Acid Stimulation in Single Carbonate Fractures with Heterogeneous Apertures

Abstract

The service life of geothermal wells is affected by the reservoir permeability. Acid fracturing has been utilized to improve productivity in fractured carbonate geothermal reservoirs recently. A modeling method for the coupled thermal-hydro-mechanical-chemical (THMC) process during acid fracturing is developed. In a benchmark test, the effect of aperture heterogeneity on the efficiency of acid fracturing in geothermal reservoirs is examined. The results show that the acid flow and aperture increase exhibit significant anisotropy in the heterogeneous characteristics. Based on a field test of acid fracturing in a geothermal well, Tongzhou, Beijing, we use the built three-dimensional fractured porous reservoir model to simulate the process of acid fracturing. The agreement between the simulation and the field test data demonstrate that the proposed modeling method is capable of simulating acid fracturing in fractured carbonate geothermal reservoirs.

1 Introduction

Geothermal wells may suffer from disappointing injectivity and productivity with time or right from the start, and hydraulic or acid fracturing treatment to improve the wells’ performance have been developed based on the experiences in the oil and gas industry. The application of the correct well stimulation technique in an optimum manner is helpful to establish and maintain the maximum energy capacity of geothermal wells. It is known that acid fracturing is an effective method for the stimulation of carbonate geothermal reservoirs. Selecting the optimal flow rate is an important task in reservoir stimulation, in order to achieve the maximum increase in permeability for a given amount of reactant. If the acid is injected too slowly, significant dissolution occurs only at the inlet, and the permeability of the system remains almost unchanged. At the other extreme of high injection velocities dissolution tends to be uniform throughout the whole fracture. Note that the optimized injection rate is affected by many factors including rock physical properties, acidizing conditions, as well as in-situ stress and temperature conditions. Numerous numerical models have been developed to simulate the reactive flow in fractured rocks, and they can consider the coupled processes of fluid flow, heat transfer, and reactive transport, as well as the heterogeneity in fracture geometrical and mineral properties (Yuan et al. 2016; Liu et al. 2017; Ma et al. 2018; Jones and Detwiler 2019). However, there are few modeling methods which can simulate the acid fracturing in fractured carbonate geothermal reservoirs considering the fully coupled thermal-mechanical-hydrological-chemical processes.

The main aim of this study is to develop a fully coupled thermo-hydro-mechanical-chemical (THMC) model for acid fracturing treatment in carbonate geothermal reservoirs containing a single fracture. The fracture is surrounded by three-dimensional low permeable rock matrix. Both a benchmark example and a field test are presented.

2 Numerical Method

Governing equations for fluid flow, reactive transport, heat transfer and solid mechanics in fracture and matrix are presented below. The fracture can be conceptualized as a set of parallel plates by assuming that the fracture midsurface is planar and that the variation in aperture is small relative to its mean value. The acid flow in rock fractures can be described as,

$$ b\rho S_{f} \frac{{\partial p_{f} }}{\partial t} + \nabla \cdot \left( {b\rho {\mathbf{u}}_{f} } \right) = bq_{m} $$

(1a)

$$ {\mathbf{u}}_{f} = - \frac{{b^{2} }}{{12\mu F_{T} }}\left( {\nabla p_{f} + \rho {\mathbf{g}}} \right) $$

(1b)

where b is the real-time aperture altered by mechanical and chemical processes, ρ is the fluid density, p_f is the aperture-averaged fluid pressure, S_f is the storage coefficient of fracture, u_f is the flow velocity vector, q_m is the lateral exchange of fluid between the fracture and the matrix, t is time, μ is the dynamic viscosity, F_T is an empirical parameter, g is the gravitational acceleration, F_T is an empirical parameter that accounts the non-laminar flow.

