# Reactive Flow and Homogenization in Anisotropic Media

## Abstract

The evolution of heterogeneous and anisotropic media in the uniform dissolution regime (low Damköhler number) is studied here using a numerical network model. The uniform dissolution extensively homogenizes the medium and therefore the flow field. The homogenization is further enhanced when the surface reaction is transport controlled—when slow diffusion of dissolved ions away from the mineral surface leads to the reduction of the global dissolution rate. Under those conditions, diffusive transport is more effective in narrow channels, which selectively enlarge, leading to an initial steep rise of the permeability. However, as dissolution proceeds, the void space widens and the overall dissolution rate drops, and permeability enhancement slows down. Finally, we review the relevance of these results to various processes in geological systems ranging from diagenesis and karst evolution, to carbon geosequestration. These findings provide fundamental insights into reactive transport processes in fractured and porous media and evolution of permeability, tortuosity, anisotropy, and bulk reaction rates in geological systems.

## Key Points

- Uniform dissolution homogenizes heterogeneous and anisotropic medium and its flow field
- Homogenization is enhanced when the reaction is transport controlled at the pore scale, due to selective dissolution of constrictions
- Under transport-controlled reaction, permeability increase is initially steep but later slows down due to a decrease in reaction rate

## Plain Language Summary

When corrosive fluids flow in an aquifer, they dissolve the rocks and change the void-space structure (e.g., acidic water invading limestone). In many geological processes and applications, the flow velocity is high compared to the chemical dissolution rate, and the corrosive fluid penetrates large distances before it loses its reactivity, resulting in relatively uniform dissolution everywhere. Here, using analog numerical model, we study how the void-space structure of porous rocks and aquifers change during uniform dissolution. In particular, we study how different initial structures and reaction conditions change the flow field and transport properties. Finally, we review the relevance of these results to various geological systems. The study improves our understanding of rock and aquifer evolution, with implications to groundwater management and other subsurface flow-related processes, such as CO_{2} geosequestration.

## 1 Introduction

Dissolution induced by reactive flow is a fundamental process in earth systems, controlling the transformation of the medium structure, from the surface (Ford & Williams, 2013) down to the mantle (Aharonov et al., 1995, 1997). It plays a pivotal role in natural processes such as rock weathering, diagenesis, metamorphism, as well as in a wide range of engineering applications (e.g., Lichtner et al., 2018; Steefel & Maher, 2009). Particularly, dissolution controls the evolution of porosity and connectivity in rocks and sediments—a key effect in any application involving subsurface flow, such as groundwater hydrology (Ford & Williams, 2013; Pinder & Celia, 2006), geothermal energy (Huenges & Ledru, 2011), CO_{2} geosequestration (Nordbotten & Celia, 2011; Orr, 2009), and hydrocarbon recovery (Agar & Geiger, 2015; Lucia, 2007).

Transport and reaction can be spatially distributed in a different manner. These inhomogeneities can be further amplified by dissolution leading to the emergence of distinct patterns. The dynamic process and the resulting patterns depend on the interplay between advection, diffusion, and reaction, which can be characterized by a pair of dimensionless numbers: (i) the Damköhler number, *Da*_{L}, describing the relative rate of reaction versus advection over the length of the system (*L*) and defined as *Da*_{L} *= Sλ*^{eff}*L/v̅*, where *S* is the specific surface area (L^{−1}), *λ*^{eff} is the effective reaction rate coefficient (L T^{−1}), and *v̅* is Darcy flux (or average superficial velocity) and (ii) Péclet number, *Pe*_{L}, the relative magnitude of advective and diffusive transport, defined as *Pe*_{L}=*v̅L/D*, where *D* is the diffusion coefficient. Equivalently, their product, *Pe*_{L}*Da*_{L}, is often used instead of *Da*_{L} (reaction vs. diffusion, termed Peclet-Damkohler or kinetic number) (Bekri et al., 1995; Golfier et al., 2002; Spiegelman et al., 2001; Steefel & Maher, 2009; Szymczak & Ladd, 2009).

In cases where advective transport is faster than diffusive transport, namely, large *Pe*_{L}, dissolution can be either homogeneous or heterogeneous, depending on *Da*_{L}. If the reaction rate is high relative to the advection rate across the system (large *Da*_{L}), dissolution is enhanced in the most conductive flow paths. This leads to the emergence of reactive-infiltration instabilities and dissolution channels (wormholes), which dramatically increase medium heterogeneity. At lower *Da*_{L}, the reactant penetration length, *l*_{p} = *L*/*Da*_{L}, is very large and the reactive fluid transverses the whole system before its reactivity is exhausted, leading to more homogeneous dissolution, usually referred to as *the uniform dissolution regime* (Golfier et al., 2002; Hoefner & Fogler, 1988; Szymczak & Ladd, 2009). These dissolution regimes are also scale dependent: Instability will develop downstream if the system length, *L*, is large enough, whereas otherwise homogeneous system dissolution is observed (Starchenko & Ladd, 2018; Steefel & Lasaga, 1990).

While the emergence of instabilities and wormholing development has been studied extensively (Aharonov et al., 1995; Daccord & Lenormand, 1987; Kelemen et al., 1995; Szymczak & Ladd, 2006; Upadhyay et al., 2015), much less attention has been given to the uniform dissolution regime, despite its relevance in many processes, like CO_{2} geosequestration (Deng, Steefel, et al., 2018; Gouze et al., 2003; Zhang & DePaolo, 2017), acidizing of sandstone reservoirs (Daccord et al., 1993; Liu & Mostaghimi, 2017; Shafiq & Mahmud, 2017), diagenetic alterations (Taylor et al., 2010; Wray & Sauro, 2017), and karst evolution (Howard & Groves, 1995; Kaufmann & Braun, 1999; Palmer, 2011). Furthermore, even at relatively low flow rates and for large length scales, many geochemical dissolution reactions (e.g., of feldspar minerals or dolomite) are sufficiently slow and expected to produce uniform dissolution (Busenberg & Plummer, 1982; Morse & Arvidson, 2002; Zhu, 2005).

