Parametric Representation of Boundary Flux in Multi-Region Fluid Flow Problems

One of the ultimate goals of reservoir engineering is to bundle the many individual factors characterizing a particular reservoir and to forecast performance for the purposes of meeting company expectations, fulfilling contracts, comparing development scenarios, etc. In particular, one is seeking the capacity of wells to inject or withdraw fluids and sustainability of production rates. The well equation tries to capture the relationship between withdrawal rate and pressure decline for the dominant time period of well operations, as a function of reservoir characteristics and well properties. The solutions to the governing equations are traditionally constructed for single phase flow and suitably modified through introduction of empirical relative permeability concepts for multiphase flow use. Early use of well equations adopted by industries is based upon numerical methods. Though numerical methods are suitable for solving the governing equations with more complex inputs, the use of numerical methods is cumbersome, and general insights are difficult to be extracted. It is noted that for most finite difference methods, if the time and the spatial discretization were not carefully conducted, results are most likely inaccurate. Also, the solutions are also dependent on grid orientation and well configuration. In this work, the parametric representation that can overcome the problems of the traditional numerical representation is proposed. Moreover, the parametric representation of flux that adapts to well positions and trajectory has been presented. Previously, semi-analytic solutions to Poisson’s Equation on two and three-dimensional rectilinear domains with Dirichlet and Neumann boundary conditions allow the role of boundary flux distribution to be examined and it has been found considerably significant in accurate forecasting of reservoir observables. A parametric representation of the flux has been proposed as a linear combination of uniform flux and uniform pressure constituents. By doing so, it replaces nonparametric boundary integral methods to solve for coupling between analytic solutions for pressure. For equal cell size problems, only two unknowns per well cell interface need to be introduced with arbitrary choice of two surface locations for pressure matching. Since rectangular grid blocks to represent a reservoir is a common scenario, this paper will focus on the adaptation of the parametric representation of cell boundary flux distribution to allow the extension to systems of either rectangular cells of different size or permeability using prolongation. Ultimately, this paper explores the parametric representation of boundary flux in multi-region fluid flow problems using Green’s Theorem. The proposed parametric representation is applicable to either isotropic systems of non-uniform cell size or heterogeneous systems. The boundary flux structure is shown to be exact for cells of identical size and permeability. It is found that problem bandwidth is significantly reduced in solving the heat equation for heterogeneous systems using parametric representation of flux in boundary integrals in comparison with nonparametric boundary integral methods, and does not increase with problem dimensionality.