Page 196 - Petroleum Geology

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(8.16)

from which the hydraulic gradient is seen to be:
(Ah/Ar) = qbrb 77 olrpog k,- (8.17)

If the total head in the borehole is hb, the total head at radius r is given by
h, = hb +

= hb -t (8.18)
Because qb
(8.19)

This equation indicates that the elevation of the potentiometric surface,
when steady production has been achieved at rate Q, increases as the natural
logarithm of the distance from the well. It also indicates that h, increases in-
definitely as the radius increases, so that steady flow is theoretically impos-
sible in a finite reservoir. Nevertheless, it is evident that h, approaches the
original total head at some radius r at any steady production rate. This is
the radius of influence of the well. When the optimum rate Q has been deter-
mined, the optimum well spacing can also be determined.
Looking at this development of a potential gradient in the context of time,
we see that it takes time to develop steady production and a stable potentio-
metric surface profile to the well. By the same token, if the well is closed in
after steady production has been achieved, it will take time for the pressure
in the well to recover its original value.
The modification of eq. 8.19 to take account of semi-steady flow, and the
development of the pressure build-up equation, are beyond the scope of this
book (see, for example, Dake, 1978, chapters 4-7), but it can be shown
that for small values of time since closing-in, the build-up equation is similar
in form to eq. 8.19:
4ntpogk,
(h - hb) = In (T + AT)/AT (8.20)
QQO
where T is the time during which the well has been produced at rate Q (for
irregular production, the cumulative production divided by time of produc-
tion is taken as Q), and AT is the closed-in time. This equation is usually
written for horizontal flow:
4ntk,
(p*-p) = ln(T + AT)/AT (8.20a)
Qq0