Coupled Fluid Flow Models And Their Finite Element Algorithms With Numerical Simulations

In this dissertation,we propose new coupled fluid flow systems for the fractured reservoir flow and solve analytically as well as numerically by using the finite element methods.More specifically,a dual-fracture-matrix system and triple-porosity-Stokes fluid flow models are proposed to study the realistic naturally fractured reservoir efficiently.The conventional and unconventional reservoir has vast applications in many areas such as petroleum engineering,shale-gas,tight-oil industry,hydrocarbon recovery,carbonate reservoir,heavy crude oil reservoir,geothermal energy,waste management,environmental remediation,purely drinkable water recovery,nuclear waste management,and many other areas.As the geometrical structures of the naturally fractured reservoir are very complicated,so it cannot be represented the real situation of the porous medium with the single porosity theory.It is worth to mention that a naturally fractured reservoir contains several porous media with multiple matrix and fractures region.Thus the triple porous medium consists of either one matrix and different fractures continua or two matrix continua and one fracture network system.As a result,Darcy's law cannot represent the multiplex porous medium accurately,which was proposed empirically160 years ago.In this contribution,we study a specialized triple-porosity model,which consists of three contiguous porous mediums with more-permeable macrofractures,lesspermeable microfractures,and stagnant-matrix medium,which is familiar as dualfracture-matrix fluid flow model.Matrix medium acts as the large fluid storage space which supplies the fluid to the fractures medium.In this model,microfractures are fed by the matrix region,and microfractures provide the fluid to the macrofractures only.Hence,the fluid flow is sequential from one medium to another medium.Since the matrix has fluid communication with less-permeable microfractures only and nodirect-communication with the macrofractures,the global domain is separated into two subdomains by taking the traditional dual-porosity region and more-permeable macrofractures region,respectively.In this model,three physically valid interface conditions are proposed for coupling the two subdomains which govern the fluid exchange among the matrix,microfractures,and macrofractures.The mass and fluid are interchanged between less-permeable microfractures,and more-permeable macrofractures are modeled by the two-fluid communication interface conditions while no-fluid interface condition is imposed between matrix and macrofractures region.The weak formulation and the well-posedness of the model are reported.We propose coupled and two decoupled schemes: implicit-explicit and data-passing-partitioned schemes.The stability and optimal convergence analysis of the implicit-explicit scheme and data-passing partitioned scheme are derived.Five numerical examples are presented to show the applicability of the dual-fracture-matrix fluid flow model.On the other hand,we propose a new fluid flow model for the unconventional fractured reservoir coupled with a free flow region.In this model,we consider three coexisting and interacting porous mediums with different intrinsic properties,which is governed by the dual-fracture-matrix equation.The dual-fracture-matrix equation consists of matrix block,large-fractures,and small-fractures,respectively.Moreover,the conduit region is described by the evolutionary Stokes equation.Five coupling conditions are utilized to capture the interfacial phenomena efficiently.The variational formulation and the well-posedness of the model are reported.The fullycoupled scheme and partitioned time-stepping algorithm are proposed.The stability and optimal convergence analysis are derived for the partitioned time-stepping scheme.The validity of the model and numerical methods are illustrated by four numerical experiments.Furthermore,we propose and analyze the two-grid finite element method for the stationary dual-permeability-Stokes fluid flow model.Two grid method is popular to solve a large multi-physics interface system by decoupling the coupled problem into several subproblems.The key idea is that on the coarse grid,the coupled model is solved through standard Galerkin finite element method.On the other hand,the coarse grid approximation is applied to the interface conditions and the mass exchange terms to solve three independent subproblems in parallel on the fine grid.The three independent parallel subproblems are the Stokes equations,the microfracture equations,and the matrix equations,respectively.Four numerical tests are presented to validate the numerical methods and illustrate the features of the dual-permeabilityStokes fluid flow model.