A Nine-Point Finite Difference Operator for Reduction of the Grid Orientation Effect

1983 ◽  
Author(s):  
P.C. Shah
Keyword(s):  
Finite Difference ◽  
Difference Operator ◽  
Orientation Effect ◽  
Grid Orientation ◽  
Grid Orientation Effect

Reduction of Grid-Orientation Effects in Reservoir Simulation With Generalized Upstream Weighting

1985 ◽  
Vol 25 (06) ◽  
pp. 902-908 ◽  
Author(s):  
James C. Frauenthal ◽  
Roland B. di Franco ◽  
Brian F. Towler
Keyword(s):  
Finite Difference ◽  
Reservoir Simulation ◽  
Computation Time ◽  
Three Dimensions ◽  
Orientation Effect ◽  
Grid Line ◽  
Orientation Effects ◽  
Grid Orientation ◽  
Upstream Weighting ◽  
Grid Orientation Effect

Abstract A generalization of upstream weighting is proposed as a method for reducing grid-orientation effects in reservoir simulation. For the two sample problems studied,. a piston-flow waterflood and a realistic gas injection, the piston-flow waterflood and a realistic gas injection, the grid-orientation effect was almost completely eliminated. The new generalized upstream weighting (GUW) method is particularly attractive because it is fast and accurate, and particularly attractive because it is fast and accurate, and can be added easily to an existing simulator that uses upstream weighting. Introduction The grid-orientation effect is a well-known phenomenon in finite-difference reservoir simulation. Numerical results are highly dependent on the orientation of the finite-difference grid imposed on the model. In practice it occurs whenever one has a strongly adverse mobility ratio. This happens when one tries to push a viscous oil with a highly mobile fluid, such as steam or hydrocarbon gas. This paper presents a technique for reducing grid-orientation effects that is fast, flexible, and easily added to an existing simulator. A good survey of the research in this area was recently published. With this in mind, we will give an published. With this in mind, we will give an idiosyncratic interpretation of some of the techniques suggested by others. The main numerical difficulty in petroleum reservoir simulation is largely a consequence of the need to estimate individual phase mobilities halfway between finite-difference gridpoints. Because averaging the values from adjacent gridpoints is numerically unstable, the midgridpoint typically is assigned the value at the next upstream point. The idea of looking upstream for information point. The idea of looking upstream for information is found throughout much of computational fluid dynamics. Many improvements on one-point upstream weighting have been proposed in the reservoir simulation literature. The principal attractions of these techniques are that they can be interchanged easily within existing computer codes and do not add significantly to computation time. We found that the upstream weighting procedures have a common feature. If the midgridpoint in procedures have a common feature. If the midgridpoint in question lies, for example, on a grid line in the x direction, these techniques consider only other points on this same grid line in the extrapolation/interpolation process. A second body of literature developed around the idea of using a nine-point (instead of the standard five-point) finite-difference scheme to represent two-dimensional (2D) second derivatives. Because the nine-point scheme is a weighted superposition of two 5-point grids with a common center point and a 45 * relative rotation, the procedure averages away the grid-orientation effect to some extent without explaining it. Nevertheless, the nine-point grid schemes include one attractive feature absent from the upstream schemes: the weighting parameter can be tuned to improve the quality of the results. parameter can be tuned to improve the quality of the results. Perhaps the biggest fault of these procedures is that they Perhaps the biggest fault of these procedures is that they do not extend easily to three dimensions. The widening of the matrix bandwidth also increases the computation time. Our proposed technique is a modification of a procedure used successfully in the convective-heat transfer literature. It amounts to a generalization of one-point upstream weighting, accomplished by the introduction of mobility values from nearby points that lie in the true upstream direction rather than along a single grid line. This is explained in more detail in the next section. Note that the technique requires very little computer time. In fact, because most reservoir simulators use an automatic timestep adjustment, the improved stability of the technique, relative to standard upstream procedures, allows larger timesteps to be taken. Also, two adjustable parameters that permit the grid-orientation effect to be almost parameters that permit the grid-orientation effect to be almost completely eliminated are introduced. Finally, because the procedure works well with the standard five-point finite-difference representation of 2D second derivatives, it generates easily to three dimensions and is completely compatible with most reservoir simulators. Governing Equations The conservation equations for multiphase fluid flow in porous media are well known. However, the porous media are well known. However, the equations for three-phase flow are listed below for completeness. The continuity equations are as follows. SPEJ P. 902

