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Abstract

Conventional pressure transient models strictly apply to areally homogeneous reservoirs. Yet, core and log data indicate this assumption is often not justified. This paper describes a model for heterogeneous reservoirs and supporting field data from the Grayburg/San Andres formations in southeastern New Mexico. Conventional models fail to match these field data. Instead, a model for a heterogeneous reservoir with a fractal structure provides a quantitative analysis. The fractal reservoir model reduces to the conventional solution in the case of a homogeneous reservoir.

Introduction

Conventional pressure transient models do not match some recent pressure transient tests on the Grayburg and San Andrea Formations pressure transient tests on the Grayburg and San Andrea Formations in southeastern New Mexico. The problem is illustrated by comparing a recent pressure buildup test in Fig. 1 with conventional type curves in Fig. 2. Since all producing zones have been sand fractured, we would expect this set of curves for a well with a vertical fracture 1 to apply. The type curves show the pressure change along with the usual pressure derivative. The pressure curve initially has a slope of unity pressure derivative. The pressure curve initially has a slope of unity if wellbore storage effects dominate the early time data. Without wellbore storage, linear flow near the vertical fracture produces an early straight pressure line of slope one half. straight lines with other slopes do not appear even for curves that fall between these two extreme cases shown. The derivative curve eventually approaches a constant value.

The field data in Fig. 1 do not follow these trends. Since the production time prior to shut-in is much larger than the shut-in time, t, a log p versus log t plot should match the drawdown curves in Fig. 2. Instead, the pressure data between 1 and 150 hours fall on a straight line with slope pressure data between 1 and 150 hours fall on a straight line with slope 0.60. substantially different front the lines in Fig. 2. The pressure derivative curve is parallel to the pressure curve. These parallel lines for over 2 log cycles of shut-in time cannot be matched with homogeneous reservoir models. After 150 hours, the pressure derivative flattens, due to interference from neighboring wells.

The shortcomings of homogeneous reservoir models become clear by looking at core photographs and well log data. The Grayburg and San Andres formations are very heterogeneous as recent outcrop and geological studies demonstrate. The present paper presents a pressure transient model for a heterogeneous reservoir that matches pressure transient model for a heterogeneous reservoir that matches the field data in Fig. 1. In the model the reservoir contains both permeable and impermeable rock. The resulting permeable areal permeable and impermeable rock. The resulting permeable areal network is assumed to have a fractal structure.

The fractal reservoir model is an extension of earlier work by Chang and Yortsos. Chang and Yortsos proposed their model for naturally fractured reservoirs without any supporting field pressure data. With some modifications, their model can also represent a reservoir with impermeable rock embedded within the permeable pay zones. For this application, we rewrite the pressure transient equations for a fractal reservoir in a form that can use available core estimates of near wellbore porosity and permeability. Then, with the proper choice of dimensionless variables, the dimensionless equations are a special case of the set previously solved by Chang and Yortsos. The solution reduces to the conventional solution in the case of a homogeneous reservoir. The Chang and Yortsos formulation is based largely on the work of physicists who studied diffusion on fractal objects.

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