Summary Pressure-rate deconvolution provides equivalent representation of variable-rate well-test data in the form of characteristic constant rate drawdown system response. Deconvolution allows one to develop additional insights into pressure transient behavior and extract more information from well-test data than is possible by using conventional analysis methods. In some cases, it is possible to interpret the same test data in terms of larger radius of investigation. There are a number of specific issues of which one has to be aware when using pressure-rate deconvolution. In this paper, we identify and discuss these issues and provide practical considerations and recommendations on how to produce correct deconvolution results. We also demonstrate reliable use of deconvolution on a number of real test examples. Introduction Evaluation and assessment of pressure transient behavior in well-test data normally begins with examination of test data on different analysis plots [e.g., a Bourdet (1983, 1989) derivative plot, a superposition (semilog) plot, or a Cartesian plot]. Each of these plots provides a different view of the pressure transient behavior hidden in the test data by well-rate variation during a test. Integration of these several views into one consistent picture allows one to recognize, understand, and explain the main features of the test transient pressure behavior. Recently, a new method of analyzing test data in the form of constant rate drawdown system response has emerged with development of robust pressure-rate deconvolution algorithm. (von Schroeter et al. 2001, 2004; Levitan 2005). Deconvolved drawdown system response is another way of presenting well-test data. Pressure--rate deconvolution removes the effects of rate variation from the pressure data measured during a well-test sequence and reveals underlying characteristic system behavior that is controlled by reservoir and well properties and is not masked by the specific rate history during the test. In contrast to a Bourdet derivative plot or to a superposition plot, which display the pressure behavior for a specific flow period of a test sequence, deconvolved drawdown response is a representation of transient pressure behavior for a group of flow periods included in deconvolution. As a result, deconvolved system response is defined on a longer time interval and reveals the features of transient behavior that otherwise would not be observed with conventional analysis approach. The deconvolution discussed in this paper is based on the algorithm first described by von Schroeter, Hollaender, and Gringarten (2001, 2004). An independent evaluation of the von Schroeter et al. algorithm by Levitan (2005) confirmed that with some enhancements and safeguards it can be used successfully for analysis of real well-test data. There are several enhancements that distinguish our form of the deconvolution algorithm. The original von Schroeter algorithm reconstructs only the logarithm of log-derivative of the pressure response to constant rate production. Initial reservoir pressure is supposed to be determined in the deconvolution process along with the deconvolved drawdown system response. However, inclusion of the initial pressure in the list of deconvolution parameters often causes the algorithm to fail. For this reason, the authors do not recommend determination of initial pressure in the deconvolution process (von Schroeter et al. 2004). It becomes an input parameter and has to be evaluated through other means. Our form of deconvolution algorithm reconstructs the pressure response to constant rate production along with its log-derivative. Depending on the test sequence, in some cases we can recover the initial reservoir pressure.
A previously published model describing pressure-buildup behavior of naturally flaetured reservoirs was combined }t'itha nonlinear, least-squares regression technique to analyze buildup data. The model adequately described the buildup response and was useful for obtaining effect ive formation permeability in the cases studied.
Summary 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. Introduction 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. Problem 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. Radial Model The drawdown problem in the radially symmetric case is described by the following equation:Equation 1 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.
This paper presents a new analytical method for an exact 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 special transformation of variables which 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 both continuous variation as well as discontinuities of the rock and fluid properties. The new solution algorithm reproduces exactly all known exact solutions for specific variation of properties including the multicomposite radial and linear model solutions. Introduction 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 cause, this 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. 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 /) and storativity (cth) coefficients and not only rock properties. Radially symmetric variations of transmissivity and storativity coefficients may develop as a result of second phase formation 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, WAG, or steam injection into an oil reservoir. Composite models which consider piecewise constant variation of properties have often been used in these situations. In these models, the space domain is divided into a number of sub domains (rings) with constant properties. The transient pressure problem effectively reduces to the problem with constant coefficients. A general radial model which allows more realistic variation of properties would be a more appropriate model in these cases. In the general case of storativity and transmissivity variations the pressure diffusivity equation is an equation with variable coefficients and there are no universal and effective analytical techniques for solving this problem exactly. For this reason, Olive 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 the small parameter. In his later work, 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 test experiment and stimulated a series of publications, beginning with Oliver himself, addressing the inverse problem of reservoir property estimation from well test data. We should note, however, that Oliver's solution is an approximate solution of the pressure transient problem which is valid only for small coefficient variation. Feitosa, Reynolds et al. found that when comparing with numerical simulation for a multicomposite model (3 zones) where the permeability varied from 10 md to 40 md Oliver's solution did not accurately match the simulation results. They even proposed some empirical correction of Oliver's solution to improve its accuracy. In this paper, we present a new analytical method for an exact solution of the 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. P. 225
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