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SPE Members Abstract This paper presents 3-D pre-processors to calculate proper well inflow relations for modeling horizontal proper well inflow relations for modeling horizontal or curved inclined wells in a finite difference reservoir simulator. The technique frequently used in the oil industry is Peaceman's analytical well model although several assumptions are violated in the case of a horizontal well. An error analysis was conducted for the well model proposed by Peaceman. Based upon this analysis, new methods are proposed to improve the accuracy of modeling horizontal and curved inclined wells. Introduction There are several methods to handle the correlation between flow rate and well pressure in standard finite difference reservoir simulators. The most simple correlation was given by Donald W. Peaceman (Ref.1,2,9) by the following equation: (1) (2) (3) Peaceman proved analytically that the finite Peaceman proved analytically that the finite difference reservoir model with the wells represented by Eqs. 1 to 3 is consistent, i.e., the solution of the model approaches the true solution as the mesh size becomes infinitely small, if the reservoir is infinitely large, the permeability is isotropic (very good approximation for anisotropic permeability), the fluid flow around a well is at pseudo steady state and most importantly, the mesh size is uniform. If Peaceman's analytical well model is applied to a horizontal well, five types of error become serious (although there are additional minor source of error). These errors are (a) boundary effect if a sealing boundary is nearby; (b) error induced by nonuniform mesh size; (c) effect of three dimensional flow due to partial penetration; (d) pressure distribution error partial penetration; (d) pressure distribution error due to elongated mesh; and (e) error due to small number of blocks through which a well passes. These errors also apply to a partially penetrating vertical well but the errors are less significant for a vertical well since the reservoir rock properties and mesh tend to be relatively uniform in the lateral direction, and the lateral flow is more dominant than the axial flow. When Peaceman originally proposed his well model, he probably did not intend to apply his model to those wells where a well is located near a sealing boundary; where flow is three dimensional; nor where the mesh size varies around a well. However, reservoir model users tend to ignore these restrictions because there are few ideal situations which can satisfy his restrictions. Hence, it is worth studying the error induced if these restrictions are ignored. The error analyses lead to proper mesh configurations which can minimize the error induced by Peaceman's well model. Peaceman's well model. This work shows that if local mesh refinement or remeshing is allowed for a well oriented parallel to the reservoir mesh, the f low rate error with Peaceman's well formula can be minimized to less than Peaceman's well formula can be minimized to less than 6 to 7 percent except for some unusual conditions. Although Peaceman's formula is useful, there are many field conditions which are not suitable to apply Peaceman's formula. For instance, it may not be Peaceman's formula. For instance, it may not be realistic to force the well orientation in the direction parallel to the mesh, if this does not match the intended orientation of the well, since the well productivity may change if the well is not oriented in productivity may change if the well is not oriented in the principal permeability directions. Actually many horizontal wells have curvature and partially penetrate into cap rocks. penetrate into cap rocks. P. 49

The problem of the reflection of SH waves at the interface between two half-space is considered. Plane waves are incident from the upper medium, which is supposed to be homogeneous. Exact solution of the equation of motion is utilized to calculate the reflection coefficient. The modulus and phase of reflection coefficient is compute for p=0, 1,2. These are compared with the corresponding results when both the media are homogeneous. It is found that the inhomogeneity has a distinct effect on the modulus as well as on the phase of the reflection coefficient