One Approach to the Grid Orientation Problem in Reservoir Simulation

1981 ◽  
Vol 21 (02) ◽  
pp. 160-161 ◽  
Author(s):  
Paul K. Vinsome ◽  
Anthony D.K. Au
Keyword(s):  
Shock Front ◽  
Single Point ◽  
Pressure Profile ◽  
Mobility Ratio ◽  
Orientation Effect ◽  
Grid Orientation ◽  
Reservoir Simulator ◽  
Phase Mobility ◽  
Line P ◽  
Grid Orientation Effect

Consider a linear, incompressible, two-phase flow in the absence of capillary pressure. Assume that a saturation shock front exists as depicted schematically in Fig. 1, and that the mobility behind the front is extremely high. During any time step of a numerical simulation, the exact position of the shock front generally is not known, so we assume that in an average sense the position of the shock may be placed midway between the nodes. In any numerical reservoir simulator that contains an upstream formulation, the value of Ss on Fig. 1 would be representative of the saturation at which the interblock phase mobility is evaluated. This would result in the calculation of the pressure profile shown by the solid Line P. There is clearly some inconsistency, as the true pressure profile is more like the dashed Line P'. Therefore one possible source of the grid orientation effect is that the upstream finite-difference formulation produces far too small a pressure drop across a shock front, and that the directionally dependent pressure field truncation error is a significant fraction of this pressure drop.Equating fluid velocities on either side of the shock front gives (1) where lambda T represents the total fluid mobility. If the pressure at the shock front p s is eliminated from these equations, we obtain (2) whereby it can be seen that the total effective mobility is given by (3) The harmonic mobility given by Eq. 3 is closer to twice the smaller value of (lambda Tu, lambda Td) than to the upstream value, so it has the capability of producing larger pressure drops across a shock front.On Fig. 1, it can be seen that the fluid fraction flowing at the shock front is more similar to the upstream value. This suggests that for the individual phase mobilities an upstream formulation should be phase mobilities an upstream formulation should be used: (4) with a similar upstream expression for the oil phase.Inspection of Eq. 4 reveals that at unit mobility ratio it reduces to the standard single-point formulation. Also, in a one-dimensional incompressible system with fixed injection rate, upsilon T in Eq. 2 is constant. Under these circumstances, the saturation profiles calculated using the harmonic mobilities are profiles calculated using the harmonic mobilities are identical to the standard single-point upstream, although the pressure profiles are different. This similarity to single-point upstream under many circumstances and the fact that the results of Todd et al. already show marked grid orientation effect at unit mobility ratio for single-point upstream suggest that the single-point harmonic formulation should be extended to two-point harmonic. The simplest way to achieve this is to replace the value of lambda wu in Eq. 4 by the two-point value. The overshoot constraints we have used are that the extrapolated value must not be less than min (lambda wu, lambda wd) nor greater than max (lambda wu, lambda wd). We shall refer to this scheme as "two-point harmonic." The net result is very simple. The normal upstream mobilities lambda in any numerical reservoir simulator are to be modified by the term 2 lambda Td/lambda Tu + lambda Td). P. 160

A New Grid Block System for Reducing Grid Orientation Effect

2007 ◽  
Vol 25 (11) ◽  
pp. 1473-1492 ◽  
Author(s):  
E. Chong ◽  
Z. Syihab ◽  
E. Putra ◽  
D. T. Hidayati ◽  
D. S. Schechter
Keyword(s):  
Orientation Effect ◽  
Block System ◽  
Grid Block ◽  
Grid Orientation ◽  
Grid Orientation Effect

scholarly journals Design of coupled finite volume schemes minimizing the grid orientation effect in reservoir simulation

2021 ◽  
Vol 425 ◽  
pp. 109923
Author(s):  
Karine Laurent ◽  
Éric Flauraud ◽  
Christophe Preux ◽  
Quang Huy Tran ◽  
Christophe Berthon
Keyword(s):  
Finite Volume ◽  
Reservoir Simulation ◽  
Orientation Effect ◽  
Finite Volume Schemes ◽  
Grid Orientation ◽  
Grid Orientation Effect
Sign in / Sign up

Export Citation Format

Share Document