A Numerical Simulator for Modeling the Coupling Processes of Subsurface Fluid Flow and Reactive Transport Processes in Fractured Carbonate Rocks

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Article A Numerical Simulator for Modeling the Coupling Processes of Subsurface Fluid Flow and Reactive Transport Processes in Fractured Carbonate Rocks

Tao Yuan 1,2 , Chenji Wei 3,*, Chen-Song Zhang 4 and Guan Qin 1,*

1 Department of Petroleum Engineering, University of Houston, Houston, TX 77204, USA; [email protected] 2 Department of Reactive Transport, Institute of Resource Ecology, Helmholtz-Zentrum Dresden-Rossendorf, 04318 Leipzig, Germany; [email protected] 3 Research Institute of Petroleum Exploration & Development (RIPED), PetroChina Company Limited, Beijing 100083, China 4 NCMIS & LSEC, Academy of Mathematics and Systems Science, Beijing 100190, China; [email protected] * Correspondence: weichenji@petrochina.com.cn (C.W.); [email protected] (G.Q.); Tel.: +1(832)496-1688 (G.Q.); Fax: +1(713)743-4323 (G.Q.) Received: 26 August 2019; Accepted: 16 September 2019; Published: 20 September 2019

Abstract: Water–rock interactions can alter rock properties through chemical reactions during subsurface transport processes like geological CO2 sequestration (GCS), matrix acidizing, and waterflooding in carbonate formations. Dynamic changes in rock properties cause a failure of waterflooding and GCS and could also dramatically affect the efficiency of the acidizing. Efficient numerical simulations are thus essential to the optimized design of those subsurface processes. In this paper, we develop a three-dimensional (3D) numerical model for simulating the coupled processes of fluid flow and chemical reactions in fractured carbonate formations. In the proposed model, we employ the Stokes–Brinkman equation for momentum balance, which is a single-domain formulation for modeling fluid flow in fractured porous media. We then couple the Stokes–Brinkman equation with reactive-transport equations. The model can be formulated to describe linear as well as radial flow. We employ a decoupling procedure that sequentially solves the Stokes–Brinkman equation and the reactive transport equations. Numerical experiments show that the proposed method can model the coupled processes of fluid flow, solute transport, chemical reactions, and alterations of rock properties in both linear and radial flow scenarios. The rock heterogeneity and the mineral volume fractions are two important factors that significantly affect the structure of conductive channels.

Keywords: reactive-transport; fracture evolution; mineral dissolution; fractured carbonate formations

1. Introduction Subsurface water–rock interactions involve the coupled phenomena of chemical reactions and fluid transport, in which the chemical reactions between minerals and water can cause mineral dissolution and precipitation. Mineral dissolution is a process in which the chemical elements in the solid phase transform into ions in the aqueous phase through various chemical reactions, and consequently lead to a decrease in mineral mass and volume. In contrast, mineral precipitation occurs when aqueous species transform to solid phases, resulting in an increase of mass and volume in solid phases. In carbonate rocks, the carbonate minerals calcite and dolomite can react with acid fluids, leading to carbonate rock dissolution and precipitation [1]. The couplings of fluid flow and mineral dissolution/precipitation may significantly change rock properties in carbonate formations. Mineral dissolution/precipitation can occur in many subsurface processes in carbonate formations, for example in waterflooding processes

Water 2019, 11, 1957; doi:10.3390/w11101957 www.mdpi.com/journal/water Water 2019, 11, 1957 2 of 18

for improving oil recovery [2], geological CO2 sequestration (GCS) for reducing CO2 emissions [3,4], and matrix acidizing for improving oil production by changing the permeability near wellbores [5,6]. In waterflooding processes in carbonate formations, the chemical reactions between hydrogen ions in water phase and carbonate minerals can cause mineral dissolution. The mineral dissolution process may be enhanced by coupling with fluid flow to form large-scale conductive channels from both natural and factitious fractures [2]. Such conductive channels can cause early water breakthrough and limited hydrocarbon production, resulting in the failure of the waterflooding program. In a GCS process, the dissolved CO2 in situ brine, via various geochemical reactions, can form a carbonate acid that may then dissolve carbonate rocks [7,8]. Such CO2–rock–brine interactions cause mineral dissolutions that could enhance the existing fracture system and form highly conductive or leakage pathways, which may present environmental risks [9–11]. In a matrix acidizing treatment, the rock–fluid interaction creates highly conductive pathways, the so-called wormholes, around well-bores for enhancing hydrocarbon flow into wells [12–15]. It is thus vitally important to well understand the underlying mineral dissolution/precipitation processes happened in the above-mentioned subsurface processes for the successful implementations. Mathematical modeling of the coupled physical processes of fluid flow, solute transport, and underline chemical reactions is vital for successful implementation of those subsurface processes, and also is challenging as rock porosity and permeability change due to fluid–rock interactions. The key challenges are the modeling of fluid flow in the free-flow regions that are dynamically changing due to the coupling effect of mineral dissolution and fluid transport. Over the last few decades, extensive studies have been performed on the coupled processes of fluid flow and mineral dissolution using the coupled reactive-transport models [16–23]. These mathematical models include conservations of momentum and solute mass [22], which developed from early studies on one-dimensional models for a single chemical reaction system [24,25] to the later studies on multi-component multi-dimensional reactive-transport systems [23,26]. The Navier–Stokes equation or the Stokes equation has been employed as a momentum equation to describe fluid flow in pore space for numerical modeling at pore-scale in which the characterization of changing mineral surface is critical in binary domains (solid and pore space) [19,22,27]. Molins [27] employed the embedded boundary and the level set methods to obtain an accurate representation of the calcite surface on the Cartesian grid. The lattice Boltzmann approach has also been used to study the reactive transport process and fracture evolution at the pore scale [28,29]. At the continuum scale and field scale, Darcy’s equation has usually been applied as a momentum equation [30–32]. Darcy’s equation can be inaccurate for modeling fluid flow in carbonate formation due to its large-scale void space features that can also be enhanced dynamically due to dissolution processes. To address such issues, Yuan et al. [33,34] presented a mathematical model that couples the Stokes–Brinkman equation and reactive-transport equations to describe calcite dissolution in both single and multiple mineral systems at the continuum scale. This model is capable of describing the fracture evolution as well due to the advantage of the Stokes–Brinkman formulation for modeling flow in fractured porous media. The Stokes–Brinkman model uses a single set of equations in the entire domain (porous media and free-flow regions) for fluid flow [35–37] and can be considered as a modified Darcy’s law by adding the viscous shear stress term to account for the effect of shear stress near the boundary layer [38–40]. In the Stokes–Brinkman equation, the effective viscosity of the fluid work as a parameter for matching the shear stress interface condition between porous media and free-flow regions [33], which can be regarded as the fluid viscosity throughout the physical domain [33,34,36,37,41]. Applying the Stokes–Brinkman equation in reactive transport process offers several advantages. First, this equation can be reduced to the Stokes or Darcy equations by choosing appropriate coefficients. In such a way, a no interface condition, for example, the Beavers–Joseph condition [42], is needed between the porous media and free-flow regions. In addition, it can model the transitional flow pattern from a Darcy-dominated region to a Stokes-dominated region, a feature that allows us to efficiently model the alterations of the rock porosity and permeability caused by the mineral dissolution. Water 2019, 11, 1957 3 of 18

