Tímarit Verkfræðingafélags Íslands - 01.08.1981, Blaðsíða 17

Pl = toDPm O)
Depending on a number of cir-
cumstances such as the flow and phase
situation in the well, the parameters A,
C and t can be taken to be constant
over a limited range of pressure. Equa-
tion (2) is then a simple differential
equation in pm and the relation (3) in-
volves only a single differentiation.
In interference testing, we can usually
take that pressure monitoring proceeds
from an equilibrium situation where
Pm = P and because of linearity, we can
then put pm(0) = p (0) = 0, that is, the
system is causal. In operational form,
the solution of (1) is then
pm = (l+t0D)ip (4)
which yields a Taylor series in the
operator tQ D.
AN INJECTION/MONITORING
WELL PAIR IN A DISTRIBUTED
RESERVOIR MODEL
Consider now a more general situa-
tion involving a porous and permeable
fluid-saturated reservoir model with a
given fixed boundary E where prescrib-
ed conditions are to be applied. Let
c = yS/y be the fluid conductivity
operator where p is the formation
permeability operator and v the
kinematic viscosity of the fluid. The
density of the fluid is p, the wet forma-
tion capacitivity or storage coefficient is
s and hence the fluid diffusivity
a = c/ps. Let P = (x,y,z) be the general
field point.
Two wells have been drilled into the
reservoir, one for injection purposes
and the other is to be applied as a
pressometer that is assumed to have a
known capacitance C taken to be cons-
tant within the pressure range of in-
terest. Moreover, the distance between
the two wells is taken to be very large
compared with the dimensions of the
íormation/well contact openings such
that for mathematical convenience the
injection zone, as seen from the
monitoring well, can be lumped into a
point Q. This situation is easily
generalized to the case of a distributed
source.
The problem setting is now the
following. The system is initially in
equilibrium and starting at t = 0 the
mass flow f(t) is being injected into the
reservoir at Q. In the case of fluid
withdrawal the sign of f(t) is negative.
The fluid injection raises the reservoir
pressure leading to the reading pm(t) at
the pressometer well. We are interested
in the pressure p (P, t) at the monitoring
well as unperturbed by the pressometer
capacitance and will therefore derive the
correction pressure p^ such that at the
pressometer p = pm + p,. As compared
with the simple situation in the
preceding section, the present case is
more complex in that the correction
pressure is a field function p, (P,t) that
has to be derived by the integration of a
diffusion type PDE.
To derive the pressure field equa-
tions, we simplify the topology of the
reservoir by neglecting the conductivity
perturbance of the pressometer and
replace it by an equivalent capacity
function su(P) such that the space in-
tegral over this function is equal to C.
Obviously, the function u(P) is localized
in that it is zero everywhere except at the
pressometer well. Moreover, let 7t(c) be
the generalized Laplacian that in the
case of a homogenous and isotropic
medium simplifies to 7t(c)= -cv In
this setting with the pressometer well in-
cluded, the total pressure field (p-p,)
satisfies the equation
ps(l +u)3t(p-p,) + 7t(c)(p—p,) = f<5(P-Q),
(5)
where ð (P-Q) is the spatial delta-func-
tion centered at Q. The pressure field p
unperturbed by the pressometer
capacitance satisfies
ps3tP + 7t(c)p = fJ(P-Q) (6)
and hence the correction field
pt satisfies
Ps3tPi + 7t(c)pi = psu3t(p—pi), (7)
To cope with this equation, we
observe that the pressure within the
pressometer is constant and equal to the
observed pressure pm. Since pm
represents the total field there, we take
that pm = (p-p t) over the support of u(P)
and equation (7) can thus be expressed
ps9(P, + 7t(c)Pi = psuDpm ( (8)
where D = d/dt and the expression on
the right is now a known function in
space and time. Moreover, it is quite ob-
vious that (p-p,) and p should satisfy
the same boundary conditions on E
and p, therefore satifies homogenous
conditions there. It follows that equa-
tion (8) can be integrated to obtain the
perturbation pressure p,. Let the diffu-
sion operator on the left of (7) be ex-
pressed
H = ps3t+ 7t(c) . (9)
and H_l be its inverse at the conditions
specified. The solution of (8) is thus for-
mally
P, (P.tJ^pH"1 (suDPm), (10)
This equation taking now the place of
equation (3) which holds only in the
much simplified lumped case of cons-
tant A. In fact, in the case of fields
varying so slowly that the time-derivative
on the right of (7) can be neglected,
equation (8) reduces to a potential equa-
tion that can be integrated to yield an
expression for Pl. For the field pressure
at the pressometer, this integral then
reduces to the same form as (3).
It is of importance to remark that to
correct the pressometer reading pm(t),
the pressure P| (t) has to be evaluated at
the source of this field. The result is
therefore strongly dependent on the
selection of a correct local model for the
pressometer system. We will refrain
from entering into a detailed discussion
of the integral (10), and limit our
remarks to perhaps the simplest case
relevant in the present context. We
assume a homogeneous and isotropic
reservoir of such an extent that as view-
ed from the pressometer, it can be taken
to be infinite. Moreover, the
pressometer system consists of, or is
equivalent to, a spherical cavity of
radius R. To investigate the response of
a system of this type, we turn to a spec-
tral type of analysis and investigate the
integral (10) when pm = pQexp (iwt)
and <o is the circular frequency. These
assumptions simplify the procedure very
considerably. Omitting details, we ob-
tain for the open cavity model the
amplitude of the correction pressure p,
at the cavity
p, = i top Cp0/As[(l + i) (R/d) + 1 ],
(11)
where C is the capacitance of the
pressometer, As = 4ticR its static ad-
mittance and d = (2c/psu)'/2 is the skin
depth of the pressure field at the fre-
quency w. This expression is useful in
that it gives the amplitude of the spec-
tral components of p,. The dominant
physical factor is the ratio R/d. Expres-
sion (11) is analog to (3) above.
SOFT SPOTS
As already stated, even small reser-
voir soft spots, that are sufficiently
close to the pressometer, can perturb
the pressure readings. For example,
there may be an inactive well with a free
liquid surface in the close vicinity of the
TÍMARIT VFÍ 1981 — 61