## Numerical Analysis of Transient Pressure Behaviors with Shale Gas MFHWs Interference

## Abstract:

After the large-scale horizontal well pattern development in shale gas fields, the problem of fast pressure drop and gas well abandonment caused by well interference becomes more serious. It is urgent to understand the downhole transient pressure and flow characteristics of multi-stage fracturing horizontal well (MFHW) with interference. Therefore, the reservoir around the MFHW is divided into three regions: fracturing fracture, Stimulated reservoir volume (SRV), and unmodified matrix. Then, multi-region coupled flow model is established according to reservoir physical property and flow mechanism of each part.

#### Authors:

#### Dapeng Gao^{1,2,3}, Yuewu Liu^{2,3}, Daigang Wang^{4}, and Guofeng Han^{2}

^{1}State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China. ^{2}Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China. ^{3}School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China. ^{4}Beijing International Center for Gas Hydrate, Peking University, Beijing 100871, China

Received: 6 November 2018 / Accepted: 9 January 2019 / Published: 15 January 2019

The model is numerically solved using the perpendicular bisection (PEBI) grids and the finite volume method. The accuracy of the model is verified by analyzing the measured pressure recovery data of one practical shale gas well and fitting the monitoring data of the later production pressure. Finally, this model is used to analyze the effects of factors, such as hydraulic fractures’ connectivity, well distance, the number of neighboring wells and well pattern arrangement, on the transient pressure and seepage characteristics of the well. The study shows that the pressure recovery double logarithmic curves fall in later part when the well is disturbed by a neighboring production well.

The earlier and more severe the interference, the sooner the curve falls off and the larger the amplitude shows. If the well distance is closer, and if there are more neighboring wells and interconnected corresponding fracturing segments, the more severe interference appears among the wells. Moreover, the well interference may still exist even without interlinked fractures or SRV. Especially, severe interference will affect production when the hydraulic fractures are connected directly, and the interference is weaker when only SRV induced fracture network combined between wells, which is beneficial to production sometimes.

When severe well interference occurs, periodic well shut-in is needed to help restore the reservoir pressure and output capacity. In the meanwhile, the daily output should be controlled reasonably to prolong the stable production time. This research will help to understand the impact of well interference to gas production, and to optimize the well spacing and achieve satisfied performance.

## Introduction

The multi-stage fracturing horizontal well (MFHW) is a crucial technology in shale gas development, and the large-scale horizontal wells pattern haves achieved remarkable performance in many fields in North America and China [1,2,3,4,5]. However, some well groups have shown increasingly dangerous well interference after producing for several years, due to the well pattern infilling and enhancement of hydraulic fracturing.

For instance, The Jiaoshiba shale gas play is the most successful shale gas reservoir in China with some wells’ cumulative production over 0.1 billion cubic meters in the first year; unfortunately, parent wells crop up jumps in water production during hydraulic fracturing processing of child wells in the later production period. Besides, North American shale plays, such as Arkoma Basin, have also shown an obvious loss of gas production because of well interference [6,7]. Fracture pressure hits and production interference are two main factors influencing the shale gas permanent development and determining the inter-well connectivity [8].

By a combination of hydraulic fracturing process and production data, the existence of well interference from the adjacent wells when well interference happens and the influence it imposes on the target well can be detected. Maintaining a high and stable gas productivity faces many challenges in the future. It is urgent to study in depth the transient pressure behaviors of shale gas MFHWs with neighboring wells, considering the complex fracture network and the multi-flow mechanism, such as the desorption and diffusion of shale gas [9,10], to recognize the well interference in time and analyze its impact on production.

Mezghani et al. [11] combined gradual deformation and upscaling techniques for direct conditioning of fine-scale reservoir models to interference test data, as consequence, both fine- and coarse-scale models are updated by dynamic data during the history matching process. Yaich et al. [12] presented a methodology to quantify the impact of well interference and optimize well spacing in the Marcellus shale. Marongiu-Porcu et al. [13] proposed a numerical simulation method for shale gas reservoirs based on geophysical, completion and development data of Eagle Ford shale gas fields, and studied the propagation of hydraulic fractures and their respective network with natural fractures.

The magnitude and orientation of in-situ stress were evaluated. Pang et al. [14] studied the effect of well interference on shale gas well SRV interpretation. Compared with the previous literature, we analyzed the interference using the numerical simulation pressure double logarithmic curve method and evaluated the influence of various factors on the interference.

To study the porous media flow and transient pressure behaviors of shale gas MFHWs, many scholars have established kinds of multi-linear flow region coupled models. Bello et al. [15] used the layered double porosity model and the Warren-Root dual-porosity model to analyze the pressure response and production dynamics of multi-stage fracturing horizontal wells in shale reservoirs. Ozkan [16] and Al-Rbeawi [17] divided the stimulated shale reservoirs into hydraulic fractures, stimulated reservoir volume (SRV), and matrix.

