Mathematical Problems in Engineering

Volume 2015, Article ID 726910, 11 pages

http://dx.doi.org/10.1155/2015/726910

## An Analytical Solution of Partially Penetrating Hydraulic Fractures in a Box-Shaped Reservoir

^{1}School of Energy Resources, China University of Geosciences, Beijing 100083, China^{2}Beijing Key Laboratory of Unconventional Natural Gas Geology Evaluation and Development Engineering, Beijing 100083, China

Received 19 August 2014; Revised 8 December 2014; Accepted 8 December 2014

Academic Editor: Shaofan Li

Copyright © 2015 He Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new method to give an analytical solution in Laplace domain directly that is used to describe pressure transient behavior of partially penetrating hydraulic fractures in a box-shaped reservoir with closed boundaries. The basic building block of the method is to solve diffusivity equation with the integration of Dirac function over the distance that is presented for the first time. Different from the traditional method of using the source solution and Green’s function presented by Gringarten and Ramey, this paper uses Laplace transform and Fourier transform to solve the diffusivity equation and the analytical solution obtained is accurate and simple. The effects of parameters including fracture height, fracture length, the position of the fracture, and reservoir width on the pressure and pressure derivative are fully investigated. The advantage of the analytical solution is easy to incorporate storage coefficient and skin factor. It can also reduce the amount of computation and compute efficiently and quickly.

#### 1. Introduction

Hydraulic fracturing technology has been a common application in the oil and gas industry during the last two decades. More and more attentions were focused on the study of pressure transient behavior of hydraulically fractured wells. In most published literatures, hydraulic fractures were assumed to be fully penetrating the formation. Limited efforts have been made to investigate the effects of partially penetrating fracture height on the performance of wells. In practice, fully penetrating fractures may lead to an early or immediate water or gas breakthrough in a reservoir with bottom water or gas cap in contact, whereas partially penetrating fractures may be the only way to prevent the early breakthrough [1–3].

No matter the problem of wells with or without hydraulic fractures, most scholars considered the fully penetrating wells or fully penetrating hydraulic fractures. However the issue of partial penetration is always ignored. In the early time, some scholars presented some methods to study partially penetrating wells. Muskat, Nisle, Brons and Marting, and Papatzacos used the method of images [4], Streltsova-Adams [5] used Laplace and Hankel transformations, and Buhidma and Raghavan [6] used Green’s function to solve the problem to partial penetration well in a reservoir. Later Yeh and Reynolds [7] used a numerical simulator to present some type curves for partial penetration, multilayered reservoirs with transient crossflow. In the late time, Ozkan and Raghavan [8] proposed a solution for a limited-entry slanted well in an infinite reservoir with closed top and bottom boundaries using the Laplace transformation and Bui et al. [9] used the double-porosity formulation of Warren and Root for naturally fractured reservoir. Fuentes-Cruz and Camacho-Velazquez [10] obtained the pressure transient behavior for partially penetrating wells completed in naturally fractured-vuggy reservoir by combination of Laplace transformation and finite Fourier transformation.

To solve the unsteady-state flow problem of fractures in the reservoir, most solutions were presented based on the using of the source solution and Green’s function provided by Gringarten and Ramey [11] which can be used in combination with Newman’s product method to generate solutions for different reservoir flow problem. At first, the pressure behavior of the partially penetrating fractures was presented by Gringarten and Ramey Jr. [12] using Green’s function. But the physical model only considered the closed upper and lower boundaries. Raghavan et al. [13] presented an analytical model that researched the effect of the vertical fracture height on the pressure transient behavior of a partially penetrated uniform-flux fractured well by evaluating the uniform-flux solution at a point in the fracture which was assumed to yield the infinite-conductivity solution. This model was an extension of the case of fully penetrating vertical fracture previously found by Gringarten et al. Rodriguez et al. [14, 15] presented semianalytical solution of the pressure transient behavior in a homogeneous and isotropic reservoir with a well intersected by a partially penetrating single vertical fracture of finite or infinite conductivity. However they did not investigate the effect of vertical fracture position on the wellbore pressure.

These previous solutions were quite significant to the later analysis of the pressure behavior of the partially penetrating fractures. Valkó and Amini [16] presented a method of distributed volume sources (DVS) to investigate a horizontal well with multiple transverse fractures in a box-shaped reservoir. The diffusivity equation considered a source term to calculate the pressure distribution and compute the production rate from a fracture. But it was only an approximate approach. Alpheus and Tiab [3] presented the analysis of the solution to the effect of partial penetration of an infinite conductivity hydraulic fracture on the pressure behavior of horizontal well extending in naturally fractured reservoirs. They founded that the duration of early linear flow regime is a function of the hydraulic fractures height. Although the mathematical model was obtained in Laplace domain with elliptical flow model, the method was complex and unclear because of the model that was obtained indirectly. Al Rbeawi and Tiab [1, 2] presented an analytical model in real time domain for the pressure behavior of a horizontal well with multiple vertical and inclined partially penetrating hydraulic fractures in an infinite homogenous reservoir to explain the pressure transient tests and forecast productivity of the well by using the instantaneous source function in three principal directions. Moreover, Lin and Zhu [17] developed a slab source method to evaluate performance of horizontal wells with or without fractures with consideration of the three-dimensional fracture geometry. However, the solution was also derived in real time domain, making it difficult to incorporate storage coefficient and skin factor that are usually obtained from the Laplace domain solution.

This study attempts to give some new insights in understanding the partially penetrating hydraulic fractures in a box-shaped reservoir. This paper presents an analytical solution that describes pressure transient behavior of partially penetrating fractures in a box-shaped reservoir and is successfully applied to examine effects of fracture half height, fracture half length, and reservoir width on performance of a fracture in a reservoir with closed boundaries based on pressure and pressure derivative concepts. Moreover the effect of the vertical position of the fracture on the pressure and pressure derivative is fully investigated. More specifically, the diffusivity equation is presented for the first time and the analytical solution of pressure transient behavior in Laplace domain is derived by using Laplace transform and Fourier transform. Then the bottomhole pressure in the real time domain can be obtained by using the inverse Laplace algorithm as proposed by Stehfest [18] subsequently. The result is validated accurately by comparing with previous results in the literature. The advantages of Laplace domain solution are that it can make it easy to incorporate storage coefficient and skin factor, can reduce the amount of computation, and improve the computational efficiently because it is unnecessary to scatter time.

#### 2. Mathematical Model

Consider a partially penetrating hydraulic fracture in a closed homogenous box-shaped reservoir as shown in Figure 1. If we assume that all fluid withdrawal will be through the fracture, the fracture is partially penetrating the formation and the fracture can be simulated as plane source [2].