Analytical methods are presented to determine pressure-transient and productivity data for deviated wells in layered reservoirs. The computational methods, which are based on Laplace transforms, can be used to generate type curves for use in direct analyses of pressure-transient data and to determine the effective skin of such wells for use in productivity computations.
Deviated wells with full or limited flow entry are very common, especially in offshore developments. The pressure-transient behavior of fully penetrating deviated wells were investigated by Cinco et al.1 for homogeneous reservoirs. Ref. 1 also contains a correlation for the pseudoradial skin factor for wells with deviation up to 75 degrees, with modification indicated for anisotropic reservoirs.
To investigate the behavior of deviated wells in layered reservoirs, the model from Ref. 1 can be used as a first approximation, modified to limited flow entry if necessary, but it has been difficult to use more exact models. It is possible, though, to generalize the methods used by Larsen2,3 for vertical wells to also cover deviated wells in layered reservoirs with and without crossflow. For reservoirs without crossflow away from the wellbore, i.e., commingled reservoirs, it is well-known how Laplace transforms can be used to handle any model with known solution for individual layers. Deviated wells fall into this category. It is therefore enough to consider systems with crossflow.
By including deviated wells with limited flow entry, horizontal wells will also be covered as a special subcategory. Analytical models of this type for horizontal wells have been considered by several authors, e.g., by Suzuki and Nanba4 and by Kuchuk and Habashy.5 Ref. 4 is based on both numerical methods and analytical methods based on double transforms (Fourier and Laplace). Ref. 5 is based on Green's function techniques.
To accurately describe flow near deviated wells, and also to capture crossflow in layered reservoirs, three-dimensional flow equations are needed within each layer. If the horizontal permeability is independent of direction within each layer, flow within Layer j can be described by the equation
under normal assumptions, where kf and denote horizontal and vertical permeability, and the other indexed variables have the standard meaning for each layer. Since an approach similar to that used in Refs. 2 and 3 will be followed, the vertical variation of pressure within each layer must be removed, at least temporarily. One way to accomplish this is to introduce the vertical average
of the pressure within Layer j, where zj-1zj=zj-1+hj and are the z coordinates of the lower and upper boundaries of the layer.