Geological media are commonly characterized by complicated void-space structure, with pore sizes or apertures often spanning several length scales (Blunt, 2017; Palmer, 1991; Sahimi, 2011) and frequently also exhibiting preferred orientation (anisotropy). Specifically, fractured media are associated with anisotropic geometry at different scales—from the surface texture of single fractures (e.g., Aydan et al., 1996; Boutt et al., 2007; Brown & Scholz, 1985; Tatone & Grasselli, 2012; Xie et al., 1999) to fracture networks (Adler et al., 2013; Bonnet et al., 2001; Sahimi, 2011). Furthermore, fault zones that comprise major preferential pathways for reactive fluids (Audra & Palmer, 2015; Klimchouk, 2017) are characterized by highly anisotropic damage zone and directional shearing texture (Candela et al., 2012; Lee & Bruhn, 1996; Renard et al., 2006), as well as anisotropic breccias in fault cores (Bense et al., 2013; Blenkinsop et al., 2007).

Pore-scale anisotropy also exists in permeable rocks (Baas et al., 2007; Benson et al., 2005; Clavaud et al., 2008; Mostaghimi et al., 2013, and references therein), in some cases very pronounced (e.g., Gueguen & Schubnel, 2003; Wright et al., 2009). In particular, in situ differential stresses can induce large anisotropy in the medium, due to the formation of dilatant microcracks aligned parallel to the maximum compressive stress direction. In addition, pores or fractures aligned oblique to the maximum compressive stress preferentially compact and close, further enhancing the anisotropy (Bruno, 1994; Kang et al., 2018; Min et al., 2004; Mitchell & Faulkner, 2008; Sayers, 1990; Simpson et al., 2001).

Yet the evolution of the complex medium structure during dissolution at low *Da*_{L} conditions and the resulting transformations of its macroscopic properties are not well understood. In addition, the implications on bulk reaction rates of dissolution itself, a core issue in fluid-rock interaction studies, are still not well quantified (Li et al., 2007; Steefel & Maher, 2009). Understanding the interplay of dissolution and the evolution of medium properties—the main objective of this paper—will improve our ability to decipher the diagenetic history of a medium, as well as to improve subsurface flow applications like groundwater management, intimately linked to the evolved void-space structure.

Of particular interest is the effect of different reaction regimes. Homogeneous dissolution at the system scale is defined to be limited by reaction (as opposed to transport-limited dissolution at higher *Da*_{L} conditions; Bekri et al., 1995; Golfier et al., 2002; Varloteaux et al., 2013). However, at the pore-scale dissolution rate can still be limited by transport under some conditions: If the rate of dissolved ion diffusion away from the mineral surface to the bulk fluid is slow (see Figure 1b), transport sustains only limited undersaturation and reactivity adjacent to the surface, and the reaction rate is set by the pore-scale transport rather than by kinetics (here referred as transport-controlled reaction). In such conditions, the characteristic time scale is given by *t*_{d} *= r/h*, where *r* is the void radius and *h* is the mass transfer coefficient (Golfier et al., 2002; Gupta & Balakotaiah, 2001; Hanna & Rajaram, 1998; Li et al., 2008). In contrast, if the kinetic rate is slower or the void space is narrow, diffusion is capable of sustaining large undersaturation adjacent to the surface, and the reaction rate is predominantly set by the reaction kinetics (referred here as surface-controlled reaction). In this case, the characteristic time scale is given by *t*_{r} *= r/λ*, where *λ* is the reaction rate coefficient (Hoefner & Fogler, 1988; Szymczak & Ladd, 2009).

In this paper, we use a network model to study the dissolution of porous and fractured media at low *Da*_{L}. We examine the effect of variable initial medium anisotropy and different reaction regimes. It is shown here that the evolving void-space structure, flow field, and macroscopic properties, as permeability, anisotropy, and reaction rates, are largely controlled by those conditions. The results illuminate the nature of the uniform dissolution regime, which leads to a homogenization of the medium and the flow field.

## 2 The Network Model

### 2.1 Network Modeling

To gain a fundamental understanding of dissolution dynamics, we use a network model, with an idealized representation of the void space as a network of discrete channels (Figure 1). This analog approach is often applied to study reactive transport processes, both at the pore scale to core scale in porous rocks (e.g., Algive et al., 2010; Budek & Szymczak, 2012; Hoefner & Fogler, 1988; Nogues et al., 2013; Raoof et al., 2012; Wang et al., 2016; Xiong et al., 2016), as well as at the field scale, in studying karst systems evolution in fractured media (e.g., Dreybrodt et al., 2005; Howard & Groves, 1995; Perne et al., 2014; Rehrl et al., 2008; Skoglund et al., 2010). The explicit but simplified representation of void space in network models allows studying the basic effect of heterogeneity and anisotropy of the medium (Friedman & Jones, 2001; Jang et al., 2011;Steefel et al., 2013; Xiong et al., 2016), while their computational efficiency enables the use of large and representative domains and repetition over multiple realizations. This is in contrast to more accurate methods, such as direct numerical simulations, that are much more restricted by their computational demand (Liu & Mostaghimi, 2017; Molins, 2015; Pereira Nunes et al., 2016).

Despite their simplicity, previous studies demonstrate that regular 2-D and 3-D pore network models can capture well the major characteristics of wormholing observed experimentally, including their structure and advancement rate, and permeability evolution including the breakthrough curve (Budek & Szymczak, 2012; Fredd & Fogler, 1998; Hoefner & Fogler, 1988; Tansey & Balhoff, 2016; Wang et al., 2016). Yet a quantitative agreement can only be expected if the pore network model is constructed based on the real pore structure (Steefel et al., 2013; Xiong et al., 2016). In such a case, the diagenetic history and the porosity-permeability evolution paths of rock samples were successfully reconstructed (De Boever et al., 2012).

