Geofluids

Volume 2019, Article ID 2046061, 10 pages

https://doi.org/10.1155/2019/2046061

## Pressure Analysis for Volume Fracturing Vertical Well considering Low-Velocity Non-Darcy Flow and Stress Sensitivity

^{1}College of Petroleum Engineering, China University of Petroleum (East China), No. 66 Changjiang West Road, Qingdao 266580, Shangdong, China^{2}School of Mining and Petroleum, Department of Civil and Environmental Engineering, University of Alberta, Edmonton, AB, Canada T6G 2R3^{3}Geological Survey of Ningxia, Yinchuan, 750021 Ningxia, China^{4}Hebei Scoilmic Petroleum Technology Co., Ltd., Chanzhou, 061001 Hebei, China

Correspondence should be addressed to Chuanzhi Cui; moc.621@8002zcc

Received 2 April 2019; Revised 23 May 2019; Accepted 21 October 2019; Published 20 November 2019

Academic Editor: Nicoló Colombani

Copyright © 2019 Zhongwei Wu 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

In general, there is stress sensitivity damage in tight reservoirs and fractures. Furthermore, the flow in tight reservoirs is the low-velocity non-Darcy flow. Currently, few researches of pressure analysis for volume fracturing vertical well are conducted simultaneously considering the low-velocity non-Darcy flow and stress sensitivity. In the paper, a novel flow model of a volume fractured vertical well is proposed and solved numerically. Firstly, the threshold pressure gradient, permeability modulus, and experimental data are, respectively, utilized to characterize the low-velocity non-Darcy flow, matrix stress sensitivity, and fracture stress sensitivity. Then, a two-region composite reservoir is established to simulate the vertical well with volume fracturing. After that, the logarithm meshing method is used to discrete the composite reservoir, and the flow model is solved by the method of finite difference and IMPES. Finally, the model verification is conducted, and the effects of the low-velocity non-Darcy flow and stress sensitivity on the pressure and pressure derivative are analyzed. The six flow regimes are identified by the dimensionless pressure and pressure derivative curve. They are, respectively, the fracture linear flow regime, early transition flow regime, radial flow regime, crossflow regime, advanced transition flow regime, and boundary controlling flow regime. The stress sensitivity and threshold pressure gradient have a great effect on the dimensionless pressure and pressure derivative. With the increase of reservoir stress sensitivity, the pressure and pressure derivative are upward at the advanced transition flow and boundary controlling regimes. However, the pressure and pressure derivative are downward at the advanced transition flow and boundary controlling regimes when the fracture sensitivity increases. An increase in the threshold pressure gradient results in a high dimensionless pressure and pressure derivative. This work reveals the effects of low-velocity non-Darcy flow and stress sensitivity on pressure and provides a more accurate reference for reservoir engineers in pressure analysis when developing a tight reservoir by using the volume fracturing vertical well.

#### 1. Introduction

Volume fracturing breaks up effective permeable reservoirs to form a fracture network, maximizes the contact area between fracture sides and reservoir matrix, minimizes the fluid seepage distance of oil and gas from the matrix in any directions to fractures, and greatly improves the entire reservoir permeability [1]. Hydraulic fracture technology is an effective method to exploit unconventional reservoirs and is reported in many works [2–5], especially for volume fracturing [6–8]. Many microfractures are developed in the tight reservoir after volume fracturing [9, 10]. During depletion, reservoir pressure changes could lead to stress variations which further alter the fracture apertures, pore, and throats, finally changing microfracture permeability and reservoir permeability [11]. In other words, the fracture and reservoir all are stress sensitivity medium [12–14]. In addition, due to the tiny porous and ultralow permeability of tight reservoirs, the flow in tight reservoirs obeys the law of low-velocity non-Darcy flow instead of Darcy flow [15], which is the fact that the flow velocity in a low pressure gradient regime is lower than what is estimated from Darcy’s law [16].