Moreover, they simplify the flow in these three regions into the one-dimensional linear flow by establishing a three-linear-flow model. Additionally, Stalgorova [18] and Zhang et al. [19] improved the three-linear-flow model by considering the un-stimulated areas between two hydraulic fracturing SRV and proposed a five-region coupled flow model. Zeng et al. [20] further subdivided the five-region coupled flow model and proposed a seven-region coupled flow model. Based on these coupled models, Wang et al. [21] and Kim et al. [22] analyzed the stress-sensitive effects of gas reservoirs and fractures.

Overall, the flow around shale MFHWs is mainly characterized by coupled linear flow models with multiple subdivided regions. In these models, hydraulic fracture (HF), SRV, and matrix are commonly applied, and they are separately discussed as follows. The flow in fracturing fractures usually satisfied Darcy law or high-speed non-Darcy law [23]. The SRV can be treated as dual media, or characterized by complex fracture network models [24]. The matrix can be regarded as a homogenous ultra-low-permeability medium. However, some scholars treat it as a dual medium with natural fracture network [25].

The equivalent permeability can be analyzed in the multiple regions coupled model to simplify the effects of desorption and diffusion in the SRV and matrix. From these three main regions, five regions or seven regions are further subdivided, but the physical parameters in the added regions are difficult to obtain. Well interference may be caused by interaction between primary hydraulic fractures and/or secondary natural fractures activated during hydraulic fracturing. Well interference has had a significant influence on the SRV interpretation. Well interference has drawn people’s attention in recent years, but its impact on transient behaviors is rarely reported.

As shown in the previous literatures, Laplace transform and Stehfest method are mainly applied to obtain the analytical solution or semi-analytical solution of transient pressure behaviors. It is principally applicable for the analysis of one single shale gas well, but it’s really difficult to solve the transient pressure with multiple well interferences. So it’s crucial to investigate a numerical method to analyze the well pattern pressure dynamic.

Therefore, to fully understand the transient pressure behaviors with well interference, a three-medium coupled numerical model is given in the paper, considering connected and unconnected HFs, SRV, and matrix. Also, we numerically solve the model by PEBI grids and the finite volume method. Different factors’ influences on transient pressure behaviors with well interference are studied. This research may help to characterize well interference and related factors and to optimize well spacing.

## Model Description

Due to the significant difference in porosity and permeability between the fracturing and un-fracturing areas in the reservoir, shale reservoirs around MFHWs are divided into sub-regions, including hydraulic fractures, SRV with abundant inducing micro-fractures, and matrix as shown in Figure 1. Among them, only SRV region is simplified as dual medium due to micro-fracture development; matrix region is single-porosity single-permeability medium, hydraulic fracture is a high permeability medium.

Assumptions: (1) Water flow is ignored, and there is only single-phase gas flow existed in each sub-region. (2) Just viscous flow exists in the HF and satisfies the Darcy law [26], neglecting the longitudinal flow. (3) SRV is treated as dual media, and each hydraulic segment’s SRV overlaps with each other in one MFHW. (4) Matrix is regarded as homogeneous ultra-low permeability media. (5) Pseudo pressure function (m) is introduced to simplify the gas composition change with temperature and pressure [27]. (6) There are three connection modes between the well and its neighboring wells, including the connection of the inducing micro-fracture clusters in the SRV and the connection of hydraulic fractures as shown in Figure 1.

**Figure 1.** Multi-region coupled shale reservoir physical model (Well-A and well-B are two multi-stage fracturing horizontal wells (MFHWs), the hydraulic fractures connected with the SRV and the SRVs are dual medium.).

## Mathematical Model

### HF Flow Model

It is assumed that fluid exchanges exist among the fracture, the SRV and the wellbore, the boundary between the well and the HFs is defined as Γ_{1}, and the boundary between the SRV and the HFs is Γ_{2}. If the HFs directly connected in one well pair, the fluid exchange between two wells’ HFs needs to be considered. Commonly, shale gas wells produce at a given production rate first according to the development scheme, and the gas supply capacity of the reservoir gradually tends to insufficient as the pressure continues to drop. The shale gas wells are converted to produce with constant pressure later. Based on this, the flow equations in the finite conductivity fractures are established as follows:

### SRV Flow Model

SRV exist around MFHWs in unconventional shale reservoirs, which leads to the flow characteristics of the fluid that are different from those in unstimulated formations. It is necessary to integrate various measurements and surveillance data to build a variable SRV reservoir model. The variable SRV model described here has the following building blocks [28]: (1) Formation evaluation: included all the reservoir characterization data derived from logs and 3D seismic inversions and structural attributes. (2) Surveillance data integration: micro-seismic data are integrated with chemical and radioactive tracer logs.

