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: (1) unsteady-state radial flow in an isotropic dual-porosity reservoir at uniform thickness, (2) negligible gravitational forces and small pressure gradients; (3) uniform initial reservoir pressure throughout the reservoir; (4) fluid production through the fracture network with the matrix blocks acting as a uniformly distributed source; (5) one-dimensional (ID) unsteady-state flow in the matrix blocks that are of regular shape; (6) gas-desorption source uniformly distributed within the matrix blocks; (7) gas-desorption rate linear with pressure; and (8) well 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

Summary Fractures are common features of many well‐known reservoirs. Naturally fractured reservoirs contain fractures in igneous, metamorphic, and sedimentary formations. Faults in many naturally fractured carbonate reservoirs often have high‐permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected‐network fracture system). New semianalytical solutions are used to understand the pressure behavior of naturally fractured reservoirs containing a network of discrete and/or connected (continuous) finite‐ and infinite‐conductivity fractures. We present an extensive literature review of the pressure‐transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual‐porosity model is a fictitious homogeneous porous medium because it does not contain any fractures. Second, by use of the new solutions, we show that for most naturally fractured reservoirs, the Warren and Root (1963) dual‐porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure‐transient well tests because it does not capture the behavior of these reservoirs. We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual‐porosity model. These examples exhibit very diverse pressure behaviors of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution and on their intensity and conductivity. We show these flow regimes with our new model for discretely and continuously fractured reservoirs. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual‐porosity reservoir‐model behavior. If we interpret them by use of this dual‐porosity model, then the estimated permeability, skin factor, interporosity flow coefficient (λ), and storativity ratio (ω) will not represent the actual reservoir parameters.

Abstract Fractures are common features of many well-known reservoirs. Naturally fractured reservoirs consist of fractures in igneous, metamorphic, and sedimentary rocks (matrix) and formations. Faults in many naturally fractured carbonate reservoirs often have high permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected-network fracture system). To understand the pressure behavior of these continuously and discretely fractured reservoirs, semianalytical solutions are presented. These solutions are used for transient well test interpretation of formations containing a network of discrete and/or connected (continuous) finite- and infinite-conductivity fractures. In this paper we present an extensive literature review of the pressure transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual-porosity model is a fictitious homogenous porous medium because it does not contain any fractures. Second, using the new solutions, we show that for most naturally fractured reservoirs the Warren and Root (1963) dual-porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure transient well tests because it does not capture the behavior of these reservoirs. We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual-porosity model. These examples exhibit the very rich pressure behavior of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution, and on their intensity and conductivity. We show these flow regimes using our new model for discretely and continuously fractured reservoirs. These derivatives will be a valuable diagnostic tool for well test interpretation. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual-porosity reservoir model behavior. If we interpret them this dual-porosity model, then the estimated permeability, skin factor, interporosity flow coefficient (λ), and storativity ratio (ω) will not represent the actual reservoir parameters.

Summary

This paper presents a new analytical solution for pressure-transient tests from layered reservoirs with or without crossflow. The analytical solution is for modeling layered-reservoir systems with unsteady-state or pseudosteady interlayer crossflow, cominingled or stratified flow, and even dual-porosity systems with pseudosteady matrix-to-fracture transfer in the presence of skin, wellbore storage, and phase segregation.

