Modeling Fracture Dissolution in Acid Fracturing Treatments with Acid Viscous Fingering

By Rencheng Dong, Mary F. Wheeler, Hang Su, and Kang Ma

The energy industry routinely employs acid fracturing—a major type of acid stimulation treatment—to improve permeability and enhance production in carbonate reservoirs. The first stage of acid fracturing involves injecting a pad fluid (commonly water or a polymer) to create an initial fracture (see Figure 1a). To break the reservoir formation, the injection pressure must be higher than the formation fracture pressure. We can model fracture propagation during pad injection with hydraulic fracturing simulators. 

The process of acid etching creates the acid fracture conductivity. One injects acid after the pad stage to etch fracture surfaces non-uniformly, which creates rough fracture surfaces (see Figure 1b). The dissolution reaction between the hydrochloric acid (HCl) and limestone (CaCO₃) is given by

\[2H^+ + CaCO_3(s) \rightarrow Ca^{2+} +CO_2+H_2O.\tag1\]

Although the acid fracture tends to close under the formation closure stress that occurs after acid fracturing, the fracture can still remain partially open with the support of asperities on fracture surfaces (see Figure 1c). The residual fracture opening therefore leads to acid fracture conductivity.

A better understanding of the coupled processes of fluid flow, reactive transport, and rock dissolution helps engineers design more effective acid fracturing operations. Researchers can evaluate and characterize the etching patterns on fracture surfaces via laboratory acid etching tests on carbonate rock samples [9]. Figure 2a displays a typical acid etching test cell and Figure 2b shows a pair of rock slabs for acid etching experiments. Researchers place the two rock slabs side-by-side in the test cell so that the open channel between the slabs mimics an open fracture. The injected acid then flows through that open fracture to etch the rock surfaces. Figure 3 represents a uniform etching pattern and Figure 4 signifies a channeling etching pattern. The channeling etching pattern in Figure 4 experiences much less conductivity decline under the closure stress than the uniform etching pattern in Figure 3 [6].

The key to success in acid fracturing treatments is the achievement of non-uniform acid etching on fracture surfaces, which can result from reservoir heterogeneity [8]. Because different minerals have different reaction rates with acid, heterogeneous mineral distribution may result in non-uniform acid etching.

One can utilize the viscous fingering mechanism to enhance non-uniform acid etching [1]. Figure 5 illustrates a typical pumping schedule that involves acid viscous fingering. When one injects the low-viscosity acid to displace the high-viscosity pad fluid, acid viscous fingering develops within the acid fracture; more carbonate rocks are dissolved in the fingering area than other areas. Since including both acid etching and acid viscous fingering in laboratory acid fracturing experiments is quite challenging, we developed a three-dimensional acid fracturing model to simulate the acid etching process with acid viscous fingering [2].

Mathematical Model for the Acid Etching Process

Our work focuses on the acid etching stage in Figure 1b. Our acid fracturing model considers fluid flow inside the fracture, acid and polymer transport, and change of fracture geometry due to mineral dissolution. We developed a numerical simulator based on the OpenFOAM library [11, 12] to solve the acid fracturing model and compute the rough acid fracture geometry that is induced by non-uniform acid etching. The simulator employs a sequential algorithm to solve the governing system (see Figure 6). The governing system for the acid etching process is as follows:

\[\textrm {Navier-Stokes  flow  inside  fracture}: \:\: \rho(\mathbf{u}\cdot \triangledown \mathbf{u}) =-\triangledown p + \mu \triangledown^2\mathbf{u}\tag2\] \[\textrm{Acid  transport  inside  fracture}: \:\: \frac{\partial c_1}{\partial t} + \triangledown \cdot (\mathbf{u}c_1)-\triangledown \cdot (\mathbf{D}_1\triangledown c_1)=0\tag3\] \[\textrm{Polymer  transport  inside  fracture}: \:\: \frac{\partial c_2}{\partial t}+ \triangledown \cdot (\mathbf{u}c_2)-\triangledown \cdot (\mathbf{D}_2\triangledown c_2)=0\tag4\] \[\textrm{Mineral  dissolution  at  fracture  surfaces}: \:\: c_s \frac{d\mathbf{r}}{dt}=\alpha R(c_1)\mathbf{n}\tag5\] \[\textrm{Polymer  viscosity  model}: \:\: \mu=\mu_w [1+A_1c_2+A_2c_2^2+A_3 c^3_2].\tag6\]

Here, \(\rho\) is the fluid density, \(\mathbf{u}\) is the fluid velocity, \(p\) is the fluid pressure, \(\mu\) is the fluid viscosity, \(c_1\) is the molar concentration of H⁺, \(\mathbf{D}_1\) is the H⁺ diffusion-dispersion coefficient, \(c_2\) is the mass fraction of the polymer, \(\mathbf{D}_2\) is the polymer diffusion-dispersion coefficient, \(c_s\) is the molar density of the carbonate mineral, \(\mathbf{r}\) is the point position vector on the fracture surfaces, \(\alpha\) is the stoichiometric conversion coefficient (defined as moles of carbonate mineral dissolved per mole of acid reacted), \(R(c_1)\) is the reaction rate at fracture surfaces, \(\mathbf{n}\) is the outward normal vector at fracture surfaces, \(\mu_w\) is the water viscosity, and \(A_1\), \(A_2\), and \(A_3\) are model parameters that one obtains by fitting the experimental data.