The existing mathematical models in the previous studies [33,34] focused on the 2-D case studies in the Cartesian coordinate system only, which limits their applications in modeling the real-world scenarios. For example, the actual matrix acidizing treatments are performed by acidic fluid injection through a wellbore, the fluid flows into porous media is radial, and the velocity decreases rapidly away from the wellbore. Therefore, a 3D mathematical model on both Cartesian coordinate and cylindrical coordinate systems is desirable for simulating the mineral dissolution process in the subsurface as well as the acidizing process at exact downhole environments. In this paper, we develop a 3D numerical simulator for modeling the coupling processes of fluid flow and mineral dissolution during the reactive transport processes. The proposed numerical model includes the Stokes–Brinkman equation for fluid flow, reactive-transport equations for solute transport and chemical reactions, and a rock property model for alterations of rock properties, which can be applied to simulate mineral dissolution under both linear flow and radial flow. The mathematical model is discretized and solved on both Cartesian coordinate and cylindrical coordinate systems sequentially. In the numerical solution procedure, the Stokes–Brinkman equation is solved by the staggered grid finite difference method, which was proposed by Harlow and Welsh [43] and was widely applied for solving the Navier–Stokes equation [44,45]. The reactive-transport equations are solved by the control volume finite difference method [23]. Effective linear solvers play a key role in scientific computing and many different types of numerical algorithms have been developed for solving linear systems. In this paper, we employ the Fast Auxiliary Space Preconditioning (FASP) package [46], which includes several efficient iterative solvers for solving linear systems of equations. In our proposed sequentially numerical procedure, we employ two linear algebraic preconditioning strategies from the FASP package for the Stokes–Brinkman and reactive transport equations, respectively. These methods are accelerated by a Krylov subspace method, the Variable-Restarting Flexible Generalized Minimal Residual (GMRES) method [47]. The numerical model for radial flow is validated using a 3D radial core-flooding experiment from the existing publication [48]. Two 3D synthetic case studies in both linear and radial core flooding are performed to investigate the alterations of rock properties and dissolution patterns. The first case study is for linear core flooding in a multi-mineral system, which is consist of calcite, quartz, and clay, and the second case study is for radial core flooding in a single calcite system. The numerical results demonstrate that the proposed numerical model is capable of modeling the coupled processes under investigation. And the rock heterogeneity and the mineral volume fractions affect the structure of conductive channels significantly. The rest of this paper is organized as follows. First, the mathematical model is briefly presented. Second, the numerical solution methods and linear algebraic solvers are discussed. The numerical experiments of synthetic core-flooding case studies are then presented. We close with the concluding remarks.

2. The Mathematical Models In our previous studies [33,34], we have developed a similar 2D mathematical model for the geochemical coupling of fluid flow in fractured reservoirs. In this paper, we focus on the 3D multicomponent flow in fractured carbonate formations on both Cartesian coordinate and cylindrical coordinate systems. In this section, we briefly present the coupled mathematical model. The readers are referred to Yuan et al. [33,34] for discussions on the coupled model and involved chemical reactions.

2.1. The Stokes–Brinkman Model The Stokes–Brinkman equation has been widely applied to model fluid flow in fractured porous media [33,34,36,37,41], as it employs a single set of equations for simulating fluid flow in the entire domain. The general formulations of the Stokes–Brinkman equation for incompressible single-phase fluid are expressed as follows [33,34,36–38,41]:

1 µKperm− v + p µ∗∆v = f in Ω, (1) ∇ − Water 2019, 11, 1957 4 of 18

v = 0 in Ω, (2) ∇ · where Ω denotes the entire computational domain, v is the physical velocity of the fluid in free-flow regions and the Darcy velocity in porous media, p is pressure, Kperm is a permeability tensor, and f is the body force, which is ignored in this study for simplicity of presentation. µ is the fluid viscosity, and µ∗ is the so-called effective viscosity, which is the critical parameter for matching the shear stress at the interface between porous media and free-flow regions [49]. The effective viscosity µ∗ is set to the fluid viscosity, µ∗ = µ in most applications [33,34,36,37,41]. The readers are referred to Yuan et al. [33] for more details of the Stokes–Brinkman equation.

2.2. The Reactive-Transport Model We use the reactive-transport equations to describe solute transport coupled with geochemical reactions in the geological subsurface. In our model, the aqueous species are classified into primary and secondary species [23]. The general mass conservation equations for the primary species α read as follows: ∂(φCTα) min + (vCT D CT ) = R , (α = 1, , Np). (3) ∂t ∇ · α − ∇ α α ··· In Equation (3), v is the fluid velocity defined in Equation (1), D is the dispersion/diffusion min coefficient, and φ is the porosity. Np is the number of primary species. Rα is the kinetic reaction rate of primary species α for mineral–water reactions. CTα represents the total concentrations of primary min species. Rα and CTα are nonlinear functions of the concentrations of primary species. For more details, the reader is referred to [26,33]. Equation (3) presents a general reactive transport model to describe solute transport and chemical reactions. In our previous studies, the reactive transport model has been applied to describe mineral dissolution processes in both a single mineral (calcite) system [33] and a multi-mineral (calcite, quartz, and clay) system [34]. In this study, we consider both scenarios in 3D. The detailed chemical reactions and related parameters can be found in [1,33,34].

2.3. The Rock Property Models The porosity of porous media can be obtained by:

XNm φ = 1 φm. (4) − m=1

The change in the individual mineral volume fraction ∅m can be calculated from [23,26]:

dφm = V r , (5) dt m m where Vm is the molar volume of the mineral. The prediction of the absolute rock permeability is another challenge because of the diﬃculties in measuring it directly. As an alternative method to direct measurement, empirical models can be applied to predict the absolute rock permeability from the porosity [50–54]. Among these models, the most widely used one is the so-called Kozeny–Carman equation [31,32,55–62], which is written as per [51]: 2 3 dm φ Kperm = , (6) 180 (1 φ)2 − where dm denotes the grain size. If we ignore changes in grain size, the permeability can then be given as: 2 3 1 φ0 φ Kperm = Kperm ( − ) ( ) , (7) 0 1 φ φ − 0 Water 2019, 11, 1957 5 of 18

where Kperm0 and ∅0 are the initial permeability and porosity, respectively.

3. Numerical Solution Strategies In this paper, the proposed 3D numerical model is solved numerically using a decoupled approach by the ﬁnite diﬀerence method. Within each time step, our numerical method contains four main steps:

(1) Firstly, the Stokes–Brinkman equation (Equation (1)) and the continuity equation (Equation (2)) are solved by a staggered grid finite difference method for velocities and pressure. (2) Secondly, the velocities at grid-cell centers are calculated by averaging the velocities of grid faces obtained from the solution of the Stokes–Brinkman equation. (3) Thirdly, based on the calculated velocities at cell centers, the concentrations of primary species are determined by solving the reactive-transport equations using an implicit control-volume upwinding finite difference scheme. (4) The last step is then to update the rock porosities and absolute permeabilities based on the concentrations of primary species.

For more details on the numerical solution procedure of the sequential method, the readers are referred to [33]. Note that the above scheme can be embedded in a nonlinear iteration until the desired accuracy. For simplicity, we do not use such nonlinear iterations in our current study. The numerical discretization and solutions for the Stokes–Brinkman equation and reactive transport equations in the 3D Cartesian coordinates are discussed as follows. The numerical discretization and solutions of Stokes–Brinkman equation in the 3D cylindrical coordinates are discussed in AppendixA.