Introduction

The earliest rigorous study of pressure behavior of layered reservoirs was performed by Lefkovits et al. in 1961. This study addressed commingled flow in a stratified system with an arbitrary number of layers. Lefkovits et al's analytical solution served as the basis for much of the work that followed. Their work, however, concerned multilayered systems with no communication in the formation other than through the well. Their work was extended to include wellbore storage effects by Tariq and Ramey, who applied Laplace transformation and Stehfest's inversion algorithm techniques. Russell and Prats, Katz and Tek, and Pendergrass and Berry studied well response in a reservoir with interlayer crossflow and showed that the early-time response of a well draining a reservoir with interlayer crossflow is similar to the response of a well in a stratified no crossflow or commingled reservoir. These studies were concerned primarily with the long-term performance of the well. Prijambodo et al presented a finite-difference model for wells producing from layered reservoirs with crossflow. Their emphasis was on the influence of the skin region on the well response. Their results included commingled and crossflow cases, but wellbore-storage effects were not considered. Bourdet presented an analytical solution to describe pressure response of layered reservoirs with pseudosteady-state crossflow. He described flow regimes and presented the effects of layered-reservoir parameters on pressure and pressure-derivative solutions. He also presented an example application with typecurve analysis and noticed the uniqueness problem associated with this method. He included the effects of wellbore storage and skin. His model is applicable only to pseudosteady-state interlayer crossflow, which is a special case of unsteady-state crossflow. Ehlig-Economides and Joseph presented an analytical model that is a generalization of Bourdet's two-layer model with pseudosteady-state interlayer cross-flow to a multilayer system. They also presented a testing and interpretation methodology for obtaining layered-reservoir descriptions. As noted by Bourdet, however, further investigation is required for better understanding of the interlayer-flow behavior and the governing layered-reservoir parameters. This paper presents a new analytical solution that describes the pressure response of a well intercepting a layered reservoir with crossflow. A detailed study of unsteady-state interlayer crossflow behavior with pressure and pressure-derivative solutions is presented. We show that the layered-reservoir model with the pseudosteady-state interlayer-crossflow solution presented by earlier investigators is a special case of the unsteady-state interlayer-flow solution presented here. We also demonstrate that the analytical solution to dual-porosity systems with pseudosteady-state matrix-to-fracture transfer is also a special case of the layered-reservoir model with unsteady-state crossflow presented here. This new analytical model can handle both commingled and crossflow-type layered reservoirs by the use of a "degree-of-crossflow parameter." We also include the effects of skin, wellbore storage, and phase segregation in the new layered-reservoir model. No published analytical model has all these capabilities. We present the effects of layered-reservoir parameters, wellbore conditions, and the drawdown period before buildup on the solution. With an automatic history-matching approach, the analytical model was applied to buildup tests, which eliminates the uniqueness problem associated with typecurve analysis. Apart from the elimination of the uniqueness problem, the advantages of our history-matching approach lie in the possibility of estimating all layer parameters-k1, k2, phi 1, phi 2, omega, lambda, and b-and wellbore parameters-s, CD, and CaD-that are impossible to obtain from type-curve analysis.

Mathematical Formulations

Fig. 1 illustrates the reservoir model considered here. It shows the diagram of a two-layer reservoir system with crossflow from the lower layer into the upper layer. The upper layer is of thickness h1, permeability k1, and porosity phi 1, while the lower layer has thickness h2, permeability k2, and porosity phi 2. Parameter b represents the fraction of the lower layer in vertical communication with the upper layer and is a measure of the degree of crossflow. Each layer of the reservoir system is assumed to be homogeneous, isotropic, and filled with a single-phase fluid. The reservoir is assumed to be horizontal, cylindrical, enclosed at the top and bottom, and of infinite extent. The initial pressure is assumed to be the same in each layer and the production rate is constant. Gravity effects are assumed negligible.

Basic concepts, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids, are discussed first. Various flow geometries are treated, including radial, linear, and spherical flow. The pseudosteady-state equations provide the basis for a brief discussion of oil well productivity, and the unsteady-state equations provide the basis for a lengthy discussion of pressure-transient test analysis. For pressure-transient test analysis, semilog techniques, type curves, damage and stimulation, modifications for gases and multiphase flow, the diagnostic plot, bounded reservoirs, average pressure in the drainage area, hydraulically fractured wells, and naturally fractured reservoirs are included. The chapter also discusses transient and stabilized flow in horizontal wells and gas-well deliverability tests. It concludes with considerations of coning in vertical and horizontal wells. Many important applications of fluid flow in permeable media involve 1D, ...