Effects of Grid Type and Difference Scheme on Pattern Steamflood Simulation Results
Authors: Coats K.H. , Ramesh A.B. 
Published: December 2007

Summary
This paper presents five-, seven-, and nine-spot pattern steamflood simulation results with parallel and diagonal grids and five- and nine-point difference schemes. The effects of difference scheme and grid orientation are examined in a two-dimensional (2D) (vertical) cross section. Effects of different types of grid spacings in cyclic steam simulation are discussed. Sample problem data sets are roughly representative of a California deposit and a more viscous Alberta crude. Various seven-spot grids (neither parallel nor diagonal) and nonuniform five-/nine-spot grids pose unique challenges to both difference schemes, with interesting results. The conclusions emphasize superiority of the nine-point difference scheme and the pitfalls of certain problem/grid/difference-scheme combinations in pattern steamflood simulation. Nine-point transmissibility alterations are used that allow rigorous use of 1/8 five-/nine-spot patterns as opposed to 1/2 or 1/4 elements with either parallel or diagonal grids. This is important because of significantly reduced cost compared with 1/4- or 1/2-pattern elements. Subject to certain conditions, the paper presents a simple procedure to calculate well-productivity indices for uniform or nonuniform grids, cross sections, or any of the three patterns, and either of the two difference schemes.

Brief Model Description
The thermal model we used is extended considerably beyond a model previously described. While the basic formulation and PVT representation are unchanged, a large number of features have been added. We will briefly discuss only the additional features pertinent to the results presented in this paper. The nine-point difference scheme is an option in addition to the conventional five-point scheme. This nine-point scheme is coded in the planes for areal or three-dimensional (3D) problems and in the x-z plane for 2D cross sections. Alterations of transmissibilities necessary to run 1/8 symmetry elements of (areally) homogeneous five- and nine-spot patterns are included in the model for five- and nine-point difference schemes with uniform or variable grid spacing. An implicit bottomhole pressure (BHP) feature exactly preserves specified rates for wells not on deliverability or constraints (e.g., maximum steam production rate). For n such wells, this feature introduces n additional variables [bottomhole flowing pressure (BHFP) for each well] and n additional constraint equations into the matrix of equations requiring solution. This feature is coded for the five-point D4 Gauss ordering, the ordinary Gauss ordering used in the nine-point scheme, and the iterative solution technique ordering.

Data Used
The results presented were obtained for the three data sets of Table 1. Various fluid and reservoir properties of these data sets are representative of various heavy-oil deposits in the U.S. and Alberta. However, no single data set should be considered representative of any particular reservoir. Injection rate is 187.5 B/D [29.8 m /d] [cold water equivalent (CWE)] of steam per pattern at 400 deg. F [200 deg. C] and 80% quality. Production rates for the pattern are reported on a full-pattern basis regardless of whether a 1/8, 1/4, 1/12, etc., pattern element is actually being simulated. Similarly, individual well production rates are reported as full well rates. Unless otherwise noted. the production wells were placed on deliverability against a BHFP of 50 psia [344.7 kPa] with no maximum or limiting production rate specified. Well-productivity indices in the pattern element runs were obtained as described in the Appendix and have units of RB-cp/day-psi. The simulator places appropriate multiplicative (time-varying) mobilities for all phases on these productivity indices. The relative permeability endpoints and exponents listed in Table 1 were used in analytical expressions for relative permeability. Most runs were performed to 4,475 days, which corresponds to about four hydrocarbon PV's of steam injected, defined as volume of steam (CWE) injected divided by initial volume of hydrocarbon-occupied pore space.

Description of Grids
Fig. 1 illustrates the block-centered parallel and diagonal grids used for symmetrical elements of repeated five- or nine-spot patterns. Fig. 1a shows a parallel grid for 1/2 of a nine-spot pattern. The inscribed Triangle 1-2-3 is the minimal symmetrical, 1/8-pattern element. Fig. 1b shows a diagonal grid for a 1/4 nine-spot pattern. The inscribed Triangle 1-1-3 is the minimal 1/8-pattern element. P. 557^