The fluid flow in porous rock matrix are normally regarded as laminar flow, which is described by the continuity equation and Darcy’s Law.

$$ \rho S_{r} \frac{{\partial p_{r} }}{\partial t} + \nabla \cdot \left( {\rho {\mathbf{u}}_{r} } \right) = q_{m} $$

(2a)

$$ {\mathbf{u}}_{r} = - \frac{k}{\mu }\left( {\nabla p_{r} + \rho {\mathbf{g}}} \right) $$

(2b)

where u_r is the Darcy velocity in the rock matrix, k is the permeability of the rock matrix, p_r is the fluid pressure in the rock matrix, S_r is the storage coefficient, which is a function of porosity ($\phi$), Biot-Willis coefficient (α_B), fluid bulk modulus (K_f), and drained bulk modulus (K_d).

$$ S_{r} = \frac{\phi }{{K_{f} }} + \left( {\alpha_{B} - \phi } \right)\frac{{1 - \alpha_{B} }}{{K_{d} }} $$

(3)

The coupling between the fluid flow and mechanical displacement is considered as,

$$ \rho S_{r} \frac{{\partial p_{r} }}{\partial t} + \nabla \left( {\rho {\mathbf{u}}_{r} } \right) = - \rho \alpha_{B} \frac{{\partial \varepsilon_{vol} }}{\partial t} + q_{m} $$

(4)

where $\varepsilon_{vol}$ is the average volumetric strain of the rock matrix. The coupling between flow through fracture and rock matrix is incorporated by,

$$ bq_{m} = \rho \left( {\left. {u_{rz} } \right|_{z = - b/2} - \left. {u_{rz} } \right|_{z = b/2} } \right) $$

(5)

where z =  ±b/2 represent the interfaces between the fracture and the rock matrix.

The rate (Rⁱ_r) of the chemical reaction between carbonate rocks and the acid is a function of acid concentration and ambient temperature,

$$ R_{r}^{i} = k_{f} C_{i}^{{n_{a} }} $$

(6a)

$$ k_{f} = A_{f} e^{{ - \frac{{E_{a} }}{RT}}} $$

(6b)

where k_f is the reaction rate coefficient that is described by the Arrhenius law (Eq. 7), C_i is the acid concentration, n_a is the order of reaction, A_f is the preexponential factor, E_a is the activation energy, R is the molar gas constant, T is temperature.

The reactive transport equation in fracture is,

$$ b \cdot [\frac{{\partial \left( {C_{i}^{f} } \right)}}{\partial t} + {\mathbf{u}}_{f} \cdot \nabla C_{i}^{f} ] - \nabla \cdot \left( {bD_{ef}^{i} \nabla C_{i}^{f} } \right) = b \cdot [R_{f}^{i} + S_{i} ] $$

(7)

where C^f_i is the solute concentration in the fracture (i = HCl or CaCl₂), Dⁱ_ef is the solute diffusion coefficient in the fracture, S_i is the solute flux from the fracture to the rock matrix, Rⁱ_f is the reaction rate in the fracture.

The reactive transport equation in porous rock matrix is,

$$ C_{i}^{r} S_{r} \frac{{\partial p_{r} }}{\partial t} + {\mathbf{u}}_{r} \cdot \nabla C_{i}^{r} - \nabla \cdot \left( {D_{er}^{i} \nabla C_{i}^{r} } \right) = R_{r}^{i} \phi $$

(8)

where C^r_i is the solute concentration in the rock matrix (i = HCl or CaCl₂), Dⁱ_er is the solute diffusion coefficient in the matrix, Rⁱ_r is the reaction rate in the rock matrix.

Dong et al. (2002) showed that almost all leakoff acid flows out of the fracture without reacting with the fracture walls, and that all acid transported by diffusion reacts with the formation at the fracture walls. With these assumptions, the change of the fracture width is represented by,

$$ \frac{{\partial b_{c} }}{\partial t} = 2k_{g} \overline{C}_{HCL}^{f} \frac{\beta }{{\rho_{r} \left( {1 - \phi } \right)}} $$

(9)

where ρ_r is the density of rock matrix, C^f_HCL is the aperture-averaged acid concentration, k_g is the apparent mass-transfer coefficient, β is acid dissolving power.

