A Conjugate Gradient-Truncated Direct Method for the Iterative Solution of the Reservoir Simulation Pressure Equation

Abstract

Abstract This paper describes a way of solving the reservoirsimulation pressure equation using preconditionedconjugate gradients. The preconditioning is based onan approximate inverse using a diagonal ordering ofthe difference equations. The new method has been tested and comparedwith the strongly implicit procedure (SIP) on anumber of problems in both two and threedimensions. In two dimensions, it is generally faster thanSIP; in three dimensions, it is much slower than SIPwhen SIP works well but can be many times faster than SIP when SIP works poorly. Use of the new method generally does not requirethe selection of an iteration parameter, which is asignificant advantage over SIP. Furthermore, it ismuch more reliable than SIP. In other words, it is farless likely to be unable to solve a given problem thanSIP is. Introduction An IMPES reservoir simulator calculates pressuresonce each time step. These pressures are calculatedby solving a matrix of simultaneous equations;thereis one equation for each cell in the system. Usuallythe solution of this set of equations is not difficult, consuming perhaps 30% of the total computing timefor the time step. However, in a difficult problem, the computation time required for this solution mayincrease dramatically, making the reservoirsimulation calculations very expensive. Whether or not the pressure equations can besolved by direct methods depends on the bandwidth.In two dimensions, the bandwidth is proportional tothe smallest dimension; in three dimensions, it isproportional to the product of the smallest twodimensions. When the bandwidth is small, directmethods are quite economical. Unfortunately, most large problems have largebandwidths, so iterative methods are used to solvethem. The most popular of the iterative techniques isprobably SIP. SIP works well in many problems, particularly those which have relativelyhomogeneous reservoir properties, but it works verypoorly in others. Furthermore, its performancedepends on a sequence of iteration parameters, theselection of which is easy in some problems but cantake a lot of time and effort in others. Recently, there has been significant progress madein the application of the preconditioned conjugategradient method. This method is quite fast, and itsuse does not require the selection of iterationparameters. However, a barrier exists to itsapplication to reservoir simulation problems. The conjugate gradient method applies directly only tosymmetric matrices, and the reservoir simulationmatrix of pressure equations is nonsymmetric. In thenew method, this is overcome by solving a symmetricapproximation to the matrix of pressure equations.The pressures so obtained, though only approximate, are usually within the accuracy achieved by iterativemethods. If not, they can be refined by solving thesymmetric matrix again, as necessary. This disposesof the symmetry problem, permitting the use of theconjugate gradient method. SPEJ P. 345^