Here, we use a 2-D regular network of cylindrical channels embedded in a soluble solid block, a simple representation of a porous media, or a well-connected fractured medium of negligible matrix permeability (Adler et al., 2013; Sahimi, 2011). In particular, the conceptual 2-D network closely resembles the network of permeable channels formed at the intersection of bedding plane with the subvertical fracture network, where karst systems evolution initiate (so-called “inception horizon”; Dreybrodt, 1988; Filipponi et al., 2009; Frumkin et al., 2017).

In the network model, conservation equations for fluid and solute are written for individual pores and nodes (pore junctions). Calculations involve solving the continuity equation for the fluid and Poiseuille's law for the pressure drop in the channels, from which fluid pressures and fluxes are calculated. Solute conservation equations are used to calculate concentrations and mass of solid dissolved. Solute transport in the channels is controlled by advection in the axial direction (flow direction) and by diffusion in the perpendicular (radial) direction, with axial diffusion neglected. This restricts the analysis to conditions where the channel Péclet number is sufficiently large (*Pe*_{l} = *v̅l/D* >> 1, where *l* is channel length), which are commonly encountered in natural processes and applications (Adler et al., 2013; Ford & Williams, 2013; Niemi et al., 2017), and often lead to uniform dissolution and low *Da*_{L} conditions, especially if the reaction is not very fast (Golfier et al., 2002; Niemi et al., 2017; Szymczak & Ladd, 2009). It is noted, however, that mechanical dispersion is inherently incorporated in network models (e.g., Bijeljic & Blunt, 2006).

The large separation of time scales between mineral dissolution and flow and solute concentration relaxation allows to adopt a quasi-static approach (Bekri et al., 1995; Detwiler & Rajaram, 2007; Lichtner, 1991; Sanford & Konikow, 1989). Specifically, the dissolution time scale, corresponding to the time needed for the doubling of the initial pore size (or fracture aperture), is of the order of months to years for the dissolution of carbonates by meteoric waters and up to thousands of years for the dissolution of silicate systems (Dove & Crerar, 1990; Ladd & Szymczak, 2017; Szymczak & Ladd, 2012). The flow and solute concentration relaxation times are several orders of magnitude smaller; thus, the porosity remains essentially frozen on the time scale of momentum and reactant transport, which can then be assumed to be at steady state at each time step. Additionally, as the relaxation time of the flow field is much faster than that of solute, these fields can be calculated sequentially at each time step. The geometrical properties are then updated at the end of each time step to be used in the subsequent step (Ameli et al., 2014; Hoefner & Fogler, 1988; Roded et al., 2018; Szymczak & Ladd, 2009).

### 2.2 Model Equations and Dimensionless Groups

### 2.3 Fluid Flow

*i*yields

*q*

_{ij}is the volumetric flow rate through channel

*ij*(

*q*

_{ij}> 0 indicates flow from node

*i*to

*j*) and the summation is over all neighboring nodes

*j*, linked to node

*i*(Figure 1a and Table 1). The channel flow rates are calculated using Hagen-Poiseuille equation

*r*

_{ij}is channel radius,

*μ*is fluid viscosity, and Δ

*p*

_{ij}=

*p*

_{i}-

*p*

_{j}is the pressure drop between two nodes, which are a distance of

*l*apart. The system of linear Equations 1 and 2 is solved for pressure values at the nodes.

Roman | U |
Amplitude, L | |
---|---|---|---|

A _{cell} |
Cell cross section, L^{2} |
u |
Interstitial velocity, L T^{−1} |

A_{sys} |
System cross section, L^{2} |
V_{cell} |
Cell volume, L^{3} |

a |
Average channel cross section, L^{2} |
V_{diss} |
Volume of channel dissolved, L^{3} |

c |
Mixing-cup solute concentration, M L^{−3} |
V_{f} |
Volume of fluid injected, T^{−1} |

c_{eq} |
Equilibrium solute concentration, M L^{−3} |
v |
Superficial velocity, L T^{−1} |

c_{sol} |
Concentration of soluble solid, M L^{−3} |
v̅ |
Average superficial velocity (Darcy flux), L T^{−1} |

c_{w} |
Concentration at the wall, M L^{−3} |
x |
Coordinate, L |

D |
Diffusion coefficient, L^{2} T^{−1} |
y |
Coordinate, L |

Da_{L} |
Damköhler number | Greek | |

f |
Function | γ |
Acid capacity number |

G |
Transport parameter | Δ | Difference |

g |
Function | η |
Flow focusing parameter |

h |
Mass transfer coefficient, L T^{−1} |
λ | Reaction rate coefficient, L T^{−1} |

J_{r} |
Reaction rate at the wall, M L^{−2} T^{−1} |
λ^{eff} |
Effective reaction rate coefficient, L T^{−1} |

J_{D} |
Diffusive flux at the channel, M L^{−2} T^{−1} |
μ |
Fluid viscosity, M L^{−1} T^{−1} |

k |
Permeability, L^{2} |
ν |
Stoichiometric coefficient |

L |
System length, L | ρ |
Coordinate, L |

l |
Channel length, L | ξ |
Axial coordinate, L |

l_{en} |
Entrance length, L | τ |
Tortuosity |

l_{p} |
Penetration length, L | χ |
Anisotropy degree |

N |
Number of channels | χ′ |
Anisotropy degree after dissolution |

n_{50} |
Fraction of longitudinal channels carrying half of the total flow | Ω_{S} |
Bulk reaction rate, mol L^{−2} T^{−1} |

Pe_{L} |
System-scale Péclet number | Ω_{St} |
Bulk reaction rate after dissolution, mol L^{−2} T^{−1} |

