Least-Squares Finite Element Method For Hydrocarbon Transport In Porous Media

Abstract

Mathematical models for the transport of constituent components of a multicomponent system in a porous medium play an important part in petroleum reservoir analysis and production. The success of such models is dependent on (1) how well the relevant physical and chemical processes controlling subsurface transport is represented by mathematical equations and their parameters and (2) how accurately and efficiently the equations are approximated with numerical methods. The existing models of multicomponent transport employ two basic sets of equations. The transport of solutes is described by a set of partial differential equations (mass conservation) with the corresponding constitutive relationships and the phase behavior is described by algebraic expressions. This study proposes a new least-squares mixed finite element method (LSFEM) for approximating multiphase flow equations and coupled multicomponent transport equations. The problem is first recast as a system of first-order partial-differential equations. Then a least-squares residual functional is constructed. The least-squares problem is then posed as a mixed finite-element model. This implies that a Co basis can be used. Also, the least-squares formulation leads to a symmetric algebraic system. Since derivatives of the field vi variables enter explicitly in the system, greater accuracy in the computed fluxes will be realized by this mixed method compared to the standard (non-mixed) Galerkin method. The accuracy and performance of the least-squares FEM for transport problems is studied. The effect of varying Peclet number on the accuracy and stability of the solutions is also investigated. Numerical dissipation and other properties of the scheme are examined. The method is verified for model problems by comparisons with analytic solutions and results in the literature. A thermodiffusion model is developed for areal compositional variations in hydrocarbon reservoirs from the fundamental equations of change in a multicomponent system. Under conditions of the stationary state and utilizing concepts of non-equilibrium thermodynamics to compute the fluxes, this model is solved using the least-squares finite-element method. Its usefulness and applicability are demonstrated by means of comparison to observed variations in a real gas field. Finally, given a system of equations describing transport, the computational cost of numerical simulation is dependent on the domain size and the resolution required to minimize numerical errors. However, the local resolution required for accurate solutions can vary over space and time as regions with rapid changes of concentration gradients propagate through the solution domain. Thus, such problems are suited for adaptive numerical strategies that seek to determine where increased numerical resolution is required and then provide more accurate approximations in those regions. Adaptive strategies based on the element residual as an error indicator in conjunction with unstructured remeshing are developed and tested for representative problems of subsurface transport.