latter phase when injection stops (rising limb of the curve in Fig. 3), the concentration at the inlet can be determined by sampling the fluid in the injection well.
3 Results
In this section, we show the benchmarks of the powerseries approach and the simulations of flowthrough experiments. Then, we discuss the results in Sect. 4.
3.1 Benchmark of the PowerSeries Approach
We benchmark the powerseries approach that obtains the reaction order, n, and the Darcyscale Damköhler number, \({\text{Da}}_{\mathrm{d}}\), using a bundle of tubes characterized by the distributions of pore sizes and effectivesurfacearea ratio described in Sect. 2.6.
Figures 4 and 5 show the benchmark of the firstorder kinetics and secondorder kinetics, respectively. In the upper part of the figures, we plot the contour lines of logscaled Jensen–Shannon distance between \(C_{\mathrm{avg}}\) and \({\overline{C}}\). The red points indicate the approximation of \({\text{Da}}_{\mathrm{d}}\) and n using the powerseries approach. In an ideal case, the red points should be in the minimum of the Jensen–Shannon distance. In the lower part of the figures, we plot the concentration over time of all scenarios.
3.2 FlowThrough Experiment
Figure 6 shows the observed concentration, \(C_{\mathrm{avg}}\), and the modeled concentration, \({\overline{C}}\). Table 1 shows the Darcyscale Damköhler number, the reaction order, and shape factor obtained by direct calculation [Eqs. (53), (58), and (68)], and by RMSE minimization of the low and high Damköhlernumber cases (\({\text{Da}}_{\mathrm{d}}=0.1\) and \({\text{Da}}_{\mathrm{d}}=40{,}000\), respectively) corresponding to the \(R_{1}\Omega _{3}\) and the \(R_{3} \Omega _{3}\) distributions. In the low Damköhlernumber scenarios, we use the leastsquares curve fitting method, Eq. (77), to obtain \({\text{Da}}_{\mathrm{d}}\) and n. The results of the leastsquares curve fitting are \({\text{Da}}_{\mathrm{d}}=0.1\), \(n=1.38\) for the \(R_{1} \Omega _{3}\) scenario, and \({\text{Da}}_{\mathrm{d}}=0.1\), \(n=1.63\) for the \(R_{3} \Omega _{3}\) scenario.
4 Discussion
4.1 Benchmark of the PowerSeries Approach
In Fig. 4, we observed that the powerseries approach obtains \({\text{Da}}_{\mathrm{d}}\) and n close to the minimum Jensen–Shannon distance. The connections between nonlinear fitting of the parameters, \({\text{Da}}_{\mathrm{d}}\) and n, and the geometric information of the porous medium are established since the powerseries approach is exact for retrieving expected values of the poresize and the effectivesurfacearea distributions.
In Fig. 5, we observed general agreement of \({\text{Da}}_{\mathrm{d}}\) and n obtained by the powerseries approach to those at the minimum Jensen–Shannon distance. As the variance of the poresize increases, the obtained \({\text{Da}}_{\mathrm{d}}\) and n (red points) stray from the minimum Jensen–Shannon distance. In the concentration plots of the \(R_{3} \Omega _{1}\), \(R_{3} \Omega _{2}\), and \(R_{3} \Omega _{3}\) cases, the modeled concentration fits well when \({{\overline{C}}<0.5}\). Some discrepancy between \({\overline{C}}\) and \(C_{\mathrm{avg}}\) is present when \({{\overline{C}}>0.5}\). Comparing to the results of firstorder kinetics, where \({\overline{C}}\) fits \(C_{\mathrm{avg}}\) well throughout all concentrations, our averaged model of secondorder kinetics can only fit reactions far from equilibrium (e.g., \({{\overline{C}}<0.5}\)).
The powerseries approach of obtaining \({\text{Da}}_{\mathrm{d}}\) and n suffers from the fact that:

1.
reaction rates at zero concentration can be hard to obtain, and

2.
numerical differentiation of higherorder derivatives can yield spurious results.
Therefore, we require other nonlinear fitting methods by minimizing the divergence between models and observations. In the next section, we discuss the intricacies of nonlinear fitting of solute concentration during flowthrough experiments.
4.2 FlowThrough Experiment
4.2.1 The \(R_{1} \Omega _{3}\) Scenario
The topleft panel of Fig. 6 shows the low Damköhlernumber, firstorderkinetics case, where both the RMSE minimization and the leastsquares fitting method fit the outlet concentration \(C_{\mathrm{avg}}\). Both methods of obtaining \({\text{Da}}_{\mathrm{d}}\) and n are accurate within 15% relative error as confirmed by direct calculation from poresize and effectivesurfacearea distributions. The good agreement can be attributed to the fact that the solute can be mostly flushed out from the porous domain, due to the low Damköhler number. Hence, the solute concentration in each tube goes down to almost zero, and spatial gradients of the solute concentration in the z direction are negligible. This creates a situation similar to the problem considering only reaction effects, which is not the case for the scenarios of high Damköhlernumber, secondorder kinetics.
Focusing on the high Damköhlernumber scenarios, we observe general agreement of fitted parameters. For the case of initial \({\text{Pe}}_{\mathrm{d}}=4000\), the error in \({\text{Da}}_{\mathrm{d}}\) can be attributed to the fact that the volume injected is not enough, such that the dissolution in smaller pores is not observed in \(C_{\mathrm{avg}}\). The case of initial \({\text{Pe}}_{\mathrm{d}}=8000\) approximated \({\text{Da}}_{\mathrm{d}}\) closer to the prescribed value than the previous case. The third case of initial \({\text{Pe}}_{\mathrm{d}}=40{,}000\), performed the best in retrieving \({\text{Da}}_{\mathrm{d}}\). This may be caused by, similar to the low Damköhlernumber case, that the initial solute concentration of all pores is pushed out from the porous domain, such that \(C_{\mathrm{avg}}\) is close to zero. The purpose of comparing these three cases is to emphasize that the inversion of parameters is influenced by how we perform the injection test, namely, by the selection of the initial Péclet number.
Furthermore, we performed a fitting in all cases considering the shape factor as unknown. For the low Damköhler number case, we have a good fit of f(r) within an order of magnitude. However, the obtained value for n exhibits a bigger error. This signals the vagueness of the reaction order and the shape factor in minimizing RMSE, where increasing or decreasing either one of the variables leads to similar RMSE. For the high Damköhler number cases, the obtained \({\text{Da}}_{\mathrm{d}}\) improved slightly comparing to the cases with a prescribed shape factor. All fitted shape factors are within an order of magnitude compared to the theoretical calculations. In particular, for the case of initial \({\text{Pe}}_{\mathrm{d}}=40{,}000\), we observe a good fit of the retrieved shape factor.
