T.A. Jelmert
Norwegian Institute of Technology
Trondheim, NorwaySven A. Vik
Saga Petroleum a.s.
Sandvika, Norway
Flow that appears bilinear may occur in homogeneous reservoirs with horizontal wells.
The importance of specific flow periods for model identification and well test interpretation has been extensively discussed in literature.1 A bilinear flow period, which is controlled by reservoir heterogeneity, is well known, but this type of flow may also appear in homogeneous reservoirs.
Failure to recognize the possibility of bilinear flow by geometry may lead to the assumption of a false reservoir model. This in turn will lead to erroneous well test interpretation.
The heterogeneity required to induce bilinear flow can be caused by fractures of finite conductivity (hydraulic and natural) or by layers of high-permeability contrast.
Bilinear flow may occur for a long horizontal well or vertical fracture of high conductivity (uniform flux or infinite conductivity). The controlling factors for the existence of the bilinear flow period are length of the well and the directional permeabilities.
The key parameter is the geometrical factor, such as the length of the well.
Horizontal well behavior
When considering flow towards a long horizontal well in a homogeneous reservoir, several possible flow periods are known. The primary flow periods may be summarized as follows:
- Initially, there is radial flow in the vertical plane.
- At a later stage, most of the flow will occur in the horizontal plane.
Usually the specific flow periods are identified from pressure derivatives. As a result, the constant pressure drop associated with the skin disappears.
Because there is no difference between the derivative responses, a horizontal well may be replaced by a source in the form of a plane for modeling purposes.
Bilinear flow period
The reservoir response to a rate change may be thought of as a pressure or potential disturbance that travels from the well and into the reservoir. The most comprehensive model for horizontal flow around a horizontal well or a high conductivity vertical fracture, is of an elliptical type. Both linear and radial flow are included as special cases.
Linear flow may occur initially when the pressure disturbance is traveling in the immediate neighborhood of the well. Radial flow occurs at a later stage when the pressure disturbance has penetrated deep into the formation. In between these times, there is a possibility of a bilinear flow period.
Origin of bilinear flow
Bilinear flow, which in the context of elliptical flow occurs immediately after the linear period, may be explained as follows:
- Opposite the long horizontal section, the flow is essentially linear.
- In addition, the areas beyond the tips of the well contribute to the production.
Van Everdingen and Hurst2 found that radial and linear flow obeys the same limiting equation for small values of time. Based on their findings, we approximate the hemiradial flow region by a linear one.
The concept is illustrated in Fig. 1b(65497 bytes). The result is bilinear flow. This concept was originally proposed by Cinco-Ley and Samaniego to explain the pressure transient behavior of finite conductivity fractures.3 The major difference is that the latter occurs on a much smaller length scale.
A schematic of bilinear flow due to a finite conductivity fracture is shown in Fig. 1b(65497 bytes).
Bilinear flow models
Contrary to predictions from model studies, the early linear-flow period apparently occurs rarely in practical tests. Du and Stewart4 investigated pressure responses of horizontal wells. According to them, field examples quoted in this paper do not show this feature (linear flow) as often as one might anticipate and unexpectedly quarter slopes have been observed in many cases.
After this statement they proceed to show that bilinear flow may occur for double porosity and layered reservoirs.
Although references to bilinear flow in homogeneous reservoirs are scarce, one may observe this behavior in published model studies. Kuchuk, et. al.,5 show an example which has segments of near bilinear behavior. Another example of the same behavior may be found in Reference 6.
Pressure signature
The analogy between the flow towards a low-conductivity fracture and a horizontal well during bilinear flow, is evident from Figs. 1b and 1c(65497 bytes) and 1c.
Because the streamlines are the same in the horizontal plane, the two cases are controlled by the same equation. The only differences are the existence of a skin factor and a different nomenclature.
Cinco-Ley and Samaniego derived Equation 1 (16514 bytes) for bilinear flow in a finite conductivity fracture.3 The k and kf denote formation and fracture permeability while b denotes fracture width.
Replacing the fracture flow parameters with the horizontal well equivalents in Equation 1 (16514 bytes) yields Equation 2 (16514 bytes). The Dps denotes pressure loss due to the pseudoskin. The length of the well is Lw, and the directional permeabilities parallel and perpendicular to the well bore axis are denoted by ky and kx, respectively.
