This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract Sequential solution of the reservoir flow equations has been made to yield the stability of calculations previously associated only with the simultaneous solution previously associated only with the simultaneous solution of the equations. Sequential solution is computationally more efficient than simultaneous solution, and it permits the use of simulation algorithms that are more dependable when applied to irregular-shaped, highly heterogeneous reservoirs. Such reservoirs are often troublesome to simulate. Coning, gas percolation, and other high-flow velocity problems have been made tractable through the use of semi-implicit saturation functions. Introduction Sequential solution of the flow equations in a reservoir simulator has been made to yield the calculational stability previously associated only with simultaneous solution of the equations. This new approach forms the basis of a general-purpose reservoir simulator, GPSIM. Sequential solution is computationally more efficient than simultaneous solution, and it permits the use of algorithms that are more dependable when predicting the performance of irregular-shaped, highly heterogeneous performance of irregular-shaped, highly heterogeneous reservoirs. Such reservoirs, which are frequently encountered in practice, are often quite troublesome to simulate. Well coning problems and fieldwide, multiwell gas percolation problems have been made tractable through the optional use of semi-implicit saturation functions including capillary pressures and portions of the Buckley-Leverett fractional flow terms. The semi-implicit approach used in GPSIM is closely related to the semi-implicit relative permeability approach previously described in the literature. Sequential solution conveniently separates the solution of the pressure equation, which is often predominantly elliptic in nature, from that of the saturation equations, which range between parabolic and hyperbolic in character. This permits approximation of the saturation equations by alternating-direction difference equations, which can be solved by direct elimination, thus avoiding an iterative solution. Optional use of the backwards-in-time difference formulation solved by strongly implicit iteration (SIP) or direct solutions methods is provided in GPSIM for use on especially difficult problems such as coning studies.

This paper presents results derived from potential flow theory for single-phase flow in the region near an injection or production well. The relations obtained are directly applicable to reservoir studies where coarse grid systems, which cannot account for near-wellbore flow, are used to describe flow within the reservoir. Near-wellbore effects have received only minor attention in the literature. This work concentrates on a general solution to this flow problem. Relations are derived for a system with one or two wells located in a rectangular area and for n wells located in an infinite area. The paper also gives a method that accounts for areal flow behavior near a wellbore in cross-sectional simulations. Comparisons of the analytical results with computer results using Peaceman's relationship for the effective wellbore radius of a simulator well show good agreement for cases where the wells are located at the block center regardless of the grid size or the position of the well block within the grid system. The paper also illustrates how pressure boundary conditions can be applied in cases where the wells are not located at the center of a grid block. The derived relationships are applicable for a simulator based on either five- or nine-point formulations. Introduction The use of pressure boundary conditions in a reservoir simulation generally involves the replacement of the well rate q in the difference equation with a relationship that is a function of the grid-block pressure. A relationship that can be used is given by Eq. 1, where p is the grid-block pressure and prw is the specified boundary condition pressure. The success of this procedure depends on a realistic interpretation of the pressures in grid blocks containing wells. In most simulation applications using rectangular grid systems, the grid-block pressure is not a good approximation of a flowing bottomhole pressure. The large size of grid blocks compared with wellbore dimensions and the radial nature of flow patterns near the wellbore are factors that must be considered in the interpretation of well-block pressures. Several authors have addressed the problem of well-block pressure interpretation. They interpret the pressure as an average pressure over a representative volume or as a flowing pressure that applies at some "effective" distance from the well location. The approach used here to analyze block pressures is similar to the technique developed by Peaceman. Peaceman interpreted the calculated block pressure as a flowing pressure at an equivalent radius from the center of the block. By assuming isotropic permeabilities, square blocks, and single-phase flow, he showed that the equivalent radius is approximately equal to 0.2 times the length of the block edge. This relationship was obtained for an interior well block in an areal model where the well is located at the center of the block. In this paper, relationships are derived for systems where the well block is a corner or edge block and for situations where the well is not centered in a grid block. The work shows that the equivalent radius of 0.2 times the length of the block edge, derived by Peaceman for an interior grid block, is a good approximation for both edge and corner well blocks. However, the relationship is different when the well is displaced from the block center to the block corner or the block edge. The interpretation of block pressures in two other commonly encountered simulation applications also is analyzed. The first deals with block pressure interpretation when a block contains more than one well, and the second addresses the effects of flow convergence near wells in cross-sectional simulations. SPEJ P. 473＾

The ma!enal is ;ub]ectto correctionby the author. Permission to copy is restricted10an abstract of not more than 300 words. Wme SPE. P O. Box 833636. Rlcharctson. resee 75463.3836. Telex: 730989 SPE DAL. \bstract simulation. By 1982, the fraction had grown to almost Advances in computing hardware and softwareone-half and currently most of our studies make use of three-d"imensional modeling in some way. Thesẽ ave provided the reservoir engineer with a wide same trends are noted throughout the industry. range of alternatives in analyzing and projecting reservoir behavior. The ability to economically runThe second factor concerns relative computing and large simulation models offers a means to generate manpower costs. Modeling efforts which require the nore accurate technical answers and to develop full-field models of larger, more complex reser-use of pseudo functions are much more manpower intensive than those that don't.Since 1990 it is #oirs. However, there are still situations, due estimated that the cost of professional engineering either to reservoir complexity or size, in which manpower has increased by about 26 percent, In the full-field models will not be practical. Therefore, detailed segment models will continue to be an same time period, computing costs have been reduced by about an order of magnitude.Thus it is a logimportant part of reservoir behavior analysis and development planning, and there is still a need for ical evolution to shift the emphasis in reservoir engineering applications toward the more computing approximate approaches, such as the use of pseudo intensive approaches. relative permeability and capillary pressure functions.However, even with the computing advances, there