4.2.2 The \(R_{3} \Omega _{3}\) Scenario
This scenario tests how effective the pore structure can be inferred in a more dispersive setup. Table 1 summarizes the obtained \({\text{Da}}_{\mathrm{d}}\) and n for both the low and high Damköhlernumber cases. For the low Damköhlernumber case, the minimum RMSE methods do not yield decent approximations of \({\text{Da}}_{\mathrm{d}}\), n, and f(r). The top right panel of Fig. 6 shows that all methods failed to fit the Vshaped region (\(t < 0.3\)), where dispersive effects matter the most. The poresize distribution \(R_{3}\) mainly consists of smaller pores, as compared to \(R_{1}\), which means that the solute to be harder to flush out. This dispersion effect is also demonstrated in the right panel of Fig. 8, where the dispersion model fails to capture the longtailed concentration profile. However, the leastsquares fitting method yields exact \({\text{Da}}_{\mathrm{d}}\), and reasonable n within 10% relative error. The leastsquares fitting method performs well for the low Damköhler number in both the \(R_{1} \Omega _{3}\) and \(R_{3} \Omega _{3}\) cases.
For the high Damköhlernumber scenarios, fitting the averaged concentration using the model, Eq. (78), does not recover porescale information in general. In Fig. 6, the concentration profile of the initial \({\text{Pe}}_{\mathrm{d}}=40{,}000\) with a prescribed shape factor does not fit the averaged concentration. Compared to the results with shapefactor fitting, the prescribed shape factor yields too much dispersion for the given Péclet number. Recall that the longitudinal dispersion coefficient is defined as \(({\text{Pe}}_{\mathrm{d}})^{2}t\,f(r)\).
The case of initial \({\text{Pe}}_{\mathrm{d}}=4000\) with shapefactor fitting gives us the best result in retrieving the model parameters. As shown in both Figs. 6 and 7, the initial Péclet number is not enough to flush out the solute in all pores. However, the dispersion effect is not as pronounced as the initial \({\text{Pe}}_{\mathrm{d}}=40{,}000\) case, due to the lower initial Péclet number. The key to better knowledge of the model parameters and porescale information is to control the initial Péclet number for less dispersion. We demonstrate that the rate of injection influences the parameter fitting of the averaged model, due to dispersion caused by the variance of pore sizes. How to properly choose the initial injection rate for inference of porescale information could be the subject of future studies.
Figure 7 shows the concentration in pores, the averaged concentration, and the modeled concentration of the high Damköhler number scenarios. When the injection stops, the increase of the inlet solute concentration is not only due to the reaction, but also due to the diffusion of solute from the reservoir to the inlet. We attribute the underestimated n to our reactive transport model, Eq. (78), not being able to capture the averaged diffusion effects in each pore, which results in a lower reaction order. Certainly, this situation is not as ideal as the low Damköhlernumber situation, which decouples transport and reaction by flushing out almost all of the residual concentration. The modeled concentration, \({\overline{C}}\), still represents the average behavior of the pore concentrations.
To summarize, although imperfect, our method of parameter estimation using a solute concentration breakthrough curve is useful for modeling the average behavior of reactive transport in porous media. The results suggest it is possible to infer porescale information using the inversion of averaged parameters.
In all of the flowthrough experiments, we considered only one observation point, either the fluid inlet or the fluid outlet. This work serves as a demonstration of the base case with only one observation. To improve the fitting of the reaction order of the high Damköhlernumber cases (especially the ones with \(R_{3} \Omega _{3}\) distribution), one can incorporate more observation points, spatially distributed within the domain.
4.3 The Applicability of the Constitutive Relations
By adding an exponent to the porescale reaction model, the Darcyscale reaction model effectively describes the averaged behavior of reactions taking place independently in different pores of the porous medium under a specific flowthrough experiment. We examine the applicability of the simple approach using the averaged concentration of firstorder kinetics as an example. The averaged concentration, Eq. (48), can be considered as a continuous mixture of exponential distributions,
where \(\lambda \) is a parameter that characterizes the exponential distributions. The finite mixture is known as the hyperexponential distribution, which is utilized for fitting longtail distributions (Feldmann and Whitt 1998; Okada et al. 2020). If \(\lambda \) is a gamma distribution, then \(C_{\mathrm{avg}}(t)\) is a Pareto distribution (Balakrishna and Lai 2009). The concentration of our proposed model, Eq. (55), has the following form when \(C_{0}=0\),
which is the cumulative distribution function of the Pareto distribution, also called the Lomax distribution. There exists a particular ratio distribution of the effective surface area and the pore sizes, \({\Omega /R}\), that satisfies \({\overline{C}}=C_{\mathrm{avg}}\). Such an existence contributes to the effectiveness of the constitutive relation for the firstorder kinetics.
In contrast, the theoretical basis of constitutive relations for secondorder kinetics is less distinct. We simply followed the derivations of firstorder kinetics and exploited the oddity of \(C_{\mathrm{avg}}\) and \({\overline{C}}\) to obtain a secondorder approximation. The solution for secondorder kinetics in a single pore, Eq. (60), can be recast into
which is a cumulative distribution function of a scaled logistic distribution. If we consider time a semiinfinite domain \([0, \infty )\), such a distribution is also known as the half logistic distribution (Balakrishnan 1985). Though we did not find or derive the relationships of the logistic distribution mixture, we denote the possibility of approximating such a mixture using the Pareto distribution, Eq. (80), since the hyperbolic tangent function can be represented by a Laurent series
which is a mixture of exponential distributions. Such expansion techniques would avoid relying on derivatives around \(C=0\) as is done in this study, which may be advantageous but requires more research.
4.4 Limitations and Outlook
The main limitation of this study is that the reaction model we considered is simple and may not be adequate to describe complex geochemical processes, e.g., a rock sample which consists of multiple dissolving minerals. We also assumed the amount of dissolved mineral is minimal such that the geometry of the pores is not affected. Though we considered two common models of first and secondorder kinetics, we treat the rate constant, \(k_{\mathrm{d}}\) and \(k_{\mathrm{p}}\), as constant, and indeed, \(k_{\mathrm{d}}\) and \(k_{\mathrm{p}}\) may change as temperature, pH, or ionic strength changes.
Moreover, the assumption that chemical reactions in each tube occur independently of the other tubes is idealized. A better approach is utilizing porenetwork models. In porenetwork modeling, the porous medium is discretized as a network of pore bodies and pore throats, and the coordination number is defined as number of connections to each pore body. Experimental studies have reported that the average coordination number of a sandstone is \(\sim 4\) (Ioannidis and Chatzis 2000; Øren and Bakke 2003; Hefny et al. 2020). Our bundleoftubes model has an average coordination number of 0 (or 1, if one considers the pore bodies reside at the fluid inlet and outlet boundaries), which is the base case for porenetwork modeling. For this base case, the reaction order does not depend on \({\text{Pe}}_{\mathrm{d}}\) or \({\text{Da}}_{\mathrm{d}}\). The dependence of the reaction order on \({\text{Pe}}_{\mathrm{d}}\) or \({\text{Da}}_{\mathrm{d}}\) for larger coordination numbers requires further studies.
Nonetheless, our simple model reveals a possible mechanistic explanation to the usage of the Darcyscale reaction order larger than one, and how it can reveal geometric information of the porous medium using the solute breakthrough curve. We suggest considerations of the aforementioned limitations as potential topics for future research.
