A Discrete Fracture Network Model for Coupled Variable-Density Flow and Dissolution with Dynamic Fracture Aperture Evolution

Abstract

Fluid flow and mass transfer processes in some fractured aquifers are negligible in the low-permeability rock matrix and occur mainly in the fracture network. In this work, we consider coupled variable-density flow (VDF) and mass transport with dissolution in discrete fracture networks (DFNs). These three processes are ruled by nonlinear and strongly coupled partial differential equations (PDEs) due to the (i) density variation induced by concentration and (ii) fracture aperture evolution induced by dissolution. In this study, we develop an efficient model to solve the resulting system of nonlinear PDEs. The new model leverages the method of lines (MOL) to combine the robust finite volume (FV) method for spatial discretization with a high-order method for temporal discretization. A suitable upwind scheme is used on the fracture network to eliminate spurious oscillations in the advection-dominated case. The time step size and the order of the time integration are adapted during simulations to reduce the computational burden while preserving accuracy. The developed VDF-DFN model is validated by simulating saltwater intrusion and dissolution in a coastal fractured aquifer. The results of the VDF-DFN model, in the case of a dense fracture network, show excellent agreement with the Henry semi-analytical solution for saltwater intrusion and dissolution in a coastal aquifer. The VDF-DFN model is then employed to investigate coupled flow, mass transfer and dissolution for an injection/extraction well pair problem. This test problem enables an exploration of how dissolution influences the evolution of the fracture aperture, considering both constant and variable dissolution rates.

1. Introduction

In this work, we consider coupled fluid flow, mass transfer and dissolution processes with variable density in fractured aquifers. Variable-density flow in fractured domains finds applications in various instances including CO₂ storage [1,2], geothermal energy [3,4] and groundwater management in fractured aquifers [5,6,7,8,9]. Furthermore, when dissolution occurs, it significantly impacts the transmissivity of the fractures, subsequently influencing the flow and the solute transport phenomena. Variable-density flow with dissolution has been shown to play a role in various processes encountered in fractured media, such as geothermal energy extraction [10], the management of oil reservoirs [11], CO₂ sequestration [12] and nuclear waste disposal [13]. Further, coupled processes of variable-density flow and dissolution in fractured aquifers can occur as a result of excessive pumping, creating voids and sinkholes that lead to land subsidence or ground collapse. This phenomenon is observed across the world and has great economic and social impacts as it can endanger the safety of human lives and properties [14].

Modeling coupled fluid flow, mass transfer and dissolution with variable density in fractured aquifers is challenging due to the high degree of nonlinearities induced by (i) the density variations and (ii) the evolution of the fracture aperture due to dissolution [15]. In the literature, the focus on dissolution processes has predominantly centered on unfractured domains and their role in fostering the creation and propagation of fractures [16,17]. The effect of dissolution on fracture evolution in fractured domains has received limited attention [18,19].

Either explicit or implicit representations of fractures can be used for modeling flow and mass transfer in fractured domains [20]. In the implicit approach, the fractures are taken into account using equivalent single- or multi-porosity models [4]. These models are based on equivalent parameters, which depend on the characteristics of the fractures as well as on the fracture network configuration [21]. The explicit approach, for its part, uses a discrete fracture representation. Among the variants of the explicit approach, the discrete fracture matrix (DFM) model allows for considering flow and mass transfer in both the fracture network and the rock matrix [22,23,24]. Fractures are represented by thin regions with distinct properties from those of the rock matrix [5]. The DFM model requires a mesh conforming to the fracture network; therefore, because of gridding problems, it is not adapted for situations with a high density of fractures. Recently, methods have emerged using non-conforming meshes to overcome gridding-related challenges [25,26].

Another variant of the explicit approach is the discrete fracture network (DFN) model, suitable for situations where matrix permeability is negligible. In this model, the matrix is considered impermeable, with flow and transport occurring exclusively through interconnected fractures [27,28]. DFN models are particularly well-suited for porous media where the permeability and porosity are entirely derived from fractures [20]. Recently, Tabrizinejadas et al. [29] showed the advantages of DFNs for modelling flow and transport in fractured domains without considering density variations. To the best of our knowledge, coupled variable-density flow and dissolution processes within a DFN framework remain unexplored. This paper aims to fill this gap by developing an efficient numerical model for this highly nonlinear problem to better investigate the effect of density variation and dissolution processes in DFNs.