Equation 2 (16514 bytes) will plot as a straight line with Slope m as defined in Equation 3 (16514 bytes) and intercept Dps with the t = 0 axis when Dp is plotted vs. t1/4.
The equation of the slope, Equation 3 (16514 bytes), provides a relationship between the three unknowns, kx, ky, and Lw. Two additional equations are required to solve for all of the unknowns. In some cases such information exists or may be estimated.
Dimensionless equations
Most model studies are presented in dimensionless form, by which the pressure response predicted by Equation 2 (16514 bytes) and results from previous model studies can be compared directly.
The dimensionless parameters are defined by Equations 4-6 (16514 bytes). The x is the coordinate in the direction perpendicular to the well.
Substitution of these into Equation 2 (16514 bytes) yields Equation 7 (16514 bytes).
In case of an isotropic reservoir (in the horizontal plane), Equation 7 (16514 bytes) will reduce to Equation 8 (16514 bytes).
The logarithmic derivative of Equation 8 (16514 bytes) becomes Equation 9 (16514 bytes) that will plot as a straight line with quarter slope and intercept 0.43 with the t = 1 axis in log-log coordinates.
Example
The numerical results in this study are calculated by a model based on Greens functions.7 It has been coded to compute the pressure response for a well located in a box-shaped reservoir.
As an illustration of the occurrence of bilinear flow, an example provided by Kuchuk, et. al.,5 has been selected. The relevant reservoir data are listed as Well 1 in Table 1 (16514 bytes). (The data given in Example 1 of the original paper have been rescaled to be consistent with Equations 4 and 5 (16514 bytes).)
We assigned large distances between the outer boundaries, LxD and LyD, to compute the infinite-acting reservoir response.
The result is shown in Fig. 2a (78143 bytes). The graph shows three curves. The upper one is the pD function. The dashed curve is the logarithmic derivative, while the straight line is the logarithmic derivative under the ideal bilinear flow assumption, Equation 9 (16514 bytes). Because Kuchuk, et. al.,5 used a different calculation approach their graph is slightly different.
The following flow periods (in chronological order) are presented in Fig. 2a (78143 bytes):
- Vertical radial
- Transition period (near bilinear)
- Hemiradial in the vertical plane
- Bilinear
- Pseudoradial flow.
- Radial flow in the vertical plane
- Transition
- Bilinear flow
- Pseudosteady flow.
References
1.Ehlig-Economides, C.A., Hegeman, P., and Vik, S.A., Guidelines to simplify well test interpretation, OGJ, July 18, 1994, pp. 33-40.
2.van Everdingen, A.P., and Hurst, W., The Application of the Laplace Transformation to Flow Problems in Reservoirs, Paper No. TP 2732, Petroleum Transactions, AIME, Vol. 186, December 1949, pp. 305-24.
3.Cinco-Ley, H., and Samaniego-V., F., Transient Pressure Analysis for Fractured Wells, JPT, September 1981, pp. 1749-66.
4.Du, K.F., and Stewart, O., Transient Pressure Response of Horizontal Wells in Layered and Naturally Fractured Reservoirs With Dual Porosity Behavior, Paper No. SPE 24682, 67th SPE Annual Conference and Exhibition, Washington D.C., Oct. 4-7, 1992.
5.Kuchuk, et al., Pressure-Transient Behavior of Horizontal Wells With and Without Gas Cap or Aquifer, SPEPE, March 1991, pp. 86-94.
6.Jelmert, T.A., Properties of the Square-Root Derivative Type Curve, Journal of Mathematical Chemistry, Vol. 8, 1991, pp. 291-305.
7.Thompson, L.G., Manrique, J., and Jelmert, T.A., Efficient Algorithms for Computing the Bounded Reservoir Horizontal Well Pressure Response, Paper No. SPE 21827, SPE Joint Rocky Mountain Meeting and Low Permeability Symposium, Denver, Apr. 15-17, 1991.
The Authors
Jelmert holds a BS in electrical engineering from Purdue University and an MS and PhD in petroleum engineering from NTH.
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