We propose another possible application of this work in view of energy conservation in a porous sample, where there is only one definition of temperature of the fluid, \(T_{\mathrm{f}}^{*}\) and the porous solid, \(T_{\mathrm{s}}^{*}\). A special case of energy conservation without pressure work and viscous heating is analogous to Eq. (13):
where \(D_{\mathrm{f}}\) is the thermal diffusivity of the fluid, and \(q_{T}\) is a heat source introduced by the porous solid. Usually, such an energy conservation model of heat tracer tests assumes thermal equilibrium between the fluid and the porous solid, \(T_{\mathrm{f}}=T_{\mathrm{s}}\), (Shook 2001; Anderson 2005; Saar 2011). However, studies and modeling on thermal disequilibrium between fluid and solid phases have gained interests lately (Karani and Huber 2017; Koch et al. 2021). If we consider a heat tracer test where we create a breakthrough curve like those in Fig. 3, we can model the behavior by
where \({\overline{D}}\) is a certain average of thermal diffusivity of the porous media, and \(L^{*}\) is a characteristic length that defines the heat flux between the solid–fluid boundary. The formulation is similar to our study of reactive transport with firstorder kinetics. Therefore, it is possible to apply the same techniques described in this work to obtain the “reaction order” and infer porescale information.
5 Conclusion
Mineral reaction kinetics defined at the pore scale are not necessarily valid at the Darcy scale. We utilize a bundleoftubes model to study the modeling of dissolution kinetics in porous media at the Darcy scale. By adding an exponent, n (i.e., the Darcyscale reaction order), to first and secondorder kinetics, the resulting constitutive relation approximates the average dissolution rate of the bundleoftubes model under specific experimental conditions. Using the poresize and the effectivesurfacearea ratio distributions to characterize the tube bundles, we expand the solute concentration of dissolving species with Taylor series and thus relate the Darcyscale Damköhler number, \({\text{Da}}_{\mathrm{d}}\), and reaction order with the distribution moments. The Taylor series expansions show that the Darcyscale reaction order of firstorder kinetics is the inverse square root of Tucker’s congruence coefficient (also known as the cosine similarity) between the pore sizes and effectivesurfacearea ratios. Therefore, an increase of reaction order indicates an increase of porescale heterogeneity. Such a relation gives a mechanistic meaning to the reaction order.
Furthermore, we simulate flowthrough experiments of dissolving porous media at the laboratory as well as the field scale and discuss how one can utilize the constitutive relation by fitting a solute concentration breakthrough curve with \({\text{Da}}_{\mathrm{d}}\) and n as unknowns. As an additional benefit, we discuss cases of the flowthrough experiments where the shape factor of longitudinal dispersivity is also considered as a fitting parameter. The inversion is successful, and the fitted parameters are close to the prescribed parameters calculated by the moments of poresize and effectivesurfacearea ratio distributions. We infer that:

1.
detailed porescale information (characterized by functions of moments) can be inferred using averaged Darcyscale quantities (such as solute concentration), and

2.
by analyzing the solute concentration of dissolving minerals over time using flowthrough experiments, we can acquire the Darcyscale reaction order and the dispersion coefficient, which represent heterogeneity at the pore scale.
The relations we derived provide us a quantitative approach to measure the spatial heterogeneity of a porous domain using the Darcyscale reaction order and reveal a mechanistic explanation for \(n>1\).
Change history
29 July 2022
A Correction to this paper has been published: https://doi.org/10.1007/s11242022018389
References
Adams, B.M., Vogler, D., Kuehn, T.H., et al.: Heat depletion in sedimentary basins and its effect on the design and electric power output of CO\(_2\) plume geothermal (CPG) systems. Renew. Energy 172, 1393–1403 (2021). https://doi.org/10.1016/j.renene.2020.11.145
AlKhulaifi, Y., Lin, Q., Blunt, M.J., et al.: Reaction rates in chemically heterogeneous rock: coupled impact of structure and flow properties studied by Xray microtomography. Environ. Sci. Technol. 51, 4108–4116 (2017). https://doi.org/10.1021/acs.est.6b06224
AlKhulaifi, Y., Lin, Q., Blunt, M.J., et al.: Reservoircondition porescale imaging of dolomite reaction with supercritical \(\text{ CO}_2\) acidified brine: effect of porestructure on reaction rate using velocity distribution analysis. Int. J. Greenh. Gas Control 68, 99–111 (2018). https://doi.org/10.1016/j.ijggc.2017.11.011
AlKhulaifi, Y., Lin, Q., Blunt, M.J., et al.: Porescale dissolution by \(\text{ CO}_2\) saturated brine in a multimineral carbonate at reservoir conditions: impact of physical and chemical heterogeneity. Water Resour. Res. 55(4), 3171–3193 (2019). https://doi.org/10.1029/2018WR024137
Algive, L., Bekri, S., Vizika, O.: Porenetwork modeling dedicated to the determination of the petrophysicalproperty changes in the presence of reactive fluid. SPE J. 15(03), 618–633 (2010). https://doi.org/10.2118/124305PA
Alnaes, M.S., Blechta, J., Hake, J., et al.: The FEniCS project version 1.5. Arch. Numer. Softw. 3, 9–23 (2015). https://doi.org/10.11588/ans.2015.100.20553
Ananthakrishnan, V., Gill, W.N., Barduhn, A.J.: Laminar dispersion in capillaries: Part I. Mathematical analysis. AIChE J. 11(6), 1063–1072 (1965). https://doi.org/10.1002/aic.690110620
Anderson, M.P.: Heat as a ground water tracer. Groundwater 43(6), 951–968 (2005). https://doi.org/10.1111/j.17456584.2005.00052.x
Arce, P.E., Quintard, M., Whitaker, S.: The art and science of upscaling. In: Galán, M.A., del Valle, E.M. (eds.) Chemical Engineering: Trends and Developments, pp. 1–39. Wiley, Hoboke (2005). https://doi.org/10.1002/0470025018.ch1
Arriaza, J.L., Ghezzehei, T.A.: Explaining longitudinal hydrodynamic dispersion using variance of pore size distribution. J. Porous Media 16, 11–19 (2013). https://doi.org/10.1615/JPorMedia.v16.i1.20
