Abstract
The numerical simulation of coning behavior bas been one of the most difficult applications of numerical analysis techniques. Coning simulations have generally exhibited severe saturation instabilities in the vicinity of the well unless time-step sizes were severely restricted. The instabilities were a result of using mobilities based on saturations existing at the beginning of the time step. The time-step size limitation, usually the order of a few minutes, resulted in an excessive amount of computer time required to simulate coning behavior. This paper presents a numerical coning model that exhibits stable saturation and production behavior during cone formation and after breakthrough. Time-step sizes a factor of 100 to 1,000 times as large as those previously possible may be used in the simulation. To ensure stability, both production rates and mobilities are extrapolated production rates and mobilities are extrapolated implicitly to the new time level. The finite-difference equations used in the model are presented together with the technique for incorporating the updated mobilities and rates. Example calculations which indicate the magnitude of the time-truncation errors are included. Various factors which affect coning behavior are discussed.
Introduction
The usual formulation of numerical simulation models for multiphase flow involves the evaluation of flow coefficient terms at the beginning of a time step and assumes that these terms do not change over the time step. These assumptions are valid only if the values of pressure and saturation in the system do not change significantly over the time step.
The design of a finite-difference model to evaluate coning behavior of gas or water in a single well usually results in a model which uses radial coordinates. A two-dimensional single-well model is illustrated in Fig. 1. This type of model will often produce finite-difference blocks with pore volumes less than 1 bbl near the wellbore while producing large blocks with pore volumes greater producing large blocks with pore volumes greater than 1 million bbl near the external radius. If one chooses to use a reasonable time-step size of, say, 1 to 10 days, then normal well rates would result in a flow of several hundred pore volumes per time step through blocks near the wellbore. Therefore the assumption that saturations remain constant, for the purpose of coefficient evaluation, is not valid. Welge and Weber presented a paper on water coning which recognized the limitation of using explicit coefficients and applied an arbitrary limitation on the maximum saturation change over a time step. While this method is workable for a certain class of problems, it is not rigorous and is not generally applicable. In 1968, Coats proposed a method to solve the gas percolation problem which is similar in that it also results from explicit mobilities. This proposal involved adjusting the relative permeability to gas at the beginning of the time step so that an individual block would not be over-depleted of gas during a time step. This method is not conveniently extended to two dimensions nor to coning problems where a block is voided many times during a time step. Blair and Weinaug explored the problems resulting from explicitly determined coefficients and formulated a coning model with implicit mobilities and a solution technique utilizing Newtonian iteration. While this method is rigorous, achieving convergence on certain problems is difficult and, in many cases, time-step size is still severely restricted. In addition to the problems resulting from explicit flow-equation coefficients in coning models, the specification of rates requires attention to ensure that the saturations remain stable in the vicinity of the producing block.
SPEJ
P. 418
The Effect of Iterative Procedures on the Robustness and Fidelity of Augmented Lagrangian SPH