Effect of Viscous Fingering on Acid Etching

Animation 1 shows two simulation cases that investigate viscous fingering’s effect on the acid etching process. The pad fluid is water in Case 1 and polymer in Case 2. Acid is injected through one perforation zone along the inlet boundary. Because of the viscous fingering mechanism, the acid-etched channel of Case 2 is narrower and longer than that of Case 1; a longer acid fracture leads to a larger overall fracture conductivity. During the early period of acid etching, acid viscous fingering occurs at the inlet. A preferential flow path for acid exists along the fracture’s center line, meaning that acid distribution is concentrated in the middle of the fracture due to the non-uniform velocity field of acid flow. Since acid concentration dictates the acid etching process, the acid-etched channel in Case 2 is narrower than in Case 1.

Effect of Perforations on Acid Etching

In Case 1 of Animation 2, we increase the number of perforations to six to investigate perforations’ effect on acid etching. Six acid-etched channels form in the fracture and the height of each one is much smaller than the channel in Case 1 of Animation 1. Proper perforation spacing constrains the height of the acid-etched channels; narrow channels are more likely to sustain fracture conductivity under closure stress than wide channels.

Acid Etching with Alternating Injection of Pad and Acid Fluids

As shown in Case 1 of Animation 2, the fingering shapes with a 1.1-millimeter-fracture-width contour coalesce near the inlet. The coalescence of acid-etched channels indicates the presence of an acid washout, which may lead to the loss of acid fracture conductivity. We apply multi-stage alternating injection of pad and acid fluids in Case 2 of Animation 2, which significantly reduces the coalescence of acid-etched channels [4]. The polymer that we inject between two stages of acid injection hinders the transverse transport of acid, thereby preventing the coalescence of acid-etched channels.

Acid Etching in a Horizontal Well

Figure 7 illustrates the multi-stage acid fracturing technique in a horizontal well, which is often applied in tight carbonate reservoirs. Engineers implement multiple stages of acid fracturing along the horizontal well. In each stage, they first inject pad fluid to create a transverse fracture and then inject acid to etch fracture surfaces. Acid flow is radial inside the transverse fracture.

Animation 3 shows two simulation cases that investigate viscous fingering’s effect on the acid etching process in a transverse fracture along a horizontal well [5]. We assume that the transverse fracture is symmetric with respect to its diameter, allowing us to reduce computational costs by modeling only one half of the fracture. The pad fluid is water in Case 1 and polymer in Case 2. We inject the acid through three perforation zones along the wellbore and find that three acid-etched channels develop radially in Case 2, while acid etching is quite uniform in Case 1.

Acid Etching in a Heterogeneous Reservoir

Figure 8 illustrates a synthetic heterogeneous carbonate reservoir model that consists of calcite (CaCO₃) and dolomite (CaMg(CO₃)₂). As Animation 4 demonstrates, we set up two simulation cases to investigate viscous fingering’s effect on the acid etching process in this heterogeneous reservoir [3]. Again, the pad fluid is water in Case 1 and polymer in Case 2. In Case 1, the acid etching pattern closely follows the spatial distribution of minerals. Since the reaction rate between HCl and calcite is much faster than the reaction rate between HCl and dolomite, most acid-etched channels form and develop in the calcite region. The channels in Case 1 may be too wide to sustain the acid fracture conductivity under a high closure stress condition. Because Case 2 introduces viscous fingering, acid etching is more non-uniform than in Case 1.

We assume that if the fracture width of one cell is less than 1.05 millimeters, this cell will serve as a contact point when the two fracture surfaces close on each other. Figure 9 compares the contact areas for Case 1 and Case 2. A larger contact area on the fracture surface means that the acid-etched channels are more likely to remain open. The contact area in Case 1 is mostly comprised of the region of dolomite mineral, while some extra contact area exists in Case 2 between fingering regions to help reduce the risk of acid-etched channel collapse.

Acknowledgments: This work was supported by the donors of the American Chemical Society Petroleum Research Fund through grant 59356-ND9.

References
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	Rencheng Dong is a Ph.D. student in the Department of Petroleum and Geosystems Engineering and the Oden Institute for Computational Engineering and Sciences at the University of Texas at Austin.
	Mary F. Wheeler is a professor and the Ernest and Virginia Cockrell Chair in Engineering at the University of Texas at Austin. She is also director of the Center for Subsurface Modeling at the Oden Institute for Computational Engineering and Sciences.
	Hang Su is a Ph.D. student in the State Key Laboratory of Petroleum Resources and Prospecting at the China University of Petroleum (Beijing).
	Kang Ma is a research engineer at China National Offshore Oil Corporation Research Institute.