Balakotaiah, V., Chang, H., Smith, F.T.: Dispersion of chemical solutes in chromatographs and reactors. Philos. Trans. R. Soc. Lond. Ser. A Phys. Eng. Sci. 351, 39–75 (1995). https://doi.org/10.1098/rsta.1995.0025
Balakrishna, N., Lai, C.D.: Univariate distributions. In: Balakrishnan, N., Lai, C.D. (eds.) Continuous Bivariate Distributions, 2nd edn., pp. 1–32. Springer, New York (2009). https://doi.org/10.1007/b101765_1
Balakrishnan, N.: Order statistics from the half logistic distribution. J. Stat. Comput. Simul. 20(4), 287–309 (1985). https://doi.org/10.1080/00949658508810784
Barton, N.G.: On the method of moments for solute dispersion. J. Fluid Mech. 126, 205–218 (1983). https://doi.org/10.1017/S0022112083000117
Battiato, I., Tartakovsky, D.M.: Applicability regimes for macroscopic models of reactive transport in porous media. J. Contam. Hydrol. 120–121(1), 18–26 (2011). https://doi.org/10.1016/j.jconhyd.2010.05.005
Battiato, I., Tartakovsky, D.M., Tartakovsky, A.M., et al.: On breakdown of macroscopic models of mixingcontrolled heterogeneous reactions in porous media. Adv. Water Resour. 32(11), 1664–1673 (2009). https://doi.org/10.1016/j.advwatres.2009.08.008
Békri, S., Thovert, J.F., Adler, P.M.: Dissolution of porous media. Chem. Eng. Sci. 50(17), 2765–2791 (1995). https://doi.org/10.1016/00092509(95)00121K
Békri, S., Thovert, J.F., Adler, P.M.: Dissolution and deposition in fractures. Eng. Geol. 48(3), 283–308 (1997). https://doi.org/10.1016/S00137952(97)000446
Bekri, S., Renard, S., DelpratJannaud, F.: Pore to core scale simulation of the mass transfer with mineral reaction in porous media. Oil Gas Sci. Technol. Rev. IFP Energies Nouv. 70(4), 681–693 (2015). https://doi.org/10.2516/ogst/2014046
Bishop, C.: Pattern Recognition and Machine Learning. Springer, New York (2006)
Blum, A.E., Lasaga, A.C.: Monte Carlo simulations of surface reaction rate laws. In: Stumm, W. (ed.) Aquatic Surface Chemistry: Chemical Processes at the ParticleWater Interface, Chap 10, pp. 255–292. Wiley, Hoboken (1987)
Boso, F., Battiato, I.: Homogenizability conditions for multicomponent reactive transport. Adv. Water Resour. 62, 254–265 (2013). https://doi.org/10.1016/j.advwatres.2013.07.014
Bourbatache, M.K., Millet, O., Moyne, C.: Upscaling diffusion–reaction in porous media. Acta Mech. 231, 2011–2031 (2020). https://doi.org/10.1007/s00707020026319
Brantley, S.L.: Reaction kinetics of primary rockforming minerals under ambient conditions. In: Holland, H.D., Turekian, K.K. (eds.) Treatise on Geochemistry, vol. 5, pp. 73–117. Pergamon, Oxford (2003). https://doi.org/10.1016/B0080437516/050751
Brantley, S.L.: Kinetics of mineral dissolution. In: Brantley, S.L., Kubicki, J.D., White, A.F. (eds.) Kinetics of Water–Rock Interaction, Chap 5, pp. 151–210. Springer, New York (2008). https://doi.org/10.1007/9780387735634_5
Brantley, S.L., Conrad, C.F.: Analysis of rates of geochemical reactions. In: Brantley, S.L., Kubicki, J.D., White, A.F. (eds.) Kinetics of Water–Rock Interaction, Chap 1, pp. 1–38. Springer, New York (2008). https://doi.org/10.1007/9780387735634_1
Bringedal, C., Berre, I., Pop, I.S., et al.: A model for nonisothermal flow and mineral precipitation and dissolution in a thin strip. J. Comput. Appl. Math. 289, 346–355 (2015). https://doi.org/10.1016/j.cam.2014.12.009
Bringedal, C., Berre, I., Pop, I.S., et al.: Upscaling of nonisothermal reactive porous media flow with changing porosity. Transp. Porous Media 114, 371–393 (2016). https://doi.org/10.1007/s1124201505309
Bringedal, C., von Wolff, L., Pop, I.S.: Phase field modeling of precipitation and dissolution processes in porous media: upscaling and numerical experiments. Multiscale Model. Simul. 18(2), 1076–1112 (2020). https://doi.org/10.1137/19M1239003
Carbonell, R.G.: Effect of pore distribution and flow segregation on dispersion in porous media. Chem. Eng. Sci. 34, 1031–1039 (1979). https://doi.org/10.1016/00092509(79)800032
Chatwin, P.C.: The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 43(2), 321–352 (1970). https://doi.org/10.1017/S0022112070002409
Crank, J.: Concentrationdependent diffusion: methods of solution. In: Crank, J. (ed.) The Mathematics of Diffusion, Chap 7, 2nd edn., pp. 104–136. Oxford University Press, Oxford (1975)
Cussler, E.L.: Homogeneous chemical reactions. In: Cussler, E.L., Cussler, E.L. (eds.) Diffusion: Mass Transfer in Fluid Systems. Cambridge Series in Chemical Engineering, 3rd edn., pp. 478–512. Cambridge University Press, Cambridge (2009). https://doi.org/10.1017/CBO9780511805134.019
Dadvar, M., Sahimi, M.: The effective diffusivities in porous media with and without nonlinear reactions. Chem. Eng. Sci. 62(5), 1466–1476 (2007). https://doi.org/10.1016/j.ces.2006.12.002
Dentz, M., de Barros, F.P.J.: Dispersion variance for transport in heterogeneous porous media. Water Resour. Res. 49(6), 3443–3461 (2013). https://doi.org/10.1002/wrcr.20288
Dokoumetzidis, A., Macheras, P.: A century of dissolution research: from Noyes and Whitney to the biopharmaceutics classification system. Int. J. Pharm. 321(1), 1–11 (2006). https://doi.org/10.1016/j.ijpharm.2006.07.011
Ebigbo, A., Phillips, A., Gerlach, R., et al.: Darcyscale modeling of microbially induced carbonate mineral precipitation in sand columns. Water Resour. Res. (2012). https://doi.org/10.1029/2011WR011714
Endres, D.M., Schindelin, J.E.: A new metric for probability distributions. IEEE Trans. Inf. Theory 49, 1858–1860 (2003). https://doi.org/10.1109/TIT.2003.813506
Esteves, B.F., Lage, P.L.C., Couto, P., et al.: Porenetwork modeling of singlephase reactive transport and dissolution pattern evaluation. Adv. Water Resour. 145(103), 741 (2020). https://doi.org/10.1016/j.advwatres.2020.103741
Ezekiel, J., Adams, B.M., Saar, M.O., et al.: Numerical analysis and optimization of the performance of CO\(_2\)Plume Geothermal (CPG) production wells and implications for electric power generation. Geothermics 98(102), 270 (2022). https://doi.org/10.1016/j.geothermics.2021.102270
Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to longtail distributions to analyze network performance models. Perform. Eval. 31(3), 245–279 (1998). https://doi.org/10.1016/S01665316(97)000035
