Browsing by Subject "Mixed finite element"
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Item Coupled flow and geomechanics modeling for fractured poroelastic reservoirs(2014-12) Singh, Gurpreet, 1984-; Wheeler, Mary F. (Mary Fanett)Tight gas and shale oil play an important role in energy security and in meeting an increasing energy demand. Hydraulic fracturing is a widely used technology for recovering these resources. The design and evaluation of hydraulic fracture operation is critical for efficient production from tight gas and shale plays. The efficiency of fracturing jobs depends on the interaction between hydraulic (induced) and naturally occurring discrete fractures. In this work, a coupled reservoir-fracture flow model is described which accounts for varying reservoir geometries and complexities including non-planar fractures. Different flow models such as Darcy flow and Reynold's lubrication equation for fractures and reservoir, respectively are utilized to capture flow physics accurately. Furthermore, the geomechanics effects have been included by considering a multiphase Biot's model. An accurate modeling of solid deformations necessitates a better estimation of fluid pressure inside the fracture. The fractures and reservoir are modeled explicitly allowing accurate representation of contrasting physical descriptions associated with each of the two. The approach presented here is in contrast with existing averaging approaches such as dual and discrete-dual porosity models where the effects of fractures are averaged out. A fracture connected to an injection well shows significant width variations as compared to natural fractures where these changes are negligible. The capillary pressure contrast between the fracture and the reservoir is accounted for by utilizing different capillary pressure curves for the two features. Additionally, a quantitative assessment of hydraulic fracturing jobs relies upon accurate predictions of fracture growth during slick water injection for single and multistage fracturing scenarios. It is also important to consistently model the underlying physical processes from hydraulic fracturing to long-term production. A recently introduced thermodynamically consistent phase-field approach for pressurized fractures in porous medium is utilized which captures several characteristic features of crack propagation such as joining, branching and non-planar propagation in heterogeneous porous media. The phase-field approach captures both the fracture-width evolution and the fracture-length propagation. In this work, the phase-field fracture propagation model is briefly discussed followed by a technique for coupling this to a fractured poroelastic reservoir simulator. We also present a general compositional formulation using multipoint flux mixed finite element (MFMFE) method on general hexahedral grids with a future prospect of treating energized fractures. The mixed finite element framework allows for local mass conservation, accurate flux approximation and a more general treatment of boundary conditions. The multipoint flux inherent in MFMFE scheme allows the usage of a full permeability tensor. An accurate treatment of diffusive/dispersive fluxes owing to additional velocity degrees of freedom is also presented. The applications areas of interest include gas flooding, CO₂ sequestration, contaminant removal and groundwater remediation.Item Multiscale mortar mixed finite element methods for flow problems in highly heterogeneous porous media(2013-12) Xiao, Hailong; Arbogast, Todd James, 1957-We use Darcy's law and conservation of mass to model the flow of a fluid through a porous medium. It is a second order elliptic system with a heterogeneous coefficient. We consider the equations written in mixed form. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period epsilon, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when epsilon is small. Moreover, we present numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces. On the other hand, we also propose to use multiscale mortars as a coarse component to construct a two-level preconditioner for the saddle point linear system arising from the fine scale discretization of the mixed finite element system. The two-level preconditioners are constructed based on the interfaces. We propose a framework to define the interpolation operators for the face based two-level preconditioners for different combination of coarse and fine scale mortar spaces for matching and nonmatching grids. In this dissertation, we show that for quasi-homogeneous problems and matching grids, the condition number of the preconditioned interface operator is bounded by (log(H/h))², which is the same as the traditional two-level preconditioners, for quasi-homogeneous problems. We show several numerical examples to demonstrate that for the strongly heterogeneous porous media, it is often desirable and even necessary to use a higher dimensional coarse mortar space to construct the coarse preconditioner to achieve convergence. We apply our ideas to study slightly compressible single phase and two-phase flow in a porous medium. We find that for the nonlinear single phase problem, the two-level preconditioners could be successfully applied to the symmetrized linear system. For the two-phase problem, using the fine scale, instead of multiscale, velocity solutions from the flow problem can greatly benefit the transport problem.