New Pressure-Transient Analysis Model for Dual-Porosity Reservoirs

Abstract

Summary This paper presents a new analytical model for interpreting pressure-transient tests for wells producing from dual-porosity reservoirs. This model includes unsteady-state matrix flow and incorporates the effects of wellbore storage, skin, and, for gas reservoirs, desorption. The model is applicable to bounded and infinite-acting reservoirs. Introduction Numerous analytical models have been presented recently to describe the transient pressure behavior of dual-porosity reservoirs. Dual-porosity or naturally fractured reservoirs are formations composed of two porous media of different porosities and permeabilities. One medium, the matrix blocks constituting the primary porosity, contains the majority of the fluid stored in the reservoir and possesses a low conductivity. The other medium, the fractured network constituting the secondary porosity, acts as the conductive medium for fluid and possesses a high flow capacity but low storativity. The storativities of the two media usually differ by several orders of magnitude: consequently, these reservoir types are referred to as dual-porosity reservoirs. These types of reservoirs are also characterized by a large permeability contrast between the two media. The basis for the study of dual-porosity media was presented by Barenblatt and Zheltov, who treated the fractured reservoir as a continuum with the fractured network superimposed on the primary porosity. Furthermore, they assumed that the flow of fluid within the matrix occurs under pseudosteady-state conditions. Warren and Root, using a formulation similar to Barenblatt and Zheitov, were the first to present analytical solutions to this model with the assumption of a pseudosteady-state matrix flow and developed a procedure for interpretation of buildup tests without wellbore storage and skin effects. Warren and Root showed that. on a semilog graph, their solution yielded two parallel straight lines with slopes related to formation flow capacity. The existence of two parallel semilog straight lines was disputed by Odeh, who used a model similar to that used by Warren and Root but who investigated different ranges of parameters. Kazemi was the first to consider the effects of unsteady-state matrix flow. He used a numerical model and assumed that the dual-porosity system can be simulated by a layered radial system. His results are similar to those of Warren and Root with the exception of a smooth unsteady-state transition zone between the two parallel semilog straight lines compared with the flat pressure profile characteristic of the pseudosteady-state transition. Later, de Swaa presented analytical unsteady-state solutions for a well producing at a constant rate in naturally fractured reservoirs. He introduced new diffusivity definitions for reservoir characterization. Kucuk and Sawyer presented a comprehensive model for gas flow in a naturally fractured reservoir of the Devonian shale. They investigated the behavior of dual-porosity gas reservoirs including the Klinkenberg effect in the tight shale matrix and the effect of gas desorption from pore surfaces of the shale matrix. Mavor and Cinco-Ley extended Warren and Root's solution to take into account the effects of wellbore storage and skin. Bourdet and Gringarten were the first to identify the existence of a semilog straight line during the transition period. They stated that this line had a slope one-half the classic parallel semilog straight lines and existed if the fracture storativity was not too large. Streltsova and Serra et al. analyzed the transition period in detail and confirmed the existence of the straight line of slope 0.5756, one-half the classic semilog straight line (1.151). Serra et al.'s solution includes unsteady-state matrix flow but not wellbore storage effects. Chen et al. presented an application of classic techniques to bounded dual-porosity systems and discussed flow regimes that may be exhibited by drawdown data. Their work, however, did not include wellbore storage, skin, or the effects of gas desorption. Cinco-Ley and Samaniego-V. presented a model based on the transient matrix flow model formulated by de Swaan-O. and demonstrated that the behavior of dual-porosity reservoirs can be correlated by use of three dimensionless parameters (i.e., w, AFD, and maD). They also established that, regardless of matrix geometry, the transition period might exhibit a straight line with a slope equal to one-half the slope of the classic parallel semilog straight lines. The purpose of this paper is to present an analytical solution for dual-porosity reservoirs, capable of modeling both pseudosteady- and unsteady-state matrix flow, for both finite and infinite-acting reservoirs. The solution includes the effect of gas desorption from the pore surfaces of shale matrix in dual-porosity gas reservoirs with sorbed gas. This is of particular application to the Devonian shale gas reservoirs and any dual-porosity "black shale" reservoir with matrix kerogen. Wellbore storage and skin effects are included in the solution. Furthermore, application of the model to analysis of field pressure-transient data with an automatic parameter-estimation technique is demonstrated. Theoretical Formulation Formulation of Flow Equations. The differential equations governing fluid flow in naturally fractured reservoirs are derived in a manner similar to de Swaan-O. formulation and are presented in Appendix A of the original version of this paper. The derivation is based on the following assumptions:unsteady-state radial flow in an isotropic dual-porosity reservoir at uniform thickness,negligible gravitational forces and small pressure gradients;uniform initial reservoir pressure throughout the reservoir;fluid production through the fracture network with the matrix blocks acting as a uniformly distributed source;one-dimensional (ID) unsteady-state flow in the matrix blocks that are of regular shape;gas-desorption source uniformly distributed within the matrix blocks;gas-desorption rate linear with pressure; andwell producing at a constant rate in a finite reservoir with wellbore storage and skin effects. The diffusivity equations describing flow in the fracture network for both oil and gas reservoirs in dimensionless form are ............(1) for oil reservoirs and ............(2) for gas reservoirs with desorption. The dimensionless pressure, PfD, is defined identically for oil and gas except that adjusted pressure instead of real pressure is used for gas to linearize the gasflow equations. SPEFE P. 384