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Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017 Chapter - p63/743
3.B.3 Build-up response and Horner plot
The MDH plot, with the simple log(∆t) time function, results directly from the log
approximation to the drawdown solution for infinite-acting radial flow. In order to use semilog
analysis for any flow period other than the first drawdown, it is necessary to take into account
superposition effects.
How to get the Horner plot:
Build-up superposition:      
t t p t p t p t p p DD DD p DD BU      ) (
IARF approximation:   





 






   S
rc
k
X
kh
qB
X p
wt
DD 8686 .0 228 .3 log log
6. 162
) ( 2


If t is large enough to reach IARF, so will t p+t. A lot of terms cancel out and we get:
Build-up superposition:  p DD
p
BU t p
t t
t
kh
qB
t p 










  log
6. 162
) (

Rearranging:    





 










  p p DD
p
p
BU t
kh
q
t p
t t
t t
kh
qB
t p log
6. 162
log
6. 162
) (
 
If the production time was too short, IARF was not reached during the flow period and ∆pDD(tp)
cannot be turned into a log approximation. The constant term on the right of the equation
becomes unknown. In this case, the analysis described below will give the permeability, but
not the skin factor. If the production time was long enough, then the term tp can also be
turned into a logarithmic approximation and we get:
IARF at tp:     





 






   S
rc
k
kh
qB
t
kh
qB
t p
wt
p p DD 8686 .0 228 .3 log
6. 162
log
6. 162
2

 
So:








 

















  S
rc
k
t t
t t
kh
qB
t p
wt p
p
BU 8686 .0 228 .3 log log
6. 162
) ( 2


We introduce the Horner Time as:
t t
t t
p
p


Infinite-acting radial flow for a build-up is characterized by linearity between the pressure
response and the logarithm of Horner time. Drawing a straight line through this point gives a
slope and an intercept:
IARF straight line: b
t t
t t
mb
t t
t t
kh
qB
Y
p
p
p
p 





















 log log
6. 162 
If the producing time tp was long enough to reach IARF, the IARF approximation for a build-up
will be similar to the drawdown relation, replacing time by Horner time, and will be given by:
IARF for a build-up:








 

















  S
rc
k
t t
t t
kh
qB
t p
wt p
p
BU 8686 .0 228 .3 log log
6. 162
) ( 2

