Geofluids

Volume 2018, Article ID 5169010, 13 pages

https://doi.org/10.1155/2018/5169010

## Multicomponent Lattice Boltzmann Simulations of Gas Transport in a Coal Reservoir with Dynamic Adsorption

^{1}School of Mechanics & Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China^{2}College of Resources & Safety Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China^{3}State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China

Correspondence should be addressed to Shenggui Liu; moc.361@2002ghsuil and Yingjun Li; nc.ca.yhpi.yhpa@jyl

Received 1 March 2018; Revised 15 May 2018; Accepted 29 May 2018; Published 12 July 2018

Academic Editor: Mandadige S. A. Perera

Copyright © 2018 Zhigao Peng 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

Gas adsorption occurs when the dynamic adsorption equilibrium conditions of the local adsorptive sites are broken. In the overall process of unconventional natural gas generation, enrichment, storage, and production, this phenomenon plays a significant role. A double-distribution Lattice Boltzmann model for solving the coupled generalized Navier-Stokes equation and advection-diffusion equation with respect to the gas-solid dynamic adsorption process is proposed for multicomponent gas migration in the unconventional reservoir. The effective diffusion coefficient is introduced to the model of gas transport in the porous media. The Langmuir adsorption rate equation is employed to control the adsorption kinetic process of gas-solid adsorption/desorption. The model is validated in two steps through fluid flow without and with gas diffusion-adsorption between two parallel plates filled with porous media, respectively. Simulation results indicate that with other parameters being equal, the rate of gas diffusion in the porous material and the area of the dynamic adsorption equilibrium-associated region increase with the matrix porosity/permeability. Similar results will happen with a greater saturation adsorption amount or a lower Langmuir pressure. The geometric effect on adsorption is also studied, and it is found that a higher specific surface area or free flow region can enhance the gas transport and the rate of adsorption.

#### 1. Introduction

Coalbed methane (CBM) is one of the most important and potential energies which has been recognized as an alternative to oil and has achieved more and more attention in recent years [1]. Different with the conventional natural gas, a coal seam is both a resource and reservoir of CBM and stores methane in nanopores in the form of adsorption, resulting in the low recovery of CBM exploitation [2]. Nowadays, understanding the mechanism of gas storage and transport in the reservoir is of vital importance for current research and practical application.

Coal is one of the most complex solids, and its physical property varies from the depth, hydrogeology, reservoir-forming process, and so on, which leads to a highly heterogeneous structure with pore size ranging from a few nanometers to over a micrometer. According to the classification scheme proposed by the International Union of Applied Chemistry (IUPAC) [3] and Hodot [4], pores are divided into three categories based on size: micropores (less than 2 nm), mesopores (from 2 to 50 nm), and macropores (from 50 to 10^{4} nm), as well as microfractures (more than 10^{4} nm). Under the existing research [2], the coal contains massive micropores, some mesopores, and little macropores. Micropores and mesopores provide space for storing most of the adsorbed molecules. Macropores are of no significance in terms of adsorption capacity in that the area contribution is very small and can usually be negligible compared to the area contributed by the micropores/mesopores; they always act as transport pores to allow adsorbate molecules to diffuse from the bulk phase into the particle interior [5].

Generally, coal is characterized as a dual-pore system [6] including the matrix porosity consisting of micropores/mesopores in the coal matrix (i.e., the porous material region) and the fracture porosity composed of nonuniformly distributed macropores and microfractures (i.e., the free flow region) [7]. The adsorption processes of methane in the reservoir can be divided into two parts: Enter the irregular macropore- microfractures system by advection-diffusion; diffuse in the nanopore network of a matrix, then adsorbed on the interior surface of pores. The heterogeneity of geometric shapes, the difference of pore size/distribution, and the involved multiple transport mechanism of fluid causing the difficulty of the research and causing the fluid flow and transport in the reservoir are usually treated at the representative elementary volume (REV) scale in experiments and numerical simulation [8–12]. Recently, the Lattice Boltzmann method (LBM) has been widely used in the field relevant to fluid flow and gas transport at the mesoscale because of its efficiency and effectiveness in the implementation of multiple interparticle interactions and complex geometry boundary conditions [13–15]. Nithiarasu et al. [16] proposed a generalized Navier-Stokes equation for isothermal incompressible flow in porous media, which ignores the detailed structure of the porous media by including an additional term to account for the presence of a porous medium. Guo and Zhao [9] developed an LB model which can be used to solve the generalized Navier-Stokes equations. Then, Chen et al. [8] employed the generalized Lattice Boltzmann (GLB) model with the Klinkenberg effect in shale gas for studying the gas slippage and its effect on apparent permeability. Ning et al. [12] and Wang et al. [11] used a similar model for studying the impact of surface adsorbed gas on apparent permeability based on 2D and 3D reconstructed shale, respectively. Liu and He [17] developed a nonorthogonal multirelaxation time (MRT) LB model with the GLB equation for investigating convection heat transfer in porous media.

