Numerical Homogenization of the Absolute Permeability Tensor Around a Well
- Zijl Wouter (Netherlands Inst. of Applied Geoscience TNO) | Anna Trykozko (Warsaw U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2001
- Document Type
- Journal Paper
- 399 - 408
- 2001. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.3.2 Multiphase Flow, 5.5.3 Scaling Methods, 4.1.2 Separation and Treating, 5.1.5 Geologic Modeling, 5.5 Reservoir Simulation, 4.3.4 Scale
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This paper presents an extension of the classical homogenization method of upscaling around a well. The use of homogenization theory has been made possible by a transform of the cylindrical homogenization cell around a well to an "equivalent rectangular cell."
First, it is shown by an introductory example for radial flow that direct use of the original homogenization cell leads to more than one coarse-scale permeability. This is at variance with the approach based on the transform to an equivalent rectangular cell, which yields one unique coarse-scale permeability that can be applied consistently both in Darcy's law and in the dissipation equation.
The main body of the paper describes the extension of this transformational approach to general 3D permeability tensors, without assuming purely radial flow. The pressure equation is solved numerically; from this solution, classical homogenization algorithms compute the "transformed homogenized permeability tensor" in the equivalent rectangular cell.
Finally, a back transform to the original homogenization cell leads to the coarse-scale permeability tensor. Validation examples compare the numerical homogenization with global upscaling.
The modeling of hydrocarbon reservoirs can be separated into two activities: geological modeling and flow modeling with reservoir simulators. The geological model focuses on the geometry and the dimensions of the subsurface layers and faults, and on their rock types and properties. The flow model focuses on quantities like pressure, pressure drop, flux, and dissipation. These quantities are related to each other by Darcy's law and the dissipation equation, in which the rock's permeability occurs.
When trying to match the geological model with the fluid-flow model, it generally turns out that the gridblocks of the flow simulator are at least a hundred times larger in volume than the units of the geological model. To counter this mismatch, the fine-scale permeabilities of the geological data model have to be upscaled to coarse-scale permeabilities that relate the gridblock-averaged pressure, flux, and dissipation to each other.
Well-known are the arithmetic and harmonic averages for flow respectively parallel and normal to layers. Equally well-known is the geometric average for the permeability's isotropic checkerboard pattern and the isotropic lognormal distribution. Kirchhoff's method, in which Darcy's law is treated in the same way as Ohm's law applied to an electric network, is popular for more complex configurations. In a reiterated form, applied repeatedly to larger cells containing smaller cells, the method is known under the name renormalization method.1 However, a disadvantage of this method is that directional effects in the form of off-diagonal coarse-scale permeability components cannot be represented.
To counter this disadvantage, the homogenization method has gained much interest.2-8 Homogenization is a method for upscaling of periodic media. Each upscaling cell requires the solution of the pressure equation with three sets of periodic boundary conditions on the boundary of the homogenization cell.
Boundary conditions that are consistent with the actual flow might appear superior above the more-or-less arbitrarily chosen periodic boundary conditions. However, this apparent superiority may be challenged. To satisfy Darcy's law, the coarse-scale permeability must also be defined as K(pf)=-<q>(d<p>/dx)-1, where <p> is the volume-averaged pressure and <q> is the volume-averaged flux.8 In addition, to satisfy the dissipation equation, the coarse-scale permeability must be defined as K(pd)=-(d<p>/dx)-2, where is the volume-averaged dissipation.2 (Of course, the full theory will be based on the 3D extension of the above equations.) Only if K(pf)=K(pd)=K, the coarse-scale Darcy's law and dissipation equation can be used together consistently. Such consistency is required not only in the determination of the pressures and fluxes in the reservoir, but also in the pumping power of a well, which is equal to the dissipation in the reservoir around the well. Also in thermal problems, where dissipation-based thermodynamic laws like Onsager's relations,9 play a role, consistency is necessary. Fortunately, it is a well-known result of homogenization theory that, for periodic boundary conditions, the two definitions, K(pf) and K(pd), lead to the same coarse-scale permeability K.2,8,10 However, when using boundary conditions derived from an actual flow pattern, this consistency is generally lost.10
Another advantage of periodic boundary conditions is that bilateral error bounds of the numerical approximations can be established rigorously following complementarity techniques.11,12
Finally, taking the actual flow boundary conditions into account means that not just the upscaling cell, but a larger domain that includes the upscaling cell, has to be part of the computations, which is a disadvantage from a computational point of view.
The homogenization method strongly relies on the mathematical theories of functional analysis and multiple asymptotic expansions.2-8 However, in this paper, homogenization will be introduced from a relatively simple physical point of view, in such a way that the final equations are exactly the same as those obtained from classical homogenization theory. In addition, our approach leads to a relatively simple extension to homogenization around a well. For that purpose, the cylindrical homogenization cell around a well is transformed to an equivalent rectangular cell.
First, the method is introduced by considering purely radial flow to a well as an example, showing that this transformational approach leads to a consistent result, while direct use of the original cell leads to a number of inconsistencies. Also, the relation with Peaceman's well model13,14 will be established. Then, the main body of the paper deals with the extension to general 3D permeabilities, without the need to postulate purely radial flow.
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