Modeling Multi-Phase Flow and Energy Transport in the Earth's Crust by Combined Finite Element and Finite Volume Methods

The challenge about modeling multi-phase flow and energy transport (e.g., in hydrothermal systems) is to define a a robust numerical solution for the highly non-linear governing equations [1]. Additionally, the numerical solution should be able to deal with flow parameters (permeability, porosity, velocity) and fluid properties (density, viscosity, compressibility) that often vary over several orders of magnitude, as well as the geometric complexity of the geologic media. We found that combining higher order finite element and total variation diminishing (TVD) finite volume methods is an efficient way to accurately model multi-phase flow and energy transport. Employing an approach that conserves the fluid volume rather than the fluid mass, we compute the fluid pressure and fluid velocities using finite elements with quadratic-barycentric interpolation functions. Fluid phases are propagated using the fluid velocities and a TVD finite volume methods Fluid properties are computed iteratively for the given temperature and pressure using an equation of state for H₂O and are fed back into the transport algorithm. Finite element and finite volume methods yield the advantage that they can be employed on unstructured and adaptively refined grids and hence deal efficiently with complex geometries. The fluid volume conservative approach allows us to track accurately changes in the fluid volume fractions due to the varying compressibility of the fluids. The quadratic-barycentric interpolation functions yield an accurate integration of the fluid pressure and fluid density field across the finite element and exact fluid velocities at the element integration points. The TVD finite volume method preserves any shock-fronts occurring during the propagation of the fluid phases while avoiding spurious oscillations and numerical diffusion. As an application of the proposed numerical method, we show the convection of water and steam around a magmatic intrusion. [1] Huyakorn, P.S. and Pinder, G.F., Computational methods in subsurface flow. Academic Press, 473 pages, 1987.