Mathematical Problems in Engineering

Volume 2018, Article ID 2582797, 10 pages

https://doi.org/10.1155/2018/2582797

## GPU-Based Computation of Formation Pressure for Multistage Hydraulically Fractured Horizontal Wells in Tight Oil and Gas Reservoirs

^{1}School of Engineering Science, University of Science and Technology of China, Hefei 230026, China^{2}Department of Basic Teaching and Experiment, Hefei University, Hefei 230601, China^{3}Engineering Technology Research Institute, Southwest Oil and Gas Field Branch, Deyang 610051, China

Correspondence should be addressed to Detang Lu; nc.ude.ctsu@ultd

Received 30 October 2017; Accepted 30 January 2018; Published 24 April 2018

Academic Editor: Suzanne M. Shontz

Copyright © 2018 Rongwang Yin 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

A mathematical model for multistage hydraulically fractured horizontal wells (MFHWs) in tight oil and gas reservoirs was derived by considering the variations in the permeability and porosity of tight oil and gas reservoirs that depend on formation pressure and mixed fluid properties and introducing the pseudo-pressure; analytical solutions were presented using the Newman superposition principle. The CPU-GPU asynchronous computing model was designed based on the CUDA platform, and the analytic solution was decomposed into infinite summation and integral forms for parallel computation. Implementation of this algorithm on an Intel i5 4590 CPU and NVIDIA GT 730 GPU demonstrates that computation speed increased by almost 80 times, which meets the requirement for real-time calculation of the formation pressure of MFHWs.

#### 1. Introduction

Horizontal well drilling and fracturing are key processes developed for unconventional oil and gas (such as shale gas and tight oil and gas).

The longer the horizontal wells, the more fractured the segments and the higher the corresponding monthly production and costs. Therefore, horizontal well fracturing optimization based on single well productivity prediction technology is of great significance.

The prediction of single well productivity is realized by solving the equation of porous flow and obtaining the bottom-hole pressure or output. The commonly used solution method is an analytic method based on the superposition principle. According to different permeabilities of the formation, single well production capacity can be divided into two types: capacity prediction based on the steady-state seepage equation and capacity prediction based on the transient seepage equation.

Because of the simplicity of the calculation of partial differential equations for steady-state capacity, it is widely used in the field.

Using the homogeneous horizontal well model, Babu et al. obtained the quasi-steady-state production formula for horizontal wells, which is applicable to horizontal wells in the presence of bottom water or gas-filled rectangular reservoirs [1, 2]. Yildiz et al. studied the characteristics of perforated wellhead wells in heterogeneous reservoirs and analyzed the variations in law of yield under the quasi-steady state of horizontal wells [3–5]. Al-Ahmadi et al. introduced a three-porosity model and established an analytical model based on the diffusion equation of the instantaneous source solution. The new technology took into account the effects of reservoir geometry, reservoir performance, fracture size, number of fractures, and spacing on production capacity [6, 7]. Xie and Li divided the horizontal well seepage area into four parts: the first and the second stage of the cylinder, the hemispherical sphere facing the heart flow, and the wellbore area flow. Based on the principle of equivalent seepage resistance, the expression of steady-state seepage capacity of fractured horizontal wells was proposed according to the formula of shale gas desorption diffusion rate [8].

The expression of steady-state capacity is simple and suitable for medium-high permeability formation prediction; however, for low-permeability oil and gas reservoirs, especially tight oil and gas reservoirs and shale gas reservoirs, the computation error is large.

With the development of computational techniques, partial differential equations, including the time factor, can be directly solved. Penmatcha and Aziz established a dynamic forecasting model of horizontal well production based on the Babu–Odeh method, which considered the influence of friction pressure drop, acceleration pressure drop, and radial flow in the horizontal well [9]. Penmatcha et al. established a model considering the basic physical properties of oil and gas reservoirs as well as some special characteristics of the shale reservoir matrix exchanging fluids with different reservoir medias and proposed a pressure transient solution for shale gas production [9, 10]. Guo et al. analyzed the effects of desorption, diffusion, viscous flow, pressure sensitivity, fracturing number of horizontal wells, and fracture spacing. Through the source function and Laplace transform combined with the numerical discretization method, the solution was obtained in the Laplace space. Furthermore, the plate curve was obtained through the Stehfest algorithm, which was applied to identify a variety of flow patterns [11–14]. Through the shale gas fracturing model, Sang et al. obtained shale gas single well production and bottom flow pressure calculation formulas, by which shale gas well production was calculated [14]. Zhao et al. studied the seepage flow behaviors of multistage fractured horizontal wells in arbitrary shaped shale gas reservoirs and got the pressure response and production performance for multifractured horizontal wells [15–18].

In the above study, both steady-state production and transient productivity were calculated for bottom pressure or flow, with no computation of formation pressure distribution. Since the pressure at any point during oil and gas formation can change over time, it is necessary to calculate the pressure distribution at each point of formation at different times. If the computation area is decomposed into 100,000 meshes at each time step, the computation is performed hundreds of thousands of times with existing method, and serial computation cannot meet the requirements of the oil and gas field. For this reason, this paper presents GPU-based model for computing formation pressure distribution.

#### 2. Model

##### 2.1. Physical Model

Suppose that there exist multistage hydraulically fractured horizontal wells (MFHWs) in a fully enclosed rectangular reservoir. Horizontal well positions, crack locations at all levels, crack shapes, and reservoir boundaries are shown in Figure 1. Figures 2 and 3 show the - plan and - plan, respectively. The basic assumptions are as follows:(1)There exists stratigraphic homogeneity, but the horizontal and vertical directions of permeability are different.(2)Fluid flow meets Darcy’s law and ignores the impact of gravity.(3)The effective thickness of the formation is the same, and the horizontal wells are parallel to the top and bottom. Assuming that the upper roof is completely closed (no gas roof and bottom water, as shown in Figure 3), the height of each crack in the horizontal well is different.(4)The stratum is a rectangular closed reservoir and the horizontal wells are parallel to both sides of the rectangle (Figure 2). Horizontal wells have cracks, their half-lengths are , and all crack spacings are changeable. The plane of the fracture is at an angle of to the horizontal wells of the wellbore. Cracks and horizontal wells are not symmetrical, and the length of crack is .(5)The fluid in the reservoir is tight oil and is mined at a constant flow rate.(6)Fluid and formation are microcompressible.