Transient Pressure Analysis for Partially Penetrating Wells in Naturally Fractured Reservoirs

Abstract

Abstract This paper presents an analytical solution that describes transient pressure behavior of partially-penetrating wells in naturally-fractured reservoirs. The solution is obtained by combining the pseudo-steady state model for naturally fractured reservoirs with the partially-penetrating well model in homogeneous reservoirs. Verification of the analytical solution is made by comparison of pressures derived analytically with those obtained from numerical simulation. The new type curves generated from the analytical solution indicate that partial penetration and double porosity effects cause a characteristic shape of the curves at early and transition time. The type curves may be used to analyze transient well test analysis for partially-penetrating wells in naturally-fractured reservoirs at early and transition time. Introduction The pressure behavior of partially-penetrating wells has been investigated extensively in the past. Many authors considered flow of the fluid in the reservoir toward the well as a radial flow with an additional pressure drop near the wellbore caused by partial penetration. This additional pressure drop is called pseudoskin due to partial penetration. Several authors developed analytical and empirical expressions for evaluating pseudoskin due to partial penetration and the time for which the assumption of radial flow becomes valid.1,2 Others developed methods for interpreting transient pressure welltest data based on spherical flow.3,4 Many authors developed solutions to the two-dimensional (2D) diffusivity equation, which included flow of fluid in the vertical direction. 5–10 The 2D governing equation for fluid flow into the well in cylindrical coordinates has the following form. Equation (1) where kz and kr are the reservoir permeabilities in the vertical and horizontal directions respectively, z is the distance in the vertical direction. The reservoir is considered to have constant thickness and properties. Eq. 1 has been solved for different boundary conditions by several methods, that included the use of the Green's function, Laplace transformation, and finite difference. Practically, all methods give identical results in terms of pressure response from the reservoir. Investigating the transient pressure behavior of partially-penetrating wells in homogeneous reservoirs revealed three periods of pressure response during build-up and drawdown well testing. These are called the early radial flow, transition flow, and late radial flow periods. Recent studies on the pressure behavior of partially-penetrating wells in homogeneous reservoirs are presented by Kucuk and Kirwan,9 and Yildiz and Bassiouni.10 Solutions are presented in the form of type curves of pressure versus time, which allow determination of permeability and skin factor. However, to-date no study has been reported in literature that deals with partially-penetrating wells in naturally-fractured reservoirs. Existing techniques for interpreting well test data for naturally-fractured reservoirs are applicable only for fully-penetrating wells.12–18 For partially-penetrating wells in naturally-fractured reservoir, the fracture-matrix transition period (which allows estimation of reservoir parameters) may occur before radial flow is established in the reservoirs. The characteristic type curve shape for fractured reservoirs is distorted by the effect of partial penetration. Consequently, existing well test interpretation techniques for naturally-fractured reservoirs based on a fully-penetrating well model can not be used to analyze transient pressure data of partially-penetrating wells during the transition period. This situation becomes more critical in cases where the reservoir is thick and the penetration ratio of the well is small.