Upscaling for Reservoir Simulation
Authors: Christie M.A. 
Published: June 2014

Distinguished Author Series articles are general, descriptive representations that summarize the state of the art in an area of technology by describing recent developments for readers who are not specialists in the topics discussed. Written by individuals recognized as experts in the area, these articles provide key references to more definitive work and present specific details only to illustrate the technology. Purpose: to inform the general readership of recent advances in various areas of petroleum engineering.

Summary
Upscaling has become an increasingly important tool in recent years for converting highly detailed geological models to simulation grids. This paper reviews and summarizes both single- and two-phase upscaling techniques.

Introduction
A principal motivation for the development of upscaling techniques has been the development of geostatistical reservoir description algorithms.1-3 These algorithms now routinely result in fine-scale descriptions of reservoir porosity and permeability on grids of tens of millions of cells. The descriptions honor the known and inferred statistics of the reservoir properties. Fig. 1 shows an example of such a reservoir description. These reservoir-description grids are far too fine to be used as grids in reservoir simulators. Despite advances in computer hardware, most full-field reservoir models still use fewer than 100,000 cells, a factor of 100 down on the geological grid.
Upscaling is needed to bridge the gap between these two scales. Given afine-scale reservoir description and a simulation grid, an upscaling algorithm as signs suitable values for porosity, permeability, and other flow functions to cells on the coarse simulation grid. Many possible choices of upscaling approach exist; see Refs. 4 through 7 for examples.

Single-Phase Upscaling
The simplest form of upscaling is single-phase upscaling. Here, the aim is simply to preserve the gross features of flow on the simulation grid. The algorithm calculates an "effective permeability," which results in the same total flow of single-phase fluid through the coarse, homogeneous block as that obtained from the fine heterogeneous block.
Pressure-Solver Methods.
In the pressure-solver method,8 we set up a single-phase-flow calculation with specified boundary conditions and then ask what value of effective permeability 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 is that no-flow boundary conditions exist on the walls of the cube. This gives rise to a diagonal tensor that can be entered directly into a reservoir simulator. 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 fine detail has been lost, the broad features are retained.

Directional Effective Permeabilities.
To calculate directional effective permeabilities, we set up calculations in the x, y, and z directions as follows.
Set up a matrix equation to solve with no-flow boundary conditions along the sides, p=1 at the inlet, and p=0 at the outlet. Solve the equation and sum the fluxes in the x direction. The effective permeability is then given by . This approach is simple and very effective in most circumstances. For example, Begget al.8 were able to obtain effective vertical permeabilities using pressure-solver techniques that agreed very closely with the values that had been obtained with a history-matching technique.
Full-Tensor Effective Permeabilities.
Alternatively, some authors9-13 assume periodic boundary conditions and calculate a full-tensor effective permeability. This is significantly more accurate, but has the disadvantage that it cannot be directly entered into a commercial reservoir simulator. Tensor effective permeabilities are still the subject 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 supported by Pickup,13 who compared the accuracy and robustness of several boundary conditions in calculating effective permeabilities.

Renormalization Techniques.
Renormalization methods offer a faster, but less accurate, method of calculating an effective permeability. For most cases, renormalization gives effective permeabilities close to direct solution of the pressure equation and allows rapid calculation of effective permeabilities from very large grid systems. The renormalization approach works by taking a large problem and breaking it down into a hierarchy of manageable problems. It has proved successful in a variety of theoretical physics areas.
The renormalization method for effective permeabilities was pioneered by King,15 who used a resister-network analogy to write down direct expressions for effective permeabilities on sequences of 2´2 cells. Fig. 4 shows the procedure. A small group of cells is extracted, then the effective permeability is calculated and put back in place of the original fine group of cells. This can be repeated for many levels and gives a fast estimation of effective permeability. Renormalization is not limited to 2´2 cells and resister-network analogies and can be coded for arbitrary changes of scale between levels through use of direct methods for matrix inversion.

Other Techniques.
Other techniques that should be mentioned include effective medium theory,15power-law averaging,16 harmonic-/arithmetic-mean techniques, and homogenization theories.17 These techniques are generally very fast, but suffer from some limitations in applicability.

Limitations in Upscaling.
One of the main limitations of upscaling is that it usually gives an answer with almost no indication of whether the assumptions made in deriving the answer hold. Limited attempts have been made to analyze the upscaling process,18 but so far, no good theory exists that unequivocally states whether an upscaled value provides a good or bad approximation.
Some areas are known to give rise to concern whether the upscaled values are good approximations; these include large-aspect-ratio gridblocks, significant transport at an angle to the grid lines, and upscaled gridblocks close in size to a correlation length of the system. The main practical advice that can be given under these circumstances is to try varying the parameter causing concern. For example, where correlation lengths are close to upscaled gridblock sizes, you can see a significant change in upscaled value for coarse-grid sizes of half or twice the original size.
Another factor is that the effective permeability values depend on the difference operator used to solve the pressure equations as well as the permeabilities on the underlying fine grid. This can be particularly important for large-aspect-ratio gridblocks typically used to calculate effective vertical permeabilities. Some recent work by Edwards19 offers potential in reducing the effects of this gridblock aspect-ratio problem.