In this study, the coupled nonlinear equations governing variable-density flow, solute transport and dissolution are simultaneously solved to prevent any potential errors associated with operator splitting. The numerical model integrates appropriate numerical schemes for both space and time discretization. The method of lines (MOL) is utilized to transform partial differential equations (PDEs) into a set of ordinary differential equations (ODEs) by discretizing spatial derivatives while maintaining time derivatives in their continuous form. The spatial discretization is based on the finite volume (FV) method, recognized for its robustness and cost-effectiveness in handling conservation laws [30]. An upwind scheme is used at the fracture intersection nodes to avoid unphysical oscillations observed with high advection transport. Temporal discretization of the obtained ODE system is performed with the DASPK [31] time solver, leveraging high-order integration methods and an efficient adaptive time-stepping scheme. DASPK has demonstrated efficiency in solving coupled flow and mass transport equations in both unfractured [32,33] and fractured domains [34]. The newly developed model for variable-density flow and dissolution with a DFN is referred to as VDF-DFN.

This paper is structured as follows. Section 2 presents the mathematical equations ruling the coupled processes of variable-density flow, solute transfer and dissolution. Section 3 provides a comprehensive explanation of the novel VDF-DFN numerical model. Section 4 is devoted to numerical experiments where the model’s validity is first investigated against the semi-analytical solution for a variant of the Henry problem involving saltwater intrusion and dissolution in a fractured coastal aquifer. Then, we apply the VDF-DFN model to simulate coupled fluid flow, mass transfer and dissolution for a horizontal injection/extraction well pair problem. Additionally, we investigate the evolution of the fracture aperture under both constant and variable dissolution rates.

2. The Mathematical Model

The mathematical model is constructed under the following assumptions. The fracture network is of codimension one within a two-dimensional bounded domain. Flow and transport in the fractures are one-dimensional, and transport is governed by advection, dispersion and dissolution processes. The fractures are filled with a porous medium through which the flow is governed by Darcy’s law. The fracture permeability is a function of the aperture, and the porosity is constant. The temperature is constant, and mechanical deformations are negligible. The fracture aperture varies solely due to dissolution, which is governed by a first-order dissolution approximation. The fluid density varies linearly with the concentration.

The flow inside a fracture with a variable aperture $b\left[L\right]$ due to dissolution is governed by the following mass conservation equation:

$${\displaystyle \frac{\partial \left(\varphi b\right)}{\partial t}}+\nabla ·\left(b\mathit{q}\right)=0$$

(1)

where the fluid velocity $\mathit{q}$ $\left[L·T^{-1}\right]$ is obtained using Darcy’s law, which is widely valid for flow in porous media [35,36]. In the case of variable density, it is written as

$$\mathit{q}=-{\displaystyle \frac{{\rho }_{0}g\kappa }{\mu }}\left(\nabla h+{\displaystyle \frac{\rho -{\rho }_0}{{\rho }_0}}\nabla z\right)$$

(2)

where $\varphi$ is the porosity [-] of the porous medium in the fracture, $t$ is the time $\left[T\right]$, $h={\displaystyle \frac{P}{{\rho }_{0}g}}+z$ is the pressure head $\left[L\right]$, $P$ is the pressure $\left[Pa\right]$, $z$ is the elevation $\left[L\right]$, $g$ is the gravity acceleration $\left[L·T^{-2}\right]$, $\rho$ is the density of mixed saltwater and freshwater $\left[M·L^{-3}\right]$, $\mu$ is the fluid viscosity $\left[M·L^{-1}·T^{-1}\right]$, ${\rho }_0$ is the freshwater density $\left[M·L^{-3}\right]$ and $\kappa$ is the permeability $\left[L^2\right]$ approximated using the cubic law $\kappa ={\displaystyle \frac{b^2}{12}}$ [37].

The transport of a contaminant inside a fracture is governed by the following reactive transport equation [38]:

$${\displaystyle \frac{\partial \left(\varphi bc\right)}{\partial t}}+\nabla ·\left(b\mathit{q}c\right)-\nabla ·\left(bD\nabla c\right)=R(c)$$

(3)

where $c$ is the solute concentration $\left[M·L^{-3}\right]$, $R\left(c\right)$ is the source/sink term $\left[M·L^{-2}·T^{-1}\right]$ representing the dissolution process and $D$ $\left[L^2·T^{-1}\right]$ is the dispersion coefficient in the fracture, given by

$$D={\alpha }_L\left|\mathit{q}\right|+D_m$$

(4)

where $D_m\left[L^2·T^{-1}\right]$ is the diffusion coefficient and ${\alpha }_L\left[L\right]$ is the longitudinal dispersivity.

Using mass conservation Equation (1), transport Equation (3) becomes

$$\varphi b{\displaystyle \frac{\partial c}{\partial t}}+b\mathit{q}·\nabla c-\nabla ·\left(bD\nabla c\right)=R(c)$$

(5)

The flow system (Equations (1) and (2)) and transport Equation (5) are coupled by (i) the dissolution process and (ii) the nonlinearity arising from the variation in density. Assuming a constant temperature and negligible fluid compressibility, the fluid density is assumed to vary linearly with the concentration as follows:

$$\rho ={\rho }_0+\left({\rho }_1-{\rho }_0\right)c$$

(6)

where ${\rho }_1$ is the density of the contaminant fluid.

