Skip Nav Destination
Filter
Filter
Filter
Filter
Filter

Update search

Filter

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

- Title
- Author
- Author Affiliations
- Full Text
- Abstract
- Keyword
- DOI
- ISBN
- EISBN
- ISSN
- EISSN
- Issue
- Volume
- References
- Paper Number

### NARROW

Peer Reviewed

Format

Subjects

Journal

Date

Availability

1-5 of 5

Keywords: weighting

Close
**Follow your search**

Access your saved searches in your account

Would you like to receive an alert when new items match your search?

*Close Modal*

Sort by

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*25 (06): 902–908.

Paper Number: SPE-11593-PA

Published: 01 December 1985

...James C. Frauenthal; Roland B. di Franco; Brian F. Towler 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...
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

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*25 (04): 565–572.

Paper Number: SPE-12566-PA

Published: 01 August 1985

...Thom Potempa The sensitivity of a numerical steamflooding model with respect to mobility

**weighting**is examined in depth. Three numerical discretization procedures are used in this investigation: a new numerical scheme. a five point finite difference method. and a procedure which, under certain...
Abstract

The sensitivity of a numerical steamflooding model with respect to mobility weighting is examined in depth. Three numerical discretization procedures are used in this investigation: a new numerical scheme. a five point finite difference method. and a procedure which, under certain assumptions, is equivalent to that introduced by McCracken and Yanosik. Three mobility weighting schemes are investigated. The first approach studied is upstream mobility weighting. The second method investigated uses harmonic total mobility weighting and upstream weighting of fractional flow terms. The approach introduced in this investigation uses the kinematic viscosity in the total mobility and the fractional flow terms. Computational results for a simulated steam drive indicate that this new mobility weighting approach is superior to the other two mobility weighting schemes. In particular, the steam displacement model formed from the combination of this new mobility weighting approach and the McCracken and Yanosik discretization procedure is shown to produce realistic simulations of an inverted seven spot pattern under a continuous steam drive.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*23 (01): 135–142.

Paper Number: SPE-10555-PA

Published: 01 February 1983

... functional evaluation of coefficients, lumping the time-derivative matrix,

**weighting**upstream permeability, and adding artificial capillary pressure in overcoming the convergence problems associated with the saturation discontinuity. Although these techniques generally smear steep gradients, Mercer and Faust...
Abstract

Hermite orthogonal collocation yields oscillatory and sometimes nonconvergent numerical solutions to convection-dominated oil reservoir flow problems. A new method of choosing collocation points emphasizes upstream values in the convective terms, yielding oscillation-free solutions to the linear convection-dispersion transport equation and convergent solutions to the Buckley-Leverett problem. Introduction Although collocation promises to be an attractive numerical procedure for the simulation of multiphase flows in porous media, efforts to use this method have had limited success. The main problem in such applications is finding a numerical method in such applications is finding a numerical method that respects the mathematical peculiarities of convection-dominated flows. Mercer and Faust, 1 for example, examined solutions to the coupled pressure and saturation equations describing immiscible displacements in oil reservoirs. They compared several approaches to solving the nonlinear saturation equation, including the Galerkin method and orthogonal collocation on linear, C 0 cubic, and C 1 (Hermite) cubic bases. They also compared the benefits of using functional evaluation of coefficients, lumping the time-derivative matrix, weighting upstream permeability, and adding artificial capillary pressure in overcoming the convergence problems associated with the saturation discontinuity. Although these techniques generally smear steep gradients, Mercer and Faust concluded that sacrificing sharp front-tracking capability is necessary in some instances to obtain convergent solutions on uniform grids. Sincovec 2 applied orthogonal collocation on Hermite basis functions to compute solutions to a nonlinear gas flow problem and to the linear convection-dispersion transport equation, demonstrating improved efficiency over finite difference methods. In applying his method to the Buckley-Leverett saturation equation, however, he used a formulation of the nonlinear term that failed correctly to propagate the saturation front. Sincovec concluded that his collocation procedure is inappropriate for the solution of the saturation equation for immiscible flow. 3

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*12 (06): 515–530.

Paper Number: SPE-3516-PA

Published: 01 December 1972

...M.R. Todd; P.M. O'Dell; G.J. Hirasaki This paper proposed the use of two-point upstream

**weighting**of fluid mobility as an alternative to the generally employed single-point approximation Use of the two-point formula results in the reduction of both numerical dispersion of flood fronts and the...
Abstract