Pe_{l} |
Channel Péclet number | ω |
Angular frequency, L^{−1} |

p |
Fluid pressure, M L^{−1} T^{−2} |
Subscripts | |

Q |
Total volumetric flow rate, L^{3} T^{−1} |
W |
Wavy channel |

q |
Channel volumetric flow rate, L^{3} T^{−1} |
in |
Inlet |

q̅ |
Inlet channel average flow rate, L^{3} T^{−1} |
out |
outlet |

r |
Channel radius, L | I |
Coordinate index, I = x or y |

S |
Specific surface area, L^{−1} |
i |
Nodes index |

Sh |
Sherwood number | j |
Neighboring nodes index |

s |
Channel surface area, L^{2} |
x, y |
Channels alignment direction |

s_{T} |
Total system surface area, L^{2} |
0 | Initial average quantity |

t |
Time, T | Superscripts | |

t_{d} |
Characteristic time scale of transport, T | ̅ | Average quantity |

t_{r} |
Characteristic time scale of reaction, T | ̂ | Scaled quantity |

#### 2.3.1 Reactive Transport and Dissolution

*c*

_{w}, and the equilibrium concentration,

*c*

_{eq}, and

*λ*is the kinetic reaction rate coefficient (L/T). Reaction products (e.g., calcium ions) then diffuse away from the mineral surface to the bulk fluid and sustain undersaturation conditions at the wall (Dreybrodt et al., 2005; Hanna & Rajaram, 1998). This diffusive flux can be expressed by

*c*is the flow-weighted average concentration (mixing-cup concentration), defined as , with

*u*the fluid velocity and

*ρ*the radial coordinate. Next,

*h*is the mass transfer coefficient, which is inversely proportional to the radius and calculated using

*D*is the molecular diffusion coefficient and

*Sh*the Sherwood number (Bejan, 2013; Gupta & Balakotaiah, 2001).

*Sh*depends on

*λ*and is bound by two asymptotic limits that are close to each other in value (the transport and kinetic limits,

*Sh =*4.364 and 3.656, respectively; Gupta & Balakotaiah, 2001; Hayes & Kolaczkowski, 1994). Here

*Sh*is approximated by a constant value,

*Sh*= 4 (Budek & Szymczak, 2012; Noiriel & Deng, 2018). This assumption is adequate as long as the solute entrance length,

*l*

_{en}, required to reach the asymptotic value is small (

*l*

_{en}≪

*l*). The entrance length increases as the radius and fluid velocity increase (see e.g., Bejan, 2013; Rohsenow et al., 1998) but remains small in our numerical simulations.

*g*, defined as

*g*(

*r*) is the ratio between the characteristic time scales of transport

*t*

_{d}

*= r/h*and reaction,

*t*

_{r}

*= r/λ*, and accounts for the extent by which dissolution rate within a single channel is hindered by transport; when

*g*≪1 reaction rate is surface controlled and limited by kinetics and when

*g*≫ 1, the reaction is transport controlled.

*ξ*, (

*Pe*

_{l}≫ 1), and radial transport effects are incorporated in the effective reaction coefficient,

*λ*

^{eff}, resulting in a 1-D advection-reaction equation

*c*(

*ξ =*0)

*= c*

_{in}, Equation 10 can be solved for the concentration at the channel outlet,

*c*(

*ξ = l*)

*= c*

_{out}

*s*= 2

*πrl*is the channel surface area. Lastly, the concentration of solute at the nodes is calculated from channel outlet concentrations,

*c*

_{out}, and solute balance, assuming full mixing at the channel intersections.

*c*

_{sol}is the molar concentration of soluble solid material and

*ν*accounts for the stoichiometry of the reaction (Algive et al., 2010; Detwiler & Rajaram, 2007). Then, the total volume of mineral dissolved from a single channel surface during a time step Δ

*t*can be calculated using

#### 2.3.2 Dimensionless Groups Characterizing Network Evolution

*c*

_{eq}

*-c*

_{out}) is determined by a function

*f*and

*g*and Equation 14

*r*

_{0}is the initial average channel radius and

*t̂*is the dimensionless time, defined as

*γ*=

*c*

_{eq}

*/c*

_{sol}

*ν*is the acid capacity number, defined as the ratio between the number of molecules in a unit volume of a mineral to the number of molecules of the solute in a unit volume of fully saturated fluid (

*c*

_{in}= 0 is assumed at the domain inlet). The functions

*f*and

*g*characterize dissolution in channels, with dependence on time arising through

*r*and

*q*. To characterize the effects of transport and reaction rates on the dissolution process, we calculate

*f*and

*g*from the initial averaged parameters of the longitudinal channels, aligned in the main flow direction,

*x*. This leads to the following definition of the dimensionless Damköhler number:

*s*

_{0}and

*S*are the initial average and specific surface area of longitudinal pores and calculated using

*s*

_{0}

*=*Σ

*s*

_{x}/

*N*

_{x}and

*S = s*

_{0}/

*V*

_{cell}, where subscript

*x*is for longitudinal channels,

*N*is the total number of channels, and

*V*

_{cell}is cell volume (comprising the channels and the associated solid volume; see the green frame in Figure 1). Next,

*q̅*and

*v̅*are the average inlet flow rate and superficial velocity and calculated using

*q̅ = Q/N*

_{in}and

*v*̅ =

*q̅/A*

_{cell}, where

*N*

_{in}is the total number of inlet nodes,

*A*

_{cell}is cell's cross-sectional area, and

*V*

_{cell}

*/A*

_{cell}=

*l*. Note that the Péclet number does not appear in these equations because the diffusive effects in the axial direction are neglected.

### 2.4 Simulation Setup

#### 2.4.1 Initial and Boundary Conditions

*Q*, from network inlet face to outlet face and impose the pressure at the boundaries

*Q*, fixed, the pressure at the inlet nodes is adjusted at every time step. Along

*y*(transverse) direction, periodic boundary conditions for flow and transport are set, and concentration at the outlet nodes is calculated from solute mass conservation (Equation 11; free-flow boundary).

#### 2.4.2 Network and Reactive Transport Settings

We use a regular rectangular network of 100 × 100 nodes and consider heterogeneity in channel volumes by sampling their diameters from a lognormal distribution with a relative standard deviation of 2.25. Such a wide pore size distribution is typical of geological media (Lucia, 2007; Sahimi, 2011). To obtain representative results for the heterogeneous media, the results were averaged using 20 realizations of statistically similar properties.

