A Two-Scale Numerical Subgrid Technique for Waterflood Simulations
- Todd Arbogast (U. of Texas at Austin) | Steven L. Bryant (U. of Texas at Austin)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2002
- Document Type
- Journal Paper
- 446 - 457
- 2002. Society of Petroleum Engineers
- 5.3.2 Multiphase Flow, 4.1.5 Processing Equipment, 5.2.1 Phase Behavior and PVT Measurements, 5.5.3 Scaling Methods, 4.3.4 Scale, 5.5 Reservoir Simulation, 5.4.1 Waterflooding, 4.1.2 Separation and Treating
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We present a two-scale numerical subgrid technique for simulating waterflooding. Local subgrid computations are combined with a coarse-grid computation to provide a fine-scale resolution of the solution. We use porosity, relative and absolute permeabilities, the location of wells, and capillary pressure curves defined on the fine scale. No explicit macroscopic coefficients nor pseudofunctions result. The method is several times faster than solving the finescale problem directly, generally more robust, and yet achieves good results as it requires no ad hoc assumptions at the coarse scale and retains all the physics of the original multiphase flow equations.
One of the more challenging problems in reservoir simulation is resolution of all the pertinent scales in both the data and the solution. Computational power limitations generally prevent one from using as fine a grid as would be desired, especially when multiple simulations for Monte Carlo or other statistical analyses are required. There are two natural approaches to address this limitation. One is to change the scale on which the data are represented by some kind of averaging procedure, and then to solve the problem on a coarse grid. The other natural approach is to improve the resolution of the numerical simulation on a coarse grid through some type of subgrid technique. In either case, the desire is to change from the fine scale on which the problem is defined to a coarser scale by defining an appropriate coarse-scale problem that captures in some way the fine-scale details. Broadly speaking, we might call either approach upscaling; however, as is common in the petroleum industry, we will use the term "upscaling" to refer to the former approach of finding averaged parameters (such as permeability) suitable for use on a coarse scale. In this paper we present an approach of the latter, subgrid type.
There is a large and growing literature on upscaling techniques. We will not attempt a literature review here (see, e.g., the excellent review by Renard and de Marsily1); we merely mention a few of the main techniques. Early techniques involved in an essential way averaging or homogenization of physical parameters such as permeability. 2-4 While such upscaling techniques can be very effective for purely linear problems, they are less satisfactory for nonlinear problems. They suffer from the elementary observation that a nonlinear function of an average is not the average of the nonlinear function. For example, over a coarse gridblock, the value of capillary pressure evaluated at the average saturation is not at all the same as the average over the gridblock of the capillary pressure.
More sophisticated upscaling techniques have been developed to circumvent the inadequacies of simple averaging,5,6 including the development of renormalization techniques to successively upscale to coarse levels, pseudofunctions, and stochastic methods. Numerical subgrid techniques have also been developed, including using modified finite element basis functions7 and explicit subgrid techniques that seek to improve the resolution of the coarse solution after it has been computed.
These techniques all attempt in some way to represent finescale information on coarse scales in an indirect way, and sometimes require at least some information about the nature of the flow that is expected under field management conditions. Although most upscaling and subgridding techniques are dynamic in that they respond to the changing state of the reservoir, many do so through anticipation of the possibilities. Often one needs some kind of closure assumption such as the imposition of local boundary conditions, the expected primary flow direction, or expected limits on certain parameters such as flow rates.
To handle the dynamic and sometimes unanticipated nature of reservoir conditions, we present in this paper an implicit numerical subgrid technique that allows us to finely resolve the pressure equation even though we end up solving the problem on a coarse grid. This technique is a locally conservative variational multiscale method.8 We also discuss its implementation in a sequential twophase waterflood research simulator. We maintain a fully implicit (as opposed to an explicit) coupling between the coarse and fine or subgrid scales, and we obtain a fine-scale representation of the reservoir state. As a consequence, we make use of the capillary pressure and relative permeability curves directly and accurately on the fine scale on which they are defined. No pseudofunctions are needed, nor do any arise in our technique. Our technique allows us to handle fine scales in the heterogeneous absolute permeability and porosity, the nonlinear functions relative permeability and capillary pressure, and even the fine-scale position of wells.
The idea is to consider the simulation as defined on a fine grid, and to de-refine this grid to form a reasonable coarse grid over which we can compute the solution. We break the solution into two parts, the coarse-scale representation of the solution plus the subgrid part. The subgrid part is defined inside the coarse gridblocks. In order to be able to compute it efficiently, it must involve only the coarse solution itself and local information. Because of this computational restriction and the need to maintain accuracy, we compute the coarse-scale Darcy velocity using a second order accurate method. Thus, even though the coarse part of the velocity has accuracy based on the coarse scale resolution H (the diameter of a coarse gridblock), the expected accuracy is actually proportional to H2. Thus, the coarse-scale velocity is accurate from the point of view of the subgrid scale, for which we use a more standard first-order accurate method such as cell-centered finite differences. The use of a low-order method for the solution on the subgrid scale is natural because heterogeneities in the permeability are likely to produce solutions with large spatial gradients. It is well known that higher-order methods do not in general improve such solutions much, certainly not enough to justify the added cost.
Because we insist on an implicit coupling between the coarse and subgrid scales, there will be a mixture of these two parts of our solution in the equations. Some kind of static condensation or Schur complement technique is needed to eliminate the subgrid unknowns from the equations. We do this using a technique involving numerical Greens functions, also called influence functions. This technique allows us to treat the subgrid and coarse scales in completely separate parts of the computer code. It is also relatively memory-efficient.
To maintain local mass conservation, our procedure is based on mixed finite element methods.9,10 It is known that the lowest order Raviart-Thomas method,11 RT0, when combined with numerical quadrature to evaluate some of the integrals that arise, is the same as cell-centered finite differences. A similar method that gives higher-order velocities uses the finite element spaces defined in 2D by Brezzi, Douglas, and Marini (BDM1),12 and generalized in 3D by Brezzi, Douglas, Duràn and Fortin (BDDF1).13
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