This paper proposed the use of two-point upstream weighting of fluid mobility as an alternative to the generally employed single-point approximation Use of the two-point formula results in the reduction of both numerical dispersion of flood fronts and the sensitivity of predicted areal displacement performance to grid orientation. Stability analysis performance to grid orientation. Stability analysis provides the time-step limitation for control of provides the time-step limitation for control of solution oscillations. This together with limitations for control of overshoot and truncation error provides a practical basis for the automatic selection of time steps. Introduction As an indication of the growing concern for controlling the total cost of large-scale reservoir simulations, the emphasis of a number of recent publications has been directed toward increasing publications has been directed toward increasing computing efficiency. In this paper, two methods to increase the computing efficiency of reservoir simulators are described. The use of two-point upstream weighting of fluid mobility is described and compared with the commonly used single-point upstream approximation. The two-point approximation generally requires fewer grid blocks to obtain a given accuracy than does the single-point approximation. In addition, the calculated performance of areal models is less sensitive to grid performance of areal models is less sensitive to grid orientation when using the two-point approximation. Computing efficiency is also increased with the use of an automatic time-step selector. Time-step limitations are described in this paper for controlling stability, overshoot (negative saturations), and truncation error. In general, these limitations change each time step as conditions change. If any of the limitations is exceeded, the results of the simulation may be meaningless. An automatic time-step selector detects and avoids running difficulties by using the proper time-step size. Using these methods, simulation proper time-step size. Using these methods, simulation results are obtained with less expenditure of engineering and computer time. TWO-POINT APPROXIMATIONS FOR FLUID MOBILITY The majority of general-purpose reservoir simulators reportedly in use today are based on the solution of finite-difference analogs to the conservation equations describing multiphase flow in porous media. Thus, the continuous domain of a reservoir is divided into a number of discrete blocks, and solutions for pressure and saturations are obtained at the grid block centers (or grid points). Central-difference approximations are normally used for the spatial derivatives in the discrete formulation of the conservation equations. As described below, this scheme necessitates the evaluation of flow coefficients (kk,/muB) at the planes separating adjacent grid blocks. As fluid and planes separating adjacent grid blocks. As fluid and reservoir properties are only defined at grid points, some method must be devised for approximating interblock flow coefficients based on values at the grid points. Of the terms that make up the flow coefficients, only the saturation-dependent relative permeability changes rapidly enough from grid block to grid block to cause significant difficulty. Although several weighting schemes have been employed in the past for evaluating the relative permeability at a block face, only single-point upstream weighting appears to be in general use. Unfortunately, use of this weighting scheme is well known to cause excessive numerical dispersion of flood fronts. In addition, areal displacement performance is found to be quite sensitive to the grid orientation for grid meshes of practical extent for large-scale reservoir simulations. This has been demonstrated qualitatively by Garrett and will be described both qualitatively and quantitatively later in this paper. As an alternative to single-point weighting of relative permeability, a two-point weighting of relative permeability, a two-point scheme is now described which results in both reduced numerical dispersion of flood fronts and decreased sensitivity of predicted areal performance to grid orientation. SPEJ P. 515

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Society of Petroleum Engineers Journal*10 (01): 25–32.

Paper Number: SPE-2374-PA

Published: 01 March 1970

... uniformity of PVT properties on a horizontal plane. Therefore, to properties on a horizontal plane. Therefore, to obtain meaningful results from reservoir engineering studies, PVT properties must be carefully

**weighted**vertically. Intensive study of the known oil recovery mechanisms within these highly...
Abstract

The high relief, fractured carbonate reservoirs of the Asmari formation in Iran have extremely thick oil columns, with a large vertical change of reservoir temperature. This large change results in a significant effect on reservoir fluid properties. In contrast, there is a remarkable uniformity of PVT properties on a horizontal plane. Therefore, to properties on a horizontal plane. Therefore, to obtain meaningful results from reservoir engineering studies, PVT properties must be carefully weighted vertically. Intensive study of the known oil recovery mechanisms within these highly fissured systems resulted in a sophisticated reservoir simulation model. The model is programmed to include these recovery processes, which occur essentially in horizontal layers or zones. It also includes a technique for volumetrically weighting the large vertical variation of the PVT data. A description of this weighting process is the primary purpose of this paper. Introduction Oilfield structures found in Iran have been described as very long, high relief, assymmetrical anticlines that contain unusually thick oil columns (see Fig. 1). The oil reservoirs generally are capped with large primary gas accumulations and are often affected by natural water drive. Producing formations include the Asmari, Bangestan, and Khami carbonates, with the Asmari being by far the most common and prolific. Although some Asmari reservoirs have been discovered that contain sandstones interbedded within the limestones and dolomites, most reservoirs in Iran contain the bulk of the oil in compact carbonates that have been contorted and highly fractured during structural deformation. The resulting anticlinal oil accumulations are produced mainly from complex fracture systems, which apparently exist quite uniformly throughout the dense matrix. Study of surface rocks, cores and well producibility show that well developed fissure producibility show that well developed fissure systems are responsible for the excellent fluid communication. Because the fissures contain relatively little oil, maintenance of such prolific rates is dependent upon rate of oil replenishment from the adjacent matrix. The various productive mechanisms determine the rate of replenishment and duration. RESERVOIR AND MODEL CONSIDERATIONS Engineering studies of the fissured Iranian reservoirs have led to a mathematical model that is even more sophisticated than an earlier one. Many complex features have been included in the current model, making it a useful aid for advising management. The model consists of a master digital computer program that encompasses 54 subprograms and runs on any of the IBM 7040, 7090 and 360/65 computers. The known oil recovery mechanisms included in the model are based on consideration of practical oil recovery observations. For example, oil recovery from the expanding gas cap is predicted by using recognized gravity drainage formations, but if necessary, retrograde condensation recovery from the gas cap can also be included. Fluid and rock expansion, solution gas drive, and water displacement are also included, and vary according to the reservoir in question. But the manner in which gas production differs from what is usually observed in a solution gas drive must be known to include a proper simulation of behavior. Experience has shown that the producing gas-oil ratio continues to decline during the entire producing life of a well unless the proximity of the gas cap results in gas coning, at which time the well is shut in. Material balance calculations show that the initial gas-oil separation singe actually occurs in the reservoir, with the fissure system readily enabling the liberated gas to segregate toward the gas cap. SPEJ P. 25