Browsing by Subject "Mixed hybrid finite element method"
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Item Simulating growth and proppant transport in non-planar hydraulic fractures(2017-11-09) Castonguay, Stephen Thomas; Mear, Mark E.; Rodin, Gregory; Ravi-Chandar, Krishnaswamy; Landis, Chad; Foster, JohnA computational model is developed to simulate the growth of non-planar hydraulic fractures. A Symmetric Galerkin Boundary Element Method (SGBEM) is used to simulate the fracture growth process. This involves only weakly singular kernels to be computed, as well as only requiring C [superscript 0,α] (Holder) continuous shape functions. Additionally special crack tip shape functions are utilized to capture the stress intensity factors, which are then used to update the geometry through a mixed mode I/II growth law. Fluid flow equations are derived for the case of flow through a thin channel defined on an arbitrarily curved 2D surface embedded in 3D space. Two separate methods for fluid flow are used to solve these equations. The first adopts the work of Rungamornrat et al. (2005) and utilizes a Galerkin Finite Element Method (GFEM) to calculate the pressure in the fluid. This, coupled with the SGBEM, allows for the investigation into the effects of material properties and wellbore boundary conditions on the simultaneous growth of multiple fractures. Several examples are presented to illustrate that the growth of cracks in three dimensions can exhibit quite complicated behavior, which could not be revealed without having the capability to fully treat the interaction between the non-planar fractures. A particularly interesting phenomenon occurs for initially parallel fractures in the presence of an anisotropic in situ stress, where the cracks petal in order to escape the stress shadows of their neighbors. The second model utilizes the Mixed Hybrid Finite Element Method (MHFEM) for fluid flow. This gives both accurate pressures and velocities by solving the mass and momentum balance equations simultaneously. The velocity fluxes are locally mass conserving over each element and are therefore well suited for use in the advection equation for the proppant. Both Raviart-Thomas and Arnold-Boffi-Falk elements are utilized for the flux degrees of freedom, which do not require the addition of any stabilization terms. The hybrid mixed method avoids the saddle point structure associated with the standard mixed method, and allows for fewer global degrees of freedom as well. A subdivision process of the SGBEM mesh is implemented through a recursive mapping to obtain a finer mesh for the fluid flow. Comparisons are made between solutions from the MHFEM model and the GFEM model, as well as analytical solutions of simplified test problems. The proppant transport process is modeled with an upwind finite volume method which takes into account gravitational settling. The finite volume method is also locally mass conservative, and by using a backwards finite difference for the temporal discretization, the solutions are unconditionally stable for any time step. One set of examples are presented to show how the number of subdivisions affects the ability to accurately track proppant fronts. Another example illustrates the models ability to handle the settling of proppant as well as its build up as the concentration reaches a maximum value.