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Exploration & Production: The Oil & Gas Review - 2003, Volume 2
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Introduction
Traditional well test analysis is based on the existence of specific flow periods, with the important flow periods being radial, linear, bilinear and spherical. The underlying conceptual models are based on simplifying assumptions regarding the geometry of the flow. This approach leads to simplified limiting equations that may be used to aid well test interpretation and to identify the most probable reservoir model. The main use of traditional (graphical) interpretation techniques is to provide good start values for automatic type curve matching (optimisation).
The interaction between a horizontal well and the exterior boundaries may lead to several specific flow periods. Apparently, most researchers expect radial and/or linear flow periods to appear. (1) Both the early radial (in the vertical plane) and the early linear flow period occur before the production from the areas beyond the tips (side regions) becomes important.
Origin of Bilinear Flow Periods
The early linear flow period depends on the assumption of inactive side regions. In some cases this assumption may not be valid, in which case a more complicated model is required. An extension of the linear flow model is to allow the side regions to have some influence. The effect of the side regions is, by our assumption, limited to feed fluid into the main flow path that is the region perpendicular to the well. If the flow is dominated by the upper and lower boundary of no-flow type, the result is linear flow in the primary flow region. If the flow in the side regions is also almost linear, the total result is bilinear flow (see Figure 1). (2)
Figure 1:Bilinear Flow in the Horizontal Plane
In a region where the majority of the flow is of linear type, a line source well may be replaced by a source in the form of a plane with minimum error. The main difference between linear flow towards a plane source and line source is that the latter has an additional pressure drop due to convergence of flow close to the well. This may be accounted for by a steady-state pseudo-skin factor. Usually, the specific flow periods are identified by use of pressure derivatives. In these instances, the constant pressure drop associated with the skin disappears. Hence, the accuracy of the simplified model may be checked against an analytical model by comparing the derivative responses. The model of Thompson, et al. (3) is used in the present study.
A plane source may be thought of as a uniform flux fracture. The height of the plane source during horizontal bilinear flow is defined by the distance between the upper and the lower boundaries, i.e. the height of the reservoir (h). The horizontal extension of plane source, however, is uncertain. By the conventional assumption (i.e. inactive side regions) the horizontal extension is equal to the length of the well (Lw). This length is also assumed to be the horizontal extension of the plane source in the bilinear flow model.
There is a possibility for bilinear flow also in the vertical plane. The occurrence of a possible early bilinear flow period may be explained (intuitively) as follows
Initially, there is elliptical (or pseudo-radial) flow in the vertical plane. Due to the directional property of the permeability, the pressure disturbance will travel further in the horizontal than the vertical direction. The elliptical geometry will change once the pressure disturbance has been influenced by the closest horizontal boundary. The no-flow boundary condition forces the isobars to intersect the boundary perpendicularly. Figure 2 illustrates that isobars may be close to rectangular. This, in turn, may induce bilinear flow.
Figure 2: Bilinear Flow in the Vertical Plane
The height of the vertical plane source (hp) is uncertain since the flow is influenced by only one physical boundary. It is expected, however, that the height of the imaginary plane source is approximately twice the distance between the well and the closest boundary, i.e.  The following definition was used in the calculation example: hp = 2 a zw, where the empirical constant ‘a’ was determined by numerical experiments to be a = 0.81. The horizontal extension of the equivalent plane source is also uncertain. Again, it was assumed that this distance is equal to the length of the well. This assumption is the same as used for the traditional conceptual model for linear flow.
Figure 3: Bilinear Flow in the Vertical Plane
Category:
Drilling & Well Services
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Tom A Jelmert is Professor of
Petroleum Engineering with the
Norwegian University of Science and
Technology (NTNU), Trondheim,
Norway. He has worked with NTNU
since 1985. From 1986 to 1997 he
also held a position as Adjunct
Professor of Mathematics and
Physics at the Academy of The
Royal Norwegian Air Force. Between
1978 and 1985 he was a research
engineer with SINTEF, Trondheim.
Professor Jelmert has been a guest
scientist with the University of
Tulsa, Oklahoma, in the period
1989 to 1990 and with the
Colorado School of Mines from
2000 to 2001. He served as a
member of the editorial board of
the Journal of Petroleum Science
and Technology from 1996 to
2002. Professor Jelmert holds a BSc
in Electrical Engineering from
Purdue University, Indiana and MSc
and Dr Ing. degrees in Petroleum
Engineering from the Norwegian
Institute of Technology (NTH),
Trondheim.
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