Abstract

This paper presents a mathematical model to describe the pressure behavior for infinite conductivity horizontal wells in naturally fractured formations. A double-porosity medium with pseudo-steady state interporosity flow is considered.

The general method of extending solutions for homogeneous reservoirs to double-porosity models is used. Wellbore storage and skin effects are included by applying the Laplace transformation to the superposition equation. This is also a general and simple method.

Methods of well testing analysis are examined, such as conventional and type-curve matching techniques.

Introduction

Recently many horizontal wells have been drilled all over the world. One of the objectives is to increase the well productivity, compared to vertical or even to vertically fractured wells. This increase in the productivity index occurs in a naturally fractured reservoir because many fissures are intercepted by the horizontal well, since its length can be much larger than that of a vertical well.

Several works on well test analysis for horizontal wells in homogeneous reservoirs have been published. published. Goode and Thambynayagam presented the solution for the pressure behavior in a horizontal well producing at constant rate without wellbore storage. producing at constant rate without wellbore storage. Analysis of drawdown and buildup tests was discussed.

Daviau et al included wellbore storage in the solution for constant rate at surface.

Clonts and Ramey considered only the uniform flux model, without wellbore storage.

Those three studies had calculated the pressure for the infinite conductivity case by using the uniform flux model at a fixed point along the wellbore. Rosa and Carvalho investigated the point along the wellbore where the uniform flux and the infinite conductivity models yield the same pressure response. They concluded that for practical purposes this point of equivalence is located at a dimensionless distance equal to 0.68 from the center of the horizontal well axis.

Ozkan et al also presented the solution for a uniform flux horizontal well, without wellbore storage.

This work presents the solution for the pressure response in a horizontal well that produces from a naturally fractured reservoir.

We consider the double-porosity model as defined by Barenblatt et al and Warren and Root. The interporosity flow is modeled with the pseudo-steady state flow equation. The well is assumed to have infinite conductivity but the pressure is calculated at a fixed point along the wellbore by using the uniform flux point along the wellbore by using the uniform flux solution.

The solution for the naturally fractured system is obtained through the application of a general methods of transforming the solution to the diffusivity equation in a homogeneous medium to the solution in a heterogeneous reservoir.

Wellbore storage and skin effects are included by applying the superposition theorem. This is also a general and simple method of including wellbore storage and skin in solutions that do not contain these effects.

Both methods previously described are applied to Laplace transformed equations. Numerical inversion is performed with the Stehfest algorithm. performed with the Stehfest algorithm. A hypothetical situation where the wellbore diameter is equal to the reservoir thickness is compared against the solution for a vertically fractured well, without wellbore storage and skin, in a double-porosity system. Good agreement is obtained.

Methods of well testing analysis are discussed, including conventional and type-curve matching techniques.

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