Accelerated Perturbation Boundary Element Model for Flow Problems in Heterogeneous Reservoirs


Kozo Sato







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The boundary element method (BEM), successfully applied to uid ow problems in porous media, owes its elegance, the computational eciency and accuracy, to the existence of the fundamental solution (free-space Green's function) for the governing equation. If there is no such solution in a closed form, the BEM cannot be applied, and, unfortunately, this is the case for ow problems in heterogeneous media. To overcome this diculty, the governing equation is decomposed into various order perturbation equations, for which the free-space Green's function can be found. At each level of perturbation, the solution is computed by the BEM and the sum- mation of various order perturbation solutions gives the complete solution for the original governing equation.

The convergence of the perturbation series depends on the distance to the nearest singularity inherent in the equation. A greater magnitude of heterogeneity makes this distance shorter, and, hence, the rate of convergence becomes slower and eventually the series diverges. However, it is possible to elicit a significant amount of information from the perturbation series and to recover an accurate approximation to the exact solution. To this end, Pade approximants are employed to accelerate the rate of convergence of a slowly convergent series and to convert a divergent series into a convergent series.

Two kinds of perturbation boundary element models are developed: one for steady-state ow problems, associated with the Laplace operator and the other for transient ow problems, associated with the modi ed Helmholtz operator in Laplace space. These models are veri ed against analytical solutions for simpli ed problems and are utilized to solve various application problems.

The perturbation BEM shows its utility in streamline tracking and well testing problems in heterogeneous media. The analytical nature of the solution is well pre- served through the free-space Green's function methodologically and through the singularity programming technically. The durability of the model to rapid spatial variability inrockproperties is established by using Pade approximants.

Through the veri cation and application problems, it is observed that if the av- erage property value within a drainage area is not much di
erent from the near-well property value, heterogeneity has little e
ect on pressure responses.

The perturbation forms for steady-state and transient ow equations derived in this study should be of value in formulating any semi-analytical scheme.

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