Geofluids

Volume 2018, Article ID 8065949, 16 pages

https://doi.org/10.1155/2018/8065949

## A Mathematical Pressure Transient Analysis Model for Multiple Fractured Horizontal Wells in Shale Gas Reservoirs

^{1}State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, China^{2}Department of Petroleum Engineering, China University of Petroleum (Beijing), Beijing 102249, China

Correspondence should be addressed to Qing Wang; nc.ude.puc@2102qw

Received 26 July 2017; Revised 2 November 2017; Accepted 5 December 2017; Published 17 January 2018

Academic Editor: Ebrahim Fathi

Copyright © 2018 Yan Zeng 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

Multistage fractured horizontal wells (MFHWs) have become the main technology for shale gas exploration. However, the existing models have neglected the percolation mechanism in nanopores of organic matter and failed to consider the differences among the reservoir properties in different areas. On that account, in this study, a modified apparent permeability model was proposed describing gas flow in shale gas reservoirs by integrating bulk gas flow in nanopores and gas desorption from nanopores. The apparent permeability was introduced into the macroseepage model to establish a dynamic pressure analysis model for MFHWs dual-porosity formations. The Laplace transformation and the regular perturbation method were used to obtain an analytical solution. The influences of fracture half-length, fracture permeability, Langmuir volume, matrix radius, matrix permeability, and induced fracture permeability on pressure and production were discussed. Results show that fracture half-length, fracture permeability, and induced fracture permeability exert a significant influence on production. A larger Langmuir volume results in a smaller pressure and pressure derivative. An increase in matrix permeability increases the production rate. Besides, this model fits the actual field data relatively well. It has a reliable theoretical foundation and can preferably describe the dynamic changes of pressure in the exploration process.

#### 1. Introduction

Shale gas is known as a key resource to meet the increasing world energy demand because of its rich reserves and extensive distribution [1]. To be more specific, many pieces of research have shown that shale gas reservoirs are characterized by extremely low permeability values. Consequently, hydraulic fracturing simulations and horizontal drilling are developed for commercial exploitation. In shale gas reservoirs, the complex pore structure, macro/microfracture network distribution, and microscale reservoir properties make the gas flow characteristics become more complicated than those in conventional reservoirs. Gas flow in organic matter, natural fractures, secondary fractures, and hydraulic fractures is controlled by different mechanisms [2]. To be more specific, 20%–80% of shale gas exists in the adsorbed state on the surface of organic particulates. Desorption of adsorbed gas in organic matter affects gas transport mechanisms significantly [3, 4]. Understanding the pressure- and rate-transient behaviors of multistage fractured horizontal wells in shale gas reservoirs is of great importance for production forecasting, well placement, and configuration optimization [5].

In recent studies [6–8], gas transport mechanisms in organic matter can be described by continuum flow, slip flow viscous flow, the Knudsen diffusion, and transition flow. Wang and Li [9] made a comparison of collision frequency in gas-gas and gas-solid interactions. They concluded that the collisions between gas molecules and the solid wall account for a large percentage of the total collisions. Therefore, the Knudsen diffusion should be considered when studying gas transport in nanopores. Darabi et al. [10] also reported that, under typical shale gas reservoir conditions, the Knudsen diffusion dominates gas transport and the contribution to cumulative production can reach 20%. Adsorbed gas desorption from the surface of organic matter also plays a significant role in gas transport [11, 12]. Swami et al. [13] developed a modified model to investigate the contribution of adsorbed gas to the final recovery. They found that desorption of adsorbed gas can increase the pore diameter, reduce tortuosity, and cause extra slippage at the solid boundary. Yu and Sepehrnoori [14] analyzed the shale gas development history for a period of 30 years in North America and concluded that different shale samples demonstrate quite different adsorption capacities.