Because of dissolution, the aperture varies with time as follows:

$${\rho }_r\delta {\displaystyle \frac{\partial b}{\partial t}}-R\left(c\right)=0$$

(7)

where ${\rho }_r$ is the rock density $\left[M·L^{-3}\right]$ and $\delta$ is the stoichiometric coefficient $\left[-\right]$ corresponding to the mass of solute for a unit mass of dissolved rock.

A first-order approximation is used for the reaction term [38,39]:

$$R\left(c\right)=k_C\left(c_S-c\right)$$

(8)

where $k_C$ is the dissolution coefficient $\left[L·T^{-1}\right]$ and $c_S$$\left[M·L^{-3}\right]$ is the concentration at saturation.

Two configurations are examined in this work. In the first configuration, the dissolution coefficient remains constant, while in the second, the rate $k_C$ increases with the fluid velocity, as observed with artificial water circulation systems within hot dry rocks [40].

The system of coupled Equations (1)–(8) governing flow, transport and dissolution must be solved with the corresponding initial and boundary conditions. The flow boundary conditions can be of the Dirichlet or Neumann type, fixing either the head or the flux for boundary fractures. The initial flow condition corresponds to the initial head in all fracture elements. The boundary conditions for transport are of the Dirichlet type, if the concentration is prescribed, or of the Neumann type if the dispersive flux is prescribed. The initial conditions for transport correspond to the initial concentration in the fracture network.

The mathematical model formed by Equations (1)–(8) cannot be solved analytically because of its high nonlinearity. In the next section, an efficient numerical model is developed for an accurate solution to the mathematical model.

3. The Numerical Model

We consider a two-dimensional fractured aquifer where the rock matrix is assumed to be completely impermeable. Thus, flow, mass transport and dissolution occur only within the connected fractures. An efficient numerical model is developed on the fracture network to address the system of coupled Equations (1)–(8) governing fluid flow, mass transfer and dissolution with variable density and fracture aperture evolution. The MOL is employed by first discretizing the spatial derivatives to obtain a system of nonlinear ODEs. The latter is then solved using high-order time integration methods. The FV method, known for its robustness and cost-effectiveness in the discretization of conservation laws, is employed for the spatial discretization. The resulting coupled system of ODEs is tackled using the DASPK solver for time integration.

3.1. Spatial Discretization of the Flow

Neglecting the variation in the porosity inside the fracture, as compared to the fracture aperture evolution, integration over the fracture element $k$ of continuity Equation (1) yields

$${\varphi }^k{\mathcal{l}}^k{\displaystyle \frac{\partial b^k}{\partial t}}+{\overline{q}}_1^k+{\overline{q}}_2^k=0$$

(9)

where ${\varphi }^k$ and ${\mathcal{l}}^k$ are the porosity and the length, assumed to be constant for fracture branch $k$. The aperture $b^k$ of $k$ is assumed to vary due to the dissolution process. The fluxes ${\overline{q}}_1^k$ and ${\overline{q}}_2^k$ at the two extremity nodes of $k$ are considered positive for outflow (see Figure 1).

With $h^k$, $z^k$ and $c^k$ respectively denoting the head, elevation and concentration of fracture branch $k$, and $T{z}_i^k$ and $T{h}_i^k$ respectively denoting the elevation and the head at the extremity node $i$ of $k$, the expression of the flux ${\overline{q}}_i^k$ is obtained from the discretization of the Darcy Equation (2) as follows:

$${\overline{q}}_i^k={\displaystyle \frac{{\rho }_{0}g{\left(b^k\right)}^2}{12\mu }}{\displaystyle \frac{2b^k}{{\mathcal{l}}^k}}\left(h^k-T{h}_i^k+r_{\rho }\left(z^k-T{z}_i^k\right)c^k\right)$$

(10)

with $r_{\rho }={\displaystyle \frac{{\rho }_1-{\rho }_0}{{\rho }_0}}$.

Plugging Equation (10) into Equation (9) yields

$${\varphi }^k{\mathcal{l}}^k{\displaystyle \frac{\partial b^k}{\partial t}}+{\displaystyle \frac{{\rho }_{0}g{\left(b^k\right)}^3}{6\mu {\mathcal{l}}^k}}\left(2h^k-T{h}_1^k-T{h}_2^k\right)=0$$

(11)

The mass conservation of water at each node $i$ of the intersection of $n$ fractures (see Figure 1b corresponding to $n=4$) is written as

$${\displaystyle \sum _{l=1}^n{\overline{q}}_i^l}=Q_i^{\ast }$$

(12)

where $Q_i^{\ast }$ corresponds to the pumping/injection rate at intersection node $i$.