3.1. Numerical Solution of the Stokes–Brinkman Model The Stokes–Brinkman equation and the continuity equation are discretized using the staggered grid method, also known as the Marker-and-Cell (MAC) scheme. In Cartesian coordinates, the idea of MAC is to place the unknowns of vx, vy, vz, and p in diﬀerent locations. More speciﬁcally, the pressure p i j k Wateis locatedr 2019, 11, at 1957 the center of each cell ( , , ), and velocities are located at the center of grid faces6 of as 20 indicated by the subscripts i 1/2, j 1/2, and k 1/2 (see Figure1). ± ± ±

Figure 1. Definition of pressure, velocities, and concentrations on a grid with indices i, j, k. Figure 1. Definition of pressure, velocities, and concentrations on a grid with indices i, j, k. With the definitions of the pressure and velocities in Figure1, the finite di fference approximations of EquationsWith the (1) and definitions (2) in the of 3D the Cartesian pressure coordinates and velocities system are in givenFigure as: 1, the finite difference approximations of Equations (1) and (2) in the 3D Cartesian coordinates system are given as: ( + ) 1 pi+1,j,k pi,j,k vx,i+3/2,j,k 2vx,i+1/2,j,k vx,i 1/2,j,k ( K ) v + − [ − − µ perm,x− i+1/2,j,k x,i+1/2,ppvvvj,k ∆x (2) ∆x2 ()[Kv1 i1, j ,, ki ,, j kx 3/2, ij kx ,, ij 1/2, kx− ij ,, k 1/2, , (8) permvx,i+,1/2,x1/2,j ,, ij+1, k 1/2,k x2 ijv , x k,i+1/2,j,k+vx,i+1/2,j 1,k vx,i+1/2,j,k+1 2vx,i+1/2,j,k+vx,i+1/22,j,k 1 + − − +xx− − ] = 0, ∆y2 ∆z2 vvvvvv 22(8) x, ij 1/2, kx ij 1,, kx ij 1/2, kx ,, ij 1/2, kx ij kx 1,, ij 1/2,k , 1, 1/2, ,, 1/2, , 1 ] 0, yz22

p p( v  2 v  v ) ()[Kv1 ijk,1, ijk ,, yij ,1,1/2,   k yij ,,1/2,  k yij ,1,1/2,   k perm,y i , j 1/2, k y ,, i j 1/2, k yx2 (9) v22 v  v v  v  v yij,, 3/2, k yij ,,  1/2, k yij ,,1/2,  k  yij ,,  1/2, k  1 yij ,,  1/2, k yij ,,  1/2, k  1 ]  0, yz22 ppvvv (2) ()[Kv1 ijkijkzi, , 1, ,, 1, jkzijkzi , 1/2, , , 1/2, jk 1, , 1/2 perm,,,z 1/2 i j kz i,,, j k 1/2 zx2

vvvvvv 22 (10) zij,, 1, kzijkzij  1/2,,  , 1/2,, kzijkzijkzijk 1, 1/2,, , 3/2,, , 1/2,, , 1/2 ] 0, yz22

vvvvvvx, ij 1/2,kx ij ,,ky i 1/2, jky ,,,i jkz 1/2,,, i j kz i 1/2,,,j k , 1/2,, , 1/2  0, (11) xyz

where the permeability at the interface is calculated as the harmonic average of the permeability values of the two neighboring blocks. In this paper, we set constant velocity on the inlet boundary, constant pressure, and free-flows at the outlet boundary, and no-slip boundary condition on the other boundaries. The boundary conditions in the Cartesian system are expressed as:

vvx in at x  0 , (12)

pp out at xL , (13) v  0 at xL , (14) n

v  0 at y  0, and yW , (15)

Water 2019, 11, 1957 6 of 18

( + ) 1 pi,j+1,k pi,j,k vy,i+1,j+1/2,k 2vy,i,j+1/2,k vy,i 1,j+1/2,k ( K ) v + − [ − − µ perm,y− i,j+1/2,k y,i,j+1/2,k ∆y ∆x2 − (9) vy,i,j+3/2,k 2vy,i,j+1/2,k+vy,i,j 1/2,k vy,i,j+1/2,k+1 2vy,i,j+1/2,k+vy,i,j+1/2,k 1 + − − + − − ] = 0, ∆y2 ∆z2 ( + ) 1 pi,j,k+1 pi,j,k vz,i+1,j,k+1/2 2vz,i,j,k+1/2 vz,i 1,j,k+1/2 ( K ) v + − [ − − µ perm,z− i,j,k+1/2 z,i,j,k+1/2 ∆z ∆x2 − (10) vz,i,j+1,k+1/2 2vz,i,j,k+1/2+vz,i,j 1,k+1/2 vz,i,j,k+3/2 2vz,i,j,k+1/2+vz,i,j,k 1/2 + − − + − − ] = 0, ∆y2 ∆z2

vx,i+1/2,j,k vx,i 1/2,j,k vy,i,j+1/2,k vy,i,j 1/2,k vz,i,j,k+1/2 vz,i,j,k 1/2 − − + − − + − − = 0, (11) ∆x ∆y ∆z where the permeability at the interface is calculated as the harmonic average of the permeability values of the two neighboring blocks. In this paper, we set constant velocity on the inlet boundary, constant pressure, and free-ﬂows at the outlet boundary, and no-slip boundary condition on the other boundaries. The boundary conditions in the Cartesian system are expressed as:

vx = vin at x = 0, (12)

p = pout at x = L, (13) ∂v = 0 at x = L, (14) ∂n v = 0 at y = 0, and y = W, (15)

v = 0 at z = 0, and z = H, (16) where vin is the injected velocity at the inlet, pout is the outlet pressure, n is the unit normal vector on the boundary, L is the length of the domain in the flow direction, W is the width of the domain, and H is the depth of the domain. The simplest way of treating the boundary is to use the reflection technique. In Figure2, Γx and Γz denote the inlet and bottom boundaries, respectively. The two dashed lines are the ghost boundaries out of the domain. On the bottom boundary Γz, the boundary condition vx = 0 is imposed exactly at the “N” point: vx,Γ = 0. Using the reflection technique, the value of vx at the “O” point on the dash line can be determined approximately: vx = v . On the inlet boundary Γr, the boundary condition is ,0 − x,1 imposed exactly at the “N” point: vz = vz . Similarly, we have vz = 2vz v . ,Γ ,0 ,Γ − z,1 In the Cartesian coordinates, when the entire domain is divided into Nx, Ny, and Nz grid blocks in the corresponding directions, respectively, the total numbers of p, vx, vy, and vz are Nx Ny Nz, × × (Nx + 1) Ny Nz, Nx (Ny + 1) Nz, and Nx Ny (Nz + 1), respectively. By imposing the defined × × × × × × boundary conditions, we have (Nx 1) Ny Nz + (Nx 1) Ny Nz + Nx (Ny 1) Nz + Nx − × × − × × × − × × Ny (Nz 1) unknown variables and the same number of discretized equations, which can be written × − as a block linear system:      g  B1x 0 0 B1p  vx   x        0 B2y 0 B2p  vy   gy     =  , (17)  B B  v   g   0 0 3z 3p  z   z       B4x B4y B4z 0 p gp where vx, vy, vz, and p are unknown vectors, and all of them go through the indices i, j, k in the same order. B1x and B1p are the submatrices resulting from the terms of vx and p in the Equation (8), respectively. B2y and B2p are the submatrices resulting from the terms of vy and p in the Equation (9), respectively. B3z and B3p are the submatrices resulting from the terms of vz and p in the Equation (10), respectively. B4x, B4y, and B4z are the submatrices resulting from the terms of vx, vy, and vz in the Equation (11), respectively. gx, gy, gz, and gp are the right-hand side vectors resulting from the defined boundary conditions. Water 2019, 11, 1957 7 of 20

v  0 at z  0 , and zH , (16)

where 푣푖푛 is the injected velocity at the inlet, 푝표푢푡 is the outlet pressure, 퐧 is the unit normal vector on the boundary, 퐿 is the length of the domain in the flow direction, 푊 is the width of the domain, and 퐻 is the depth of the domain.

