Journal of Chemistry

Volume 2015, Article ID 596597, 8 pages

http://dx.doi.org/10.1155/2015/596597

## A Mathematical Model for the Analysis of the Pressure Transient Response of Fluid Flow in Fractal Reservoir

^{1}State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China^{2}Institute of Applied Mathematics, Xihua University, Chengdu 610039, China

Received 12 August 2014; Accepted 14 October 2014

Academic Editor: Jianchao Cai

Copyright © 2015 Jin-Zhou Zhao 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 study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractal reservoir which can avoid the complicated mathematical deduction. The models are presented in three kinds of outer boundary conditions (infinite, constant pressure, and closed). The influence of wellbore storage effect, skin factor, and variable flow rate production is also involved in the inner boundary conditions. The analytical solutions are constructed in the Laplace space and presented in a pattern with one continued fraction—the similar structure of solution. The pattern can bring convenience to well test analysis programming. The mathematical beauty of fractal is that the infinite complexity is formed with relatively simple equations. So the relation of reservoir parameters (wellbore storage effect, the skin factor, fractal dimension, and conductivity index), the formation pressure, and the wellbore pressure can be learnt easily. Type curves of the wellbore pressure and pressure derivative are plotted and analyzed in real domain using the Stehfest numerical invention algorithm. The SCMS and type curves can interpret intuitively transient pressure response of fractal spherical flow reservoir. The results obtained in this study have both theoretical and practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS.

#### 1. Introduction

The mechanics of oil and gas seepage is a discipline which researches the law and state of fluid flow in porous media. The practical development of oil and gas reservoirs shows that reservoir distribution and its space structure are awfully complicated. In 1982, Mandelbrot and Blumen first proposed the fractal geometry theory which used self-similar to characterize the complexity of things [1]. By combining the theories of fractal and seepage mechanics, it is able to describe fluid flow paths in porous media effectively. Acuna et al. explained the fractal characters of natural porous media with extremely complex pore structure [2]. Because flow paths of fluid in porous media can be seen as tortuous capillaries, Cai and Yu applied a fractal dimension to describe the tangle of capillary pathways and derived calculation formula of actual length of tortuous flow paths [3]. And spontaneous imbibition of wetting liquid in fractal porous media including gravity was studied [4]. The researches were regarded as a crucially important driving mechanism for enhancing oil recovery in naturally fractured reservoir. In the early 1990s, Chang and Yortsos applied the fractal theory in reservoir models and then set up new mathematical models [5, 6]. Tian and Tong established radial fluid flow models of fractal reservoirs [7]. The models have shown that the order of the fractional dimension has influence on the whole pressure behavior. Particularly, the effect on pressure behavior is larger in the early time stage. Based on fractal geometry and the semiempirical Kozeny-Carman equation which is the most famous permeability-porosity relation in the field of flow in porous media, Xu and Yu [8] developed a new form of permeability and Kozeny-Carman constant. After that, they showed that fractal dimension had significant effect on permeability, which can enhance the effective permeability [9, 10]. Santizo discussed the effect of fractal dimension and fractal conductivity index on pressure derivative in finite-conductivity fractures [11]. The papers obtained some numerical or analytical solutions of the formation pressure and the wellbore pressure in different fractal reservoirs or plotted pressure-time curves to analyze the influence of reservoir parameters [12–17].

Different versions of cylindrical flow model have been studied on the assumption that wells were completely opened and fully penetrated the productive formation. However, the assumption may seem too restrictive for practical application. Reservoirs have vertical permeability and the length of injection or extraction region was small compared to thick formations; spherical flow model can provide a good approximation in practice [18]. Schroth and Istok [19] illustrated the applicability of the derived spherical flow solution and provided a comparison with its cylindrical flow counterpart. They showed the spherical flow solution increased with increasing anisotropy in hydraulic conductivities. Joseph and Koederitz [20] presented short-time interpretation methods for radial-spherical flow in homogeneous and isotropic reservoirs inclusive of wellbore storage, wellbore phase redistribution, and damage skin effects. Liu [21] studied the transient spherical flow behavior in porous media and derived its nonlinear partial differential equation and obtained its analytical, asymptotic, and approximate solutions by using the methods of Laplace transform.

In the above proposed studies, the solution processes of their mathematical models are very complicated. Based on some study in a boundary value problem of the second-order linear homogeneous differential equation, Li and Liao recently introduced a new idea, similar construction method. The similar structure theory and its application were developed maturely [22]. Chen and Li presented similar construction method of solution (SCMS) for the boundary value problem of Bessel equation and gave the method to analyze the structure characteristics of solution [23]. After that, Sheng et al. proposed SCMS for a fractal reservoir and listed its steps [17]. They verified that the SCMS is a straightforward method to solve mathematical models in reservoir engineering, but they just did a theoretical study. SCMS of the boundary value problems of Airy equation, Weber equation, Euler hypergeometric equation, and composite first Weber system were studied [24–28]. SCMS in fractal dual-porosity reservoir and dual-permeability reservoir were presented [29–31].

This paper presents a mathematical model for the analysis of the pressure transient response of fluid spherical flow in fractal reservoir, which considers wellbore storage and skin effect in inner boundary conditions. According to a mathematical method, called SCMS, the expressions of the dimensionless formation pressure and the dimensionless wellbore pressure are constructed in the Laplace space, which avoid complicated calculations. Type curves of the wellbore pressure and pressure derivative responses are plotted and analyzed in real domain using the Stehfest numerical invention algorithm [32]. The results obtained in this study have essential significance to understand the pressure characteristics and provide theoretical basis in such a reservoir. This paper embodies the relationship between engineering and mathematics.

#### 2. Materials and Methods

##### 2.1. Reservoir Characteristics

Nonpenetrating wells that occur in a thick formation can be treated as spherical systems, as shown in Figure 1. Using fractal theory to describe the pore nature and transport properties, we assume that dimensional fractal network is embedded in the dimensional Euclidean rock . The fluid only flows in the fractal network, and the flow will obey Darcy’s law from the reservoir into the wellbore. The reservoir has a uniform thickness of and original formation pressure is . The fractal reservoir is mined with a single small opening hole in the top part. The fluid flow rate is . The gravity and the capillary pressure are ignored.