Second Comparative Solution Project: A Three-Phase Coning Study

Abstract

Second Comparative Solution Project: A Three-Phase Project: A Three-Phase Coning Study Summary Eleven companies participated in the Second Comparative Solution Project. The problem to be solved is a three-phase coning problem that can Project. The problem to be solved is a three-phase coning problem that can be described as a radial cross section with one central producing well. The oil and water densities are nearly equal, so the oil/water capillary transition zone extends high up into the oil column. Wide variations in rates occur, and the solution GOR is unusually high for oil with such high density. These problem characteristics make the problem difficult to solve, thus increasing its value as a test of simulation techniques. Various aspects of the numerical solutions obtained are compared in this paper. In general, the solutions agree reasonably well. paper. In general, the solutions agree reasonably well. Introduction The previous effort in this series generated considerable interest. On request, on April 30, 1981, Aziz Odeh presented the paper for a second time to a full house of presented the paper for a second time to a full house of the London Petroleum Section of SPE. The interest in continuing a project such as this is great--both in the U.S. and abroad. Following his London success, Aziz Odeh wrote in a letter to SPE, "Extension of the cooperative effort started with this publication to cover more complex models and problems would be very beneficial to the industry. I recommend that the SPE undertake such an endeavor. I believe the industry will welcome such a project." During the organization of the 1982 SPE project." During the organization of the 1982 SPE Symposium on Reservoir Simulation, the program chairman, Khalid Aziz, therefore suggested that we organize a comparison of results on another test problem. Because the previous problem had been a field-scale simulation, it was suggested that a coning study might be of interest. A problem drawn from an actual field case was simplified somewhat to provide a challenging test problem. It is a single-well radial cross section that problem. It is a single-well radial cross section that involves gas and water coning as well as gas repressuring. It is a difficult problem that provides a good test of the stability and convergence behavior of any simulator. Invitations were mailed to a number of oil companies and consulting organizations. Eleven companies chose to participate in the project (Table 1). The organizers tried participate in the project (Table 1). The organizers tried to invite every company that had a distinct simulation capability. Doubtless, some were inadvertently omitted, and we apologize to such groups. We feel that the numerical solutions obtained agree surprisingly well, considering the diversity of discretization and solution techniques used. Whether the consensus is actually close to the mathematical solution is, of course, still an open question. The paper presents the text of the problem, a comparison of results in graphical and problem, a comparison of results in graphical and tabular form, and a brief description of each model. We have included a display of data concerning simulator performance (e.g., number of timesteps, number of Newtonian performance (e.g., number of timesteps, number of Newtonian iterations, and timing). Calculations were performed on a number of different computers. Because the problem is small, it is difficult to draw any general conclusions from the data; in this paper, we point out a few ideas that have occurred to us. As remarked by some participants, the problem is rather artificial in that it involves rate variations that would not be likely to occur in practice. Furthermore, the solution GOR is unusually high for oil with such a high density. Both of these characteristics, however, make the problem more difficult to solve, increasing its value as a test problem more difficult to solve, increasing its value as a test of simulation techniques. Description of the Simulators ARCO Oil and Gas Co. ARCO's two coning simulators are implicit, three-phase, black-oil simulators. The numerical formulation in both versions is a linearized semi-implicit scheme with upstream weighting for phase mobilities. Within a timestep, only the nonlinear accumulation term is updated if necessary. The algebraic equations are solved directly. One version treats the wells as distributed sources and sinks. Phase and layer production rate allocation are based on the phase mobility of each layer. The D4 reordering scheme is used to improve efficiency. Three-phase relative permeabilities are calculated by Stone's first method. The second version strongly couples a well equation with the reservoir equations and solves simultaneously for the reservoir variables. as well as the well rate or bottomhole pressure (BHP). Pressure gradient in the wellbore is assumed hydrostatic. Standard ordering of gridblocks is used for solution of linear equations. Three-phase relative permeabilities are calculated by Stone's second method. P. 345