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Exploration & Production: The Oil & Gas Review - 2003, Volume 2
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Methodology
It has been pointed out that bilinear flow often occurs unexpectedly for horizontal wells. (4) It was demonstrated that this effect could be caused by heterogeneities. Classic bilinear flow responses (straight line of quarter slope on a log-log plot) for horizontal wells could also be explained by rescaling of Cinco-Ley and Samaniego’s limiting equation5 for a finite conductivity fracture. (2) The straight lines, however, represent long-term solutions that may or may not be visible. In this article, the solution to the bilinear flow model is generalised to accommodate small time values. By use of the solution procedure of Cinco-Ley and Samaniego,5 the Laplace space solution for bilinear flow in the vertical plane was found to be:
Equation 1
for the wellbore pressure. The Laplace transformation of the time derivative may be obtained by s-multiplication.
Equation 2
Equations 1 and 2 may be inverted numerically by Stehfest’s algorithm. (6) For small time values, which correspond to large values of the Laplace variable s, the left-hand term within the parenthesis may be neglected in comparison with the right-hand term. Under the assumption that hpD = 2 a zwD, the following equation applies:
Equation 3
Equation 3 may be inverted to yield:
Equation 4
where 
The skin factor has been added to account for the difference between a well in the form of a line and a plane.
The logarithmic derivative becomes:
Equation 5
Equation 5 will present as a straight line with a quarter slope in a log-log co-ordinate system. From Equation 5 it may be concluded that the pressure derivatives from all reservoirs that are characterised by the same dimensionless co-efficient Cv will approach the same quarter slope straight line. The co-efficient will be referred to as the correlation factor of bilinear flow in the vertical plane.
Equation 6
In the same way, the Laplace space solution to bilinear flow in the horizontal plane was obtained as:
Equation 7
The Laplace transform of the pressure derivative may be obtained by s-multiplication.
Equation 8
For small values of s, which correspond to large time values, the left-hand term within the parenthesis may be neglected in comparison with the right-hand term,
Equation 9
which may easily be inverted to yield:
Equation 10
The logarithmic derivative becomes:
Equation 11
From Equation 11, the correlation factor of bilinear flow in the horizontal plane is obtained. The y-direction is parallel to the axis of the well and the x-direction is parallel to it.
Equation 12
It is advantageous to have the highest permeability perpendicular to the well. Hence, the correlation factor is usually < 1 The horizontal permeability ratio is the only parameter in Equation 8. Hence, Ch is the correlation factor during the entire bilinear flow period. Cv does not have this property.
The correlation factor will present as a parameter on a type curve designed for bilinear flow. Hence, the horizontal permeability ratio may be estimated by graphical type curve matching. The anisotropy ratio may be improved by subsequent automatic type curve matching (optimisation).
For large values of the Laplace variable s, which corresponds to small values of time, the second term in the parenthesis Equation 7 may be neglected. Subsequent inversion yields the equation for the conventional linear flow period.
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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