Combining Equations (10) and (12) gives

$${\displaystyle \sum _{l=1}^n\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}\left(h^l-T{h}_i^l+r_{\rho }\left(z^l-T{z}_i^l\right)c^l\right)}=Q_i^{\ast }$$

(13)

The continuity of the head at intersection node $i$ $\left(T{h}_i=T{h}_i^l,\forall l=1,\dots ,n\right)$ yields

$$T{h}_i={\displaystyle \frac{{\displaystyle \sum _{l=1}^n\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}\left(h^l+r_{\rho }\left(z^l-T{z}_i^l\right)c^l\right)-Q_i^{\ast }}}{{\displaystyle \sum _{l=1}^n\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}}}}$$

(14)

Assuming that the first node of fracture $k$ is shared by $n_1$ fractures and the second node is shared by $n_2$ fractures, the substitution of Equation (14) for $T{h}_1^k$ and $T{h}_2^k$ in mass conservation Equation (11) yields the final flow equation to solve:

$${\varphi }^k{\mathcal{l}}^k{\displaystyle \frac{\partial b^k}{\partial t}}+{\displaystyle \frac{{\rho }_{0}g{\left(b^k\right)}^3}{6\mu {\mathcal{l}}^k}}\left(2h^k-{\displaystyle \frac{{\displaystyle \sum _{l=1}^{n_1}\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}\left(h^l+r_{\rho }\left(z^l-T{z}_1^l\right)c^l\right)-Q_1^{\ast }}}{{\displaystyle \sum _{l=1}^{n_1}\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}}}}-{\displaystyle \frac{{\displaystyle \sum _{l=1}^{n_2}\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}\left(h^l+r_{\rho }\left(z^l-T{z}_2^l\right)c^l\right)-Q_2^{\ast }}}{{\displaystyle \sum _{l=1}^{n_2}\frac{{\rho }_{0}g{\left(b^l\right)}^3}{6\mu {\mathcal{l}}^l}}}}\right)=0$$

(15)

In this equation, the unknowns are the heads, the apertures and the concentrations at fracture branch $k$ and at its sharing fracture branches $l$. The flow system is constructed by writing Equation (15) for all the fracture branches forming the fracture network.

3.2. Discretization of Transport and Dissolution

Integrating advection–dispersion transport Equation (5) over $k$ yields

$${\displaystyle \underset{0}{\overset{{\mathcal{l}}^k}{\int }}\varphi b\frac{\partial c}{\partial t}}+{\displaystyle \underset{0}{\overset{{\mathcal{l}}^k}{\int }}b\nabla ·\left(qc\right)}-{\displaystyle \underset{0}{\overset{{\mathcal{l}}^k}{\int }}bc\nabla ·q}-{\displaystyle \underset{0}{\overset{{\mathcal{l}}^k}{\int }}\nabla ·\left(bD\nabla c\right)}={\displaystyle \underset{0}{\overset{{\mathcal{l}}^k}{\int }}R\left(c\right)}$$

(16)

which can be rewritten as

$${\varphi }^k{b}^k{\mathcal{l}}^k{\displaystyle \frac{\partial c^k}{\partial t}}+q_1^{t,k}+q_2^{t,k}-\left({\overline{q}}_1^k+{\overline{q}}_2^k\right)c^k={\mathcal{l}}^{k}R\left(c^k\right)$$

(17)

The total flux $q_m^{t,k}$ at the extremities $m=1,2$ of fracture branch $k$ is defined by

$$q_m^{t,k}=q_m^{adv,k}+q_m^{disp,k}$$

(18)

where $q_m^{adv,k}$ and $q_m^{disp,k}$ are, respectively, the advective and dispersive flux at extremity $m$ of $k$.

To avoid unphysical oscillations, an upwind scheme is used for the advective flux as follows:

$$q_m^{adv,k}={\overline{q}}_m^k\left[\underset{c_m^{k,\ast }}{\underbrace{{\lambda }_m^k{c}^k+\left(1-{\lambda }_m^k\right)c_m}}\right]$$

(19)

where $c^k$ is the concentration at the center of fracture branch $k$, $c_m$ is the concentration at its extremity node $m$, and ${\lambda }_m^k$ is the upwinding parameter, defined as ${\lambda }_m^k=1$ if $\left({\overline{q}}_m^k\ge 0\right)$ or ${\lambda }_m^k=0$ otherwise.

Thus, the advective flux at node $m$ is calculated using the upstream concentration $c_m^{k,\ast }={\lambda }_m^k{c}^k+\left(1-{\lambda }_m^k\right)c_m$, which corresponds to the concentration of fracture $k$ $\left(c_m^{k,\ast }=c^k\right)$ in the case of an outflow $\left({\overline{q}}_m^k\ge 0\right)$ and to the concentration of intersection node $m$ $\left(c_m^{k,\ast }=c_m\right)$ in the case of an inflow $\left({\overline{q}}_m^k<0\right)$.

