Page 363 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 298

9.2 THE UNSTEADY STATE WATER INFLUX THEORY OF HURST AND VAN
EVERDINGEN

The flow equations for oil into a wellbore are identical in form to the equations
describing flow from an aquifer into a cylindrical reservoir; only the radial scale is
different. In the former case, when an oil well is opened up on production at a constant
rate q, the pressure response at the wellbore can first of all be described under
transient flow conditions, before the reservoir boundary effects are felt, followed
possibly by a period of late transient flow and finally by stabilized, semi-steady state
flow. Irrespective of the flow condition the general equation for calculating the pressure
in the wellbore at any time was presented in Chapter 7, sec. 6, as

4t
t
p D () = 2 t π DA + 1 2 ln D − p ( DA ) (7.42)
t
D
(
γ DMBH )
where
2kh
π
p D (t D ) = (p i – p wf)
qµ

which is the dimensionless pressure function describing the constant terminal rate
case. That is, it determines the pressure drop at r = r w due to a rate change from zero
to q applied at the inner boundary at time t = 0.

In the description of water influx from an aquifer into a reservoir there is greater interest
in calculating the influx rate rather than the pressure drop. This leads to the
determination of the influx as a function of a given pressure drop at the inner boundary
1
of the system. In this respect Hurst and van Everdingen solved the radial diffusivity
equation for the aquifer-reservoir system by applying the Laplace transformation to the
equation, expressed in terms of dimensionless variables as

1 ∂ r ∂ p ∂ p D (7.18)
D
=
r ∂ r D D t ∂ D t ∂ D
D
where in this case

r
r = (9.1)
D
r o
and

kt
t = (9.2)
D
φµ cr 0 2

in which r o is the outer radius of the reservoir and all the other parameters in both
equs. (9.1) and (9.2) refer to aquifer rather than reservoir properties, which will be the
case for all the equations in this chapter unless specifically stated otherwise.

Instead of obtaining constant terminal rate solutions of equ. (7.18), Hurst and van
Everdingen derived constant terminal pressure solutions of the form