Fischer, C., Luttge, A.: Beyond the conventional understanding of water–rock reactivity. Earth Planet. Sci. Lett. 457, 100–105 (2017). https://doi.org/10.1016/j.epsl.2016.10.019
Fischer, C., Kurganskaya, I., Schäfer, T., et al.: Variability of crystal surface reactivity: What do we know? Appl. Geochem. 43, 132–157 (2014). https://doi.org/10.1016/j.apgeochem.2014.02.002
Fogler, H.S.: Elements of Chemical Reaction Engineering. Prentice Hall, Hoboken (2016)
Gabrovšek, F., Dreybrodt, W.: A model of the early evolution of karst aquifers in limestone in the dimensions of length and depth. J. Hydrol. 240(3), 206–224 (2001). https://doi.org/10.1016/S00221694(00)003231
Gärttner, S., Frolkovič, P., Knabner, P., et al.: Efficiency and accuracy of micro–macro models for mineral dissolution. Water Resour. Res. 56(8), e2020WR027,585 (2020). https://doi.org/10.1029/2020WR027585
Goltz, M.N., Roberts, P.V.: Using the method of moments to analyze threedimensional diffusionlimited solute transport from temporal and spatial perspectives. Water Resour. Res. 23(8), 1575–1585 (1987). https://doi.org/10.1029/WR023i008p01575
Gong, L., Nie, L., Xu, Y.: Geometrical and topological analysis of pore space in sandstones based on Xray computed tomography. Energies 13, 3774 (2020). https://doi.org/10.3390/en13153774
Gostick, J., Aghighi, M., Hinebaugh, J., et al.: Openpnm: a pore network modeling package. Comput. Sci. Eng. 18(4), 60–74 (2016). https://doi.org/10.1109/MCSE.2016.49
Gray, W.G.: A derivation of the equations for multiphase transport. Chem. Eng. Sci. 30(2), 229–233 (1975). https://doi.org/10.1016/00092509(75)800108
Guo, J., Quintard, M., Laouafa, F.: Dispersion in porous media with heterogeneous nonlinear reactions. Transp. Porous Media 109, 541–570 (2015). https://doi.org/10.1007/s1124201505354
Hefny, M., Qin, C.Z., Saar, M.O., et al.: Synchrotonbased porenetwork modeling of twophase flow in Nubian Sandstone and implications for capillary trapping of carbon dioxide. Int. J. Greenh. Gas Control 103(103), 164 (2020). https://doi.org/10.1016/j.ijggc.2020.103164
Heße, F., Radu, F., Thullner, M., et al.: Upscaling of the advection–diffusion–reaction equation with Monod reaction. Adv. Water Resour. 32(8), 1336–1351 (2009). https://doi.org/10.1016/j.advwatres.2009.05.009
Hussaini, S.R., Dvorkin, J.: Specific surface area versus porosity from digital images. J. Pet. Sci. Eng. 196(107), 773 (2021). https://doi.org/10.1016/j.petrol.2020.107773
Ioannidis, M.A., Chatzis, I.: On the geometry and topology of 3d stochastic porous media. J. Colloid Interface Sci. 229(2), 323–334 (2000). https://doi.org/10.1006/jcis.2000.7055
Jeschke, A.A., Dreybrodt, W.: Dissolution rates of minerals and their relation to surface morphology. Geochim. Cosmochim. Acta 66(17), 3055–3062 (2002). https://doi.org/10.1016/S00167037(02)008931
Jiang, W.Q., Chen, G.Q.: Environmental dispersion in layered wetland: moment based asymptotic analysis. J. Hydrol. 569, 252–264 (2019). https://doi.org/10.1016/j.jhydrol.2018.12.005
Jung, H., NavarreSitchler, A.: Physical heterogeneity control on effective mineral dissolution rates. Geochim. Cosmochim. Acta 227, 246–263 (2018a). https://doi.org/10.1016/j.gca.2018.02.028
Jung, H., NavarreSitchler, A.: Scale effect on the time dependence of mineral dissolution rates in physically heterogeneous porous media. Geochim. Cosmochim. Acta 234, 70–83 (2018b). https://doi.org/10.1016/j.gca.2018.05.009
Karani, H., Huber, C.: Role of thermal disequilibrium on natural convection in porous media: insights from porescale study. Phys. Rev. E 95(033), 123 (2017). https://doi.org/10.1103/PhysRevE.95.033123
Kaufmann, G., Dreybrodt, W.: Calcite dissolution kinetics in the system CaCO\(_3\)H\(_2\)OCO\(_2\) at high undersaturation. Geochim. Cosmochim. Acta 71(6), 1398–1410 (2007). https://doi.org/10.1016/j.gca.2006.10.024
Koch, T., Weishaupt, K., Müller, J., et al.: A (dual) network model for heat transfer in porous media. Transp. Porous Media 140, 107–141 (2021). https://doi.org/10.1007/s11242021016025
Kozeny, J.: Über kapillare leitung des wassers im boden. Wien Akad. Wiss. 136, 271 (1927)
Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951). https://doi.org/10.1214/aoms/1177729694
Kumar, K., van Noorden, T.L., Pop, I.S.: Effective dispersion equations for reactive flows involving free boundaries at the microscale. Multiscale Model. Simul. 9(1), 29–58 (2011). https://doi.org/10.1137/100804553
Kumar, K., NeussRadu, M., Pop, I.S.: Homogenization of a pore scale model for precipitation and dissolution in porous media. IMA J. Appl. Math. 81(5), 877–897 (2016)
Lasaga, A.C.: Kinetic Theory in the Earth Sciences. Princeton University Press, Princeton (1998). https://doi.org/10.1515/9781400864874
Lee, J., Rolle, M., Kitanidis, P.K.: Longitudinal dispersion coefficients for numerical modeling of groundwater solute transport in heterogeneous formations. J. Contam. Hydrol. 212, 41–54 (2018). https://doi.org/10.1016/j.jconhyd.2017.09.004
Levene, M., Kononovicius, A.: Empirical survival Jensen–Shannon divergence as a goodnessoffit measure for maximum likelihood estimation and curve fitting. Commun. Stat. Simul. Comput. 50, 3751–3767 (2019). https://doi.org/10.1080/03610918.2019.1630435
Levenson, Y., Emmanuel, S.: Porescale heterogeneous reaction rates on a dissolving limestone surface. Geochim. Cosmochim. Acta 119, 188–197 (2013). https://doi.org/10.1016/j.gca.2013.05.024
Li, L., Salehikhoo, F.: The role of magnesite spatial distribution patterns in determining dissolution rates: When do they matter? Geochim. Cosmochim. Acta 155, 107–121 (2015). https://doi.org/10.1016/j.gca.2015.01.035
Li, L., Peters, C.A., Celia, M.A.: Effects of mineral spatial distribution on reaction rates in porous media. Water Resour. Res. 43(W01), 419 (2007). https://doi.org/10.1029/2005WR004848
Li, L., Salehikhoo, F., Brantley, S.L., et al.: Spatial zonation limits magnesite dissolution in porous media. Geochim. Cosmochim. Acta 126, 555–573 (2014). https://doi.org/10.1016/j.gca.2013.10.051
Li, R., Yang, C., Ke, D., et al.: The scaling of mineral dissolution rates under complex flow conditions. Geochim. Cosmochim. Acta 274, 63–78 (2020). https://doi.org/10.1016/j.gca.2020.01.048
