| Coalbed methane (CBM) is a kind of abundant, low cost unconventional natural gas energy source, and mainly composed of CH4. CBM reserves is very huge in China, next only to Russia and Canada, and only shallow reserves has reached 36.81×1012m3 [31].Thus, the effective development of coalbed methane (CBM), not only can provide efficient, clean energy, but also can reduce coal seam gas concentration and eliminate potential safety hazard during coal mining, having practical significance and strategic importance. CBM migration simulation is an important basic work for CBM development, being able to provide theoretical and data support. Comparing to conventional oil and water flow in porous media reservoirs, CBM migration has some characteristics as follows.CBM reservoir is a dual porosity system consisting of coal matrix and fracture network, and the coalbed gas is stored mainly in the matrix in ad-sorbed state and a small amount in the fracture system as free phase [32]. As coalbed gas is released from the CBM reservoir, the pressure of frac-ture system drops and the adsorption capacity of the coal matrix decreases, resulting in gas diffusion in matrix and desorption from matrix surface to fracture system [41]. Therefore, the gas flow process in CBM reservoirs in-cludes gas desorption-diffusion in coal matrix and gas seepage in fracture systems [10,45,2]. Usually, the gas diffusion process in coal matrix is gov-erned by Fick’s law and the seepage flow in fracture system is described by Darcy’s law.Different from the incompressibility or slight compressibility of oil and water, gas compressibility is significant, and need to be considered in the CBM migration model through gas state equation. In addition, the gas flow in the fracture network is driven not only by the conventional pressure field but also by the gas concentration field, so we also need to consider Darcy seepage and Fick’s diffusion in the mutil-fields for the gas migration model [50,51].Comparing to sandstone and carbonate reservoirs, the hardness and strength of coal seam is low, resulting in the fracture deformation being significant. Therefore, it is very necessaryt to consider the influences of the fracture deformation on the fracture permeability in the gas flow model [24,33,42]. For this purpose, by introducing permeability modulus 7, we use exponential function to describe the relation between permeability and pressure.In recent years, more and more field and experimental studies indicate that the non-Darcy behaviors of gas flow exist in the high permeable frac-ture system such as the zone near the wellbore[56]. Forchheimer model has been applied in various oil, water and coalbed gas non-Darcy flow problems, and has obtained some useful computational and practical results. Neverthe-less, due to the assumption of a constant non-Darcy factor βin Forchheimer model, Forchheimer model could not describe the flow characteristic in transi-tion region bewteen Darcy and non-Darcy, causing some limitations of using Forchheimer model [64]. In 2004, Barree and Conway [6] proposed a new non-Darcy flow model, which could describe the entire range of relationships between flow velocity and pressure gradient from low to high flow velocity through porous media, including those in transitional zones.In addition, injection of CO2, so-called enhanced coalbed methane (ECB-M) recovery method, has been conducted to enhance the ultimate recovery [55,27]. Therefore, in fact, coalbed gas is a kind of binary mixture gas (CH4 and CO2), and it is very necessary to study the coalbed gas CO2-displacement migration problem in ECBM reservoirs.Based on the above understandings about coalbed gas flow mechanism, we could derive and establish the coalbed methane migration model, which is time-dependent strong coupling nonlinear partial differential equations and need to be solved by numerical method.A series of conventional mathe-matical and numerical models have been developed, obtaining some useful computational and simulation results [65,54,14,25,56,46,52,57,11,58]. However, there are lack of the design, analysis and test of numerical schems for coalbed methane migration model.In 1960’s, K. Feng [22] proposed the finite element method, which is based on the classical Ritz-Galerkin variational method, widely used in ship-building, machinery, architecture, seepage mechanics and so on [28,15,16, 17]. In 1970’s, the general theory of the mixed finite element method was introduced by Babuska[7] and Brezzi [8], and then Falk and Osborn [23] proposed an improved method. By introducing the intermediate variable (velocity or flux), the order of differential equation is reduced, which can decrease the smooth requirements of finite element space, this is the main advantage