Currently, there are many researches of the volume fracturing. Wang et al. [17] and Lu et al. [18] proposed a semianalytical model to analyze the productivity and pressure of a volume fractured vertical well in tight reservoirs. In their research, the reservoir consists of two regions, one of which is the main fracture. Another is a circular stimulated reservoir volume (SRV), which is idealized to a Warren-Root model [19]. The unstimulated reservoir volume (un-SRV) is not contributed to the flow. That is different from the actual situation, and the stress sensitivity and low-velocity non-Darcy flow are ignored. Zhu et al. [20] gave an analytical pressure solution for the volume fractured vertical well with a rectangular SRV. In their research, the reservoir is composed of multiple linear flow regions, which include SRV and un-SRV. Unfortunately, they do not consider the effect of stress sensitivity and low-velocity non-Darcy flow. Su et al. [21] proposed a fractal analytical model of volume fractured vertical wells. In their research, the whole reservoir is divided into two regions, one of which is an SRV and another is an un-SRV. The flow obeys the Darcy, and the permeability is not a function of effective pressure, which deviates the actual situation in the tight reservoir. Zhang et al. [22] proposed a semianalytical model for vertically fractured wells with stimulated reservoir volumes (SRV). The fractal porosity and permeability are employed to describe the heterogeneous distribution of porous media in the SRV. But the stress sensitivity and low-velocity non-Darcy flow are ignored. The researches mentioned above are all about volume fracturing, but they do not take the effect of the stress sensitivity and low-velocity non-Darcy flow into consideration.

Guo et al. [23] proposed a transient mathematical model for horizontal wells with consideration of the threshold pressure and nonlinear flow. In their research, the nonlinear flow is high-velocity nonlinear flow, which is different from that caused by low-velocity mentioned in this paper. And their work is just applied to the multistage fractured horizontal well. Wang et al. [24] conducted research of the transient pressure of the volume fractured vertical well. In his research, the two-region composite model is built. The inner region represents a circle SRV and is idealized as a dual-porosity medium, while the outer zone represents an un-SRV and is idealized as a single porosity reservoir. However, the low-velocity non-Darcy flow and the stress sensitivity of fracture medium are ignored. Ji et al. [25] present a new analytical model for MFHW in tight oil reservoirs where the reservoir is subdivided into five continuous linear flow regions. The effects of non-Darcy flow and stress sensitivity are both considered. However, his research is just applied to the multistage fractured horizontal well. By considering the effect of the stress sensitivity and low-velocity non-Darcy flow, Wu et al. [26] proposed a pressure analysis model for a multistage fractured horizontal well in the tight reservoir. In his research, the tight reservoir is divided into 4 regions, which are region I to region IV. The medium of each region is stress sensitivity, and the flow in the unstimulated reservoir region obeys the low-velocity non-Darcy law. But his research is just for the multistage fractured horizontal well instead of the volume fracturing vertical well. Wu et al. [27] conduct another research of transient pressure. In this work, the fracture is discrete into many subsections. The superposition principle is utilized to build the semianalytical pressure analysis model of a multistage fracturing horizontal well considering the stress sensitivity of fracture medium. However, the reservoir is an ordinary medium instead of stress sensitivity medium in their research, and the low-velocity non-Darcy flow is ignored.

As stated above, few works simultaneously considering the effect of the low-velocity non-Darcy flow and stress sensitivity are conducted for the volume fracturing well. This paper fills this gap. To this end, the researches of the flow model of the volume fracturing well and its verification are conducted in Section 2. After that, the result and discussion are presented in Section 3. In Section 4, conclusions are presented.

#### 2. The Flow Model and Verification

##### 2.1. Physical Model of Volume Fracturing Vertical Well

According to microseismic monitoring mapping [24] and the result of indoor core fracturing experiment [28, 29], we can know that the reservoir after volume fracturing can be divided into two regions, one is the SRV and another is the un-SRV. The SRV is circular, and many fractures are developed in the SRV (Figure 1).