The mass conservation of the contaminant at intersection node $m$, shared among $n$ fractures, is written as

$${\displaystyle \sum _{l=1}^n{q}_m^{t,l}=0}$$

(20)

The dispersive flux at node $m$ of $k$ is approximated by

$$q_m^{disp,k}={\displaystyle \frac{2b^k{D}_k}{{\mathcal{l}}^k}}\left(c^k-c_m\right)$$

(21)

where $D_k$ is the dispersion at fracture element $k$, expressed using Equation (4) as

$$D_k={\alpha }_L^k\left|q_k\right|+D_m^k$$

(22)

where ${\alpha }_L^k$, $\left|q_k\right|={\displaystyle \frac{\left|q_2^k-q_1^k\right|}{2b^k}}$ and $D_m^k$ are, respectively, the dispersivity coefficient, the norm of the velocity and the molecular diffusion in fracture branch $k$.

Substituting Equations (19) and (21) into Equation (20) provides the expression of the concentration at node $m$ as follows:

$$c_m={\displaystyle \frac{{\displaystyle \sum _{l=1}^n\left({\lambda }_m^l{\overline{q}}_m^l+\frac{2b^l{D}_l}{{\mathcal{l}}^k}\right)c^l}}{{\displaystyle \sum _{l=1}^n\left(\frac{2b^l}{{\mathcal{l}}^l}{D}_l-{\overline{q}}_m^l\left(1-{\lambda }_m^l\right)\right)}}}$$

(23)

Plugging Equations (19), (21) and (23) into Equation (18), which we substitute into transport Equation (17), yields the final transport equation:

$$\begin{array}{l}{\varphi }^k{b}^k{\mathcal{l}}^k{\displaystyle \frac{\partial c^k}{\partial t}}+\left(1-{\lambda }_1^k\right){\overline{q}}_1^k\left[{\displaystyle \frac{{\displaystyle \sum _{l=1}^{n_1}\left({\lambda }_1^l{\overline{q}}_1^l+\frac{2b^l{D}_l}{{\mathcal{l}}^k}\right)c^l}}{{\displaystyle \sum _{l=1}^{n_1}\left(\frac{2b^l}{{\mathcal{l}}^l}{D}_l-{\overline{q}}_1^l\left(1-{\lambda }_1^l\right)\right)}}}-c^k\right]+\left(1-{\lambda }_2^k\right){\overline{q}}_2^k\left[{\displaystyle \frac{{\displaystyle \sum _{l=1}^{n_2}\left({\lambda }_2^l{\overline{q}}_2^l+\frac{2b^l{D}_l}{{\mathcal{l}}^k}\right)c^l}}{{\displaystyle \sum _{l=1}^{n_2}\left(\frac{2b^l}{{\mathcal{l}}^l}{D}_l-{\overline{q}}_2^l\left(1-{\lambda }_2^l\right)\right)}}}-c^k\right]\\ +{\displaystyle \frac{2b^k{D}_k}{{\mathcal{l}}^k}}\left[2c^k-{\displaystyle \frac{{\displaystyle \sum _{l=1}^{n_1}\left({\lambda }_1^l{\overline{q}}_1^l+\frac{2b^l{D}_l}{{\mathcal{l}}^k}\right)c^l}}{{\displaystyle \sum _{l=1}^{n_1}\left(\frac{2b^l}{{\mathcal{l}}^l}{D}_l-{\overline{q}}_1^l\left(1-{\lambda }_1^l\right)\right)}}}-{\displaystyle \frac{{\displaystyle \sum _{l=1}^{n_2}\left({\lambda }_2^l{\overline{q}}_2^l+\frac{2b^l{D}_l}{{\mathcal{l}}^k}\right)c^l}}{{\displaystyle \sum _{l=1}^{n_2}\left(\frac{2b^l}{{\mathcal{l}}^l}{D}_l-{\overline{q}}_2^l\left(1-{\lambda }_2^l\right)\right)}}}\right]={\mathcal{l}}^k{k}_C\left(c_S-c^k\right)\end{array}$$

(24)

Finally, dissolution Equation (7) is integrated over fracture k and written as follows using Equation (8):

$${\rho }_r\delta {\displaystyle \frac{\partial b^k}{\partial t}}-k_C\left(c_S-c^k\right)=0$$

(25)

To sum up, the final nonlinear flow–transport–dissolution system with variable density and variable aperture is formed by the coupled Equations (15), (24) and (25). This system has heads $h^k$, concentrations $c^k$ and apertures $b^k$ for all fracture branches k as its unknowns. Hence, the size of the nonlinear flow–transport–dissolution system is triple the number of fractures.