As for the coal gas adsorption process, in general, it is considered as a physisorption process because the molecular forces involved are normally of the Van der Waals type [2, 7]. Many studies, including both experiment and numerical simulation, have focused on (isothermal) adsorption equilibrium without concerning the intermediate states or time in recent years [11, 18]. However, the isothermal adsorption equilibrium can only provide static information about the adsorption process at a specific stage, which is insufficient to reflect the kinetic behavior of the coal gas stored in the reservoir. In adsorption kinetics, the concentration changes associated with adsorption were related to the time variable. Therefore, it is for the benefit of studying the effect of various factors, such as the gas content, capacity, porosity, and permeability, on the adsorption process. Recently, more and more researchers concentrated on developing an LB model for multiple transport mechanisms with chemical reaction in the porous media. He et al. [19] proposed an LB model for fluid diffusion-convection with a chemical kinetic reaction using the double distribution function for controlling the fluid flow and diffusion, which introduced a source/sink term in the diffusion equation to govern the reaction process. Tian et al. [10] proposed a coupled LB model using the double distribution function with the GLB equation to describe the geochemical reactions during CO_{2} injection. It is essential that a novel LB model be developed not only for multiple transport mechanism investigation on methane migration in the reservoir but also for practical reservoir upscaling techniques used in unconventional natural gas exploitation.

In the present work, we developed a double-distribution Lattice Boltzmann model to solve a coupled generalized Navier-Stokes equation, and the advection-diffusion equation is proposed for unconventional natural gas migration in the reservoir that includes gas-solid adsorption and desorption, which is based on the REV model of Guo and Zhao [9] and the diffusion-reaction model of He et al. [19]. The model mainly contains three parts: (1) the fluid flow at the REV scale which is governed by the generalized Navier-Stokes equations, (2) the mass transfer in porous media with respect to the effective diffusivity and a source/sink term solved by the advection-diffusion equation, and (3) the gas-solid adsorption process in the porous matrix which is governed by the typical Langmuir adsorption rate equation. The remaining parts are organized as follows: In Section 2, a generalized LB model for fluid flow in porous media is revisited briefly. A passive scalar LB equation with adsorption is presented in Section 3. Numerical results and discussion are given in Section 4. Finally, a brief conclusion is made in Section 5.

#### 2. Generalized Model for Fluid Flow in Porous Media

For isothermal flows of incompressible fluids in porous media at the REV scale, the generalized Navier-Stokes equation which was proposed by Nithiarasu et al. is capable for the simulation.

##### 2.1. Generalized Navier-Stokes Equation

The governing equations of mass and momentum for the generalized Navier-Stokes equations can be given by
where is the fluid velocity, in m/s; is the porosity; is the fluid density, in kg/m^{3}; is the pressure, in Pa; is the effective kinematic viscosity, in m^{2}/s, where and is the viscosity ratio; and represents the total body force including both medium resistance and external forces and can be given by
where is the external body force, in N; is the geometric function, in N; and is the permeability of porous media, in m^{2}.

Both and are related to the porosity. For a porous medium composed of solid particles, the Ergun correlation gives (Ergun 1952) where is the diameter of the solid particle, in m.

##### 2.2. LB Model for Generalized Navier-Stokes Equations

Guo and Zhao [9] constructed an LBE model which can be used to solve the generalized Navier-Stokes equations; the corresponding evolution equation of the particle distribution function is where is the discrete density distribution function, is the local equilibrium equation, is the discrete velocity of particles, is the time step, is the relaxation time, is the unit matrix, is the sound velocity of the lattice, and is the weighting coefficient.

As in the standard LBE, the density and velocity of flow are defined as

Because the force also contains the flow velocity, the velocity can be explicitly given by where is a temporary velocity defined by and the parameters and are given by

By using the Chapman-Enskog technique with the pressure and the effective viscosity , the generalized LB model can recover to (1) and (2) in the incompressible limit.

