Summary
Analytical methods are presented to determine the pressure-transient behavior of multibranched wells in layered reservoirs. The computational methods are based on Laplace transforms and numerical inversion to generate type curves for use in direct analyses of pressure-transient data. Any number of branches with arbitrary direction and deviation can in principle be handled, although the computational cost will increase considerable with increasing number of branches. However, due to practical considerations, a large number of branches is also unlikely in most cases.
Introduction
With increased interest in multibranched wells as a means to improve productivity, it is important to have computational methods for predictions and analyses of such wells. Ozkan et al.1 presented such solutions for dual lateral wells in homogeneous formations. The present paper extends these results to multibranched wells in layered reservoirs. The approach covers reservoirs both with and without formation crossflow, but cases without crossflow can also be handled similar to homogenous reservoirs. Boundary effects are not included, but can be added from an equivalent homogeneous model if pseudoradial flow is reached within the infinite-acting period.
The methods used in this paper are direct extensions of methods presented by Larsen2 for deviated wells in layered reservoirs. The results in Ref. 2 apply for any deviation, and hence, also for horizontal segments within different layers. The approach was restricted, however, to cover at most one segment within each layer with no overlap vertically. In the approach used in the present paper, these restrictions have been removed.
Mathematical Approach
Except for simple cases with only vertical branches, general multibranched wells will require a three-dimensional flow equation within individual layers to capture the flow geometry. 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. Copying Ref. 2, an approach similar to Refs. 3 and 4 will be followed with vertical variation of pressure within each layer removed by passing to the vertical average. For Layer j, the new pressure
p a j ( x , y , t ) = 1 h j ∫ z j − 1 z j p j ( x , y , z , t ) d z , ( 2 )
is then obtained, where z j?1 and zj= zj?1+hj are the z coordinates of the lower and upper boundaries of the layer.
There is one major problem with the direct approach above. It cannot handle the boundary condition at the wellbore for nonvertical segments. To get around this problem, each perforated layer segment will be replaced by a uniform-flux fracture in the primary solution scheme. This approach is illustrated in Fig. 1 for a two-branched well in a three-layered reservoir, with Branch 1 fully perforated through the reservoir and Branch 2 fully perforated in Layers 1 and 2 and partially completed with a horizontal segment in Layer 3. Since an infinite-conductivity wellbore (consisting of the branches) will be assumed, a time-dependent skin factor is added to each fracture to get the actual branch (i.e., deviated well) pressure from the fracture solution. This is identical to the approach used in Ref. 2 for individual branches.
With branch angle ?j (as a deviation from the vertical) and completed branch 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 branch length Lwj is assumed to consist of a single fully perforated interval. The fracture half-length in layers with vertical branch segments will be set equal to the wellbore radius rwa To capture deviated branches with more than one interval within a layer, the model can be subdivided by introducing additional layers.
If Eq. 1 is integrated from zj?1 to z j, as shown in Eq. 2, then the new flow equation
k j h j ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) p a 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 a j ∂ t ( 4 )
is obtained. The two gradient terms remaining in Eq. 4 represent 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 expressions in the form
k j ′ ∂ p j ∂ z | z j = k j + 1 ′ ∂ p j + 1 ∂ z | z j = λ j ′ ( p a , j + 1 − p a j ) ( 5 )
for each j, where λj′ is a constant determined from reservoir parameters or adjusted to fit the response of the well. For details on how to choose crossflow parameters, see Refs. 3 and 4, and additional references cited in those papers. Additional fracture to well drawdown is assumed not to affect this approach.
Since vertical flow components are important for deviated branches, the crossflow parameters in Eq. 5 will be important elements of the mathematical model. If, for instance, the standard choice from Refs. 3 and 4 is used, then vertical flow will be reduced even in isotropic homogeneous formations, but doubling the default ? is sufficient in many cases to remove this error. However, since these parameters will be quite uncertain in field data anyway, the modeling should be more than adequate.
Study and Application of Intelligent Sliding Mode Control for Voltage Source Inverters