Lin, J.: Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory 37, 145–151 (1991). https://doi.org/10.1109/18.61115
LorenzoSeva, U., ten Berge, J.M.F.: Tucker’s congruence coefficient as a meaningful index of factor similarity. Methodology 2, 57–64 (2006). https://doi.org/10.1027/16142241.2.2.57
Luhmann, A.J., Tutolo, B.M., Bagley, B.C., et al.: Permeability, porosity, and mineral surface area changes in basalt cores induced by reactive transport of CO\(_2\)rich brine. Water Resour. Res. 53(3), 1908–1927 (2017). https://doi.org/10.1002/2016WR019216
Ma, J., Querci, L., Hattendorf, B., et al.: Toward a spatiotemporal understanding of dolomite dissolution in sandstone by CO\(_2\)enriched brine circulation. Environ. Sci. Technol. 53(21), 12,45812,466 (2019). https://doi.org/10.1021/acs.est.9b04441
Ma, J., Ahkami, M., Saar, M.O., et al.: Quantification of mineral accessible surface area and flowdependent fluidmineral reactivity at the pore scale. Chem. Geol. 563(120), 042 (2021). https://doi.org/10.1016/j.chemgeo.2020.120042
Maher, K.: The dependence of chemical weathering rates on fluid residence time. Earth Planet. Sci. Lett. 294, 101–110 (2010). https://doi.org/10.1016/j.epsl.2010.03.010
Mahmoodlu, M.G., Raoof, A., Bultreys, T., et al.: Largescale pore network and continuum simulations of solute longitudinal dispersivity of a saturated sand column. Adv. Water Resour. 144(103), 713 (2020). https://doi.org/10.1016/j.advwatres.2020.103713
Meile, C., Tuncay, K.: Scale dependence of reaction rates in porous media. Adv. Water Resour. 29(1), 62–71 (2006). https://doi.org/10.1016/j.advwatres.2005.05.007
Meng, X., Yang, D.: Dynamic dispersion coefficient of solutes flowing in a circular tube and a tubebundle model. J. Energy Resour. Technol. (2017). https://doi.org/10.1115/1.4037374
Menke, H.P., Andrew, M.G., Blunt, M.J., et al.: Reservoir condition imaging of reactive transport in heterogeneous carbonates using fast synchrotron tomography  effect of initial pore structure and flow conditions. Chem. Geol. 428, 15–26 (2016). https://doi.org/10.1016/j.chemgeo.2016.02.030
Menke, H.P., Reynolds, C.A., Andrew, M.G., et al.: 4D multiscale imaging of reactive flow in carbonates: assessing the impact of heterogeneity on dissolution regimes using streamlines at multiple length scales. Chem. Geol. 481, 27–37 (2018). https://doi.org/10.1016/j.chemgeo.2018.01.016
Mikelić, A., Devigne, V., van Duijn, C.J.: Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damkohler numbers. SIAM J. Math. Anal. 38(4), 1262–1287 (2006). https://doi.org/10.1137/050633573
Moore, J., Lichtner, P.C., White, A.F., et al.: Using a reactive transport model to elucidate differences between laboratory and field dissolution rates in regolith. Geochim. Cosmochim. Acta 93, 235–361 (2012). https://doi.org/10.1016/j.gca.2012.03.021
Nancollas, G.H.: Kinetics of crystal growth from solution. J. Cryst. Growth 3–4, 335–339 (1968). https://doi.org/10.1016/00220248(68)90170X
Natarajan, N., Kumar, G.S.: Spatial moment analysis of multispecies contaminant transport in porous media. Environ. Eng. Res. 23(1), 76–83 (2018). https://doi.org/10.4491/eer.2016.147
NavarreSitchler, A., Brantley, S.: Basalt weathering across scales. Earth Planet. Sci. Lett. 261, 321–334 (2007). https://doi.org/10.1016/j.epsl.2007.07.010
Naviaux, J.D., Subhas, A.V., Rollins, N.E., et al.: Temperature dependence of calcite dissolution kinetics in seawater. Geochim. Cosmochim. Acta 246, 363–384 (2019). https://doi.org/10.1016/j.gca.2018.11.037
Nielsen, A.E.: Electrolyte crystal growth mechanisms. J. Cryst. Growth 67(2), 289–310 (1984). https://doi.org/10.1016/00220248(84)901891
Nielsen, F.: On a generalization of the Jensen–Shannon divergence and the Jensen–Shannon centroid. Entropy 22, 221 (2020). https://doi.org/10.3390/e22020221
Noiriel, C., Steefel, C.I., Yang, L., et al.: Upscaling calcium carbonate precipitation rates from pore to continuum scale. Chem. Geol. 318–319, 60–74 (2012). https://doi.org/10.1016/j.chemgeo.2012.05.014
Okada, M., Yamanishi, K., Masuda, N.: Longtailed distributions of interevent times as mixtures of exponential distributions. R. Soc. Open Sci. 7(2), 191,643 (2020). https://doi.org/10.1098/rsos.191643
Oldham, K.B., Myland, J.C., Spanier, J.: The hyperbolic tangent tanh(x) and cotangent coth(x) functions. In: Oldham, K., Myland, J., Spanier, J. (eds.) An Atlas of Functions, pp. 289–296. Springer, US, New York (2009). https://doi.org/10.1007/9780387488073_31
Øren, P.E., Bakke, S.: Reconstruction of Berea sandstone and porescale modelling of wettability effects. J. Pet. Sci. Eng. 39(3), 177–199 (2003). https://doi.org/10.1016/S09204105(03)000627
Österreicher, F., Vajda, I.: A new class of metric divergences on probability spaces and its applicability in statistics. Ann. Inst. Stat. Math. 55, 639–653 (2003). https://doi.org/10.1007/BF02517812
Paine, M.A., Carbonell, R.G., Whitaker, S.: Dispersion in pulsed systemsI: heterogenous reaction and reversible adsorption in capillary tubes. Chem. Eng. Sci. 38(11), 1781–1793 (1983). https://doi.org/10.1016/00092509(83)850350
Palmer, A.N.: Origin and morphology of limestone caves. GSA Bull. 103(1), 1–21 (1991). https://doi.org/10.1130/00167606(1991)103<0001:OAMOLC>2.3.CO;2
Plummer, L.N., Wigley, T.M.L.: The dissolution of calcite in CO\(_2\)saturated solutions at 25\(\circ \)C and 1 atmosphere total pressure. Geochim. Cosmochim. Acta 40(2), 191–202 (1976). https://doi.org/10.1016/00167037(76)901769
Plummer, L.N., Wigley, T.M.L., Parkhurst, D.L.: The kinetics of calcite dissolution in CO\(_2\)–water system at 5 degrees to 60 degrees C and 0.0 to 1.0 atm CO\(_2\). Am. J. Sci. 278, 179–216 (1978). https://doi.org/10.2475/ajs.278.2.179
Pogge von Strandmann, P.A.E., Burton, K.W., Snæbjörnsdóttir, S.O., et al.: Rapid CO\(_2\) mineralisation into calcite at the CarbFix storage site quantified using calcium isotopes. Nat. Commun. (2019). https://doi.org/10.1038/s41467019100038
Qin, C.Z., Hassanizadeh, S.: Porenetwork modeling of solute transport and biofilm growth in porous media. Transp. Porous Media 110, 345–367 (2015). https://doi.org/10.1007/s1124201505461
