Mixed Hybrid Finite Element Formulation for Subsurface Flow and Transport
Authors: Abushaikha Ahmad S. , Voskov Denis V. , Tchelepi Hamdi A. 
Published: July 2019

We present a mixed hybrid finite element formulation for modelling subsurface flow and transport. The formulation is fully implicit in time and employs tetrahedron elements for the spatial discretization for the subsurface domain. It comprises all the main physics that dictate the flow behaviour for subsurface flow and transport, since it is developed on, and inherits them from the Automatic Differentiation General Purpose Research Simulator (AD-GPRS) of Stanford University Petroleum Research Institute (SUPRI-B).

Traditionally, the finite volume formulation is the method employed for the computation of fluid dynamics and reservoir simulation, thanks to its local conservation of mass and energy, and straight-forward implementation. However, it requires the use of structural grids and fails in handling high anisotropy inside the material properties of the domain. Also, the method is of a low computational order; the computed local solution in the gird is piecewise constant.

Here, we use the mixed hybrid finite element formulation which is of high order and can handle the high anisotropy for the material properties. It solves the momentum and mass balance equations simultaneously, hence the name mixed. This strongly coupled scheme facilitates the use of unstructured grids which are important for modelling the complex geometry of the subsurface reservoirs. The Automatic Differentiation library of AD-GPRS automatically differentiates the computational variables needed for the construction the Jacobian matrix which consists of the momentum and mass balance unknowns and any presence of wells.

We use two types of tetrahedron elements, Raviart Thomas (RT0) and Brezzi-Douglas-Marini (BDM1), low and high order respectively. The RT0 has one momentum equation per interface, and the BDM1 has three momentum equations per interface assuring second-order flux approximation. Therefore, when compared to the finite volume approach where the Jacobian consists of the mass balance and well unknowns only, the mixed hybrid formulation will eventually have a larger Jacobian (a one order of magnitude for the high order element) which is computationally expensive. However, none the less, the formulation converges numerically and physically better than the finite volume approach, as we show.

The full system is solved implicitly in time to account for the non-linear behaviour of the flow and transport at the subsurface level which is highly pressure, volume, and temperature (PVT) dependent. Therefore, we make use of the already robust PVT formulations in AD-GPRS. We present a carbon dioxide (CO2) sequestration case for the Johnson formation and discuss the numerical and computational results.

This is of crucial important for Qatar and the Middle East where effective reservoir modelling and management requires a robust representation of the flow and transport at the subsurface level using state of the art formulations.

In the literature, Wheeler et al. (2010) employ a multipoint flux mixed finite element approach to eliminate the momentum balance equation form the Jacobian and substitute it by the well-established multipoint flux approximation (MPFA) in the mass balance equation. Since it is based on MPFA, it will still suffer from convergence issues where high anisotropy is present in the material properties. They have recently expanded their work to compositional modelling of fluid, Singh & Wheeler (2014), however they solve the system sequentially in time, where in our method we solve the system fully implicit in time. Sun & Firoozabadi (2009) solve the pressure implicitly and the fluid properties explicitly in time by further decoupling the mass balance equations which decreases the physical representation of the non-linear behaviour of the flow and transport at the subsurface level.