*average*size of transverse channels (perpendicular to the main flow direction,

*y*), whereas the initial

*average*size of the longitudinal channels is initially the same in all simulations. Anisotropy degree is defined as

*a*is the average cross section of the cylindrical channels, calculated using

*a*

_{I}

*= Σπr*

_{I}

^{2}

*/N*

_{I}, where subscript index

*I*is

*y*or

*x*, denoting transverse and longitudinal channels, respectively.

*χ*is negative for wide transverse channels and positive for narrow ones. In some instances, anisotropy can be very pronounced, for example, in the case of unidirectional tensile joints, with permeability ratio of orders of magnitude between orthogonal directions (Adler et al., 2013; Barton, 2006). Consequently, we study a wide range of anisotropy degree,

*χ*, in between −0.98 and 0.98, corresponding to a ratio of 100 between cross sections of the channels aligned in orthogonal directions.

Each time step begins with calculation of the flow field by solving the system of Equations 1 and 2. Next, using the channel flow rates, *q*, the solute concentration at the nodes is calculated by solving Equation 11 and solute balance in nodes. Lastly, the change in pore radii is calculated using Equation 14. The time step is small enough so that the changes in pore radii will constitute a small fraction of the original radius. Additionally, a sensitivity test of the effect of time step was performed: Simulations were run with decreasing time step until the volume of reactive fluid (*V*_{f}) needed to achieve 100-fold permeability increase stabilized.

We focus here on relatively uniform dissolution, where the reaction rate is low compared to advective transport, setting *Da*_{L} = 0.1. The Damköhler number is calculated based on the average radius of longitudinal channels, which is initially the same in all the simulations (incorporated in Equation 18 through *s* and *G*). To study the effect of different reaction regimes, *G* is altered (100, 1, and 0.01) by modifying the reaction rate coefficient, *λ*. In order to maintain *Da*_{L} constant while varying *λ*, the flow rate *q̅* is changed accordingly (cf. Equations 18 and 19). The Péclet number is kept large in all simulations (*Pe*_{l} > 10^{5}) corresponding to the assumption of the dominance of advective transport considered here.

The model was implemented in Matlab computer code (MATLAB, 2018). The simulations were running on PC, with quad intel (R) Core (TM) i7-7700 CPU, 3.600 GHz, and 32 GB RAM, and run time was 10–40 s for a single simulation.

## 3 Results and Discussion

### 3.1 Media and Flow Field Homogenization

Prior to dissolution, the flow regime is governed by the imposed initial anisotropy, which controls flow focusing into preferential pathways and flow tortuosity (Jang et al., 2011; Sahimi, 2011; Thompson & Brown, 1991). This can be seen in Figures 2a.1–2a.3, which show initial flow for networks of different anisotropy. The presence of wide transverse channels (*χ* = −0.98) promotes bypassing narrow constrictions in the longitudinal direction. In this case, flow focuses in highly conductive tortuous pathways with dominant transverse flow components (Figure 2a.1), resulting in relatively high overall permeability of the medium, *k* (Figure 3a) (calculated using Darcy's law, *Q =* (*A*_{sys}*k/μ*)(*p*_{in}*/L*), where *A*_{sys} is the system cross-sectional area, *A*_{sys} *= N*_{in}*A*_{cell}).

In contrast, the presence of narrow transverse channels (*χ* = 0.98) leads to a more direct and relatively uniform flow in the longitudinal direction (Figure 2a.3), which also limits the overall permeability (Figure 3a), because any narrow constrictions along the longitudinal paths are difficult to bypass. Theoretical bounds for permeability are provided by considering the extreme cases of (i) no transverse channels, *χ* = 1, and (ii) transverse channels of infinite width, *χ* → −1 (blue circles in Figure 3a; Hunt & Ewing, 2009; Jang et al., 2011). The effect of anisotropy on flow is also manifested through the probability density function (PDF) of the velocities (Figure 2c), where *v*_{x} is the superficial velocity in the longitudinal channels, calculated using *v*_{x} *= q*_{x}*/A*_{cell}. Before dissolution takes place (at *t* = 0, black line) for wide transverse channels, *χ* = −0.98, flow heterogeneity is large with high number of low velocity values, spanning many orders of magnitude. No clear maximum in the distribution can be detected for *χ* = −0.98 and 0. Conversely, for narrow transverse channels, *χ* = 0.98, the flow is much more uniform with a well-defined maximum in the distribution.

Following dissolution, velocity distribution and the medium structure become more uniform with a clear maximum appearing in all distributions (Figures 2b and 2c). Both very low and very high velocities disappear from the distributions. The velocity distribution after dissolution depends considerably on the extent by which transport hinders reaction, characterized by the parameter *G* (Equation 19). When the reaction is surface-controlled (low *G*), dissolution increases channel sizes uniformly in the medium, homogenizing the transport properties and hence also the flow field. Under transport-controlled reaction conditions (large *G*), the homogenization process is even further enhanced, due to a more intense dissolution in the narrow channels compared to the slower dissolution in wider conduits (see supporting information Figure S1 for a comparison of the resulting flow field with *G* = 0.01, 1, and 100). For large *G*, the dissolution rate is mostly controlled by the diffusion rate of dissolved ions from the mineral surface to the bulk fluid, which is faster in narrower conduits (Varloteaux et al., 2013). This effect is manifested in the velocity distribution, with a smaller fraction of lower velocities (corresponding to flow in the narrow channels) at large *G*, compare Figure 2c.