(3) Well performance data integration: Production data is used to determine different flow regimes during the well history and to set bounds for stimulation parameters, such as HF half-length and permeability. (4) Numerical simulation: Micro-seismic attributes (density and magnitude) are converted to a permeability model after being calibrated with tracer logs and production flow regime parameters. pressure, volume, temperature (PVT) data is matched against an Equation of State and input into the model. Due to the abundant micro-fractures induced by hydraulic fracturing in SRV, SRV is regarded as a dual medium containing the matrix and fracture systems.

(1) Matrix system flow model

Assuming that gas desorbed from the SRV matrix system, the desorption gas satisfies the Langmuir isothermal adsorption equation on the surface of the matrix bedrock. The migration of gas includes viscous flow, Knudsen diffusion, and surface diffusion. So, the matrix system flow model is Equation (2):

(2) Fracture system flow model

Because the micro-fractures in the SRV region are very developed, how to characterize the fracture network equivalently in the seepage model has been a difficult problem to solve. For this reason, many scholars hypothesize that the development and spread of fracture networks satisfy the fractal characteristics and propose a fractal model that characterizes natural fracture networks [29,30].

However, the critical parameters such as the fractal dimension in the model are difficult to determine. Also, considering complex networks will greatly increase the complexity of meshing and numerical calculations. Therefore, the equivalent permeability is used to characterize the comprehensive permeability of the fracture system in the SRV region. The gas in the fracture medium in the SRV is mainly in the form of free gas. Therefore, only the viscous gas flow and Knudsen diffusion are considered in the fracture medium, and the apparent permeability ks,f is used to represent the permeability of the fracture medium [31]:

Assume that the wellbore only has fluid exchange with the fracture, neglecting the direct fluid exchange between the fracture system and the wellbore in the SRV, and defining the interface between the fracture and the matrix systems in the SRV as Γ_{3}. Then the flow model in the fracture system of the SRV is presented as Equation (4):

### Matrix Flow Model

The shale gas reservoir is rich in kerogen organic matter, and the hydrocarbon gas generated in the kerogen satisfies the saturation adsorption and then spreads from the kerogen pores to the inorganic matrix pore space where the hydrocarbon concentration is relatively reduced. The gas in the kerogen occurs in two forms: free gas and adsorbed gas. The pores in the kerogen have the same order of magnitude as the gas molecules in the shale gas.

Therefore, the free gas will generate Knudsen Diffusion in the kerogen nanoporous network. At the same time, the kerogen is saturated with a large amount of adsorbed gas, and the adsorbed gas on the surface of the skeleton will produce surface diffusion. Assuming that the shale gas reservoir is isothermally developed, the Langmuir isotherm adsorption equation is used to describe the adsorption and desorption of kerogen.

The apparent permeability of the unmodified Matrix region proposed by Singh et al. [32] and Civan et al. [29,30] is:

Thus, the kerogen-medium continuity equation considering Knudsen diffusion, adsorption-desorption and surface diffusion is obtained as Equation (6):

### Model Solution

Accuracy and efficiency of reservoir simulators in complex systems depend highly upon a proper grid selection. Grids based on a cartesian coordinate system have been widely used, but have some disadvantages: (a) Flexibility in the description of faults, pinchouts, hydraulic fractures, horizontal wells and general discontinuities presented in reservoirs; (b) inflexibility in representing well locations; and (c) suffer from grid orientation effects. PEBI grids have been applied to the oil industry for about a decade.

On the other hand, generation and construction of PEBI grids are not as easy as cartesian grids. The construction of PEBI grids for a reservoir is feasible only if it is done by a numerical grid generation procedure. These PEBI grids are locally orthogonal. It means the block boundaries are normal to lines joining the nodes on the two sides of each boundary. This allows a reasonable accurate computation of inter-block transmissibility for heterogeneous but isotropic permeability distribution.

The irregular geologic body boundary can be depicted by PEBI grids. In this paper, the unstructured PEBI grid is applied to mesh the solution area and carry on local grid refinement around MFHWs, in which the connection between the center node of each grid and the adjacent grid center node is perpendicular to the interface, as shown in Figure 2.

**Figure 2.** Perpendicular bisection (PEBI) meshing grids of HFs and matrix.

Finding a pressure solution for large-scale reservoirs that takes into account fine-scale heterogeneities can be very computationally intensive. One way of reducing the workload is to employ multi-scale methods that capture local geological variations using a set of reusable basis functions. One of these methods, the multi-scale finite-volume (MsFV) method is well studied for 2D Cartesian grids, but has not been implemented for stratigraphic and unstructured grids with faults in 3D.

With reservoirs and other geological structures spanning several kilometers, running simulations on the meter scale can be prohibitively expensive in terms of time and hardware requirements. Multiscale methods are a possible solution to this problem, and extending the MsFV method to realistic grids is a step on the way towards fast and accurate solutions for large-scale reservoirs.