|
|
|
Exploration & Production: The Oil & Gas Review - 2003, Volume 2
|
|
|
|
|
Order high-quality repints of any articles on this website
|
|
|
|
|
|
Case Studies
A numerical study was presented that showed results approaching straight lines with quarter slopes. (7) Such lines could be a coincidence or could represent bilinear flow. To resolve this issue, their data was changed in such a way that the correlation factors remained unchanged. The height of the reservoir, (hD) was kept constant. Then the parameters were varied such that the correlation factors were constant, i.e. Cv = 1.778 and Ch = 1. This was achieved by changing the permeability ratio kz/kx and adjusting the distance between the well and the closest boundary, zwD, according to Equation 6. The horizontal permeability was isotropic, which, according to Equation 12, leads to Ch = 1. The relevant data is listed in Table 1. The result is presented in Figure 4. The broken line curves are from the bilinear flow models. The others were obtained by the analytical model.3 In all of the runs, the pressure was calculated at a dimensionless distance 0.68 from the midpoint of the well axis.
Figure 4: Pressure Derivatives
Figure 5: ffect of Reservoir Height (hD)
Figure 4 Figure 5
Comparison of limiting equation against analytical solution
Bilinear flow in the vertical plane is controlled by the closest boundary and bilinear flow in the vertical plane is controlled by both the upper and lower boundaries. Hence, the former flow period appears first.
The interpretation from the analytical flow model is as follows: radial flow in the vertical plane (horizontal line), bilinear flow in the vertical plane (line of quarter slope), hemi-radial flow (horizontal line), bilinear flow in the horizontal plane and, finally, radial flow in the horizontal plane (horizontal line). It is hard to believe that the match between the simplified model, Equation 8, and the analytical model obtained during the bilinear flow period in the horizontal plane is coincidental.
Table 1: Well and Reservoir Data, Case 1
The predicted response of the vertical bilinear model (Equation 2) did not match the analytical result. Interestingly, the extrapolated long-term solution (Equation 5) matched (see Figure 4). It was found that the adjustment constant a should be 0.81 for the present case. Hence, the height of the equivalent plane source is hp = 2 0.81 zw.
Table 2: Well and Reservoir Data, Case 2
Category:
Drilling & Well Services
|
|
|
|
|
|
|
|
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.
|
|
|
|
Download PDF
References
