3D Transport in Acid-Fracturing Treatments: Theoretical Development and Consequences for Hydrocarbon Production

Abstract

Summary While the effect of 2D proppant transport during hydraulic fracturing has been studied extensively and debated frequently, relatively little attention has been given to acid-fracturing treatments. In addition to fluid-density differences and gravity-driven segregation, spatial variation of the acid temperature and of the rock chemical properties results in more complicated physical phenomena for acid fracturing. Acid diffusion in the fracture has been estimated empirically with no considerations made to model correctly the acid flow across the fracture width. The etched-width profile resulting from 3D flow could differ significantly from that predicted by the 1D, piston-like displacement normally included in acid-fracture simulators. An important consequence of this variation in the etched pattern is a substantially different prediction of the fracture conductivity, and hence the post-stimulation hydrocarbon production. Acid flow in the fracture can affect the outcome of reservoir stimulation and must be considered when designing acid-fracturing treatments. The equations governing fluid flow are developed initially for a 2D pressure profile. These equations are coupled with the formulations for acid wall reaction, heat transfer, and diffusion within the framework of a P3D hydraulic fracturing simulator. The theoretical formulation for acid transport across the fracture width is presented then and coupled with the equations for 2D acid flow. The equation of acid diffusion is solved across the fracture width, hence solving the equations of fluid flow for three dimensions in the fracture. The resulting fracture geometry and conductivity are used subsequently with a reservoir simulator to illustrate the consequences of the 3D acid formulation on hydrocarbon production. Simulated and field examples are presented to illustrate the effects of 3D acid flow on the etched-width distribution and post-stimulation production. Introduction Hydraulic fracturing with acid is an alternative to propped fractures in acid-soluble formations such as dolomite and limestone. In such cases, fracture conductivity is obtained by etching the fracture faces instead of using proppant to prevent the fracture from closing. The acid treatment, however, is more complicated to design because of the difficulty in controlling both the fracture length and conductivity. The former is governed by the chemical reaction between the rock and the fracturing fluid, and the latter by etching patterns formed by the reacting acid in the formation. To design and evaluate such acid treatments, numerical simulators have been developed based on several physical mechanisms. The first mechanism is the acid reaction at the wall that creates the etched width and decreases the acid concentration at the wall. An acid concentration profile is created across the fracture width, with the acid diffusing from the center of the fracture to the fracture wall because of the concentration differences and leakoff velocity (second mechanism). Finally, because of fluid flow, the acid is transported along the fracture. The two limiting cases in modeling these mechanisms are the "diffusion-limited" case and the "reaction-limited" case. The diffusion-limited case occurs when the reaction is very fast, so that the rate of diffusion is limited by the rate at which acid can be transported to the surface and the acid concentration at the wall is apparently zero. The other case is obtained when the acid can be brought to the face as rapidly as it is consumed by reaction. This is called reaction-limited and implies that the wall concentration is approximately equal to the average acid concentration. Several simulators for treatment design and evaluation have been developed1–3 during the past decade based on acid-reaction models and 1D fluid flow. The basic assumptions of these models are the following:Fluid flow in the fracture is assumed to be plug flow. The fluid in the fracture has vertical fronts; i.e., the effects of the fracture width profile and leakoff variation along the fracture height are neglected.Acid diffusion from the center of the fracture to the fracture wall is calculated based on empirical formulations that assume a constant fracture width, infinite reaction rate, and no entrance effects. A new acid model is proposed to eliminate such limitations by solving the fluid-flow equation in three dimensions. This simulator has been developed based on a P3D hydraulic fracturing simulator that has been tested widely and validated.4,5 Two fundamentally different but intrinsically coupled principles are involved in developing a 3D acid fracturing simulator. The first is a 2D fluid transport along the fracture length and height. This formulation requires calculating the pressure gradients and velocities along the fracture length and height. The second concept involves a rigorous calculation for acid diffusion across the fracture width. This is achieved by superimposing a 3D finite difference mesh within the fracture width and tracking the acid concentrations at every cell across the fracture. Finally, the standard principles of acid reaction and fracture-height growth are coupled within this framework to develop the implicit 3D acid simulation. Various physical principles implemented in the development of the simulator that are central to acid fracturing are presented. Secondary considerations in hydraulic fracturing are briefly discussed in the Appendix. The equations describing the acid-reaction model are presented first, followed by the algorithm for 2D acid transport. The 3D formulation that solves for the acid concentration across the width is presented next. Results obtained by applying this simulator to different examples and field cases are given; simple benchmark problems are presented to validate these results against known solutions. Finally, field examples are shown to illustrate its application for real life situations.

Acid Fracture Treatment Design with Three-Dimensional Simulation

“…Other researchers have investigated how to quantify the acid fracturing process with an emphasis on resolving the acid concentration profile and etched width by simulation (Lo and Dean, 1989;Settari, 1993;Romero et al, 2001;Settari et al, 2001). For the transport of acid in the fracture, the velocity profile is typically assumed to follow a known analytical solution (e.g., flow between parallel plates) and only varies in the length direction of the fracture.…”

“…The Settari et al (2001) model domain is one half of the fracture width but does not include any modeling along the fracture height. Romero et al (2001), realizing the need for three-dimensional (3D) simulation, expanded the model to include changes along the fracture height. However, both the more advanced Settari et al (2001) and Romero et al (2001) models use analytical velocity profiles in the solution of the acid convection and diffusion, failing to couple fundamentally these processes.…”

“…Romero et al (2001), realizing the need for three-dimensional (3D) simulation, expanded the model to include changes along the fracture height. However, both the more advanced Settari et al (2001) and Romero et al (2001) models use analytical velocity profiles in the solution of the acid convection and diffusion, failing to couple fundamentally these processes. Settari et al (2001) and Romero et al (2001) include comparisons of their multidimensional models to earlier models of acid fracturing.…”

“…However, both the more advanced Settari et al (2001) and Romero et al (2001) models use analytical velocity profiles in the solution of the acid convection and diffusion, failing to couple fundamentally these processes. Settari et al (2001) and Romero et al (2001) include comparisons of their multidimensional models to earlier models of acid fracturing. Settari et al (2001) model laboratory experiments and one field case with one-and two-dimensional (gridding across the fracture width and length) simulations.…”

“…The one-dimensional (1D) model consistently under predicts the etched width across the fracture length for both the laboratory studies and the one field example. Romero et al (2001) also compare their 3D model to a previous 1D model developed within that research group, similar to Settari et al (2001), but Romero et al (2001) use realistic field cases instead of laboratory scale simulations. Interestingly, the 1D model over predicts the acid-etched width for the first presented case study and under predicts the acid-etched width for the second presented case study (as compared to the 3D model output).…”