Page 54 - A Practical Companion to Reservoir Stimulation

PRACTICAL COMPANION TO RESERVOIR STIMULATION

EXAMPLE D-3
Equation D-7 suggests that this derivative function would
Use of a Derivative Function in the be parallel to the net pressure function, also forming the same
Interpretation of Pressure During Pumping log-log slope. The distance between the two lines would be
log C,. C, is the pressure value at t = 1, and C, is the slope of
Develop a pressure derivative function to interpret and con- the net pressure curve. In fact, if and only if the two curves are
firm pressure patterns during pumping. Apply this to the PKN parallel, the model (implied by Eqs. 7-46 to 7-5 I) is in effect.
model for 17 = I, n' = 0.5, and Ap, (lmin) = 200 psi. In dealing with real data, the derivative function in

Solution (Ref. Section 7-3) Eq. D-6 is simply the slope of the net pressure data at any
All observed net pressures (irrespective of the model) are point multiplied by its corresponding value of time.
Applying this technique to the PKN model for = 1 and
powers oft of the form n'=Q.5, then Eq. 7-47 becomes
Ap/ = C,tC-', 03-51 Ap/ = 200t"". (D-8)
where C, and Cz are constants. The derivative function would be
As given by Eqs. 7-46 to 7-5 I, log-log plots of Aprvs. t not
only reveal the type of model in effect but also give a notion
of the fluid efficiency (as shown in Example D-2). (D-9)
These patterns can be identified and corroborated through
the use of a derivative function as in pressure transient Figure D-2 is a plot of both net pressure and its derivative
analysis given by Eq. 1-56. function. This type of analysis is helpful in dealing with real
Applying this function to Eq. D-5, data, especially to detect deviations from the patterns pre-
dicted by the model. Derivatives respond faster for the visual
db, dAP, inspection of data trends, as has been the case for pressure
~ =t- (D-6) transient analysis.
dlnt dr
and
d AP,
dlnt
~ = tc,c2tC+ = C,CztC'.

D-4