Considering fluid flow and chemical reactions, heat transport in the fracture can be described as,

$$ b[\rho C\frac{\partial T}{{\partial t}} + {\mathbf{u}}_{f} \rho C \cdot \nabla T] - \nabla \cdot \left( {b\lambda \nabla T} \right) = bQ_{chem}^{f} + {\mathbf{n}} \cdot {\mathbf{q}} $$

(10)

where C is the specific heat capacity of the fluid, λ is the thermal conductivity of the fluid, n∙q is the heat from the matrix into the fracture, Q^f_chem is the heat source induced by chemical reactions.

Heat transport in the rock matrix can be described as,

$$ \left( {\rho_{i} C_{i} } \right)_{eff} \frac{\partial T}{{\partial t}} + {\mathbf{u}}_{r} \rho C \cdot \nabla T - \nabla \cdot \left( {\lambda_{eff} \nabla T} \right) = Q_{chem}^{r} $$

(11)

where (ρ_iC_i)_eff is the volumetric mean of the heat capacity, λ_eff is the volumetric mean of the heat conductivities of the rock and the acid, Q^r_chem is the heat source derived from chemical reaction.

Shear dilation and elastic displacement are considered to affect the fracture aperture. Elastic layer is complemented to simulate the fracture displacement, given as,

$$ \sigma^{\prime}_{n} = \sigma_{n} - p_{f} = k_{n} u_{n} $$

(12a)

$$ \sigma_{s} = k_{s} u_{s} $$

(13b)

where σ_n and σ_s are the normal and shear stresses, σ′_n is the effective normal stresses, k_n and u_n are the normal stiffness and normal displacement, k_s and u_s are the shear stiffness and shear displacement.

The shear-induced dilation can be described as,

$$ u_{d} = \left\{ {\begin{array}{*{20}l} 0 \hfill \\ {u_{s} \tan \varphi_{d} } \hfill \\ {u_{cs} \tan \varphi_{d} } \hfill \\ \end{array} \begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} } \right.\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} \begin{array}{*{20}l} {u_{s} \le u_{p} } \hfill \\ {u_{p} \le u_{s} \le u_{cs} } \hfill \\ {u_{s} \ge u_{cs} } \hfill \\ \end{array} $$

(13)

where u_p is the shear displacement at the peak shear stress, u_cs is the shear strain at the critical stress, φ_d is the dilation angle.

Therefore, the fracture aperture mediated by the mechanical and chemical processes becomes,

$$ b = b_{0} + b_{c} + u_{n} + u_{d} $$

(14)

Thermo-poro-elasticity typically describe the linked interactions among fluid, temperature and deformation in porous rock matrix. The relationship between stress tensor (σ), strain tensor (ε), and pore fluid pressure (p_f) is,

$$ {{\varvec{\upsigma}}} = {\mathbf{C}}_{kr} \left( {{{\varvec{\upvarepsilon}}} - \varepsilon_{T} } \right) - \alpha_{B} p{\mathbf{I}} $$

(15a)

$$ \varepsilon_{T} = \alpha_{T} \left( {T - T_{ref} } \right) $$

(15b)

$$ p = M\left( {\zeta - \alpha_{B} \phi } \right) $$

(15c)

where C_kr is the elasticity matrix measured under drained condition. Thermal strain (ε_T) is expressed in terms of the coefficient of thermal expansion (α_T) and the change from an initial temperature (T_ref), M is the Biot modulus, which is the inverse of the storage coefficient, ζ is the variation of fluid content.

Navier's equations for a solid in equilibrium under purely gravitational load is,

$$ - \nabla \sigma = \rho_{av} {\mathbf{g}} = [\rho_{f} \phi + \rho_{r} \left( {1 - \phi } \right)]{\mathbf{g}} $$

(16)

where ρ_av represents the average density.

The reservoir is simulated as a three-dimensional (3D) domain including one single fracture, and the 3D domains are discretized into finite tetrahedron elements. When the domain discretization using tetrahedron elements and the time discretization using backward Euler time integration scheme are completed, the linear system is solved using parallel direct sparse solver interface in COMSOL.

