This paper presents a new analytical method for solution of the pressure transient problems for general radial and linear models with heterogeneities. In these models, the storativity and the transmissivity coefficients are considered functions of the distance from the well. The solution method is based on a special transformation of variables that reduces the problem to a diffusivity equation with constant coefficients. This equation is then solved in the Laplace transform domain, and the solution is inverted into the real-time domain by the Stehfest algorithm. The solution method allows continuous variation as well as discontinuities of the rock and fluid properties. The new solution algorithm reproduces all known exact solutions for specific variation of properties including the multicomposite radial and linear model solutions.
Radial and linear homogeneous reservoir models provide the characteristic behaviors of radial and linear flow regimes one often identifies (at least during certain time periods) in well tests. It is, therefore, natural to extend these models to heterogeneous conditions. Of course, this assumption implies that the heterogeneities with only radial or linear symmetry are being considered. One may argue that this is too restrictive and that such a symmetric heterogeneity variation around a well is very unlikely.1 This argument may have some merit if one associates heterogeneities with rock properties only. In transient well-test analysis, however, the heterogeneous pressure behavior is very often identified with deviation from the single-phase homogeneous pressure behavior. From this point of view, the heterogeneities are associated with spatial variation of the transmissivity (kh/µ) and storativity (fcth) coefficients and not only rock properties.
Radially symmetric variations of transmissivity and storativity coefficients may develop as a result of the formation of second phase in the vicinity of a producing well (gas evolution in oil reservoirs or condensate formation in gas condensate reservoirs), or as a result of gas, water, water alternating gas (WAG), or steam injection into an oil reservoir. Composite models that consider piecewise constant variation of properties have often been used in these situations.2–4 In these models, the space domain is divided into a number of subdomains (rings) with constant properties. The problem effectively reduces to a problem with constant coefficients. A general radial model that allows more realistic variation of properties would be more appropriate in these cases.
In the general case of storativity and transmissivity variations, the pressure diffusivity equation is an equation with variable coefficients. There are no universal and effective analytical techniques for solving this problem exactly. For this reason, Oliver5 used a perturbation method to solve the drawdown problem. Actually, he studied a simpler case of small permeability variation near some characteristic value. Oliver found the first term in the infinite series expansion of the solution with respect to this small parameter. In his later work,6 Oliver extended the perturbation technique to the problem of both transmissivity and storativity variations. Oliver's solution provided remarkable insight into the nature of well testing and stimulated a series of publications, beginning with one by Oliver himself,7 addressing the inverse problem of reservoir property estimation from well-test data.8,9 We should note, however, that Oliver's solution is an approximate solution of the pressure transient problem that is valid only for small coefficient variation. Feitosa et al.8 found that when comparing with numerical simulation for a multicomposite model (5 zones) in which the permeability varied from 10 to 40 md, the Oliver's solution did not accurately match the simulation results. They even proposed an empirical correction for Oliver's solution to improve its accuracy.
In this paper, we present a new analytical method for an exact solution of the heterogeneous pressure transient problem with no limitation on the degree of property variation. The method works for both radial and linear problems. The solution algorithm is simple and efficient. The problem is solved in the Laplace transform domain. We then rely on the Stehfest algorithm to invert the Laplace space solution to real time. We present the method for the pressure drawdown problem at constant production rate. The solution can easily be generalized for any rate history by the method of superposition.
We consider a forward problem of predicting single-phase flow of a slightly compressible fluid in an areally heterogeneous singlelayer reservoir with radial or linear symmetry. In addition to areal heterogeneities, we also allow areal variation of reservoir thickness h. Averaging the 3D diffusivity equation in the vertical direction, taking into account no-flow conditions at the top and at the bottom of the reservoir, and assuming a slow variation of h, one can show that the flow problem in this case is reduced to a 2D diffusivity equation. The requirement of radial (linear) symmetry implies that the reservoir boundary is radially (linearly) symmetric, and that the storativity and transmissivity coefficients are functions of the distance from the well only. In this case, the fluid flow governing equation is further simplified, and it becomes a 1D diffusivity equation with variable coefficients. In the forward fluid-flow problem, the reservoir boundary and the coefficients fcth and kh/µ are considered known.
The methods for solving the radial and the linear problems are slightly different. We first present the solution of the radial problem.
The drawdown problem in the radially symmetric case is described by the following equation:
Here, the dimensionless storativity and transmissivity coefficients fdctdhd and kdhd/µd are defined based on the values of rock and fluid properties near the well at rd = 1. We also assume that the fluid viscosity µd and the total compressibility ctd vary areally.