Qiu, T., Wang, Q., Yang, C.: Upscaling multicomponent transport in porous media with a linear reversible heterogeneous reaction. Chem. Eng. Sci. 171, 100–116 (2017). https://doi.org/10.1016/j.ces.2017.05.018
Randolph, J.B., Saar, M.O.: Combining geothermal energy capture with geologic carbon dioxide sequestration. Geophys. Res. Lett. (2011). https://doi.org/10.1029/2011GL047265
Raoof, A., Nick, H., Wolterbeek, T., et al.: Porescale modeling of reactive transport in wellbore cement under CO\(_2\) storage conditions. Int. J. Greenh. Gas Control 11, S67–S77 (2012). https://doi.org/10.1016/j.ijggc.2012.09.012
Raoof, A., Nick, H.M., Hassanizadeh, S.M., et al.: PoreFlow: a complex porenetwork model for simulation of reactive transport in variably saturated porous media. Comput. Geosci. 61, 160–174 (2013). https://doi.org/10.1016/j.cageo.2013.08.005
Ray, N., Oberlander, J., Knabner, P.: Numerical investigation of a fully coupled micro–macro model for mineral dissolution and precipitation. Comput. Geosci. 23, 1179–1192 (2019). https://doi.org/10.1007/s1059601909876x
Reddy, M.M.: Kinetics of calcium carbonate formation. In: Internationale Vereinigung für theoretische und angewandte Limnologie: Verhandlungen, vol 19, pp. 429–438. Taylor & Francis (1975) https://doi.org/10.1080/03680770.1974.11896082
Reddy, M.M.: Crystallization of calcium carbonate in the presence of trace concentrations of phosphoruscontaining anions: I. Inhibition by phosphate and glycerophosphate ions at pH 8.8 and 25\(\circ \)C. J. Cryst. Growth 41(2), 287–295 (1977). https://doi.org/10.1016/00220248(77)900574
Redeker, M., Rohde, C., Pop, I.S.: Upscaling of a triphase phasefield model for precipitation in porous media. IMA J. Appl. Math. 81(5), 898–939 (2016). https://doi.org/10.1093/imamat/hxw023
Saar, M.O.: Review: geothermal heat as a tracer of largescale groundwater flow and as a means to determine permeability fields. Hydrogeol. J. 19, 31–52 (2011). https://doi.org/10.1007/s1004001006572
Salehikhoo, F., Li, L., Brantley, S.L.: Magnesite dissolution rates at different spatial scales: the role of mineral spatial distribution and flow velocity. Geochim. Cosmochim. Acta 108, 91–106 (2013). https://doi.org/10.1016/j.gca.2013.01.010
Sallés, J., Thovert, J.F., Adler, P.M.: Deposition in porous media and clogging. Chem. Eng. Sci. 48(16), 2839–2858 (1993). https://doi.org/10.1016/00092509(93)80031K
SantosSánchez, R.D.L., ValdésParada, F.J., Chirino, Y.I.: Upscaling diffusion and reaction processes in multicellular systems considering different cell populations. Chem. Eng. Sci. 142, 144–164 (2016). https://doi.org/10.1016/j.ces.2015.11.031
Shi, D., Brown, P.W., Ma, W.: Lognormal simulation of pore size distributions in cementitious materials. J. Am. Ceram. Soc. 74, 1861–1867 (1991). https://doi.org/10.1111/j.11512916.1991.tb07800.x
Shook, G.M.: Predicting thermal breakthrough in heterogeneous media from tracer tests. Geothermics 30(6), 573–589 (2001). https://doi.org/10.1016/S03756505(01)000153
Sjöberg, E.L., Rickard, D.T.: Calcite dissolution kinetics: surface speciation and the origin of the variable pH dependence. Chem. Geol. 42(1), 119–136 (1984). https://doi.org/10.1016/00092541(84)900093
Subhas, A.V., Rollins, N.E., Berelson, W.M., et al.: A novel determination of calcite dissolution kinetics in seawater. Geochim. Cosmochim. Acta 170, 51–68 (2015). https://doi.org/10.1016/j.gca.2015.08.011
Svensson, U., Dreybrodt, W.: Dissolution kinetics of natural calcite minerals in CO\(_2\)–water systems approaching calcite equilibrium. Chem. Geol. 100(1), 129–145 (1992). https://doi.org/10.1016/00092541(92)90106F
Taghizadeh, E., ValdésParada, F.J., Wood, B.D.: Preasymptotic Taylor dispersion: evolution from the initial condition. J. Fluid Mech. 889, A5 (2020). https://doi.org/10.1017/jfm.2020.56
Teng, H.H., Dove, P.M., De Yoreo, J.J.: Kinetics of calcite growth: surface processes and relationships to macroscopic rate laws. Geochim. Cosmochim. Acta 64(13), 2255–2266 (2000). https://doi.org/10.1016/S00167037(00)003410
Tutolo, B.M., Luhmann, A.J., Kong, X.Z., et al.: CO\(_2\) sequestration in feldsparrich sandstone: coupled evolution of fluid chemistry, mineral reaction rates, and hydrogeochemical properties. Geochim. Cosmochim. Acta 160, 132–154 (2015). https://doi.org/10.1016/j.gca.2015.04.002
ValdésParada, F.J., AguilarMadera, C.G., Álvarez Ramírez, J.: On diffusion, dispersion and reaction in porous media. Chem. Eng. Sci. 66(10), 2177–2190 (2011). https://doi.org/10.1016/j.ces.2011.02.016
ValdésParada, F.J., Lasseux, D., Whitaker, S.: Diffusion and heterogeneous reaction in porous media: the macroscale model revisited. Int. J. Chem. React. Eng. 15(6), 20170,151 (2017). https://doi.org/10.1515/ijcre20170151
van Duijn, C.J., Pop, I.S.: Crystal dissolution and precipitation in porous media: pore scale analysis. J. Reine Angew. Math. 577, 171–211 (2004). https://doi.org/10.1515/crll.2004.2004.577.171
van Duijn, C., Mikelić, A., Pop, I.S., et al.: Effective dispersion equations for reactive flows with dominant Péclet and Damkohler numbers. In: Marin, G.B., West, D., Yablonsky, G.S. (eds.) Advances in Chemical Engineering, Chap 1, vol. 34, pp. 1–45. Academic Press, Cambridge (2008). https://doi.org/10.1016/S00652377(08)00001X
van Noorden, T.L.: Crystal precipitation and dissolution in a porous medium: effective equations and numerical experiments. Multiscale Model. Simul. 7(3), 1220–1236 (2009a). https://doi.org/10.1137/080722096
van Noorden, T.L.: Crystal precipitation and dissolution in a thin strip. Eur. J. Appl. Math. 20(1), 69–91 (2009b). https://doi.org/10.1017/S0956792508007651
Varloteaux, C., Békri, S., Adler, P.M.: Pore network modelling to determine the transport properties in presence of a reactive fluid: from pore to reservoir scale. Adv. Water Resour. 53, 87–100 (2013a). https://doi.org/10.1016/j.advwatres.2012.10.004
Varloteaux, C., Vu, M.T., Békri, S., et al.: Reactive transport in porous media: porenetwork model approach compared to porescale model. Phys. Rev. E 87(023), 010 (2013b). https://doi.org/10.1103/PhysRevE.87.023010
Vikhansky, A., Ginzburg, I.: Taylor dispersion in heterogeneous porous media: extended method of moments, theory, and modelling with tworelaxationtimes lattice Boltzmann scheme. Phys. Fluids 26(2), 022,104 (2014). https://doi.org/10.1063/1.4864631