*τ*and (ii) the degree of flow focusing

*η*. Here

*τ*is defined as (Koponen et al., 1996; Matyka & Koza, 2012)

*τ = Σ*(|

*v*

_{x}|

*+ |v*

_{y}

*|*)

*/Σv*

_{x}, where the summation is performed over all the channels. Flow focusing,

*η = n*

_{50}

^{−1}, is evaluated from the fraction of longitudinal channels that carry half of the total flow,

*n*

_{50}(Jang et al., 2011). The changes in

*τ*and

*η*are especially pronounced when the transverse channels are initially more conductive (

*χ*< 0), and flow is focused and highly tortuous (Figure 2a.1). At those conditions, the initial highly heterogeneous flow field is homogenized by the dissolution in narrow longitudinal channels, which creates flow shortcuts that reduce tortuosity and flow focusing (Figures 2b.1, 4a, and 4b). After extensive dissolution (

*k/k*

_{0}= 10

^{3}; not shown in Figure 4), the flow field is almost perfectly homogenized, with

*τ*and

*η*approaching their minimum values,

*τ*= 1 and

*η*= 2, corresponding to completely direct and uniform flow, respectively.

The homogenization by the uniform dissolution also acts to decrease system-wise anisotropy, reducing the difference in the sizes of the channels in orthogonal directions (anisotropy degree following dissolution *χ′* approaches zero, Figure 4c). Similarly, when the reaction is inhibited by transport, *G* = 100 and *G* = 1, the homogenization is enhanced and leads to a more isotropic system, due to intensified growth of the narrow channels. We note the small differences in anisotropy degree following dissolution, *χ′*, between fully and partially hindered reaction, *G* = 100 and *G* = 1. This is due to a competition: For *G* = 100 dissolution effectively acts to reduce anisotropy due to focused dissolution in the narrow channels; however, as the channels widen, strong transport inhibition leads to a decline of reaction rate, which in turn decreases the rate of homogenization compared to *G* = 1, leading eventually to a similar outcome.

The reduction in anisotropy is also evident in the permeability anisotropy following dissolution: As the dissolution progresses, the transverse permeability approaches that in the longitudinal direction, *k*_{x}*/k*_{y} → 1 (Figure 4d). Interestingly, also when anisotropy degree, *χ*, is initially large and the injection is in the direction of the more conductive channels, the system becomes more isotropic and both anisotropy degree, *χ*, and permeability anisotropy, *k*_{x}*/k*_{y}, decrease. Another observation is that an initially isotropic system (*χ* → 0) remains so following dissolution.

In the case of 3-D networks, the initial flow field tends to be more heterogeneous and focused in preferential pathways, relative to 2-D (Fiori & Jankovic, 2012). In such a case, it is expected that the homogenization effect and the associated change in properties will be even more intense. That is because the flow field homogenization is more pronounced when the initial flow heterogeneity is large.

### 3.2 Permeability Evolution

Pronounced effects of anisotropy degree, *χ*, and transport parameter, *G*, on network structure and flow field are also manifested in the longitudinal permeability evolution as shown in Figure 5. Here, *V*_{f} is defined as the volume of reactive fluid injected, normalized by the system volume. For narrow transverse channels, *χ* = 0.98, and low initial permeability, relatively small amount of reactive fluid injection, *V*_{f}, leads to a steep enhancement of permeability (Figures 3b and 5). In contrast, with wide transverse channels, *χ* = −0.98, and focused preferential flow, significant reactive fluid injection, *V*_{f}, and dissolution are required to attain the same permeability increase.

The effect of the reaction regime on permeability is milder than that of anisotropy, however, more intricate and evolving with time. At the initial stage of dissolution, permeability increases faster with larger *G* (Figure 5a). This is because at transport-controlled conditions (large *G*) dissolution primarily takes place at narrow channels that affect the flow considerably and their dissolution leads to steep permeability enhancement. As dissolution proceeds, this trend flips and *k/k*_{0} = 10^{3} is reached earlier under surface-controlled conditions (low *G*; Figure 5b). This change is attributed to the steady growth of the channels under the surface-controlled dissolution, and the corresponding slowdown of the growth when the reaction is hindered by transport as the dissolution proceeds.

### 3.3 Theoretical Results of Simplified System: A Wavy Channel

To analyze and demonstrate further the effect of the transport parameter *G*, we consider a simplified model system: a single channel with a wavy profile and repeated constrictions and dilations (black line in Figure 6a). This allows us to calculate the geometry evolution analytically and numerically (details of these methods are described in Appendix A). The characteristic dimensionless numbers for the channel are and *G* (the latter given by Equation 19).

When the reaction is transport controlled (*G* = 100), dissolution is enhanced at the channel constrictions (Deng, Molins, et al., 2018; Dykaar & Kitanidis, 1996), flattening the channel profile and making it more uniform (red lines in Figure 6a). Conversely, for surface-controlled reaction (*G* = 0.01), the channel erodes more uniformly, preserving the original profile shape (blue lines). The permeability evolves in a similar manner to that in the network model: At an early stage of the dissolution, for large *G*, channel permeability increase is steep due to enhanced dissolution of the constrictions, which are the main factor limiting the flow (Figure 6b). But as dissolution proceeds and the channel grows wider the transport limitation has an increasing effect, the reaction rate declines, and as a result, permeability enhancement is larger for *G* = 0.01 (see the wider channel for *G* = 0.01 for *V*_{f} = 10.5·10^{5} in Figure 6a). The calculation of channel permeability, *k*, is described in Appendix B.

### 3.4 Reaction Rate

The macroscopic (bulk) reaction rate depends on the intrinsic (batch) rate and the transport conditions, both at the pore scale and system scale. There is an intricate feedback between the void-space structure, transport, and reaction rate—transport closely depends on the detailed structure, which can evolve during the dissolution process, and consequently, reaction rate also evolves in time. Here, we study the effect of changing the anisotropy degree and the mode of transport, *G*, on the time-dependent bulk reaction rates.

^{2}T) along the channel can be calculated using

*c*(

*ξ*) is obtained by solving Equation 10 using

*c*(

*ξ =*0)

*= c*

_{in}; this leads to

*Ω*

_{S}

*=*1

*/s*

_{T}

*Σ*(

*sΩ*), where

*s*

_{T}is the total geometric surface area in the network and the summation is done over all the channels. To quantify the effect of anisotropy on reaction rate (even before any dissolution), we calculate separately the contribution to the reaction rates from longitudinal and transverse channels,

*Ω*

_{x0}and

*Ω*

_{y0}, using

*Ω*

_{I}

*=*(1

*/Σ*

_{I}

*s*)

*Σ*

_{I}(

*sΩ*) where the summation is performed over all the longitudinal (

*I = x*) or transverse channels (

*I = y*) at

*t*= 0.