The simplest way of treating the boundary is to use the reflection technique. In Figure 2, Γ푥 and Γ푧 denote the inlet and bottom boundaries, respectively. The two dashed lines are the ghost boundaries out of the domain. On the bottom boundary Γ푧 , the boundary condition 푣푥 = 0 is imposed exactly at the “” point: 푣푥,Γ = 0. Using the reflection technique, the value of 푣푥 at the “O” point on the dash line can be determined approximately: 푣푥,0 = −푣푥,1. On the inlet boundary Γ푟, the boundary condition is imposed exactly at the “” point: 푣푧 = 푣푧,Γ. Similarly, we have 푣푧,0 = 2푣푧,Γ − Water푣푧,12019. , 11, 1957 7 of 18

Figure 2. Projection formulation of the Marker-and-Cell (MAC) scheme. Figure 2. Projection formulation of the Marker-and-Cell (MAC) scheme. Therefore, the velocity and pressure can be approximated simultaneously by solving the linear systemsIn the (Equations Cartesian (17) coordinates, and (A9) when in Appendix the entireA) domain for the is finite divided di ff intoerence 푁푥, approximations 푁푦, and 푁푧 grid of blocks the Stokes–Brinkmanin the corresponding equation directions, and continuity respectively, equation the total in Cartesian number ands of cylindrical푝, 푣푥, 푣푦, and coordinates 푣푧 are 푁 systems,푥 × 푁푦 × respectively.푁푧, (푁푥 + 1 The) × 푁 saddle-point푦 × 푁푧, 푁푥 × systems(푁푦 + 1) from× 푁푧 the, and MAC 푁푥 discretization× 푁푦 × (푁푧 + 1 of), the respectively. Stokes–Brinkman By imposing equation the aredefined usually boundary large-scale, conditions, ill-conditioned, we have and (푁 symmetric.푥 − 1) × 푁푦 E×ffi푁cient푧 + (푁 preconditioning푥 − 1) × 푁푦 × 푁푧 methods+ 푁푥 × ( for푁푦 − such1) × problems푁푧 + 푁푥 × include푁푦 × (푁 the푧 − geometric1) unknown multigrid variables schemes and the with same the distributednumber of Gauss–Seideldiscretized equations, smoother which [63] andcan the be blockwritten preconditioners as a block linear [64 ].system In this: study, we employ the block lower triangular preconditioner in the following form: B 0 0 B v 1  g  11x B1x 0 0 p 0 −x x     0 B0 B2 0y 0 B 0 v g  22y p  y ,  y  (18)    , (17)  0 0 B3z 0    0 0 B33z B pvz g z B4x B4y B4z S    B B B 0p g 1 1 4x 4 y1 4 z   p  where S = B4xB1−x B1p + B4yB2−y B2p + B4zB3−z B3p. whereIn our 퐯푥, simulations,퐯푦, 퐯z, and the퐩 are inverses unknown of B 1vectors,x, B2y, and andB all3z areof them obtained go through by a few the fixed indices steps 푖 of, 푗, classical 푘 in the algebraicsame order. multigrid 퐁ퟏ푥 and (AMG) 퐁ퟏ푝 are methods the submatrices as they behave resulting like from the Poisson’s the terms equation. of 푣푥 and Furthermore,푝 in the Equation the Schur(8), respectively. complement can퐁ퟐy beand replaced 퐁ퟐ푝 are by the submatrices resulting from the terms of 푣푦 and 푝 in the 퐁 퐁 푣 푝 Equation (9), respectively. ퟑ푧 and ퟑ푝 are the submatrices 1 resulting from the1 terms of 푧 and in 1 − − S B4xdiag(B1x)퐁− B1퐁p + B4ydiag퐁 (B2y B2p + B4zdiag(B3z B3p, (19) the Equation (10),≈ respectively. ퟒ푥, ퟒ푦, and ퟒ푧 are the submatrices resulting from the terms of 푣푥, 푣푦, and 푣푧 in the Equation (11), respectively. 퐠푥, 퐠푦, 퐠푧, and 퐠푝 are the right-hand side vectors or a better approximation (see for example [65]). resulting from the defined boundary conditions. 3.2. NumericalTherefore, Solution the velocity of the Reactive and pressure Transport can Model be approximated simultaneously by solving the linear systems (Equations (17) and (A9) in Appendix A) for the finite difference approximations of the StokesWe– followedBrinkman ourequation previous and continuity numerical equation method in to Cartesian solve the and reactive-transport cylindrical coordinates equations systems, [33]. The advective mass flux term of Equation (3) is discretized using a first-order upwinding method. min The nonlinear terms CTα and Rα are linearized using the classical Newton–Raphson method. A fully implicit finite difference scheme is employed for the reactive-transport equations. For more details on numerical solving reactive-transport equations, the readers are referred to see Yuan et al. [33]. The reactive transport equations give rise to nonsymmetric linear equations upon discretization. When the advection term is relatively small, the equations are Poisson-type diffusion-dominated equations. Hence, we can apply classical AMG as a preconditioner to solve such problems. The multilevel hierarchy in AMG is constructed based on the coefficient matrix only and can be applied to general meshes. The original AMG idea was developed under the assumption that the coefficient matrix being a symmetric M-matrix; see the recent review paper [66]. When the advection term dominates the equations, we apply the Incomplete LU factorization ILU(k) method as a preconditioner [67]. Water 2019, 11, 1957 8 of 18

4. Numerical Experiments The numerical solution method of the coupled model in a 2D Cartesian system has been extensively tested and validated in the previous studies [33,34]. In this paper, we present numerical experiments to validate the proposed numerical method in 3D.

4.1. Model Validation In this numerical experiment, we compare our numerical results with the published experimental results by Tardy et al. [48]. The 15% HCL solution is injected into an Indiana Limestone core through the central borehole at an injection rate of 12 mL/min. The core has an average porosity of 0.11 and an average permeability of 19.6 mD, with dimensions of 2.77 in (0.07 m) in the outer radius, 0.125 in (0.0032 m) in the inner radius, and 2.25 in (0.057 m) in height, as indicated in [48]. The domain is Water 2019, 11, 1957 9 of 20 partitioned into a mesh of 100 100 100 in the cylindrical coordinate system. The initial porosity × × values at grid nodes are uniformly distributed in the interval of [0.01, 0.21] with a mean value of 0.11 as at the inletindicated during in [68 ]. acid The initial injection, permeability and is the calculated simulation from the result initial porosity shows via good the Kozeny–Carman agreement with the experimentalequation result (Equations. (6)). Figure3 shows the transient variations of the pressure at the inlet during acid injection, and the simulation result shows good agreement with the experimental results.