3 Case Studies

3.1 Benchmark Example

A benchmark reservoir model containing a vertical stimulation well and a single horizontal fracture with was built (Fig. 1), and the effect of aperture heterogeneity on acidizing efficiency was examined. The numerical reservoir model was 300 m long, 300 m wide, and 50 m high, which was used to represent the carbonate geothermal reservoir at a depth of about 3 km. To study the acidizing process in heterogeneous fractured geothermal reservoir, the fracture was divided into 900 domains. We built two models with different initial fracture apertures: 1) heterogeneous model: the initial fracture apertures followed the log-normal distribution with the mean fracture aperture of 1 mm, the standard deviation of 0.25 mm, and the spatial correlation length of 15 m; 2) uniform model: the uniform initial aperture was 1mm. A tetrahedron mesh is used in this study, which has a characteristic element length of about 0.1 to 20 m, resulting in about 165,000 elements and 30,000 nodes. The initial and boundary conditions pertain to the general condition of the carbonatite reservoir at the depth of 3 km. In-situ stresses σ_v = 90 MPa, σ_H = 50 MPa, and σ_h = 40 MPa were applied on the model boundaries. The initial temperature of the reservoir was 80 °C, and the temperature of the injected acid was 20 °C. Acidizing was performed by injecting 28% acid solution (acid concentration of 8075 mol/m³) through the stimulation well in the fractured reservoir. Under the pressure difference between the stimulation well and the reservoir, the reactive acid flowed through the fracture and induces aperture alteration. The period for acidizing was 120 min and was divided into three stages, each of which lasted for 40 min. The pumping rates of the three stages were 2.0, 3.0 and 4.0 m³/min. Respectively. The model parameters are listed in Table 1.

With the injection, the aperture altered by chemical reaction increased gradually in both heterogeneous model (Fig. 2 a, b, c) and uniform model (Fig. 2 d, e, f). The heterogeneous model and the uniform model had significantly different aperture enhancement patterns, i.e., the acid flow and aperture increasement exhibit significant anisotropy in the heterogeneous fractures (heterogeneous model). The distribution of H⁺ concentration in the fractures of heterogeneous model and uniform model also showed different patterns (Fig. 3b, e). Neither the fluid viscosity (Fig. 3a and d) nor the temperature (Fig. 3c and f) was significantly affected by the heterogeneous apertures.

Fig. 1.

The numerical model (a) and the mesh (b) of the benchmark example.

Table 1. Model parameters

Fig. 2.

Comparison of the aperture enhancement patterns between models with the heterogeneous apertures (upper row) and a uniform initial aperture (low row) at different stages. Note that the central area of 100 × 100 m² is plotted.

Fig. 3.

Comparison of the acid temperature (a and d), H+ concentration (b and e) and viscosity (c and f) between models with the heterogeneous apertures (upper row) and a uniform initial aperture (low row) at 120 min. Note that the central area of 100 × 100 m² is plotted.

3.2 Case Study

A geothermal well was drilled in Tongzhou, Beijing from March to December 2018 at a depth of 3000 m. From the field test data, the injection pressure of the acid was 5–6 MPa, and was lowered to 1 MPa before injecting the after-pad fluid. The fluid utilized in acid fracturing is 20% hydrochloride acid. As the diameter of the downhole tube is 73 mm, wellbore diameter is simplified as 100 mm in the simulation. The reservoir is mainly composed of dolomite, and the acid concentration at the surface of the rock was calculated using the following surface concentration model (Li et al. 1993),

$$ \frac{{\partial C_{HCL}^{wall} }}{\partial t} = k_{g} (\overline{C}_{HCL}^{f} - C_{HCL}^{wall} ) - k_{f} (C_{HCL}^{wall} )^{{n_{a} }} $$

(17)

Under the same pump rate as the field test, the simulation gave the similar surface pressure as the field test (Fig. 4). After 139 min of fracturing construction the maximum fracture half-length is about 65 m (Fig. 4).

Fig. 4.

The modeled pressure (a) and apertures (b) for the case study of acid fracturing in Tongzhou, Beijing.

4 Conclusion

A modeling method for the coupled thermal-hydro-mechanical-chemical (THMC) process during acid fracturing is developed, and a benchmark example and a field test are presented, which demonstrate that the proposed modeling method is capable of simulating acid fracturing in fractured carbonate geothermal reservoirs.