Modelling of Fluid Flow in a Petroleum Reservoir Using an Engineering Approach

Abstract The mathematical approach is the most commonly used approach in reservoir simulation. The classical mathematical approach considers numerous impractical assumptions leading toward the development of unrealistic reservoir simulator. In contrast, recently developed engineering approach is much promising as it has numerous advantages, such as – scope of bypassing the formulation of partial differential equations and discretization of partial differential equations, the ability to avoid rigorous and complex mathematics, and capability of realistic representation of reservoir behaviour through eliminating spurious assumptions. The present study outlines the route map for developing a reservoir simulator using an engineering approach. Major challenges encountered in reservoir simulation and the fundamentals of various available modelling approaches are addressed in this paper. The outlook for both classical mathematical approach and engineering approach are reviewed along with their strengths and weaknesses. Fluid flow equations are derived based on the proposed engineering approach. To do that, a set of non-linear algebraic flow equations in the time integral form is developed using the mass balance equation, an equation of state, and a constitutive equation without going through the formulation of partial differential equations and discretization step. The time integral is then approximated to obtain the non-linear algebraic flow equations for all the gridblocks of the reservoir. The significance of the engineering approach for describing the accurate fluid flow through porous media is compared to the to conventional mathematical approach. The engineering approach provides the same fluid flow equations as the classical mathematical approach for both the radial cylindrical and cartesian coordinate system but, without going through the formulation of partial differential equations and discretization step. Much simpler ordinary differential equation solvers, e.g., Runge-Kutta method or Euler method can be used in the engineering approach to obtain the solution, whereas the classical mathematical approach does not have this advantage. Both the classical mathematical approach and the engineering approach treat the initial conditions in the same way. If classical mathematical approach uses second-order approximation then the same accuracy is obtained for both approaches in treating the boundary conditions. The engineering approach provides more precise dealing to the constant pressure boundary condition for block-centred gridding system in case of using the first-order approximation. The engineering approach gives the justification of using the central difference approximation for second order space derivative in classical mathematical approach. Results show that the proposed engineering approach based fluid flow model provides better flow prediction than the conventional mathematical approach based flow model. The outcome of this study will help engineers and researchers to develop more transparent simulator instead of creating a black box where the natural chaotic behaviour of the underground reservoir will be more understandable.

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