Upscaling for Reservoir Simulation
- M.A. Christie (BP Exploration)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- November 1996
- Document Type
- Journal Paper
- 1,004 - 1,010
- 1996. Society of Petroleum Engineers
- 5.5.8 History Matching, 5.5 Reservoir Simulation, 5.1 Reservoir Characterisation, 4.3.4 Scale, 5.1.5 Geologic Modeling, 5.5.3 Scaling Methods
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Distinguished Author Series articles are general, descriptiverepresentations that summarize the state of the art in an area of technology bydescribing recent developments for readers who are not specialists in thetopics discussed. Written by individuals recognized as experts in the area,these articles provide key references to more definitive work and presentspecific details only to illustrate the technology. Purpose: to informthe general readership of recent advances in various areas of petroleumengineering.
Upscaling has become an increasingly important tool in recent years forconverting highly detailed geological models to simulation grids. This paperreviews and summarizes both single- and two-phase upscaling techniques.
A principal motivation for the development of upscaling techniques has beenthe development of geostatistical reservoir description algorithms.1-3 Thesealgorithms now routinely result in fine-scale descriptions of reservoirporosity and permeability on grids of tens of millions of cells. Thedescriptions honor the known and inferred statistics of the reservoirproperties. Fig. 1 shows an example of such a reservoir description. Thesereservoir-description grids are far too fine to be used as grids in reservoirsimulators. Despite advances in computer hardware, most full-field reservoirmodels still use fewer than 100,000 cells, a factor of 100 down on thegeological grid.
Upscaling is needed to bridge the gap between these two scales. Given afine-scale reservoir description and a simulation grid, an upscaling algorithmassigns suitable values for porosity, permeability, and other flow functions tocells on the coarse simulation grid. Many possible choices of upscalingapproach exist; see Refs. 4 through 7 for examples.
The simplest form of upscaling is single-phase upscaling. Here, the aim issimply to preserve the gross features of flow on the simulation grid. Thealgorithm calculates an "effective permeability," which results in the sametotal flow of single-phase fluid through the coarse, homogeneous block as thatobtained from the fine heterogeneous block.
In the pressure-solver method,8 we set up a single-phase-flow calculationwith specified boundary conditions and then ask what value of effectivepermeability yields the same flow rate as the fine-grid calculation. Fig. 2shows a schematic. The results we obtain depend on the assumptions we make,particularly with regard to boundary conditions. The most common assumption isthat no-flow boundary conditions exist on the walls of the cube. This givesrise to a diagonal tensor that can be entered directly into a reservoirsimulator. Fig. 3 shows an example two-dimensional calculation that has scaleda 128´128 fine grid up to an 8´8 coarse grid. Although almost all the finedetail has been lost, the broad features are retained.
Directional Effective Permeabilities.
To calculate directional effective permeabilities, we set up calculations inthe x, y, and z directions as follows.
Set up a matrix equation to solve with no-flow boundary conditions along thesides, p=1 at the inlet, and p=0 at the outlet. Solve the equation and sum thefluxes in the x direction. The effective permeability is then given by . Thisapproach is simple and very effective in most circumstances. For example, Begget al.8 were able to obtain effective vertical permeabilities usingpressure-solver techniques that agreed very closely with the values that hadbeen obtained with a history-matching technique.
Full-Tensor Effective Permeabilities.
Alternatively, some authors9-13 assume periodic boundary conditions andcalculate a full-tensor effective permeability. This is significantly moreaccurate, but has the disadvantage that it cannot be directly entered into acommercial reservoir simulator. Tensor effective permeabilities are still thesubject of active research, particularly in the area of symmetry.12,14Durlofsky14 gives a good summary of scaleup involving tensor permeabilities. Hefavors application of periodic boundary conditions. His approach is supportedby Pickup,13 who compared the accuracy and robustness of several boundaryconditions in calculating effective permeabilities.
Renormalization methods offer a faster, but less accurate, method ofcalculating an effective permeability. For most cases, renormalization giveseffective permeabilities close to direct solution of the pressure equation andallows rapid calculation of effective permeabilities from very large gridsystems. The renormalization approach works by taking a large problem andbreaking it down into a hierarchy of manageable problems. It has provedsuccessful in a variety of theoretical physics areas.
The renormalization method for effective permeabilities was pioneered byKing,15 who used a resister-network analogy to write down direct expressionsfor effective permeabilities on sequences of 2´2 cells. Fig. 4 shows theprocedure. A small group of cells is extracted, then the effective permeabilityis calculated and put back in place of the original fine group of cells. Thiscan be repeated for many levels and gives a fast estimation of effectivepermeability. Renormalization is not limited to 2´2 cells and resister-networkanalogies and can be coded for arbitrary changes of scale between levelsthrough use of direct methods for matrix inversion.
Other techniques that should be mentioned include effective medium theory,15power-law averaging,16 harmonic-/arithmetic-mean techniques, and homogenizationtheories.17 These techniques are generally very fast, but suffer from somelimitations in applicability.
Limitations in Upscaling.
One of the main limitations of upscaling is that it usually gives an answerwith almost no indication of whether the assumptions made in deriving theanswer hold. Limited attempts have been made to analyze the upscalingprocess,18 but so far, no good theory exists that unequivocally states whetheran upscaled value provides a good or bad approximation.
Some areas are known to give rise to concern whether the upscaled values aregood approximations; these include large-aspect-ratio gridblocks, significanttransport at an angle to the grid lines, and upscaled gridblocks close in sizeto a correlation length of the system. The main practical advice that can begiven under these circumstances is to try varying the parameter causingconcern. For example, where correlation lengths are close to upscaled gridblocksizes, you can see a significant change in upscaled value for coarse-grid sizesof half or twice the original size.
Another factor is that the effective permeability values depend on thedifference operator used to solve the pressure equations as well as thepermeabilities on the underlying fine grid. This can be particularly importantfor large-aspect-ratio gridblocks typically used to calculate effectivevertical permeabilities. Some recent work by Edwards19 offers potential inreducing the effects of this gridblock aspect-ratio problem.
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