Pressure-Transient Response of an Infinite-Conductivity Vertical Fracture in a Reservoir with Double-Porosity Behavior

Abstract

Summary. This paper describes the behavior of a naturally fractured reservoir when a well is producing through a vertical fracture with uniform flux or infinite conductivity. The reservoir model is a double-porosity medium, and both pseudo-steady-state and transient-interporosity-flow models are studied. The derivative vs. time for each solution is shown, and an analysis method for matching field data to such a model is presented. presented. This study combines a reservoir and a well model, both of which have been extensively studied separately. Included are a quick review of the main characteristics of these models and the methods used to solve the problems. At the end of the study, a general method is suggested to extend problems. At the end of the study, a general method is suggested to extend models with constant boundary conditions-e.g., homogeneous and double-porosity cases. Introduction Most transient-flow solutions concern the behavior of homogeneous formations. This paper is an extension of one of these solutions to fissured systems. The double-porosity medium. a representation of naturally fractured reservoirs, was first introduced by Barenblatt and Zheltov and Barenblatt et al. Warren and Roots introduced the ratio of storativities and the interporosity-flow parameter and showed that these two parameters are sufficient to describe the medium. Further studies were made to adapt double-porosity solutions to practical cases encountered in petroleum engineering-e.g., wellbore- storage and skin effects and interference tests. Other fundamental aspects of interporosity-flow models, including layering and inter-porosity flow, were developed. Gringarten et al. studied the behavior of a homogeneous reservoir totally penetrated by a thin vertical fracture. The problem was first solved by assuming a uniform flux from the reservoir into the fracture, then assuming that the fracture was of infinite conductivity (pressure uniform within the fracture). To our knowledge, no study has been done to extend the fractured-well solution to heterogeneous formations, despite the growing number of cases where heterogeneous models are needed to perform interpretations of well test data. It is unusual to fracture a well when the reservoir is known to have heterogeneities, because fracturing is difficult owing to the high permeability of the fissures. In recent times, some stimulation service companies have been fracturing heterogeneous formations by filling the fissures with a tight material. Furthermore, it is common to fracture layered formations, and the double-porosity model is a limit of the double-permeability model 11 when the permeability contrast between the layers increases. This paper discussespresentation of the separate models,basic tools used to solve the problem,solution for the uniform-flux fracture model,extension of the homogeneous solution to solve the infinite-conductivity model,suggestion of an analysis method, anda general method to extend homogeneous solutions to the double-porosity model. Presentation of the Separate Models Presentation of the Separate Models We consider a double-porosity medium as it was defined by Barenblatt and Zheltov, Barenblatt et al., and Warren and Root. A double-porosity reservoir, or naturally fractured reservoir, consists of two distinct porous media of separate porosities and permeabilities: the matrix medium (with a high storativity but a low permeabilities: the matrix medium (with a high storativity but a low permeability) and the fissures (high permeability and limited storativity). permeability) and the fissures (high permeability and limited storativity). At each point of the reservoir, we assign two pressures: the matrix pressure, pf, and the fissure pressure, pf. Before production, the pressure, pf, and the fissure pressure, pf. Before production, the double-porosity reservoir is in equilibrium (pm, =pf =pi). When we start flowing a well, the fluid will first be produced from and through the fissures, creating a difference of pressure between the two media. At this time, the reservoir reacts as a homogeneous medium corresponding to the characteristics of the fissures, as if the matrix did not exist. Because of this difference of pressure, the matrix will start to feed the fissures with fluid, allowing the production to continue. This period is called the transition period, during which the radius of investigation does not change. When this transition period is finished, the reservoir reacts as a homogeneous medium with the permeability of the fissures and the storativity of the total system. The transition flow yields an S shape on a log-log graph. which is similar to heterogeneous-system behavior. If we assume that the fluid is slightly compressible, the diffusivity equation related to the fissures isNeglecting the diffusion within the matrix, the equivalent equation for the matrix pressure isIn Eqs. 1 and 2, q* is the volumetric interporosity-flow rate from the matrix to the fissures per unit of bulk volume. The pseudo-steady-state model of interporosity flow giveswhere is the interporosity-flow shape factor, proportional to the inverse of the matrix block- size. SPEFE P. 510