Relaxation Methods Applied to Oil Field Research

Abstract

Abstract A numerical method for solving partial differential equations in steadystate fluid flow is described. This method, known as the "relaxationmethod," has two advantages over analytical methods:practically anyproblem can be solved, anda solution can be obtained quickly. Adisadvantage is that the solution is not general. The method is applied to coreanalysis and relative permeability measurement to calculate constrictioneffects and to calculate the true pressure drop measured by a center tap in a Hassler type relative permeability apparatus. Further applications aresuggested. Introduction Many problems in fluid flow cannot be solved analytically because of thenature of the boundary conditions. For many problems, however, an exact answeris not necessary because boundary conditions are not exactly defined or theparameters describing the porous medium are not accurately known. Therelaxation method can be used to obtain an approximate answer easily andquickly for the flow of incompressible fluids in porous media. The method canalso be used for other types of problems, such as determining the stress in ashaft under load, or the temperature distribution during steady state heatflow. In this discussion only calculations concerned with the flow of fluids inporous media will be considered. The method was introduced by R. V. Southwellin 1935. Theory The treatment given here follows that given by Emmons. Consider a porousmedium to be replaced entirely by a net of tubes of equal length and uniformcross-sectional area as shown in part in Fig. 1. Assume that the net of tubesbehaves exactly like the porous medium which it replaces; that is, the net canbe made fine enough to reproduce exactly the porous medium. T.P. 3145