Mayerhofer et al. [15] studied the difference between conventional double wings symmetric fractures and hydraulic fractures of Barnet shale and were the first to propose the simulated reservoir volume (SRV) conception. In SRV, multistage hydraulic fracturing connects existing natural fractures, which generates a large fracture network [16]. Kucuk and Sawyer [17] firstly developed an analytical model to investigate transient pressure in shale gas reservoirs. However, the model ignored diffusion flow and the desorption effect. Lee and Brockenbrough [18] analyzed production data coming from vertical wells and developed an analytical solution to describe transient pressure. Ozkan et al. [19] proposed a dual-mechanism, that is to say, a dual-porosity horizontal well model for shale gas reservoirs, which includes hydraulic fractures, as well as the inner region and the outer region. The model can adopt the diffusive flow mechanism but fail to take adsorbed gas into consideration. Stalgorova and Mattar [20–22] extended the trilinear flow model to a five-region model for MFHWs in homogeneous shale gas reservoirs. In order to investigate the effect of natural fractures on matrix permeability, Apaydin et al. [23] combined composite blocks into an analytical trilinear flow model. Zhao et al. [24, 25] analyzed the pressure response and production performance of MFHWs in shale gas reservoirs and they introduced the source function theory to characterize SRV. Liu et al. [26] discussed the desorption and diffusion effects in the analytical model and studied the productivity-decline law. Wang [27] considered the stress sensitive effect of natural fractures and hydraulic fracture angles in the semianalytical model. His research implied that the stress sensitive effect has a significant impact on bottom hole flowing pressure.

In recent years, many general semianalytical models, considering microscopic seepage mechanisms, have been proposed with the rapid development of computer technology [24, 25, 28–30]. These models can describe predominant flow regimes for shale well but fail to consider all of the typical characteristics and possible situations in shale gas reservoirs. In the present study, a novel apparent permeability model was put forward by taking into consideration viscous flow, slip flow, transition flow, gas adsorption/desorption [31], and the poromechanical response [32, 33]. The novelty of this study concentrates on the improvement of the semianalytical model to conduct pressure- and rate-transient analysis in SRV, by considering microscale percolation mechanisms and heterogeneity. The rest of this study can be described in the following way. In Section 2, the apparent permeability model for gas transport in shale nanopores is introduced. The semianalytical model for MFHWs in a heterogeneous shale gas reservoir, by considering microscale percolation mechanisms in a dual-porosity formation, is introduced in Section 3. Log-log dimension pressure, dimension pressure derivative, and production type curves are plotted firstly, and then a sensitivity analysis of formation parameters is conducted in Section 4. An actual field case is studied in Section 5. Finally, some conclusions are drawn in Section 6.

#### 2. Apparent Permeability Model for Gas Transport in Shale Nanopores

##### 2.1. Viscous Flow

The total shale gas flux is composed of a viscous flow flux and a Knudsen diffusion flux component. Viscous flow is strongly prominent when the short molecule free path is far less than the pore diameter. In other words, the transport mechanism is governed by viscous flow in macropores (where the pore diameter > 50 nm). The Hagen-Poiseuille equation can describe the molar flux [34] as stated below:where is the viscous flow flux, mol/(m^{2}s); is porosity, dimensionless; is tortuosity, dimensionless; is the pore radius, m; is viscosity, Pas; is the universal gas constant, J/(molK); is temperature, K; is pressure, Pa; and is the shale gas transport distance, m.

Gas viscosity gradually deviates from the traditional viscosity definition for high Knudsen numbers. Karniadakis et al. [35] modified the viscosity definition in the Knudsen layer by considering the rarefaction effect.where is the viscosity considering rarefaction effect, Pas; is the rarefaction coefficient, dimensionless; and Kn is the Knudsen number, dimensionless.

According to the Darcy equation, the apparent permeability of viscous flow can be expressed as [36]:where is the apparent permeability, taking into account the effect of rarefaction, m^{2}; is the gas mole volume at standard conditions, 22.414 10^{−3} m^{3}/mol; and is the permeability of viscous flow, m^{2}.

##### 2.2. The Knudsen Diffusion

When there is shale gas transport through micropores under low pressure, the molecule free path is long and equal to the pore diameter. The Knudsen diffusion is prominent because collisions between molecules and the micropores wall are more frequent than the intermolecular collisions. The flux of the Knudsen diffusion can be expressed as [34]where is the Knudsen diffusion flux, mol/(m^{2}s); is the Knudsen diffusion coefficient, m^{2}/s; and is the gas concentration, m^{3}/mol. The Knudsen diffusion coefficient is obtained aswhere is the gas molar mass, kg/mol. The rough wall has a significant effect on the Knudsen diffusion. Darabi et al. [10] proposed a function to describe the roughness effect as follows:where is the Knudsen diffusion coefficient which takes into consideration the roughness effect, m^{2}/s; is the mean free path of gas molecules, m; and is the fractal dimension of the rough wall, dimensionless.