Figure 3. Comparison of pressure at the inlet during acid injection. Figure 3. Comparison of pressure at the inlet during acid injection. 4.2. Numerical Studies 4.2. NumericalNow Studies we present two 3D synthetic core flooding case studies to investigate alterations of rock properties and dissolution patterns with different heterogeneities, volume fractions, and flow conditions. NowIn thewe first present case, the two proposed 3D synthetic numerical core model flooding is applied forcase 3D stud linearies core to flooding investigate in a multi-mineral alterations of rock propertiessystem, and which dissolution is consist of patterns calcite, quartz, with and different clay [34]. In heterogeneities, order to examine the volume effects of heterogeneityfractions, and flow conditions.of porous In the media first andcase, mineral the proposed volume fractions numerical on the model alterations is applied of rock propertiesfor 3D linear and dissolution core flooding in a multi-mineralpatterns, syst sensitivityem, which studies is haveconsist been of performed. calcite, quartz, In the second and clay case [34] study,. In the order proposed to examine numerical the effects of heterogeneitymodel is applied of porous for 3D radialmedia core and flooding mineral in a volume single calcite fractions system, on which the can alterations be applied toof simulate rock properties the matrix acidizing process at exact downhole environments. and dissolutionCase Study patterns, 1: 3D s linearensitivity flow core stud flooding.ies have In been thiscase, performed. we use a 3D In synthetic the second core casesample, study, the proposedwhich numerical is 10-cm model long, 5-cm is applied wide, and for 5-cm 3D tall,radial as shown core flooding in Figure4 in. The a single minerals calcite are assumed system, to which can be appliedbe homogeneously to simulate the distributed, matrix acidizing which indicates process the at initial exact mineral downhole volume environments. fractions are the same Casethroughout Study 1: the 3D entire linear sample. flow As core indicated flooding in [30. ],In the this initial case, porosity we use values a 3D at gridsynthetic nodes incore the poroussample, which = + = is 10-cmmedia long, are 5 defined-cm wide, as: φ andφ0 5-fcm, where tall,φ 0 as shown0.2 is the in mean Figure value of4. porosity, The minerals the fluctuations are assumedf are to be uniformly distributed in the interval [ ∆φ0, ∆φ0], and the magnitude of heterogeneity a is defined as homogeneously distributed, which indicates− the initial mineral volume fractions are the same throughout the entire sample. As indicated in [30], the initial porosity values at grid nodes in the porous media are defined as: 휙 = 휙0 + 푓 , where 휙0 = 0.2 is the mean value of porosity, the fluctuations 푓 are uniformly distributed in the interval [−∆휙0, ∆휙0] , and the magnitude of ∆휙 heterogeneity 푎 is defined as 푎 = 0. We consider three scenarios with the different magnitudes of 휙0 heterogeneity in the porous media. In this case study, three disconnected fractures representing natural fractures are embedded in the porous media. We set 휙 = 1 and choose large 퐊퐩퐞퐫퐦 values in fractures. Acidic fluid with a pH value of 3 is injected from the inlet boundary at 25 o C at a constant rate. The outlet pressure is fixed at 100 kPa. The initial and injected values of primary species concentrations are given in Table 1. Heterogeneity of the porous media is an important factor that affects structures of the flow channel during reactive transport processes. In a realistic case, natural heterogeneities could trigger very complicated flow paths, which in turn causes complex structures in the transport of acid fluid. The various flow paths lead to different dissolution patterns. The initial mineral volume fraction is another important parameter that can affect alterations of rock properties significantly due to the different chemical reactivity with acidic fluid among the minerals.

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∆φ a = 0 . We consider three scenarios with the different magnitudes of heterogeneity in the porous φ0 media. In this case study, three disconnected fractures representing natural fractures are embedded in the porous media. We set φ = 1 and choose large Kperm values in fractures. Acidic fluid with a pH value of 3 is injected from the inlet boundary at 25 ◦C at a constant rate. The outlet pressure is fixed at Wate100r kPa.2019, 11 The, 1957 initial and injected values of primary species concentrations are given in Table1. 10 of 20

Figure 4. The dimension of the core sample and the initial porosity distribution with ﬂuctuation in the Figureinterval 4. [The0.15, dimension 0.15]. of the core sample and the initial porosity distribution with fluctuation in − the interval [−0.15, 0.15]. Table 1. Initial and injected concentrations of aqueous species.

Table 1. Initial and injectedCa2+ (a)concentrationsNa+(a) of aqueous species3+(a) . (a) pH Al SiO2 ퟐ+(퐚) 4 +(퐚) 4 ퟑ+(퐚) 6 (풂5) Initial Condition (mol/L) 푪풂 6.05 10− 푵풂 1.38 10− 푨2.93풍 10 − 7.38푺풊10푶ퟐ− 7 pH −4 × −4× × −6 × −5 InitialInjected Condition CO -saturated (mol/L) brine6 (mol.05 /×L)10 1.57 101.382 × 10 1.03 2.934.08× 1010 7 1.217.38 10× 106 3 7 2 × − × − × − Injected CO2-saturated 1.57 ×Note:10−2 (a) represents1.03 aqueous species.4.08 × 10−7 1.21 × 10−6 3 brine (mol/L) Heterogeneity of the porousNote: media (a) re isprese annts important aqueous species factor. that affects structures of the flow channel during reactive transport processes. In a realistic case, natural heterogeneities could trigger very complicated flow paths, which in turn causes complex structures in the transport of acid fluid. 4.2.1. The Effect of Heterogeneity The various flow paths lead to different dissolution patterns. The initial mineral volume fraction is anotherIn this important section, we parameter have three that different can aff intervalsect alterations of fluctuations of rock properties 푓 of [−0.05 significantly, 0.05], [−0 due.10, 0 to.10 the], anddiff erent[−0.15 chemical, 0.15], respectively, reactivity with to investigate acidic fluid the among effects the of minerals. heterogeneity on dissolution patterns and variations of rock properties. The corresponding magnitudes of heterogeneity are chosen to be 0.25, 0.5,4.2.1. and The 0.75, Eff respectively.ect of Heterogeneity The initial mineral volume fractions of 90% calcite, 5% quartz, and 5% clay is fixed.In The this other section, initial we haveconditions three and different boundary intervals conditions of fluctuations are the samef of [as 0.05,shown 0.05 in] ,Table[ 0.10, 1. 0.10], − − andWe[ 0.15, consi 0.15der] ,three respectively, disconnected to investigate fractures the with eff ectsthe same of heterogeneity aperture of on1 mm dissolution as shown patterns in Figures and − 5a,variations 6a and 7 ofa. rockThe computational properties. The domain corresponding is meshed magnitudes using a 100 of × heterogeneity 100 × 50 Cartesian are chosen grid to system. be 0.25, Figures0.5, and 5– 0.75,7 illustrate respectively. the initial The porosity initial mineral distributions volume and fractions wormhole of 90% struc calcite,tures 5% at quartz,the breakthrough. and 5% clay Theis fixed. definition The other of a initialbreakthrough conditions is the and situation boundary when conditions the ratio are of the the same pressure as shown drop in to Table the 1initial. pressureWe drop consider decreases three disconnectedto 1% [18,30,33, fractures34]. Figures with the5b, 6b same and aperture 7b demonstrate of 1 mm that as shown the heterogeneity in Figures5a, of6 aporous and7a. media The computationalhas significant effects domain on is the meshed structure using of the a 100 conductive100 pathways50 Cartesian in the grid fractured system. × × porousFigures media.5–7 illustrate Without the heterogeneity, initial porosity the distributions dissolution andfront wormhole should propagate structures uniformly. at the breakthrough. In Figure 5Theb with definition a lower ofmagnitude a breakthrough of heterogeneity, is the situation the wormhole when the structures ratio of the tend pressure to be more drop straight to the initial and smoothpressure at drop breakthrough, decreases tobecause 1% [18 ,the30,33 permeability,34]. Figures is5b, almost6b and 7uniformb demonstrate throughout that the the heterogeneity core and the flowof porous direction media of injected has significant acidic fluid effects remains on the almost structure the of same. the conductive As the magnitude pathways of in heterogeneity the fractured increases, the dominant wormholes become highly branched and irregular, as shown in Figure 7b.