Calculation of the initial reaction rate, *Ω*_{S0}, in the network shows that when the reaction is transport controlled (large *G*), it varies by an order of magnitude, depending on the anisotropy degree (black solid line in Figure 7a). The reaction rate is low for negative *χ*, due to significant diffusive inhibition of the dissolution in transverse channels, *Ω*_{y0}, and increases with increasing *χ* (red line). In contrast, the reaction rate in longitudinal channels, *Ω*_{x0}, is almost independent of *χ* (blue line). Although *Ω*_{y0} increases dramatically for large *χ*, it is only partially reflected in the bulk reaction rate *Ω*_{S0}, due to the low surface area of the longitudinal channels compared to the transverse ones. Following dissolution, as the channels widen, the bulk rate, *Ω*_{St}, decreases due to the diffusive limitation (see the dashed black line in Figure 7a which shows *Ω*_{St} at the time when *k/k*_{0} = 10^{3}). Intermediate conditions (*G* = 1) show a similar trend as the fully transport-controlled regime but with weaker effects (Figure 7b).

In contrast, under surface-controlled reaction conditions (low *G*), the reaction rate is nearly independent of anisotropy degree and also does not change appreciably following dissolution (Figure 7c). Slightly slower reaction rates are observed for low *χ*, due to the large surface area of the transverse channels, leading to increased dissolved concentration which reduces the driving force for reaction.

## 4 Applicability to Field Conditions

Lastly, we discuss the relevance of the results presented here to geological systems. In particular, we characterize the conditions for which geological media are subjected to similar reactive flow conditions and alterations as the ones studied here (Figure 8). The large variability in mineralogy characteristics of geological media results in a large range of reaction rates, *Ω*_{S}, and kinetic coefficients, *λ*. Examples for common rapid reactions (*λ* ~ 10^{−4} m/s) include limestone dissolution by acidic solutions (pH ~ 3; Peng et al., 2015; Plummer et al., 1978) or of halite by water (Alkattan et al., 1997). Intermediate rates occur in gypsum (*λ* ~ 10^{−6} m/s; Colombani, 2008) and limestone karst (pH ~ 6, *λ* ~ 10^{−7} m/s; Dreybrodt et al., 2005; Palmer, 1991), in all of which dissolution is commonly described using first-order kinetics (except close to saturation). Slower reaction is observed when dolomite lithology is involved (Busenberg & Plummer, 1982; Morse & Arvidson, 2002) or if the groundwater composition is close to saturation, where higher-order kinetics is observed (Dreybrodt et al., 2005; Palmer, 2011). An even slower dissolution reaction occurs in feldspar minerals (Li et al., 2006; Zhu, 2005).

This large variability in reaction rates suggests that low *Da*_{L} conditions and uniform dissolution can persist under a wide range of flow conditions or length scales (Steefel & Lasaga, 1990). However, for highly reactive systems a field-scale uniform dissolution can only take place under relatively high flow rates. For example, let us consider a range of system reactivities, *λ* = 10^{−4}, 10^{−7}, and 10^{−10} m/s, with a characteristic length scale, *L*, of 0.1 km and of specific surface areas, *S* = 0.1 m^{−1} for fractured media (Pacheco & Alencoão, 2006; Pacheco & Van der Weijden, 2014) to 10^{4} m^{−1} for porous rocks (Mostaghimi et al., 2013; Noiriel et al., 2012). A relatively uniform dissolution, characterized by *Da*_{L} = 0.1 (Equation 18), requires Darcy flux of *v̅*>10^{−2} m/s for highly reactive systems (*λ* ~ 10^{−4} m/s), *v̅*>10^{−5} m/s for moderate reactivity (*λ* ~ 10^{−7} m/s), and *v̅*>10^{−8} m/s for *λ* ≈ 10^{−10} m/s. High fluxes can occur in permeable shallow aquifers, especially in fractured (Kolditz, 2001; Palmer, 1991) and karstic media (Ford & Williams, 2013; White, 2002) and near pumping wells (Bear & Cheng, 2010; Niemi et al., 2017). However, in most cases, the flow rates are not high enough for uniform alterations to take place in highly reactive systems (Szymczak & Ladd, 2012) unless the system size is relatively small (~cm; Golfier et al., 2002; Menke et al., 2017). Consequently, nonuniform dissolution patterns (channeling) are typically observed in fast-reacting systems at the field scale (Bosák et al., 2016; Frumkin, 1994). In fact, even for high rate CO_{2} injections involving high reactivity, uniform dissolution is only expected at a large distance from the injection well, after pH sufficiently increases and reactivity drops (Luquot & Gouze, 2009; Niemi et al., 2017).

There is however evidence that field-scale uniform dissolution in carbonates can take place under natural conditions (moderate reactivity). Some karstic maze-like caves are formed by rapid floodwater flow into the entire fracture network, which leads to their uniform enlargement into an extensive network of passages, over time scales of 10^{4}–10^{5} years (Howard & Groves, 1995; Palmer, 1991, 2011). Exceptionally, a similar phenomenon is even documented to take place in highly reactive salt rock (Frumkin, 2013). It is noted that modeling of developed speleogenesis and large passage formation is beyond the scope of this work. Yet maze-cave formation is a good example of homogeneous dissolution at the system scale, which is limited by reaction, compatible with the uniformly dissolved systems analyzed in the present study.

Furthermore, carbonate dissolution can be remarkably slow, if induced by groundwater with a composition close to calcium saturation, producing significant karst and large speleological structures over time scales of millions of years (Dreybrodt et al., 2005; Frumkin et al., 2017; Palmer, 2011). In field-scale systems (*L* > 0.1 km) with groundwater close to saturation, a uniform dissolution may control karst evolution in fractured media even under the low groundwater velocities expected in deep aquifers (>1 km), where Darcy flux may be of the order of ~10^{−7} m/s (Garven, 1995; Gvirtzman et al., 1997), or under higher flow rates in porous rocks where specific surface area, *S*, is large (Mostaghimi et al., 2013; Noiriel et al., 2012).

