Page 197 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 135

since it was assumed in deriving equ. (5.1) that the porous medium was completely
saturated with a single fluid thus implying the use of the absolute porosity.
Alternatively, allowing for the presence of a connate water saturation, the φc product
can be interpreted as

(c S + c S + c )
φ absol ute (1 − S wc) × o o w wc f (5.24)
(1 S )
−
wc
in which φ absolute (1 − S wc) is the effective, hydrocarbon porosity, and the
compressibility is equivalent to that derived in Chapter 3, equ. (3.19), which is used in
conjunction with the hydrocarbon pore volume. In either event, the products expressed
in equs. (5.23) and (5.24) have the same value, the reader must only be careful not to
mix the individual terms appearing in the separate equations.

Equation (5.20) is the radial diffusivity equation in which the coefficient k/φµc is called
the diffusivity constant. This is an equation which is frequently applied in physics, for
instance, the temperature distribution due to the conduction of heat in radial symmetry
would be described by the equation

1 ∂ ∂ T 1 T
∂
r =
rr ∂ r ∂ K t ∂

in which T is the absolute temperature and K the thermal diffusivity constant. Because
of the general nature of equ. (5.20) it is not surprising that many reservoir engineering
papers, when dealing with complex solutions of the diffusivity equation, make reference
3
to a text book entitled "Conduction of Heat in Solids", by Carslaw and Jaeger , which
gives the solutions of the equation for a large variety of boundary and initial conditions
and is regarded as a standard text in reservoir engineering.

REFERENCES

1) Matthews, C.S., Brons, F. and Hazebroek, P., 1954. A Method for Determination
of Average Pressure in a Bounded Reservoir. Trans. AlME. 201: 182-191.

2) Dranchuk, P.M. and Quon, D., 1967. Analysis of the Darcy Continuity Equation.
Producers Monthly, October: 25-28.

3) Carslaw, H.S. and Jaeger, J.C., 1959. Conduction of Heat in Solids. Oxford at the
Clarendon Press, (2nd edition).