#### 3. Governing Equation for Gas Transport with Adsorption/Desorption

##### 3.1. The Advection-Diffusion Equation for Mass Transfer

Generally, gas transport in porous media can be considered as a mass transfer process [20–22], which can be expressed by the advection-diffusion equation
where is the concentration of the gas phase, in mol/m^{3}; is the diffusion coefficient, in m^{2}/s; is the resource/sink term, which is involved in the adsorption-desorption process, in mol/(m^{3}·s).

The diffusion coefficient is a key variable describing the gas transport capacity in porous media and is determined by the physical property of the porous material. Since the pore size included in the organic matrix is in the order of nanometers, the diffusion process which is generally called Knudsen diffusion is quite complicated compared to the gas diffusion in the free flow region and should be considered separately. By considering diffusive scattering for gas transport in a simple geometric capillary with a length and radius , Kang et al. [24] estimated the Knudsen diffusion coefficient where is the mean pore radius of the coal matrix, in m; is the gas constant, in J/(mol·K); is the temperature, in K; and is the gas molar mass, in kg/mol.

However, the measurement of diffusivity is the outcome of an expression derived from a single capillary pore, which will be somewhat larger than the experimentally obtained diffusion coefficient. Typically, for gas transport in porous media, the porosity/tortuosity ratio should be related to diffusivity. Therefore, we can roughly obtain the effective diffusion coefficient [24] where is the organic/total volume ratio and is the tortuosity.

##### 3.2. The Rate Equation for Adsorption-Desorption Kinetics

Gas migration through solid media is always accompanied with gas-solid adsorption-desorption due to the local concentration fluctuant breaking down the adsorption equilibrium. In order to reveal the gas transport behavior with adsorption in a coal gas reservoir, this paper employed the typical Langmuir adsorption rate equation which is widely used in unconventional natural gas industries and is applicable for gas adsorption at the nanoscale [19, 25, 26]:
where is the adsorption rate constant, in m^{3}/(mol·s); is the desorption rate constant, 1/s; is the absorbed amount, in mol/m^{3}; and is the saturated adsorption capacity, in mol/m^{3}.

Adsorption equilibrium information is the most important part of the adsorption process, when ; (13) can be recovered as the equivalent Langmuir isothermal adsorption equilibrium equation
where is the pressure of the field, where , in Pa; *P*_{L} represents the Langmuir pressure, where , in Pa.

In the Langmuir kinetic equation, the magnitude of and fixes the adsorption/desorption rate; meanwhile, determines the slope of the isothermal adsorption curve, thus any two of them can determine the third. It is obvious that the rate of adsorption is a concentration-dependent parameter and the rate of desorption is in proportion to the adsorbed amount.

In order to incorporate the gas-solid adsorption, the approach of He et al. [19] was modified to include the self-adaptive conversion between free gas and adsorbed gas. Under this situation, each lattice node corresponding to the porous media is considered a well-adsorptive site. Therefore, the Langmuir rate equation can be integrated into the source/sink term in (10) as follows:

In the evolution process, the adsorbed amount was updated over time; the new amount can be obtained using the first-order difference scheme

It must be mentioned that a broad variety of pore size and distribution exists in the coal matrix, which increases the anisotropic and complexity of fluid flow, gas diffusion, and gas-solid physisorption. However, for upscaling and computational efficiency purposes, we can simplify the porous media with homogenization theory due to the material of porous media, the pressure, and the concentration which will just fluctuate in a few ranges at the REV scale.

##### 3.3. LB Model for Gas Transfer with Adsorption in Porous Media

For 2D simulation of the species transfer process, by ignoring velocities at the D2Q9 diagonals, the model can be reduced to the D2Q5 LB model and will not lose accuracy; thus, the following D2Q5 LB equation is employed to control mass transfer: where is the discrete density distribution function of mass transfer, is the corresponding local equilibrium equation, and is the relaxation time of mass transfer.

Similarly, the concentration of (18) is given by and with , (17) can be recovered as (10).

#### 4. Results and Discussion

Before the investigation, a schematic diagram is displayed in Figure 1 to explain the adsorption process of coalbed methane in the reservoir. Generally, gas stored in the coal matrix mainly contains three forms: free gas, adsorbed gas, and dissolved gas.