Virtanen, P., Gommers, R., Oliphant, T.E., et al.: SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Methods 17, 261–272 (2020). https://doi.org/10.1038/s4159201906862
von Wolff, L., Pop, I.S.: Upscaling of a Cahn*Hilliard Navier–Stokes Model with Precipitation and Dissolution in a Thin Strip. arXiv. (2021) https://doi.org/10.48550/ARXIV.2103.06780
Vrentas, J.S., Vrentas, C.M.: Dispersion in laminar tube flow at low Peclet numbers or short times. AIChE J. 34(9), 1423–1430 (1988). https://doi.org/10.1002/aic.690340904
Wen, H., Li, L.: An upscaled rate law for magnesite dissolution in heterogeneous porous media. Geochim. Cosmochim. Acta 210, 289–305 (2017). https://doi.org/10.1016/j.gca.2017.04.019
Wen, H., Li, L.: An upscaled rate law for magnesite dissolution in heterogeneous porous media: the role of time and length scales. Geochim. Cosmochim. Acta 235, 1–20 (2018). https://doi.org/10.1016/j.gca.2018.04.024
White, A.F., Brantley, S.L.: The effect of time on the weathering of silicate minerals: Why do weathering rates differ in the laboratory and field? Chem. Geol. 202, 479–506 (2003). https://doi.org/10.1016/j.chemgeo.2003.03.001
Wood, B.D., Radakovich, K., Golfier, F.: Effective reaction at a fluid–solid interface: applications to biotransformation in porous media. Adv. Water Resour. 30(6), 1630–1647 (2007). https://doi.org/10.1016/j.advwatres.2006.05.032
Xiong, Q., Baychev, T.G., Jivkov, A.P.: Review of pore network modelling of porous media: experimental characterisations, network constructions and applications to reactive transport. J. Contam. Hydrol. 192, 101–117 (2016). https://doi.org/10.1016/j.jconhyd.2016.07.002
Zhang, Y., Benson, D.A., Baeumer, B.: Moment analysis for spatiotemporal fractional dispersion. Water Resour. Res. 44(4), W04,424 (2008). https://doi.org/10.1029/2007WR006291
Zhang, L., Hesse, M.A., Wang, M.: Transient solute transport with sorption in Poiseuille flow. J. Fluid Mech. 828, 733–752 (2017). https://doi.org/10.1017/jfm.2017.546
Zhao, Z., Skelton, A.: An assessment of the role of nonlinear reaction kinetics in parametrization of metamorphic fluid flow. J. Geophy. Res. Solid Earth 119, 6249–6262 (2014). https://doi.org/10.1002/2014JB011016
Acknowledgements
This work is supported by the Swiss National Science Foundation project, entitled “Analysing spatial scaling effects in mineral reaction rates in porous media with a hybrid numerical model". We also thank the Werner SiemensStiftung (Werner Siemens Foundation) for its support of the Geothermal Energy and Geofluids (GEG.ethz.ch) Group at ETH Zurich, Switzerland. We further thank the anonymous reviewers for their critical comments, which greatly improved this paper.
Funding
Open access funding provided by Swiss Federal Institute of Technology Zurich. This work is supported by the Swiss National Science Foundation project (Number 175673), entitled “Analysing spatial scaling effects in mineral reaction rates in porous media with a hybrid numerical model”.
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Appendices
Appendix 1: Derivation of the Longitudinal Dispersivity Using Spatial Moment Analysis
We introduce an advection equation of solute concentration in a pore
The velocity, \(\langle u_{z} \rangle \), is defined using Eq. (26)
When injecting a solute pulse at \(z=0\), the center of mass of the solute is located at
Similar to the volume averaging procedure, Eq. (47), we define the center of mass in a bundle of tubes system
Following the procedures of spatial moment analysis (Goltz and Roberts 1987; Dentz and de Barros 2013; Lee et al. 2018; Natarajan and Kumar 2018), the mean velocity is
which is the volume averaged velocity, Eq. (32). The change of spatial variance over time is
The longitudinal dispersion coefficient is
which concludes the derivation of Eq. (35). The nondimensional solute transport equation is
Following Crank (1975), the fundamental solution of the aforementioned transport equation is
We test the validity of the dispersion coefficient by comparing with the volumeaveraged solute concentration. We consider two cases of pore size distributions \(R_{1}\) and \(R_{2}\) with initial injection at \(z_{0}=0\) and \({\text{Pe}}_{\mathrm{d}}=10\).
Figure 8 shows the comparison between \({\overline{C}}\) and \(C_{\mathrm{avg}}\) at different time steps. When the travel distance increases, the difference between \({\overline{C}}\) and \(C_{\mathrm{avg}}\) increases. Such effect is more pronounced when \(\sigma \) is larger, as shown in the right panel of Fig. 8. The higherorder method of moments is employed for better modeling of dispersion effects (Chatwin 1970; Zhang et al. 2008; Vikhansky and Ginzburg 2014; Jiang and Chen 2019). However, we limit our analysis to second order to focus on the topic of Darcyscale reaction order.
Appendix B Taylor Series of the Averaged SecondOrder Kinetics Model
The averaged secondorder kinetics model, Eq. (65), is
We show derivations of expanding Eq. (97) using a Taylor series. A Taylor series expansion of \({\overline{C}}\) around \(t=0\) is
which consists of derivatives of \({\overline{C}}\). Differentiate Eq. (97)
The initial condition, \({\overline{C}}(0)=0\), leads to \({\overline{C}}^{\prime \prime }(0)=0\), which corresponds to the secondorder term in Eq. (64). Instead of performing further differentiation, we rearrange Eq. (97)
Applying the binomial approximation to \((1+{\overline{C}})^n\) yields
Since we are interested in derivatives around \(t=0\) and given the initial condition \({\overline{C}}(0)=0\), and since \(n=O(1)\), the constraint of \(n{\overline{C}} \ll 1\) is valid. Differentiate Eq. (101)
which retains the property \({\overline{C}}^{''}(0)=0\). Further differentiate and omit \({\overline{C}}^{''}\)
Hence,
Therefore, the Taylor series with an approximated thirdorder derivative is
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Huang, PW., Flemisch, B., Qin, CZ. et al. Relating DarcyScale Chemical Reaction Order to PoreScale Spatial Heterogeneity. Transp Porous Med 144, 507–543 (2022). https://doi.org/10.1007/s11242022018170
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DOI: https://doi.org/10.1007/s11242022018170