Modeling of Acid Fracturing

Abstract

Summary The paper presents a theoretical framework for modeling acid fracturing stimulations. Starting from the fundamental equations of fluid mechanics, fracture mechanics, convection, and diffusion, the paper outlines the steps necessary to derive simplified equations for an acid fracturing model. Unlike some existing models, the coupled problem of fracture geometry, acid transport, and diffusion is solved simultaneously in this paper. Although an infinite reaction rate is assumed in the solution of the problem, an empirical correlation is used to account partially for finite reaction rates. Errors in the governing equations of some of the existing models are identified. To assess the accuracy of the approximations used in the present model, exact solutions are used for comparison. Predictions from the present model are compared with a model in the literature, and the results are found to be in reasonable agreement. As in all existing acid fracturing models, not all the phenomena of the acid fracturing process have been incorporated into the present model. Nevertheless, the present model is improved over existing models because it is derived from fundamental equations and thus forms a basis from which further improvements can be made. Introduction Acid fracturing is a common well stimulation technique in the petroleum industry for limestone and dolomite formations. In an acid fracturing treatment, an inert fluid (known as the pad) is injected into a well under high pressure, creating a fracture in the formation. As the fracture length is increased with continued fluid injection, acid is injected into the formation, reacting with the formation on the fracture surface, The acid is transported along the fracture by convection during fracturing. At the same time, the acid is transferred to the reactive surface by diffusion and by fluid leakoff into the formation. Once the acid reaches the fracture face, it reacts with the formation. Because the acid fracturing process is complicated, simplifying assumptions have to be made to make the problem of modeling the process tractable. On the other hand, several important features have to be retained to model the physics of the process properly, including fracture geometry (fracture length, width, and height), fluid leakoff rate, convection along the fracture, mass transfer of the acid to the rock surface, and the acid reaction rate on the surface. Of course, these processes occur simultaneously during the acid fracturing treatment, so they are not independent of one another. In limestone formations, the acidizing process is limited by the rate of acid transport, not by the reaction rate. As one of the simplifying assumptions in this paper, acid reaction rates are ignored at the well and during the stimulation. This paper describes a model of acid fracturing based on a two-fluid generalization of the Perkins and Kern model and a one-dimensional (1D) approximation of the general two-dimensional (2D) diffusion-convection problem. Several authors previously presented acid fracturing models. The model described in this paper, although in many ways similar to those described in previous publications, is derived directly from the 2D model, and the mass-transfer rate comes directly from the analysis of the 2D diffusion-convection problem. The model described here consists of two parts: a fracturing model and an acid transport model. We first write the governing equations for a Perkins-Kern fracture model derived by Nordgren. We then describe the Perkins-Kern approximation and generalize it to two fluids. The two-fluid generalization of the Perkins-Kern approximation is the basis of the fracture model proposed in this paper. For the acid transport model, the 2D convection-diffusion equation is used as a starting point for the derivation of the ID approximation averaged over the fracture width. The mass-transfer rate obtained from such an approximation is compared with the full 2D mass-transfer rate obtained from solving the 2D equation. Governing Equations Fracture Model. In Nordgren's fracture model, the fracture height, h, is assumed to be constant. The rate of fluid leakoff per unit length into the formation at any point in the fracture can be approximated by ................................ (1) where t=time and CL=leakoff coefficient that is usually measured in a static filtration test or a minifracture test. The factor 2 in Eq. 1 accounts for fluid leakoff rates from both fracture surfaces. In general, the coefficient is a function of the reservoir properties, fracturing-fluid properties, and filter-cake buildup. Note that = is the time it takes for the fracture to reach Point x. The governing equations for Nordgren's fracture model consist of the continuity equation and the fracture-width/pressure-elasticity relationship: ...................... (2) ...................... (3) where h = fracture height, q = fluid flux rate, qw = fluid-loss rate at the fracture surface, bmax = maximum opening at the center of the fracture cross section, E = Young's modulus, = Poisson's ratio, and [ ] = net pressure acting on the fracture surface. In addition, many fracturing fluids approximately obey a power-law relationship between the shear stress, s, and the shear strain rate, ......................... (4) a form commonly assumed for non-Newtonian fluids. In Eq. 4, u is the velocity down the fracture, n and K are fluid constants, and y is in the direction normal to the fracture wall. The fracture fluid is Newtonian if n = 1. Guillot and Dunand showed that fracture fluids can exhibit Newtonian behavior (n = 1) at low shear rates and power-law behavior (n less than 1) at high shear rates. They found that an Ellis model produced a reasonable fit of their experimental data for a wide range of shear rates (0.01 to 2,000 seconds -1). In addition, for the high shear rates normally encountered in fracturing applications, they found that the power-law model was a suitable approximation for calculating fracture shapes. The power-law model in Eq. 4 is used in the next section to calculate fracture shapes. Perkins and Kern Approximation. As Nordgren observed, an approximate solution of Eqs. 1 through 4 with fluid loss can be obtained from the zero leakoff solution. That is, we first obtain the zero leakoff solution by setting the time derivative and the fluid-loss rate, qw, in Eq. 2 to zero and by integrating to obtain an expression for the fracture width, bmax: ................................... (5) where i = injection rate, L = fracture length, and the fracture width is required to be zero at the fracture tip. We then account for the fluid loss by modifying the fracture volume. In deriving Eq. 5, we made use of the velocity profile and its relationship to i to give an expression for the pressure gradient down the fracture; the volumetric flow rate is equal to the velocity u integrated over the cross section between the fracture surfaces. SPEPE P. 94^