Water 2019, 11, 1957 10 of 18 porous media. Without heterogeneity, the dissolution front should propagate uniformly. In Figure5b with a lower magnitude of heterogeneity, the wormhole structures tend to be more straight and smoothWater 2019 at, 11 breakthrough,, 1957 because the permeability is almost uniform throughout the core and the11ﬂow of 20 directionWater 2019,of 11, injected 1957 acidic ﬂuid remains almost the same. As the magnitude of heterogeneity increases,11 of 20 theWate dominantr 2019, 11, 1957 wormholes become highly branched and irregular, as shown in Figure7b. 11 of 20

Figure 5. Porosity profile of fractured rock with fluctuations in the interval [−0.05, 0.05]. (a) Initial condition; (b) dominant wormhole. FigureFigure 5.5. PorosityPorosity proﬁleprofile ofof fracturedfractured rockrock withwith ﬂuctuationsfluctuations inin thethe intervalinterval [[−0.05,0.05, 0.05].0.05]. ((aa)) InitialInitial Figure 5. Porosity profile of fractured rock with fluctuations in the interval −[−0.05, 0.05]. (a) Initial condition;condition; ((bb)) dominantdominant wormhole.wormhole. condition; (b) dominant wormhole.

Figure 6. Porosity profile of fractured rock with fluctuations in the interval [−0.10, 0.10]. (a) Initial Figure 6. Porosity proﬁle of fractured rock with ﬂuctuations in the interval [ 0.10, 0.10]. (a) Initial condition; (b) dominant wormhole. − condition;Figure 6. Porosity (b) dominant profile wormhole. of fractured rock with fluctuations in the interval [−0.10, 0.10]. (a) Initial Figurecondition; 6. Porosity (b) dominant profile wormhole of fractured. rock with fluctuations in the interval [−0.10, 0.10]. (a) Initial condition; (b) dominant wormhole.

Figure 7.7. Porosity proﬁleprofile of fracturedfractured rockrock withwith ﬂuctuationsfluctuations inin thethe intervalinterval [[−0.15,0.15, 0.15].0.15]. (a) InitialInitial − condition; (b) dominant wormhole. condition;Figure 7. Porosity(b) dominant profile wormhole of fractured. rock with fluctuations in the interval [−0.15, 0.15]. (a) Initial Figurecondition; 7. Porosity (b) dominant profile wormhole of fractured. rock with fluctuations in the interval [−0.15, 0.15]. (a) Initial 4.2.2.condition; The Effect (b of) dominant the Mineral wormhole Volume. Fraction 4.2.2.In The this Effect section, of the the Mineral mineral Volume volume Fractionfractions are initially set to 80% calcite, 15% quartz, and 5% 4.2.2. The Effect of the Mineral Volume Fraction clay, In and this 70% section, calcite, the 25% mineral quartz, volume and fractions 5% clay, are respectively. initially set The to 80% fluctuations calcite, 15% 푓 are quartz, fixed and in the5% intervalclay, In and this [− 70%0 section,.15 , calcite,0.15 ]the, and 25%mineral the quartz, corresponding volume and fractions 5% clay,magnitude are respectively. initially of heterogeneity set The to 80% fluctuations calcite, 훼 is 0.75. 15% 푓 Otherare quartz, fixed initial and in and 5% the clay, and 70% calcite, 25% quartz, and 5% clay, respectively. The fluctuations 푓 are fixed in the interval [−0.15,0.15], and the corresponding magnitude of heterogeneity 훼 is 0.75. Other initial and interval [−0.15,0.15], and the corresponding magnitude of heterogeneity 훼 is 0.75. Other initial and

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Water 20194.2.2., 11, 1957 The Effect of the Mineral Volume Fraction 12 of 20 In this section, the mineral volume fractions are initially set to 80% calcite, 15% quartz, and boundary conditions are the same among these three scenarios. The changes in average porosity are 5% clay, and 70% calcite, 25% quartz, and 5% clay, respectively. The fluctuations f are fixed in the comparedinterval to investigate[ 0.15, 0.15] ,the and effects the corresponding of the mineral magnitude composition of heterogeneity on theα isalterations 0.75. Other of initial rock and properties − (Figure boundary8). Figure conditions 8 illustrates are the a comparison same among these of the three average scenarios. porosity The changes of the in fra averagecture porosityporous aremedia from the beginningcompared to tothe investigate breakthrough the effects among of the these mineral three composition scenarios. on the The alterations calcite has of rock the properties highest chemical reactivity(Figure with8). theFigure hydrogen8 illustrates ion a comparison among calcite, of the average quartz, porosity and of clay, the fracture which porous consequently media from cause the the beginning to the breakthrough among these three scenarios. The calcite has the highest chemical highest chemical reaction rate. Therefore, an increased volume fraction of calcite at the beginning can reactivity with the hydrogen ion among calcite, quartz, and clay, which consequently cause the highest lead to anchemical expanded reaction volume rate. Therefore, of dissolved an increased calcite volume and can fraction augment of calcite the at themineral beginning dissolution can lead to rate. As a result, aan larger expanded average volume porosity of dissolved and calcitean earlier and can breakthrough augment the mineralcan be dissolutionobtained, rate.as shown As a result, in Figure 8. a larger average porosity and an earlier breakthrough can be obtained, as shown in Figure8.

Figure 8. Comparison results of the average porosity. Figure 8. Comparison results of the average porosity. Case Study 2: 3D radial flow core flooding. In this case, we present the synthetic 3D radial Caseflow Study core flooding2: 3D radial case studies flow core to simulate flooding a matrix. In this acidizing case, process.we present A 3D the synthetic synthetic core sample 3D radial flow core floodinghas a dimension case studies of 2 cm to insimulate inner diameter, a matrix 20 cmacidizing in outer process. diameter, andA 3D 10 synthetic cm in height. core Several sample has a disconnected fractures with a uniform aperture of 2 mm are randomly distributed inside of the porous dimension of 2 cm in inner diameter, 20 cm in outer diameter, and 10 cm in height. Several media, as shown in the low two figures of Figure9. The initial porosity in the porous media is uniformly disconnecteddistributed fractures in the interval with of a [ uniform0.05, 0.35] with aperture a mean of value 2 mm of 0.2. are The randomly acidic fluid with distributed a pH value inside of 1 of the porous ismedia, injected as into shown the porous in the media low throughtwo figures the central of Figure borehole 9. atThe a constant initial rate.porosity The pressure in the porous at the media is uniformlyouter boundarydistributed is fixed in the at 10 interval Mpa. The of core [0. sample05, 0.35 contains] with only a mean one mineral, value of calcite, 0.2. withThe aacidic specific fluid with 2 3.73 2 a pH valuesurface of area1 is ofinjected 0.039 m into/g the[69], porous and the reactionmedia ratethrough constant thekrm centralwas adjusted borehole to 10 −at a molconstant/(m s) rate. The based on the results of model validation. pressure at the outer boundary is fixed at 10 Mpa. The core sample contains only one mineral, calcite, Figures 10 and 11 represent the porosity profiles at different times with 3 and 6 originally 2 with a specificdisconnected surface fractures area under of 0.039 the radial(m / floodingg) [69] condition,, and the respectively.reaction rate During constant the flooding 푘푟푚 was process, adjusted to 10−3.73 molthe injected/(m2s) acidbased fluid on flows the preferentiallyresults of model through validation. the fractures due to the good fluid conductivity. FigureChemicals 10 reactions and 11which represent occur at the the mineralporosity surface profiles under at the different action of the times acidic with fluid then 3 and leads 6 to originally disconnectedthe dissolution fractures of theunder mineral the rock.radial As flooding a consequence, condition, the conductive respectively. pathways During are formed the flooding nearby process, the initial fractures (Figures 10b–d and 11b–d). Comparing Figures 10 and 11, the more initially the injected acid fluid flows preferentially through the fractures due to the good fluid conductivity. disconnected fractures are embedded into the porous media, the greater the number of ramified Chemicalstructures reactions will which be, and occur the formation at the ofmineral conductive surface pathways under will the be obtainedaction of more the quickly.acidic fluid then leads to the dissolution of the mineral rock. As a consequence, the conductive pathways are formed nearby the initial fractures (Figure 10b–d and Figure 11b–d). Comparing Figures 10 and 11, the more initially disconnected fractures are embedded into the porous media, the greater the number of ramified structures will be, and the formation of conductive pathways will be obtained more quickly.