3.3. Temporal Discretization

The resulting nonlinear system is integrated with respect to time using high-order methods due to their efficiency for nonlinear systems compared to lowest-order methods [32]. A single system of ODEs is constructed as follows:

$$G\left(t,\mathit{y},{\mathit{y}}^{\prime }\right)=0$$

(26)

where the vector $\mathit{y}={\left[h^k,c^k,b^k\right]}_{k=1,\dots ,nb\_fractures}$ corresponds to the unknown head, concentration and aperture at all the fracture branches of the network.

For the solution of system (26), we use DASPK, a sophisticated solver for ODE systems and differential–algebraic equation (DAE) systems [31]. DASPK employs the Fixed Leading Coefficient Backward Difference Formulas (FLCBDFs) [41,42,43] of varying orders up to five. DASPK incorporates an efficient automatic time-stepping scheme. It dynamically adjusts both the order of integration and the time step length to improve efficiency and, at the same time, ensure a small temporal error. The Newton Method is used to linearize system (26), where the Jacobian matrix J is approximated by the finite difference method. To reduce the computational cost, the same Jacobian is utilized for multiple time steps, with updates only occurring when necessary (when convergence fails). The efficiency of the DASPK time solver has been demonstrated for nonlinear coupled processes in porous media [33].

4. Numerical Experiments

This section presents numerical experiments conducted using the developed VDF-DFN model for the simulation of two test problems. The first example is a variant of the Henry problem involving saltwater intrusion with dissolution in a fractured coastal aquifer. In this scenario, a constant fracture aperture is assumed, and the results obtained using different fracture networks are compared to the semi-analytical solution established in the literature [44] for saltwater intrusion with dissolution in a coastal aquifer. In the second example, the VDF-DFN model is utilized to simulate coupled fluid flow, mass transfer and dissolution for an injection/extraction well pair problem. Here, we explore the impact of dissolution on the evolution of the fracture aperture, considering both constant and variable dissolution rates.

4.1. The Henry Saltwater Intrusion Problem with Dissolution

The Henry problem [45] serves as a widely recognized illustration of saltwater intrusion within a hypothetical rectangular coastal aquifer 2 m in length and 1 m high (Figure 2). The original problem deals with a confined coastal aquifer where saltwater intrudes from the right boundary due to its higher density. This intrusion persists until equilibrium is attained with the freshwater flowing in from the opposite left vertical boundary, as illustrated in Figure 2. Henry’s original semi-analytical solution, developed in 1964, determined steady-state isochores using the Fourier Galerkin (FG) method. The FG method is based on the expansion of the stream function and the concentration with Fourier series. However, the initial FG semi-analytical solution derived by Henry was deemed insufficiently accurate, primarily due to its low number of terms in the Fourier series [46]. Subsequent studies introduced more precise semi-analytical solutions, incorporating a greater number of terms [46,47].

In the following, we consider the semi-analytical solution developed by Younes and Fahs [44], addressing the complex scenario of saltwater intrusion with dissolution and low diffusion. The chosen diffusion coefficient was 10 times smaller than the original coefficient used by Henry. To ensure the stability of the FG solution, a substantial increase in the number of coefficients in the Fourier expansion was necessary, ultimately reaching 6195.

Table 1. Model parameterizations for the Henry problem with dissolution and low dispersion.

Permeability	$k_0=1.0204\times {10}^{-9}$ m2
Porosity	$\varphi =0.35$
Dispersivities	${\alpha }_L={\alpha }_T=0$ m
Molecular diffusion	$D_m=6.6\times {10}^{-7}$ m2/s
Dissolution rate	$k_C={10}^{-6}$ m/s
Dimensioless concentration at saturation	$c_S=10$
Flow boundary conditions	- Hydrostatic distribution at right vertical boundary - Fixed flux at left vertical side: $Q_0=6.6\times {10}^{-5}$ m2/s - Top and bottom are impermeable boundaries
Transport boundary conditions	- Zero concentration c = 0 on left vertical side $\left({\rho }_0=1000\mathrm{kg}/{\mathrm{m}}^3\right)$. - Dimensionless concentration c = 1 on right-hand side $\left({\rho }_1=1025\mathrm{kg}/{\mathrm{m}}^3\right)$. - Zero normal diffusion along top and bottom

shows the parameterization and boundary conditions. The computed concentration is relative to that of the sea; hence, it is dimensionless.

In the DFN simulation of this problem, the coastal aquifer is considered fractured with a uniform orthogonal distribution of the fractures, such as in [7]. The network contains $\left(n_f+1\right)$ horizontal fractures and $\left(2n_f+1\right)$ vertical fractures (Figure 2). Density-driven flow and dissolution processes exclusively take place within the fracture network. To be equivalent to the unfractured case, the injection of freshwater flux is fixed to $Q_0/\left(nf+1\right)$ for the intersection nodes that touch the left vertical boundary of the domain. All fractures maintain fixed permeability $k_f=k_0$ and constant aperture $b={\displaystyle \frac{1}{\left(nf+1\right)}}$.

