Summary
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 types curves for use in direct analyses of pressure-transient data and to determine the effective skin of such wells for use in productivity computations.
Introduction
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. Reference 1 also contains a correlation for the pseudoradial skin factor for wells with deviation up to 75°, 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 Reference 4 is based on both numerical methods and analytical methods based on double transforms (Fourier and Laplace). Reference 5 is based on Green's function techniques.
Mathematical Approach
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
k j ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) p j + k j ′ ∂ 2 p j ∂ z 2 = μ ϕ j c t j ∂ p j ∂ t ( 1 )
under normal assumptions, where kj and kj′ 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
P j ( x , y , t ) = 1 h j ∫ z j − 1 z j p j ( x , y , z , t ) d z ( 2 )
of the pressure within layer j, where zj−1 and zj=zj−1+hj are the z coordinates of the lower and upper layer boundaries.
There is one apparent problem with the approach above, it cannot handle the boundary condition at the wellbore directly. For each perforated layer, the well segment will therefore be replaced by a uniform flux fracture in the primary solution scheme, as illustrated in Fig. 1 for a fully perforated deviated well and in Fig. 2 for a partially perforated well with variable angle, with a transient skin effect used to correct from a fractured well solution to a deviated well solution. With well angle θj (as deviation from the vertical) and completed well length Lwj in layer j, the associated fracture half-length will be given by the identity
x f j = 1 2 L w j s i n θ j ( 3 )
for each j. The completed well length Lwj is assumed to be a single fully perforated interval. The fracture half-length in layers with vertical well segments will be set equal to the wellbore radius rw. To capture deviated wells with more than one interval within a layer, the model can be subdivided by introducing additional layers.
Although the well deviation is allowed to vary through the reservoir, the well azimuth will be assumed constant. The projection of the well in the horizontal plane can therefore be assumed to follow the x axis, and hence assume that y=0 along the well. Keeping the well path in a single plane is actually not necessary, but it simplifies the mathematical development and the computational complexity.
If Eq. (1) is integrated from zj−1 to as shown in Eq. (2), then the flow equation
k j h j ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) P j + k j ′ ∂ p j ∂ z | z j − k j ′ ∂ p j ∂ z | z j − 1 = μ ϕ j c t j h j ∂ P j ∂ t ( 4 )
is obtained, with the gradient terms representing flux through the upper and lower boundaries of layer j. In the standard multiple-permeability modeling of layered reservoirs, the gradient terms are replaced by difference terms in the form
k j ′ ∂ p j ∂ z | z j = k j + 1 ′ ∂ p j + 1 ∂ z | z j = λ j ′ ( P j + 1 − P j ) ( 5 )
for each j, where λj′ is a constant determined from reservoir parameters or adjusted to fit the well response. For details on how to choose crossflow parameters, see Refs. 2 and 3 and additional references cited in those papers. Additional fracture to well drawdown is assumed not to affect this approach.
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