A Triangular Element-Based Finite Volume Formulation For Solving Problems Of Heat Transfer In Oil Reservoirs

Currently, most of reservoir simulators have been developed using a finite volume method (FVM) as the numerical scheme to discretize the domain. However, FVM faces some issues to handle appropriately complex domains and boundaries. An element-based finite volume method (EbFVM) numerical scheme combines the FVM advantages and the ability of finite element methods to tackle complex reservoir domains.

The purpose of this work is to obtain a numerical formulation where EbFVM is applied to discretize the differential equation that describe diffusive situation for incompressible flow in a two-dimensional domain with problems of nonlinear characteristics in a transient regime.

The spatial discretization was performed by using a structured grid with triangular elements, which are very convenient to represent any two-dimensional complex domain with good accuracy. The conservation laws are locally applied in a secondary control volume grid, which was built around a node by connecting the centroid of each triangle with the midpoints of the triangle’s sides ( Minkowycs, 2006 ). The equation of the element was obtained from interpolation function depending on element coordinates and nodal values, as proposed in the work of Baliga and Patankar. Fluid and rock properties remain constant inside each element, but these properties may vary from element to element, and can be calculated according to the pressure and temperature values prevailing in each element. In the case of single-phase flow, the equation of state used was the fluid compressibility definition. The discretization of the time was developed with the implicit scheme, which is more numerically stable when solving problems with larger time steps, resulting in less computer time. The algorithm employed to discretize the conservation equation, was used to handle all conserved properties, in a sequential manner.

In this work, two examples were compared with solutions obtained from commercial simulation programs that employ the traditional FVM. One example involves a single phase flow and the other consists of heat injection by using a bottom hole heater. Numerical performance were studied with good accuracy in results.