As for the reaction regime (characterized by the transport parameter *G*), we note that the reaction can be transport controlled if the reaction rate coefficient, *λ*, is high or if the pore sizes or apertures are large, leading to inhibition of reaction by diffusive transport that becomes substantial for *G* > ~0.1. For example, considering *G =* 0.1 (Equation 19), *Sh* = 4, *D* ≈ 10^{−9} m^{2}/s, and initial pore size of *r* ~ 10^{−5}–10^{−3} m (Blunt, 2017; Dreybrodt et al., 2005; Peng et al., 2015) provides *λ* ~ 10^{−7}–10^{−5} m/s. This implies that relatively fast geochemical systems, as well as limestone karstic systems, can be within the transport-controlled reaction regime.

## 5 Summary and Conclusions

We have studied uniform dissolution in heterogeneous and anisotropic media using a 2-D network model. In this regime, dissolution takes place throughout the medium, leading to a homogenization of the medium and consequently of the flow field, which is expected to be even more pronounced in 3-D networks. When transverse channels are wide compared to longitudinal ones, the flow field is initially highly heterogeneous and the flow is focused in a few tortuous pathways. In such media, the dissolution-induced homogenization effect is very pronounced and the preferential tortuous flow paths are eliminated quickly. Furthermore, uniform dissolution always leads to a reduction in anisotropy, even when the injection is in the direction of the wider, more conductive channels. In the transport-controlled regime, the homogenization is further enhanced because the diffusive hindrance is weaker in the narrow channels and leads to their selective enlargement.

The initial anisotropy degree also controls the permeability evolution: For wide transverse channels and heterogeneous flow field, initial permeability is high and its enhancement by dissolution is relatively slow. In the opposite case of narrow transverse channels, the ability of the flow to bypass narrow constrictions is limited; therefore, the initial permeability is low and its enhancement is much more rapid. At the uniform regime, the underlying mechanism for permeability enhancement is the enlargement of the less conductive channels. This is in contrast to the wormholing regime at higher *Da*_{L}, where preferential pathways are enhanced (Golfier et al., 2002; Szymczak & Ladd, 2009). The reaction regime, characterized by the transport parameter *G*, has an intricate effect on the permeability evolution: This has been demonstrated for an idealized model medium of a channel with a wavy profile. For transport-controlled dissolution (large *G*), permeability initially increases rapidly due to selective dissolution of the narrower constrictions, whereas at later stages transport inhibition gains importance and permeability enhancement becomes slower compared to surface-controlled reaction conditions.

Further, it is shown that the pore-scale reaction regime and anisotropy degree control the macroscopic reaction rates, which depend on the intrinsic kinetic rate and the transport conditions. At a transport-controlled regime, bulk reaction rates change by almost an order of magnitude depending on initial anisotropy degree and following dissolution and widening of the channels, whereas in the surface-controlled regime bulk reaction rates barely change. Finally, we discuss the relevance of the results presented here to geological systems and characterize the conditions for which geological media are subjected to uniform alterations. We also discuss field-scale examples of karstic homogeneous dissolution.

Our findings show that within the uniform dissolution regime the medium becomes more homogenous and isotropic, in contrast to the higher *Da*_{L} regime when heterogeneity increases with the dissolution and wormholes develop (Aharonov et al., 1997; Upadhyay et al., 2015). The results provide insights into the link between void-space geometry and the evolution of macroscopic properties of geological media due to the dissolution. These findings contribute to our understanding of geological processes such as diagenesis or karst evolution in fractured media, as well as of engineering applications such as carbon geosequestration and groundwater management.

## Acknowledgments

R. Holtzman acknowledges partial support from the Israeli Science Foundation (ISF-867/13). P. Szymczak was supported by the National Science Centre (Poland) under research Grant 2016/21/B/ST3/01373. We acknowledge Israel Water Authority student's scholarship. We also thank S. Emmanuel for valuable discussions.

## Appendix A: Dissolution in Wavy Channel

*L*, of initially wavy profile (black line in Figure 6a)

*x*is the axial coordinate and

*U*and

*ω*are amplitude and angular frequency. The derivation below is analogous to that in Budek and Szymczak (2012) but for the different initial conditions considered. Here, we assume low Reynolds number and small amplitude of the corrugations. Under these conditions the velocity profile can be assumed to be locally parabolic. At higher Reynolds number the flow separation takes place and vortices appear (Deiber & Schowalter, 1979; Esquivel et al., 2012) which affect mass transport (Dykaar & Kitanidis, 1996).

*γ*=

*c*

_{eq}

*/c*

_{sol}

*ν*,

*G*=

*λ*2

*r*

_{0}

*/ShD*,

*ĉ =*(

*c*

_{eq}

*-c*)

*/c*

_{eq}, and the scaled radius is

*r̂ = r/r*

_{0}. With those definitions Equation 10 in the main text gives

*t̂ = λγt/r*

_{0}and axial coordinate

*x̂ = x/L*, Equation A2 becomes

*x̂*and using Equation A5 leads to

*ĉ*through the derivative

*∂r̂/∂t̂*using Equation A4 provides equation in terms of

*r̂*alone,

*W*(

*x̂*) can be solved using the initial condition of the scaled Equation A1

*r̂*(

*x̂ =*0,

*t̂*) =

*r̂*

_{in}(

*t̂*), attained by solving Equation A4 with

*ĉ*= 1

*G →*∞, Equation A12 reduces to

## Appendix B: Permeability of a Channel of Variable Cross Section

*a = πr*

_{0}

^{2}, leads to

## Open Research

# Data Availability Statement

Data produced in this study are available online (https://doi.org/10.5281/zenodo.3988934).