Figure 3a shows the long-time steady-state distribution of the concentration for calculations using a coarse network corresponding to $nf=10$. As depicted in Figure 3a, the saltwater intrusion occurs at the right side, moving to the left and ultimately reaching equilibrium. Remarkably, the intrusion is more prominent towards the bottom, predominantly influenced by density effects. Notice that the substantial amount of saltwater in the domain is attributed to both dissolution and intrusion.

Figure 3b shows a comparison between the FG semi-analytical solution and the VDF-DFN numerical solution obtained using a coarse fracture network corresponding to $nf=10$ and a dense fracture network corresponding to $nf=40$. The results of this figure show very good agreement between the semi-analytical solution and the VDF-DFN results obtained on the dense fracture network. This strong agreement serves as a validation for the developed VDF-DFN numerical model for density-driven flow with dissolution.

The developed VDF-DFN can be used with more general fracture networks (not necessarily orthogonal) involving different permeabilities and/or apertures. As an example, Figure 4 shows the concentration distribution on a general fracture network formed by the combination of a first orthogonal network corresponding to $nf=20$ with a second network formed of a random distribution of fractures. The random fractures are twice as permeable as orthogonal fractures, and their aperture is 5 times smaller than that of regular fractures. The number of fracture elements of the original orthogonal fracture network corresponding to $nf=20$ is 1660. Adding the random fractures yields a general fracture network formed of 3304 fracture elements. The steady-state results of the Henry saltwater intrusion problem with dissolution on the general fracture network show a greater amount of salt in the domain (Figure 4), reflecting an increase in saltwater intrusion and dissolution induced by the added random fractures.

4.2. Fracture Evolution Due to Dissolution in an Injection/Extraction System

For this test case, we consider a $100\mathrm{m}\times 100\mathrm{m}$ square fractured horizontal aquifer with a uniform fracture network formed of $\left(n_f+1\right)$ longitudinal fractures and $\left(n_f+1\right)$ transverse fractures (Figure 5). The flow field within this aquifer is generated by an extraction and injection well system, with uniform regional flow from the left boundary, where the hydraulic head is imposed to 101 m, to the right boundary, where the head is imposed to 100 m. The separation distance between the extraction and injection wells is 12 m, with the injection well located at (44 m, 50 m) and the pumping well located at (56 m, 50 m). For the transport, a Dirichlet boundary (c = 1) is prescribed at the left side, and a Neumann condition corresponding to zero diffusive flux is prescribed on the right side. Initially, all fractures are saturated with the solute (c = 1). Freshwater (c = 0) intrudes from the injection well. A uniform dense fracture network based on $nf=50$ is employed.

The permeability of the fractures is ruled by the cubic law, and it changes during the simulation due to the evolution of the aperture induced by dissolution. The simulation was conducted over a time span of $T_f={10}^6$ s using the parameters depicted in

Table 2. Parameters for the injection/extraction well pair problem.

Aperture of Fractures	$b=1$ mm
Porosity	$\varphi =0.5$
Pumping/injection rate	$Q^{\ast }={10}^{-3}$ m2/s
Dispersivity	${\alpha }_L=0$ m
Molecular diffusion	$D_m={10}^{-8}$ m2/s
Dissolution rate	$k_C^0={10}^{-6}$ m/s
Dimensionless concentration at saturation	$c_S=1$
Density of rock	${\rho }_r=2700$ kg/m3
Stoichiometric coefficient	$\delta =0.4$

. To examine the impact of the dissolution coefficient on the flow behavior and final concentration distribution, simulations were carried out for the following three cases: (a) without dissolution, (b) with a fixed dissolution rate $k_C^0$ and (c) with a dissolution rate k_c that increases with the fluid velocity, as suggested in [40]. For this last case, the dissolution coefficient in the fracture is ruled by $k_C=k_C^0\left(1+10\left|q^k\right|\right)$, where $\left|q^k\right|={\displaystyle \frac{1}{2b^k}}\left|q_2^k-q_1^k\right|$ is the norm of the velocity in fracture branch k. For the studied problem, k_c varies from $k_C^0$ (for fractures far from the wells) to $2k_C^0$ for fractures with high velocity (near the wells).

Figure 6 shows the head, concentration and aperture distributions for the three investigated cases. Although these results were obtained on the discrete fracture network, all distributions are represented as continuous 2D plots. In the case without dissolution (case a), where the aperture of all fractures remains unchanged and the corresponding permeability is relatively small according to the cubic law, significant hydraulic effects are observed. This includes a strong depression corresponding to a head of 99.1 m, observed in the pumping well, and a strong overpressure corresponding to a head of 101.9 m in the injection well. As a consequence, a large zone of influence of the injection/extraction well pair is observed (Figure 6(a1)). The concentration distribution shows that the injected clean water occupies a large zone of an almost circular shape around the wells (Figure 6(a2)). Notably, not all the freshwater from the injection well is captured by the pumping well, and a portion of the injected freshwater escapes, reaching the right boundary (Figure 6(a2)).

