Numerical Simulation of Coning Using Implicit Production Terms

This paper describes the use of a multipbase, multidimensional mathematical model to predict twoand three-phase coning bebavior. Severe computational instability in. the /orm of saturation .,, osczllaiions in grid idocks 7EW ih .we[[~ore ~~ commonly encountered in tbe matbernatical simulation of coning. This instability is due to the explicit (dated at the beginning of a time step and held constant {or that time step) handling o/ saturation dependent transmissibilities and production terms in the finite-dif/erence solution of the {low equations. An analysis of stability with respect to explicit handling of saturatiorr-dep endent transmissibilities is presented in this paper. This analysis shows why explicit transmissibilities can ----.1. +;--.~oh ?e~t?i~ti~n ,fQT CQn@ rc>ull ;T2 c Se-”’e?e ..rr. G--.-y simulation. The use of implicit production terms in the dij~erence equaiions to redfLCe i?lsidiii:i~~ is discussed and examples are given. These examples show that the implicit handling of production terms alone can result in a fivejold increase in permissible time step /or a coning simulation with virtually no increase in computing time per time step. A laboratory water-coning experiment was simulated and excellent agreement was obtained between computed and observed results. A three-phase coning example for a gravity-segregation reservoir is also p ~esen’ted. INTRODUCTION Sim.u]ation ~f coning behavior is normally done by numerically solving the flow equations expressed in cylindrical (r, z, d) coordinates with symmetry in n.id”-l manuscript received in Society of Petroleum Engineers -.. s...office Aug. 29, 1969. Revised manuscript received March 10, 1970. Paper (SPE 2595) presented at SpE qqth Annual Fall Meeting, held in Denver, Colo., Sept. 28-Ott. 1, 1969. @ Copyright 1970 American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. *Presently with international Computer Applications Ltd. in Houston, Tex. lReference~ given ~~end of paper. This paper will be printed in Transactions volume 249, which will cover 1970. SEPTEMBER, 1970 5%=rthe 0 direction. The finite-difference technique of numerical solution of differential equations requires that the portion of the reservoir being simulated be divided into grid blocks as shown in Fig. 1. Since coning is a well phenomenon and not a gross reservoir phenomenon, the grid blocks must necessarily be relatively smaii in the vicinity of the wellbore because both pressures and saturations vary rapidly in this region. Severe computational instability is commonly encountered in the simulation of coning due to the relatively small grid-block sizes and high flow velocities in the vicinity of the wellbore. During a time step that would be considered normal for most reservoir simulation problems, a block near the wellbore is required to pass a volume of fluid many times its pore volume. Computational instability resuits when saturation-dependent quantities hi tile finite-difference solution of the flow equations are set at the beginning of a time step and held constant