Water 2019, 11, 1957 13 of 20 Water 2019, 11, 1957Water 2019, 11, 1957 12 of 18 13 of 20

Figure 9. The dimension of the core sample, initial porosity profile, and randomly distributed fractures. Figure 9. The dimension of the core sample, initial porosity proﬁle, and randomly distributedfractures.

Figure 9. The dimension of the core sample, initial porosity profile, and randomly distributed fractures.

Figure 10. Temporal and spatial evolution of porosity. (a) initial condition; (b) 14.4 min; (c) 28.8 min; Figure 10(.d Temporal) 48 min (breakthrough). and spatial evolution of porosity. (a) initial condition; (b) 14.4 min; (c) 28.8 min; (d) 48 min (breakthrough).

Figure 10. Temporal and spatial evolution of porosity. (a) initial condition; (b) 14.4 min; (c) 28.8 min; (d) 48 min (breakthrough).

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Figure 11. Temporal and spatial evolution of porosity. (a) initial condition; (b) 14.4 min; (c) 28.8 min; Figure 11(.d Temporal) 47.4 min (breakthrough). and spatial evolution of porosity. (a) initial condition; (b) 14.4 min; (c) 28.8 min; (d) 47.4 min (breakthrough). 5. Conclusions 5. ConclusionsIn this study, we developed a 3D mathematical model that couples the Stokes–Brinkman, reactive-transport, and rock properties equations to simulate fluid flow, solute transport, chemical In thisreactions, study, and we alterations developed of porosity a 3D and mathematical permeability in a modelfractured that porous couples media. The the coupled Stokes model–Brinkman, reactive-transport,can be applied and to rock simulate properties the mineral equations dissolution to processes simulate in bothfluid linear flow, and solute radial flow.transport, We have chemical reactions,developed and alterations and implemented of porosity a sequential and permeability numerical solution in a procedure fractured to solve porous this media.coupled model.The coupled model canWe be applied applied the to newly simulate developed the numericalmineral dissolution simulator for processes three dimensional in both simulation linear and study radial on the flow. We coupled processes of fluid transport and mineral dissolution both on Cartesian and cylindrical grids. have developedThe preliminary and implemented simulation results a sequential can be highlighted numerical as follows: solution procedure to solve this coupled model. We applied the newly developed numerical simulator for three dimensional simulation study Firstly, the proposed numerical model is applied for 3D linear core flooding in a multi-mineral on the coupled• processes of fluid transport and mineral dissolution both on Cartesian and cylindrical system, which consists of calcite, quartz, and clay. Sensitivity studies clearly show the effects of grids. The preliminaryheterogeneity simulation of porous media results and mineralcan be volumehighlighted fractions as on follows: the rockproperties and dissolution  Firstly, thepatterns. proposed The numerical numerical results model demonstrate is applied that the for magnitude 3D linear of heterogeneitycore flooding has in a significanta multi-mineral impact on the structure of the dominant wormholes: Without heterogeneity, the dissolution front system, which consists of calcite, quartz, and clay. Sensitivity studies clearly show the effects of propagates uniformly. The higher initial calcite content has a significant impact on porosity and heterogeneityfracture ofevolution porous via anmedia increase and in the mineral reactive surfacevolume area fractions and the subsequent on the augmentation rock properties of and dissolutionthe chemicalpatterns. reaction The numerical rate. results demonstrate that the magnitude of heterogeneity has a significantIn the impact second case on study,the structure the proposed of numericalthe dominant model iswormholes: applied for 3D Without radial core heterogeneity, flooding in the • dissolutiona single front calcite propagates system. Several uniformly. disconnected The higher fractures initial are randomly calcite content distributed has inside a significant of porous impact on porositymedia. and During fracture the flooding evolution process, via an the increase fractures propagate in the reactive radially surface within the area porous and media the subsequent and eventually form the conductive channels, wormholes. Therefore, the proposed numerical model augmentationcan be appliedof the tochemical simulate thereaction matrix rate. acidizing process in fractured carbonate formations at exact  In the seconddownhole case environments. study, the proposed numerical model is applied for 3D radial core flooding in a single calcite system. Several disconnected fractures are randomly distributed inside of However, it should be noted that the model developed in this work is mainly focused on the porousreactive media. transport During processes the flooding at the core process, scale. Challenges the fractures arise whenpropagate simulating radially this process within at the the porous mediafield and scale eventually due to the treatment form th ofe fractures conductive in this model.channels, More wormholes. efficient numerical Therefore, models for the field proposed numericalapplications model will can be developedbe applied in theto simulate future. the matrix acidizing process in fractured carbonate formations at exact downhole environments. However, it should be noted that the model developed in this work is mainly focused on the reactive transport processes at the core scale. Challenges arise when simulating this process at the field scale due to the treatment of fractures in this model. More efficient numerical models for field applications will be developed in the future.

Author Contributions: T.Y. conducted the research work under the supervision of G.Q., T.Y. developed the numerical model and numerical solution procedure, performed the numerical experiments, and wrote the paper. C.W. discussed the results, and reviewed and edited the paper. C.-S.Z. developed the linear solver Package

