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Abstract

A number of authors have questioned the assumption of pseudosteady-state matrix-fracture flow in the Warren and Root model. The assumption of uniform block size has not been investigated extensively, however. The consensus is that fracture spacing, rather than being uniform, follows a distribution similar to an exponential distribution.

This paper examines the effects of an exponential distribution of fracture spacing on the distribution of block size, interporosity flow coefficient, and the resulting pressure distribution. The paper also discusses a number of alternative models for naturally fractured reservoirs that give rise to pressure behavior very similar to that predicted by the Warren and Root model.

Introduction

A number of authors^1,2,3 have questioned the assumption of pseudosteady-state (PSS) matrix-fracture flow in the Warren and Root⁴ dual porosity model. Others have tried to justify this assumption by pointing out that the presence of a skin between the matrix and fracture system can give rise to PSS behavior.

However, another assumption of the Warren and Root model, that of uniform block size, has not been examined extensively. Cinco, Samaniego, and Kucuk⁵ developed a model to predict the pressure transient response for a naturally fractured reservoir having a distribution of different sizes. They found that the shape of the distribution had a significant effect on the shape of the pressure response. In particular, if there is a broad distribution of block sizes, the classic V-shaped minimum in the derivative predicted by the Warren and Root model will not occur. However, their paper did not address the issue of the form of distribution to be expected for field data.

Other authors have conducted field studies of the distribution of fracture and joint spacing; the consensus is that spacing follows an exponential distribution⁶. This paper examines the consequences of an exponential distribution of fracture spacing on the distribution of block size and interporosity flow coefficient and on the resulting pressure transient response.

In spite of the difficulties with the assumptions in the Warren and Root model, many pressure transient tests in naturally fractured reservoirs exhibit pressure transient behavior consistent with that predicted by the model^7,8,9,10,11. Indeed, Gringarten⁹ reports that a majority of tests that he has personally observed appear to exhibit pseudosteady-state interporosity flow behavior.

How can these observations be reconciled? We believe that many pressure responses that have previously been attributed to the pseudosteady-state model may in fact have been caused by other mechanisms.

In the next section, we briefly review the literature on pressure transient responses in naturally fractured reservoirs. In the following section, we discuss the consequences of assuming an exponential distribution of joint spacing on the distribution of matrix block size, and on the resulting pressure response. In the third part of this paper, we discuss the pressure responses predicted by a number of alternative models.

Literature Review

One of the earliest attempts to model a naturally fractured reservoir was presented by Pollard¹². In Pollard's model, the pressure drop between the matrix and the fracture system, the pressure drop within the fracture system, and the pressure drop between the fracture system and the wellbore were each assumed to follow a pseudosteady-state relationship. Pollard's method provided estimates of each of these pressure differences from buildup test data, as well as the volume of the fracture system. Pollard used his method to select candidates for acidization with the goal of removing either completion skin or resistance to flow within the fracture system. Pirson and Pirson¹³ later extended Pollard's model to allow the pore volume associated with the matrix to be estimated, and hence the “radius of influence” or effective drainage radius of the well.