Geofluids

Volume 2018, Article ID 2186194, 18 pages

https://doi.org/10.1155/2018/2186194

## Apparent Permeability Model for Shale Gas Reservoirs Considering Multiple Transport Mechanisms

College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

Correspondence should be addressed to Shijun Huang; nc.ude.puc@jhsh

Received 11 April 2017; Revised 1 June 2017; Accepted 12 March 2018; Published 4 June 2018

Academic Editor: Alexandra Amann-Hildenbrand

Copyright © 2018 Shijun Huang 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

Shale formation is featured in nanopores and much gas adsorptions. Gas flow in the shale matrix is not a singular viscous flow, but a combination of multiple mechanisms. Much work has been carried out to analyze apparent permeability of shale, but little attention has been paid to the effect of unique gas behavior in nanopores at high pressure and adsorbed layer on apparent permeability. This work presents a new model considering multiple transport mechanisms including viscous flow (without slip), slip flow, Knudsen diffusion, and surface diffusion in the adsorption layer. Pore diameter and mean free path of gas molecules are corrected by considering the adsorption layer and dense gas effect, respectively. Then the effects of desorption layer, surface diffusion, and gas behavior on gas apparent permeability in nanopores of shale are analyzed. The results show that surface diffusion is the dominant flow mechanism in pores with small diameter at low pressure and that the effect of adsorbed layer and dense gas on apparent permeability is strongly affected by pressure and pore diameter. From the analysis results, the permeability value calculated with the new apparent permeability model is lower than in the other model under high pressure and higher than in the other model under high pressure, so the gas production calculated using the new permeability model will be lower than using the other model at early stage and higher than using the other model at late stage.

#### 1. Introduction

Much attention has been paid to shale due to the considerable volume of natural gas trapped in it. Over the past decades, technology advances in horizontal drilling and hydraulic fracturing have enabled profitable production of shale gas. However, because of the unique deposit character and flow mechanisms of shale gas, controversy still exists on how much gas can be produced from shale [1].

Compared with conventional reservoirs, shales are characterized with pores between 1 and 100 nm, of which the dominant diameter is in nanoscale [2], making it difficult to get accurate shale permeability. Both experimental methods and theoretical methods were proposed to solve the problem. Due to overlarge time consumption, constant-pressure steady-state-flow measurement is not applicable to shale [3]. As a result, other approaches such as TED [4] and crushed rock method [5] were commonly adopted. However, flow regimes could not be well characterized with experimental methods. Therefore, different theoretical models were proposed to investigate gas flow behavior in shale, among which apparent permeability models are the most prevailing and can be divided into two categories. The first category is correcting intrinsic permeability with a function of Knudsen number. Empirical parameters are included in most of the models, and gas flow regimes (viscous flow, slip flow, transition flow, and Knudsen diffusion) in a single capillary are classified based on Knudsen number and viscous flow slip flow and diffusion are mutually exclusive. Beskok and Karniadakis [6] conducted many experiments and presented a correction factor, and Florence et al. [7] simplified the correction factor. Many other correlations are also provided in apparent gas permeability in tight porous media [8–11]. The second category is weighted models, in which flow and diffusion are not mutually exclusive and their weight in apparent permeability is different at different Knudsen numbers. When the Knudsen number is small, collision between gas molecules is dominant and gas flow is characterized by viscous flow and slip flow. When the Knudsen number is large, however, collision between gas molecules and the pore surface becomes dominant and gas flow is characterized by transition flow or Knudsen diffusion. In most cases, these two regimes coexist in nanopore in shale and account for different weights. Ertekin et al. [12] provided the weights for slip flow and Knudsen diffusion; apparent permeability is calculated by a weighted average of slip flow and Knudsen diffusion. Many new weighted models are also proposed by Liu et al. [13], Javadpour et al. [14, 15], Darabi et al. [16], and Shahri et al. [17]. These models have a relatively comprehensive analysis of gas flow in nanopores with the pore scale effect; however, some other unique properties of shale affecting gas apparent permeability are not considered.

The first characteristic of shale formation affecting apparent permeability is that there are a lot of organics and clay minerals with much gas adsorption in the shale matrix [18]. Gas adsorption in shale is generally believed to be single-layer physical adsorption [19]. It is suggested by Akkutlu and Fathi [20] that surface diffusion exists in the adsorption layer, and it is driven by a concentration gradient. Experiments of core permeability tests, pressure decay tests, and numerical calculation show that the permeability measured and calculated with adsorptive gas is much larger than that with nonadsorptive gas and indicate that the high permeability is caused by surface diffusion of adsorptive gas [21–23]. Many theories have been presented to describe surface diffusion in porous media, in which the hopping model is most widely used for surface diffusion [24–27]. If enough energy is acquired by the adsorbed gas molecule and it bounces to the nearest adsorption site, activation process and surface diffusion happen. It is verified that it is reasonable to use Langmuir isotherms to study surface diffusion [28]. Wu et al. [29–31] proposed a surface diffusion model which takes account of coverage of adsorbed gas at high pressure. Although surface diffusion is incorporated in these models, the effect of the adsorbed layer on gas retention and flow capacity is often neglected, which is not negligible as the radius of shale nanopores is in the same order with that of methane molecules. Another impact factor of shale formation is that gas in shale nanopores is dense gas rather than rarefied gas [32]. In shale nanopores, the storage and transport space for gas molecules is in the same order with size of gas molecules and therefore makes the assumption of rarefied gas no longer valid and need correction. The intrinsic size of molecules and the interaction between molecules are negligible for ideal gas. Wang et al. [33, 34] build a multiscale scheme to capture both slip and nonideal gas effects and considered the Enskog equation to cover the dense gas effect in the nanochannel. However, the effects of the adsorption layer, surface diffusion, and dense gas in formation conditions are not all considered in most of the existing models, and an apparent permeability model is needed to incorporate all these impact factors.

This paper presents a comprehensive apparent permeability model in which all the flow mechanisms, including viscous flow, slip flow, Knudsen diffusion, adsorption layer, surface diffusion, and dense gas effect, are taken into account. Then the effects of the desorption layer, surface diffusion, and dense gas in shale nanopores with different diameters are analyzed.

#### 2. Apparent Permeability Model for Shale Nanopores

##### 2.1. Adsorption Layer

The Langmuir isothermal adsorption equation is widely used to describe the adsorption and desorption of coal bed methane [32] and also introduced into the study of the transport regime of shale gas. It is based on instantaneous phase equilibrium; that is, adsorption or desorption due to pressure change is completed instantaneously. As the permeability of the shale matrix is extremely low, desorption time is negligible compared with gas flow in nanopores and the assumption made by Langmuir is also valid in shale gas reservoirs. The equation is as follows: in which is the volume of adsorbed gas per unit mass of shale, m3/t; is the Langmuir volume, m3/t, denoting the maximum adsorption capacity of the shale matrix at a certain temperature; and is the Langmuir pressure, MPa, denoting the pressure at which the actual adsorption is half of the maximum adsorption on the adsorption isotherm, as shown in Figure 1.