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Author Contributions: T.Y. conducted the research work under the supervision of G.Q., T.Y. developed the (FASP), wrote the content of FASP, and reviewed and edited the paper. G.Q. conceived this study, discussed the numerical model and numerical solution procedure, performed the numerical experiments, and wrote the paper. results,C.W. discussed and reviewed the results, and edited and reviewed the paper. and All edited authors the give paper. the C.-S.Z. final approval developed of the linearmanuscript solver to Package be submitted. (FASP), wrote the content of FASP, and reviewed and edited the paper. G.Q. conceived this study, discussed the results, Funding: This research is partially supported by the US National Science Foundation through NSF Grant DMS- and reviewed and edited the paper. All authors give the final approval of the manuscript to be submitted. 1209124, the Society of Petroleum Engineers—Gulf Coast Section through its distinguished professorship endowment,Funding: This the researchInstitute of is Petroleum partially supported Exploration by &the Development US National, Langf Scienceang Branch, Foundation the PetroChina through NSF Company Grant DMS-1209124, the Society of Petroleum Engineers—Gulf Coast Section through its distinguished professorship Limited, CNPC Chuanqing Drilling Engineering Company Limited, Sinopec Tech Houston LLC, the Key endowment, the Institute of Petroleum Exploration & Development, Langfang Branch, the PetroChina Company ResearchLimited, CNPCProgram Chuanqing of Frontier Drilling Sciences Engineering of CAS, and Company the National Limited, Science Sinopec Foundation Tech Houston of China LLC, the(11971472). Key Research The authorsProgram thank of Frontier PetroChina Sciences Company of CAS, andLimited the Nationaland Sinopec Science Corp Foundation for their perm of Chinaission (11971472). to publish The this authors paper. thank PetroChina Company Limited and Sinopec Corp for their permission to publish this paper. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Appendix A In this paper, the unknowns of 푣 , 푣 , 푣 , and 푝 in 3-D cylindrical coordinates are defined in In this paper, the unknowns of v푟 , v휃 , v푧, and p in 3-D cylindrical coordinates are defined in Figure A1. The pressure 푝 is located inr theθ centerz of each cell (푖, 푗, 푘), and velocities are located in the Figure A1. The pressure p is located in the center of each cell (i, j, k), and velocities are located in the center of grid faces as indicated by the subscripts 푖 ± 1/2, 푗 ± 1/2, and 푘 ± 1/2. center of grid faces as indicated by the subscripts i 1/2, j 1/2, and k 1/2. ± ± ±

Figure A1. Definition of pressure, velocities, and concentrations on a grid with indices i, j, k. Figure A1. Definition of pressure, velocities, and concentrations on a grid with indices i, j, k. With the definitionsWith the definitions of the pressure of the pressure and velocitie ands velocities in Figure in A1 Figure, the A1 finite, the difference finite diff approximationerence approximationss of of EquationsEquations (1) (1)and and (2)(2) in the the 3D 3D cylindrical cylindrical coordinate coordinate system system are given are given as: as:

pi+1,j,k pi,j,k 1 ri+1(vr,i+3/2,j,k vr,i+1/2,j,k) ri(vr,i+1/2,j,k vr,i 1/2,j,k) ( 1) + − [ − − − − µKperm,r− i+1/2,j,kvr,i+1/2,j,k ∆r r 2 1 pijk1,,− p ijk ,,i+1/2 1 r i  1()() v ri ,3/2,,  jk ∆r v ri ,1/2,,  jk  r iri v ,1/2,,  jk  v ri ,1/2,,  jk 2 ()[Kv v r,i+1/2,j,k perm,r i 1/2,,vr,i+ j1 k/2, rj+ ,1/2,, i1,k 2 jv kr,i+1/2,j,k+vr,i+1/2,j 1,k vr,i+1/2,j,k+1 2vr,i+1/2,j,k+vr,i+1/2,j,k 12 1 − − − − 2 + 2 2 r+ ri1/2 2 r (A1) − r i+1/2 r i+1/2 ∆θ ∆z vθ2,i,j+1/2,k vθ,i,j 1/2,k 2 v− − v2 v v v2 v v 2 ri,1/2,, jk1 ] = ri ,1/2,1,0,  jk  ri ,1/2,,  jk ri ,1/2,1,  jk  ri,1/2,, jk  1 ri ,1/2,,  jk ri ,1/2,,  jk  1 r i+1/2 ∆θ − 2 2 2  2 rrii1/2 1/2  z (A1) pi,j+1,k pi,j,k ri+1/2(vθ,i+1,j+1/2,k vθ,i,j+1/2,k) ri 1/2(vθ,i,j+1/2,k vθ,i 1,j+1/2,k) 1 1 − 1 − − − − − (µKperm,θ− ) + vθ,i,j+1/2,k + r ∆θ [ r 2 2 i,jvv1/,,i2, jk 1/2, k ,, i j 1/2,i k − i ∆r 2 v θ,i,j+1/2,k vθ,i,j+3/2,k 2vθ,i,j+1/2,k+v]θ,i,j 1 0,/2,k vθ,i,j+1/2,k+1 2vθ,i,j+1/2,k+vθ,i,j+1/2,k 1 2 1 − − − − + 2 + 2 2 + 2 (A2) r ir i1/2 r i  ∆θ ∆z v + + v + 2 r,i 1/2,j 1,k− r,i 1/2,j,k 2 ] = 0, − r i ∆θ

pi,j,k+1 pi,j,k 1 ri+1/2(vz,i+1,j,k+1/2 vz,i,j,k+1/2) ri 1/2(vz,i,j,k+1/2 vz,i 1,j,k+1/2) ( 1) + − [ − − − − − µKperm,z− i j k+ vz,i,j,k+1/2 ∆z r 2 , , 1/2 − i ∆r (A3) vz,i,j+1,k+1/2 2vz,i,j,k+1/2+vz,i,j 1,k+1/2 vz,i,j,k+3/2 2vz,i,j,k+1/2+vz,i,j,k 1/2 1 − − − − + 2 2 + 2 ] = 0, r i ∆θ ∆z

ri+1/2vr,i+1/2,j,k ri 1/2vr,i 1/2,j,k vθ,i,j+1/2,k vθ,i,j 1/2,k vz,i,j,k+1/2 vz,i,j,k 1/2 − − − + − − + − − = 0, (A4) ri∆r ri∆θ ∆z

Water 2019, 11, 1957 15 of 18 where the permeability at the cell interface is calculated as the harmonic average of the permeabilities of the two neighboring cells. In this paper, we set constant velocity at the inlet boundary, constant pressure and free-ﬂows at the outlet boundary, and no-slip boundary condition at the other boundaries. The boundary conditions in the cylindrical system are expressed as:

vr = vin at r = rw, (A5)

p = pout at r = re, (A6) ∂v = 0 at r = re, (A7) ∂n v = 0 at z = 0, and z = H, (A8) where rw denotes the wellbore radius, re denotes the radius of the outer boundary, vin is the injected velocity at the wellbore, pout is the outlet pressure, n is the unit normal vector on the boundary, and H is the depth of the domain. In the cylindrical coordinates system, when the entire domain is divided into Nr, Nθ, and Nz grid blocks in the corresponding directions, respectively, the total number of p, vr, v , and vz is Nr N Nz, θ × θ × (Nr + 1) N Nz, Nr N Nz, and Nr N (Nz + 1), respectively. By imposing the deﬁned × θ × × θ × × θ × boundary conditions, we have (Nr 1) N Nz + (Nr 1) N Nz + Nr N Nz + Nr N − × θ × − × θ × × θ × × θ × (Nz 1) unknown variables and the same number of discretized equations, which can be expressed as − a system of linear equations:    B B 0 B  v  g  1r 1θ 1p  r   r        B2r B2θ 0 B2p  vθ   gθ     =  , (A9)  B B  v   g   0 0 3z 3p  z   z       B4r B4θ B4z 0 p gp where vr, vθ, vz, and p are unknown vectors, and all of them go through the indices i, j, k in the same order. B1r, B1θ, and B1p are the submatrices resulting from the terms of vr, vθ, and p in the Equation (A1), respectively. B2r, B2θ, and B2p are the submatrices resulting from the terms of vr, vθ, and p in the Equation (A2), respectively. B3z and B3p are the submatrices resulting from the terms of vz and p in the Equation (A3), respectively. B4r, B4θ, and B4z are the submatrices resulting from the terms of vr, vθ, and vz in the Equation (A4), respectively. gr, gθ, gz, and gp are the right-hand side vectors resulting from the deﬁned boundary conditions.

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