In case (b), where dissolution occurs at a fixed rate, $k_C^0$, the injected freshwater induces the enlargement of the fractures (Figure 6(b3)). Figure 7 shows the evolution of the aperture of the horizontal fracture located at the center between the injection and extraction wells. This figure shows an almost linear increase in the aperture over time due to dissolution. Close to the injection well, the aperture of the fractures nearly doubles, reaching 1.9 mm. As a result, the permeability of the fractures where dissolution is important experiences a significant increase (due to the cubic law). Hence, the fluid moves more easily from the injection well to the pumping well. The depression observed in the pumping well and the overpressure in the injection well are less important than for case (a). The pressure in the injection well is 100.7 m, while in the pumping well, it reaches 100.2 m (Figure 6(b1)). The zone of influence of the injection/extraction is reduced. The freshwater distribution exhibits an almost circular shape with a smaller diameter around the wells (Figure 6(b2)). It is important to note that the aperture distribution shows more significant enlargement in the vicinity of the injection well compared to the region near the pumping well (Figure 6(b3)). This observed phenomenon aligns with the underlying physics. Indeed, the dissolution is more significant around the injection well, where the concentration is nearly zero, as compared to the area near the extraction well. In the latter case, the concentration of the fluid is non-zero since it has been contaminated by dissolution during its travel from the injection well. In case (c), where the dissolution rate $k_C$ varies with the velocity, the enlargement of the fracture apertures is even more important than in the previous two cases, with values reaching 3.7 mm near the injection well. The aperture of the horizontal fracture located at the center between the injection and extraction wells increases linearly over time, with a more significant slope than in case (b) (see Figure 7). Figure 6(c3) shows that the aperture is significantly enlarged for the fractures between the injection and pumping wells, where the velocity is significant. This results in an aperture distribution that takes on an almost tunnel-like shape between the two wells (Figure 6(c3)). The presence of this tunnel with a significantly high aperture (and, consequently, high permeability) allows a more efficient connection between the injection and pumping wells. The overall pressure distribution exhibits minimal impacts from the well pair (Figure 6(c1)), and the zone of influence for the pair of wells is reduced. The concentration distribution shown in Figure 6(c2) reveals that the freshwater occupies a smaller area with an ellipsoid shape situated between the two wells.

5. Conclusions

In this work, we considered coupled fluid flow, mass transfer and dissolution in fractured media with variable density and dynamic fracture aperture evolution. We adopted the discrete fracture network (DFN) model, considering that flow, mass transfer and dissolution processes occur only within the connected fractures. The rock matrix was assumed to be completely impermeable. Modeling these three coupled processes in a DFN can present significant computational challenges due to the high nonlinearities stemming from density variations and aperture modification. In this study, we developed an efficient numerical simulator that simultaneously solves flow, transport and dissolution equations with density variations. The spatial discretization uses the robust and cost-effective finite volume method. To prevent unphysical oscillations for advection-dominated cases, we devised an upwind scheme tailored specifically for the fracture network. Upstream concentrations at intersection nodes are computed by considering all connected fracture concentrations with outflows at that node. Integration with respect to time is conducted using sophisticated ODE solvers, which enable the utilization of high-order methods. This allows for dynamic adjustment of the size of the time step and order of integration during the simulation, crucial for optimizing computational efficiency and, at the same time, having good accuracy.

The validity of the developed VDF-DFN model was established by comparing it with the semi-analytical solution of a variant of the Henry problem, which incorporates saltwater intrusion and dissolution in a fractured coastal aquifer. The comparison demonstrated excellent agreement between the semi-analytical solution and the VDF-DFN results obtained on a dense fracture network, thereby validating the accuracy of our approach.

The developed model was then employed to explore the evolution of fracture apertures induced by dissolution in a problem involving both injection and extraction wells. The flow field in this problem is generated by the extraction and injection system, with uniform regional flow. If no dissolution occurs, a large zone of influence of the injection/extraction well pair is observed. In contrast, when dissolution occurs at a fixed rate, the injected freshwater triggers an enlargement of the fractures. The depression observed in the pumping well is less important, and the zone of influence of the injection/extraction wells is reduced. In the final case, where the dissolution rate varies with velocity, the aperture enlargement is even more important compared to the previous two cases. The aperture is significantly enlarged for the fractures between the injection and the pumping wells, where the velocity is significant. The aperture distribution shows an almost tunnel-like shape, characterized by high permeability, which connects the two wells. The overall pressure distribution is weakly affected by the pair of wells, and the concentration distribution reveals a reduced zone with an ellipsoid shape located between the two wells. This test case points out the benefit of efficient DFN models, which can be suitable and cost-effective tools for investigating coupled transfer processes in highly fractured aquifers.