of mixed method. Therefore, mixed finite element method has more practicability in solving engineering problems, and obtains quick ad-vancement [47,38,39,9,12,21,60,34]. Mixed finite element method for quasi-linear second-order elliptic equations is researched by Milner[36]. In [29,37,43], the mixed finite element method for nonlinear elliptic equation-s is considered. For the Forchheimer non-Darcy flow problem, Girault and Wheeler [26] proposed a mixed finite element method, in which the veloc-ity and the pressure are approximated by Crouzeix-Raviart element. Pan and Rui [44] introduced a mixed element approximation for the stationary Darcy-Forchheimer equation using the Raviart-Thomas mixed element and the Brezzi-Douglas-Marini mixed element. Cell-centered difference method (CCFDM) is a relatively accurate difference method, which could be regard-ed as the lowest RT mixed finite element method in the special numerical integral rule. Weiser [59] introduced the block-centered difference method (BCFDM) for the linear elliptic problems. Arbogast [3,4] studied the cell- centered difference method for elliptic problem with tensor coefficients under quadrilateral mesh. Rui [48] considered the block-centered finite difference method for the Darcy-Forchheimer model, they all gave the second-order accuracy error estimates.In this thesis, for the coalbed methane migration problem, based on the mass,momentum and energy conservation laws, we derive and estab-lish coalbed methane migration model, in which the effects of fracture de-formation and non-Darcy flow have been considered. Futhermore, consid-ering the time-dependent,nolinear and convection-diffusion characteristics of migration model, we use finite element, mixed finite element and cell-centered difference methods to establish numerical schemes, give error esti-mates. Through numerical simulation, the error estimate results are verified, and the dynamic characteristics of pressure, velocity, component concentra-tion and gas production have been studied.The dissertation is divided into five chapters. The outline is as follows.In chapter 1, we describe the coalbed gas state, desorption-diffusion-seepage mechanism and basic assumptions, including:the definition of coalbed gas concentration, desorption-diffusion process, Darcy and non-Darcy flow description, and the relation between permeability and pore pressure. Fur-thermore, the Sobolev space, corresponding norms, and several useful lemmas are recommended.In chapter 2, for the coalbed methane migration problem, we derive and establish coalbed methane migration model, in which the effects of fracture deformation and non-Darcy flow have been considered. Considering non-linear characteristic of migration model, we give the Galerkin finite element approximation scheme, prove the existence and uniqueness of numerical solu-tion, and establish the error estimates for the numerical solution. Further, we consider the mixed element approximation scheme, in which the gas pressure and velocity are approximated using RTo. Through numerical examples, the error estimate results are verified, and analyse the dynamic characteristics of pressure, velocity and absorbed gas concentration.In chapter 3, introducing discrete inner products and associated norms, we consider the cell-centered difference scheme for the coalbed gas migration model. The existence and uniqueness of numerical solution is proved, and the error estimates are established through introducing auxiliary problem. Through numerical simulation, the error estimate results are verified, and parameter sensitive analysis are carried out, so as to describe the dynamic characteristics of pressure, velocity and adsorbed gas concentration.In chapter 4, introducing the Barree-Conway equation, the coalbed gas Barree-Conway non-Darcy flow model is established. Considering the nonlin-ear characteristic of Barree-Conway model, we present mixed finite element approximation scheme, and establish error estimates for the approximation results. Numerical results verify the error estimate results. Also, for the two production modes of constant production and constant wellbore pressure, we analyse the effects of non-Darcy parameters such as characteristic length τ, minimum permeability ratio et al.In chapter 5, considering the CO2-injection production process, a CO2-CH4 binary gas non-Darcy flow model is proposed, based on the Barree-Conway non-Darcy flow equation. The mixed element approximatin scheme and Galerkin finite element scheme are alternately used to solve the gas flow equation and component equation. Through numerical simulation, the non-Darcy pressure,velocity and component concentration are analysed. |
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