Well Test Analysis of Naturally Fractured Reservoirs

Abstract

Summary. Methods are presented for matching observed pressure data during drawdown or buildup tests. The methods, illustrated with step-by-step examples, allow calculation of fracture transmissivity, storage capacity coefficient, skin, size of matrix blocks, fracture porosity, fracture storage. and radius of investigation. The effect of matrix block shapes in the transition period has been investigated by use of a stratum model, a model made of cubes with spaces between the cubes, and a model made of "matchsticks" separated by two orthogonal fracture planes. Consideration has been given to a gradient flow model as well as to unsteady-state and pseudosteady-state interporosity flow. Because many naturally fractured reservoirs are fault-related, the effect of single and intercepting sealing faults hav been investigated. Even after a match of observed pressure data is obtained, there is uncertainty about calculated parameters. Consequently, a synergetic approach integrating geologic models, logs, cores, outcrops, and well testing is the only sound procedure for evaluating naturally fractured reservoirs. Introduction Naturally fractured reservoirs have been studied intensively during the last 10 years in the geologic and engineering fields. Transient pressure analysis has received particular attention. particular attention. Barenblatt and Zheltov assumed pseudosteady-state interporosity flow in a model made of orthogonal, equally spaced fractures. Warren and Root used the same assumption and concluded that a conventional semilog plot of pressure vs. time should result in two parallel straight lines with a transition period in between. The separation of the two lines allowed calculation of the storativity ratio-i.e.. the fraction of the total storage within the fracture system. Kazemi used a numerical model of a finite reservoir with a horizontal fracture under the assumption of unsteady-state interporosity flow, substantiating Warren and Root's conclusion concerning the two parallel straight lines. The transition period, however, was different because of the unsteady-state rather than pseudosteady-state interporosity flow assumption. A breakthrough in the analysis of naturally fractured reservoirs was provided by de Swaan-O., who developed a diffusivity equation and analytic solutions to handle unsteady-state interporosity flow. This method, however, could not analyze the transition period between the two parallel straight lines. Najurieta developed approximate analytic solutions of de Swaan's radial diffusivity equation that could handle the transition period, as well as the first and late straight lines. More recently, Streltsova used a gradient flow model and indicated that the transition period yielded a straight line with a slope equal to one-half the slope of the early and late straight lines. Her examples showing the one-half slope yielded values of storativity ratio equal to 0.37, 0.26, and 0.48. Serra et al. reached the same conclusion using a stratum model for the cases in which the storativity ratio was smaller than 0.0099. Various type curves have been developed to analyze naturally fractured reservoirs with unsteady-state and pseudosteady-state interporosity flow. The curves, pseudosteady-state interporosity flow. The curves, including the pressure derivative, are valuable but must be used carefully to avoid potential errors resulting from multiple matches, especially when working by hand. This paper presents straightforward equations for well test analysis of naturally fractured reservoirs. The equations can be handled analytically and allow matching of measured pressure points. The equations are approximate but have the advantage of encompassing pseudosteady-state, unsteady-state, and gradient flow models, as well pseudosteady-state, unsteady-state, and gradient flow models, as well as matrix blocks of any shape. Comparison with results from a numerical simulator indicates that the equations presented here are valid for most cases. presented here are valid for most cases. Theory The models in this study are shown in Figs. 1 through 3. Fig. 1 shows a uniformly fractured, stratified reservoir with the distance between fractures equal to hm. Fig. 2 shows a uniformly fractured reservoir made of cubes with spaces in between. The size of each matrix block is hm. The spaces represent the fractures. Fig. 3 shows a uniformly fractured reservoir made of rectangular parallelepipeds separated by two orthogonal fractured parallelepipeds separated by two orthogonal fractured planes, or a matchstick model. In a microscopic sense. planes, or a matchstick model. In a microscopic sense. both matrix and fractures are homogeneous and isotropic. The matrix blocks have a uniform distribution throughout the reservoir. Geologically, Fig. 1 would represent a shallow reservoir (less than about 2,500 ft 176- m]) or a deep reservoir dominated by thrust faulting. Fig. 2 would be an idealization of a reservoir with regional or tectonic shear fractures cut by horizontal fractures, and Fig. 3 would represent a reservoir with regional or tectonic shear fractures that is not cut by horizontal fractures. Fluid movement toward the wellbore occurs only in the fractures. SPEFE P. 239