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

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## Abstract

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_{2}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.

## 1. Introduction

_{2}sequestration (GCS) for reducing CO

_{2}emissions [3,4], and matrix acidizing for improving oil production by changing the permeability near wellbores [5,6].

_{2}in situ brine, via various geochemical reactions, can form a carbonate acid that may then dissolve carbonate rocks [7,8]. Such CO

_{2}–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 [9,10,11]. 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 [12,13,14,15]. 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.

## 2. The Mathematical Models

#### 2.1. The Stokes–Brinkman Model

#### 2.2. The Reactive-Transport Model

#### 2.3. The Rock Property Models

## 3. Numerical Solution Strategies

- (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.

#### 3.1. Numerical Solution of the Stokes–Brinkman Model

#### 3.2. Numerical Solution of the Reactive Transport Model

## 4. Numerical Experiments

#### 4.1. Model Validation

#### 4.2. Numerical Studies

#### 4.2.1. The Effect of Heterogeneity

#### 4.2.2. The Effect of the Mineral Volume Fraction

## 5. Conclusions

- Firstly, the proposed numerical model is applied for 3D linear core flooding in a multi-mineral system, which consists of calcite, quartz, and clay. Sensitivity studies clearly show the effects of heterogeneity of porous media and mineral volume fractions on the rock properties and dissolution patterns. The numerical results demonstrate that the magnitude of heterogeneity has a significant impact on the structure of the dominant wormholes: Without heterogeneity, the dissolution front propagates uniformly. The higher initial calcite content has a significant impact on porosity and fracture evolution via an increase in the reactive surface area and the subsequent augmentation of the chemical reaction rate.
- In the second case 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 porous media. During the flooding process, the fractures propagate radially within the porous media and eventually form the conductive channels, wormholes. Therefore, the proposed numerical model can be applied to simulate the matrix acidizing process in fractured carbonate formations at exact downhole environments.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 4.**The dimension of the core sample and the initial porosity distribution with fluctuation in the interval [−0.15, 0.15].

**Figure 5.**Porosity profile of fractured rock with fluctuations in the interval [−0.05, 0.05]. (

**a**) Initial condition; (

**b**) dominant wormhole.

**Figure 6.**Porosity profile of fractured rock with fluctuations in the interval [−0.10, 0.10]. (

**a**) Initial condition; (

**b**) dominant wormhole.

**Figure 7.**Porosity profile of fractured rock with fluctuations in the interval [−0.15, 0.15]. (

**a**) Initial condition; (

**b**) dominant wormhole.

**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; (

**d**) 48 min (breakthrough).

**Figure 11.**Temporal and spatial evolution of porosity. (

**a**) initial condition; (

**b**) 14.4 min; (

**c**) 28.8 min; (

**d**) 47.4 min (breakthrough).

$\mathit{C}{\mathit{a}}^{2+\left(a\right)}$ | $\mathit{N}{\mathit{a}}^{+\left(a\right)}$ | $\mathit{A}{\mathit{l}}^{3+\left(a\right)}$ | $\mathit{S}\mathit{i}{\mathit{O}}_{2}^{\left(\mathit{a}\right)}$ | pH | |
---|---|---|---|---|---|

Initial Condition (mol/L) | $6.05\times {10}^{-4}$ | $1.38\times {10}^{-4}$ | $2.93\times {10}^{-6}$ | $7.38\times {10}^{-5}$ | 7 |

Injected CO_{2}-saturated brine (mol/L) | $1.57\times {10}^{-2}$ | 1.03 | $4.08\times {10}^{-7}$ | $1.21\times {10}^{-6}$ | 3 |

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**MDPI and ACS Style**

Yuan, T.; Wei, C.; Zhang, C.-S.; Qin, G.
A Numerical Simulator for Modeling the Coupling Processes of Subsurface Fluid Flow and Reactive Transport Processes in Fractured Carbonate Rocks. *Water* **2019**, *11*, 1957.
https://doi.org/10.3390/w11101957

**AMA Style**

Yuan T, Wei C, Zhang C-S, Qin G.
A Numerical Simulator for Modeling the Coupling Processes of Subsurface Fluid Flow and Reactive Transport Processes in Fractured Carbonate Rocks. *Water*. 2019; 11(10):1957.
https://doi.org/10.3390/w11101957

**Chicago/Turabian Style**

Yuan, Tao, Chenji Wei, Chen-Song Zhang, and Guan Qin.
2019. "A Numerical Simulator for Modeling the Coupling Processes of Subsurface Fluid Flow and Reactive Transport Processes in Fractured Carbonate Rocks" *Water* 11, no. 10: 1957.
https://doi.org/10.3390/w11101957