By combining (4) and (6), the apparent permeability of the Knudsen diffusion can be obtained and written as follows:

##### 2.3. Weight Factor

In the actual shale gas reservoirs, different transport mechanisms exist at the same time. Therefore, the total shale gas flux should be a weighted summation of the viscous flow flux and the Knudsen diffusion flux, based on their different contributions. The weight factors of the viscous flow and the Knudsen diffusion flow are defined as the ratios of intermolecular collisions and molecular/pore-wall collisions to total collisions, respectively. Based on the definitions for a gas molecule free path [11] and collision numbers [37], the weight factors of viscous flow and Knudsen diffusion can be approximated as follows:

Therefore, the total apparent permeability in nanopores is

With a decrease in pressure, the gas molecule free path increases immediately. The weight factor of the Knudsen diffusion also increases. The piece of research carried out by Wu et al. [38] showed that, under the 10 nm radius condition, viscous flow dominates when the pressure is larger than 4.02 10^{5} Pa; otherwise, the Knudsen diffusion dominates.

##### 2.4. Sorption-Induced Swelling Response

Organic matter is described by weak strength and strong sensitivity to stress change. With an increase in reservoir pressure, adsorbed gas begins to desorb. Gas desorption results in shrinkage of the organic matrix and to an increased effective hydraulic diameter [39]. According to the solid deformation theory [40] and the Langmuir isotherm equation [41], the relationship between the degree of solid deformation and the reservoir pressure can be written as follows:where is the solid deformation degree, dimensionless; is the Langmuir volume, m^{3}/kg; is the density of the shale matrix, kg/m^{3}; is the shale matrix Young modulus, Pa; is the Langmuir pressure, Pa; is the initial pressure, Pa; and is the rock compressibility, Pa^{−1}.

Meanwhile, based on Seidle’s model [42], the approximation relation between the porosity and the solid deformation degree is given by

Assume that the pore volumes are proportional to the gas flow channels. In accordance with the capillary model, the effective hydraulic diameter can be obtained as

##### 2.5. Gas Adsorption/Desorption

Because of the large surface area and oil-wet characteristic, nanopores have a strong adsorbed gas capacity. The mass balance equation, which considers the adsorbed gas, can be expressed aswhere is the shale gas density, kg/m^{3}; is the gas compressibility, Pa^{−1}; and is the mass of adsorbed gas per unit volume of shale, kg/m^{3}. The apparent permeability, which considers desorption of absorbed gas, can be expressed as

Cui et al. [12] defined the effective adsorption porosity as

According to the Langmuir isotherm equation, the effective adsorption porosity can be expressed as

Finally, the apparent permeability model for gas transport in shale nanopores can be expressed as

#### 3. Productivity Prediction Model for MFHWs

##### 3.1. Physical Model

Figure 1 shows the diagram of MFHWs in a shale gas reservoir. The reservoir is divided into seven contiguous regions: two upper-reservoir regions (regions 6 and 7), two outer-reservoir regions (regions 4 and 5), two inner-reservoir regions (regions 2 and 3), and a hydraulic region (region 1). Not all reservoir zones between primary hydraulic fractures are stimulated by induced fractures. 1D linear flow is assumed within unstimulated region and the flow direction depends on the location. As shown in Figure 1, a region of higher permeability around each fracture, the so-called SRV area, is introduced to represent fracture branching (region 2). The SRV region occupies part of the space between fractures and the flow in the other part (region 3) towards region 2, parallel to the wellbore. Regions 4 and 5 are in the outer reservoir beyond the tips of the hydraulic fractures and they connect with regions 2 and 3 in direction. Because of the higher permeability in region 2, the pressure decreases in this area first. Therefore, the flow direction in regions 4 and 5 is towards regions 2 and 3, perpendicular to the wellbore. Although the main extending direction of the hydraulic fractures is horizontal ( direction in Figure 1), we also need to consider fractures height in vertical direction ( direction in Figure 1). Based on this consideration, regions 6 and 7 are defined in the upper reservoir beyond the tips of the hydraulic (connecting with regions 2 and 3 in direction) and the flow direction is vertical to the wellbore, towards the lower reservoir.