Summary
The common approach of using refined grids and small timesteps usually does
not provide accurate and efficient numerical models to simulate horizontal-well
pressure-transient tests. In most cases, this approach requires impracticably
long computational times or creates artifacts on pressure and derivative
responses that may be confused with the characteristics of certain flow
regimes. This paper presents an investigation of the grid and timestep
requirements for the conventional well index (WI) to provide accurate
simulations of horizontal-well pressure-transient responses. The results of
this study indicate that the conventional WI should be used with
log-distributed grid and timesteps to obtain a closer match to the analytical
solution at early times. There is a limit to grid refinement beyond which the
results do not improve. For these cases, the transient well index (TWI)
introduced in this paper allows larger grids while improving the accuracy of
simulations, especially at early times.
Introduction
The need for numerical simulation of the pressure-transient behavior of
horizontal wells often arises in the interpretation of complex structures and
the calibration of static data used in reservoir models. To compute the
bottomhole pressures, reservoir simulators require a relationship between the
wellblock and wellbore pressures. This relationship is obtained by considering
an appropriate well-deliverability equation, which expresses the flow rate as a
function of the difference between the wellblock and wellbore pressures. The
pressure calculated in numerical simulation is uniform within the wellblock,
whereas deliverability equations consider the change of pressure as a function
of distance from the wellbore. However, it is possible to find an equivalent
distance in the wellblock at which the pressure corresponds to the simulated
wellblock pressure. Therefore, the choice of the appropriate deliverability
relationship and equivalent wellblock radius is essential for accurate
numerical calculation of wellbore pressures. The selected deliverability
relationship is invoked into the numerical algorithm in the form of WI.
Van Poollen et al. (1968) and Peaceman (1978, 1983) presented the first
comprehensive discussions of the WI to simulate wellbore pressures. Peaceman
(1978, 1983) introduced the probe radius formula to represent the equivalent
wellblock radius in the WI. Peaceman’s approach has been commonly accepted and
widely used in most numerical simulators.
Steady-state and radial-flow assumptions are inherent in Peaceman’s
numerical WI. When this conventional WI concept is applied to wells with
nonradial-flow regimes (e.g., wells off-centered in the grid and inclined or
horizontal wells at late times) and used to calculate pressure-transient
responses, the accuracy of wellbore pressures may not be satisfactory. The
common approach for the solution of this problem is to reduce the grid size
(and the timesteps) so that the flow convergence within smaller wellblocks
becomes closer to the radial and steady-state assumption is more reasonable
because of the smaller storativity of the wellblocks.
Peaceman’s probe radius formula (Peaceman 1978, 1983) has been commonly used
for horizontal wells without much critical assessment of success for early
times. As an alternative, Babu et al. (1991a, 1991b) presented another WI and
equivalent wellblock radius formula for horizontal wells. Their formula
extended Peaceman’s conventional WI to wellblocks with high aspect ratio and
high anisotropy.
Blanc et al. (1999) proposed techniques including the use of a TWI to
improve the grid-system requirements in modeling pressure responses of vertical
wells, especially at early times. They implied the extension of their technique
to horizontal wells but did not provide conclusive information.
In this paper, our objective is to provide a means of improving
pressure-transient modeling in standard simulators. We intend to respond to two
different categories of the modeler’s needs. If the modeler has the option of
modifying WI in the simulator, then we recommend the use of the TWI as
introduced for horizontal wells in this paper. If an existing (commercial)
simulator is to be used without revising WI, then we provide recommendations
for the selection of the grid system and timesteps.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
13 September 2004
- Meeting paper published:
8 November 2004
- Revised manuscript received:
10 February 2007
- Manuscript approved:
24 February 2007
- Version of record:
20 April 2007
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