The Impact of Upscaling Errors on Conditioning a Stochastic Channel to Pressure Data
- Fengjun Zhang (U. of Tulsa) | A.C. Reynolds (U. of Tulsa) | D.S. Oliver (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2003
- Document Type
- Journal Paper
- 13 - 21
- 2003. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 1.2.3 Rock properties, 5.5.8 History Matching, 5.1 Reservoir Characterisation, 5.1.5 Geologic Modeling, 5.5.3 Scaling Methods, 5.6.4 Drillstem/Well Testing
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This work discusses an automatic history-matching procedure to efficiently condition the location and geometry of a stochastic channel to well-test pressure data. Our emphasis is on computational difficulties that are related to the nonlinearity of the objective function, or arise from the definition of effective permeability on a reservoir simulation gridblock that is intersected by one or more channel boundaries. We show that upscaling gridblock permeabilities for gridblocks intersected by channel boundaries introduces spurious local minima in the objective function that must be optimized to obtain a realization of the model.
We show that an automatic history-matching procedure can be used to define upscaled permeabilities. When the permeability contrasts across facies boundaries are extreme, however, the resulting properties may be influenced by the difference in local truncation errors between the fine- and coarse-grid systems. Moreover, the upscaled permeabilities obtained by history matching depend on the flow regimes reflected in the pressure data history matched.
The problems we observed are not specific to channels but will occur whenever boundaries of facies with large contrasts are estimated by history matching. One important objective of this work is to provide computational insight for the general problem of conditioning a reservoir facies model to production data by automatic history matching.
Several authors have previously considered the estimation of a channel's location, size, and shape from pressure data. One of the most general papers on this problem is that of Bi et al.1 In their procedure, channel geometric parameters, channel and nonchannel permeability and porosity are all modeled as random variables or fields. The authors present an automatic history-matching procedure to simultaneously generate realizations of all model parameters. The stochastic channel used in Bi et al. 1 is essentially identical to a model for simulating river beds within a channel belt that was introduced by Georgsen and Omre. 2 A lengthy and fairly comprehensive review of many relevant publications (e.g., Landa,3 Landa and Horne, 4 and Rahon et al. 5) related to the problem of conditioning a channel to pressure data was given by Bi et al. 1 and will not be presented here.
In the automatic history-matching procedure employed in this work, channels are moved and deformed within a 3D reservoir simulation grid to minimize an objective function that includes the sum of squares of pressure mismatch terms. The method requires that we implement a procedure to define effective permeability values for simulation gridblocks that are intersected by one or more channel boundaries. For 2D channels, Landa3 used an average permeability based on the proportion of the area of the gridblock occupied by the channel. Rahon et al., 5 who modeled a channel by the triangulation of a 3D surface, used homogenization to define permeability and porosity in gridblocks intersected by the surface of the channel. Bi et al.1 used a volumetric (or arithmetic) average to define effective gridblock permeabilities and porosities. Intuitively, it seems more appropriate to use a combination of arithmetic and harmonic averages, depending on the flow direction. Unfortunately, this type of upscaling can introduce spurious local minima in the objective function. The effect of upscaling on the objective function is more significant when effective gridblock permeability is defined as a harmonic average. As neither the harmonic nor the volumetric average is strictly correct in all situations, we introduce an automatic history-matching procedure to define upscaled gridblock permeabilities.
In this work, we simulate a single channel within a rectangular 3D box and condition the stochastic channel to pressure data obtained at one or more wells by automatic history matching. Several authors have previously considered the estimation of a channel's location, size, and shape from pressure data. One of the most general papers on this problem is that of Bi et al.1 The stochastic channel model used by Bi et al.1 and in this paper is a simplified version of the model that was introduced by Georgsen and Omre2 for simulating river beds within a channel belt.
The channel center and boundaries are defined by a marked point process that associates four Gaussian random fields with a principal direction line (PDL). The PDL is defined by its intercept at x=0 and by the slope of the projection of the PDL onto the x-y and x-z planes. The location of the channel center is defined by the horizontal and vertical deviation from the PDL. Two additional stationary Gaussian random fields, the width and aspect ratio, are used to define the channel boundaries. (The aspect ratio is equal to the width divided by the thickness.)
To embed a simulated channel within the reservoir simulation grid, it is necessary to determine the location of channel boundaries within the simulation grid. Knowledge of these intersection surfaces is used to define permeabilities and porosities for gridblocks that contain both channel and nonchannel facies. For any x such that xi-1/2 <x i+1/2, the thickness, the width, and the deviation from the PDL are constant. The top and bottom boundaries of a channel panel lie in two parallel horizontal planes and the left and right boundaries lie in two parallel vertical planes. Thus, as shown in Bi et al.1 and Bi,6 it is easy to write down the equations that define the location of these boundaries relative to the reservoir simulation grid. The complete channel is obtained by gluing a sequence of channel panels together as indicated in Fig. 1.
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