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Figure 5 illustrates the effect of the dimensionless reservoir height (h_D) (see Table 2). Bilinear flow in the horizontal plane depends only on the permeability ratio (Equation 12). Again, the horizontal permeability ratio was kept constant. The well is located halfway between the upper and lower boundaries. The height of the reservoir was changed from h_D = 0.5 (the upper curve) to h_D = 0.2 (lower curve). The interpretation is: radial flow in the vertical plane, bilinear flow in the horizontal plane and pseudo-steady flow (straight line of unit slope). Again, the bilinear flow model is satisfactory. The non-occurrence of the pseudo-radial flow period indicates that the drainage area is too small for this period to appear.

Figure 6: Effect of Area

Comparison of limiting equation against analytical solution

The effect of the dimensionless drainage area is shown in Figure 6. The data provided in Table 3 corresponds with a correlation factor Ch = 0.666. The interpretation is: radial flow in the vertical plane, bilinear flow in the horizontal plane and finally pseudo-steady flow. The effect of increasing drainage area is to push the unit slope straight lines in the right-hand direction. A pseudo-radial flow period (horizontal line) appears between the bilinear and the pseudo-steady-state period for the larger drainage areas. The simplified model cannot predict pseudo-steady flow since the assumption of an infinite-acting reservoir is implied in the mathematical formulation.

Table 3: Well and Reservoir data, Case 3

Conclusion

Flow period diagnostics are an important aid in selecting the most probable reservoir model. Well test interpretation depends on the use of a correct model. This article supports the proposition that bilinear flow may occur for horizontal wells in a homogeneous reservoir. The interpretation is that the interaction between the directional properties of the permeability tensor and external boundaries separates the reservoir into distinct flow regions. The occurrence of a bilinear flow period may also be caused by heterogeneities. Both possibilities should be considered.

Limiting equations for bilinear flow in a homogeneous reservoir have been derived. The equations may explain unexpected pressure signatures of bilinear appearance. The conventional (well-known) linear flow model is included in the bilinear flow model.

The validity of the correlation factors seems to be confirmed. The correlation factor for bilinear flow in the horizontal plane remains unchanged during a linear to bilinear flow transition. The permeability ratio will present as a parameter on a type curve designed for bilinear flow in the horizontal plane.

The model for bilinear flow in the horizontal plane captures the transition from essentially linear flow to pure bilinear flow. The model for bilinear flow in the vertical plane does not have this capability, hence, experimental support for the latter is weaker.

Nomenclature

a adjustment parameter
h height (m)
k permeability (m²)
L length (m)
p pressure (Pa)
S skin factor (dimensionless)
s Laplace variable t time (s)
x Distance along the reservoir x-axis
y Distance along the reservoir y-axis
z Distance along the reservoir z-axis

Subscripts

D dimensionless
h horizontal direction
p equivalent plane source
w wellbore
x x-direction
y y-direction
v v-direction

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