Summary
In this paper, we construct approximate analytical solutions for the
injection wellbore pressure at vertical and horizontal water injection wells
using the Thompson- Reynolds steady-state theory. The solutions are based on
adding to the single- phase solution, a two- phase term which represents the
existence of the two-phase zone and the movement of the water front. We first
present the solutions for an isotropic reservoir and then show that we can
obtain the solution to an anisotropic problem by introducing a coordinate
transformation to convert an anisotropic system to an equivalent isotropic
system.
The analytical solutions provide insight into the behavior of injectivity
tests at horizontal and vertical wells. For example, for a restricted-entry
case, it is shown that the pressure derivative may be negative throughout an
injection test even when the duration of the test exceeds ten or more days. We
also show that for a well near a fault, the ratio of slopes reflected by
derivative data will not in general be equal to two.
Introduction
We consider water injection at a constant rate through a vertical or
horizontal well into a homogeneous oil reservoir above bubblepoint pressure. We
provide approximate analytical solutions for the injection pressure change at
the injection well under isothermal conditions. Wellbore storage effects are
not considered.
In past work (Peres and Reynolds 2003), we have used a steady-state theory
to derive solutions for the pressure response at a water injection well. In the
vertical well case, the solution assumed a complete-penetration well; in the
horizontal well case, it assumed that the well is equidistant from the top and
bottom of the formation and that the formation is isotropic kz = k. Here, we
construct approximate analytical pressure solution for the restricted-entry
vertical well case for k = kz and for a horizontal well for the case where
the well's axis is not equidistant from the top and bottom boundaries and
the permeability field is anisotropic. The solutions are based on adding to the
single-phase solution, a two- phase term which represents the existence of the
two- phase zone and the movement of the water front. We present models for the
movement of water based on a combination of Buckley-Leverett equations that
allow us to accurately approximate the two-phase flow component of the
analytical solution. The accuracy of results generated from approximate
solutions are checked by comparing them to solutions generated from a black-oil
simulator (IMEX 2000).
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
27 May 2005
- Meeting paper published:
26 September 2004
- Revised manuscript received:
12 June 2006
- Manuscript approved:
5 July 2006
- Version of record:
20 March 2007
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