Tímarit Verkfræðingafélags Íslands - 01.12.1983, Blaðsíða 12

nient to use the expression (45) as the
standard definition of the formation
capacitivity. The matrix factor
represents then a quantification of the
influence of the various pore
geometrics.
6. ELEMENTARY TEST
MECHANICS IN POROUS
MEDIA WITH DARCY TYPE
FLOW
(6.i) tícisic relations. A geothermal
reservoir has essentially three
capacitances that can respond to
testing, that is, (1) fluid/rock com-
pressibility, (2) free liquid surface
mobility and (3) intergranular vaporiza-
tion. Most frequently, a light testing
load will activate only the compressi-
bility and we will therefore turn our at-
tention to this case. Moreover, a light
test load generates only small flow
velocities and the flow will then be
linear. Although a great deal of caution
has to be exercised in assuming
homogeneous flow parameters in real
field cases, we will, nevertheless, resort
to discussing the test flows due to com-
pressibility on the basis of Darcy type
llow in homogeneous and isotropic
media.
Let p (P,t) be the fluid pressure and
c = k/v the fluid conductivity where k is
the formation permeability and v the
kinematic viscosity of the fluid.
Moreover, let p be the density of the
Huid, s be the formation capacitivity
defined by equation (40) and q be the
fluid mass flow density. Darcy’s equa-
tion reads then
q = -cvp (57)
and the resulting equation for the
pressure in a homogeneous and
isotropic formation is
satp + cnp = F (58)
where 7i = -V2 is the Laplacian operator
and F is a fluid source density. Equation
(58) is a simple parabolic equation
which is conveniently restated in terms
of the fluid diffusivity a = c/ps,
(1 /a.) a tp + np = (F/c) (59)
The formation admittivity
a = (psc) ^ (60)
is another parameter of interest in
pressure diffusion theory.
Most reservior testing involves the
simultaneous observation of flow and
pressure variables. It is convenient to in-
troduce the following terminology that
is in common use in electrical engineer-
ing. When pressure and flow variables
are measured at the same fluid position
or port, we will refer to this arrange-
ment as a driving port test. When
measurement at different ports, the test
is of the transfer type. The well pressure
buildup and drawdown tests that are
now important field procedures are thus
of the driving port type. Well in-
terference tests are, on the other hand,
transfer type tests.
(6.ii) Steady state cases. In a steady
state situation, equation (58) reduces to
the well known potential equation
np = (f/c) (61)
In the present context, we are
interested in steady state flow fields
around cavities and, in particular, in the
cavity admittance. This quantity is
defined in the following way. Consider
a fluid-filled cavity of some general
form having an internal fluid pressure
p. The ambient formation fluid pressure
at some distance from the cavity is
taken to be zero. Under steady state
conditions there will be an outflow of
fluid from the cavity, Q, that can be ex-
pressed
Q = A$P (62)
where As is the steady state cavity ad-
mittance. This quantity is obtained by
solving (61) with the proper boundary
conditions and deriving the total flow
Q. Steady state admittance have been
worked for a few simple cavity
geometrics (Sunde, 1968) and are given
in Table II below.
Table II
Steady State Cavity Admittances
Admittance As
Sperical cavity of
radius R 47tcR
Penny-shaped cavity of
radius R 8cR
Cylindrical cavity of
radius R and length
L»R 27tcL/ln(L/R)
In driving point well testing where
quasi steady state conditions can be
achieved, we observe p and Q and ob-
tain A by equation (62). If the cavity has
a simple form, we can then use the rela-
tions given in Table II to derive the for-
mation fluid conductivity c. Hence,
knowing the cavity geometry, a simple
pump test can yield direct data on this
parameter.
(6.iii) Oscillatory pressure/flow tests.
In many field situations, it is not possi-
ble to arrive at a steady state and some
type of transient type testing will then
have to be carried out. Obviously, an
oscillatory test is the simplest non-
steady state procedure. Below we will
consider two very simple such cases.
First, consider a homogeneous and
isotropic half-space of unknown
capacitivity s and conductivity c that are
to be derived by testing. For this pur-
pose we carry out a driving port test by
injecting at the surface a uniform
oscillatory flow density q = q0exp(i w t),
where q0 is the amplitude and u the
angular frequency. We observe the re-
quired injection pressure p0exp(i u t).
Solving the homogeneous one-
dimensional form of equation (59) for
this case, but omitting details, we obtain
for the amplitudes
p o = “opo (63)
where a0, the complex oscillatory sur-
face admittivity is
a0 = c(iw/a)^ = (iupsc)* = (1+i)a(w/2)^ (64)
and a is the formation caracteristic ad-
mittivity as given by equation (60)
above. Defining the formation skin
depth
d$ = (2c/psm)J (65)
We can also express
a0 = (1+i)c/ds (66)
The admittivity has the dimension of
(s/m) and the absoltute value
Similar results are easily obtained for
the spherical cavity of radius R. Solving
(58) for this case, we obtain the complex
cavity admittance
A = As[l + (l+i)(R/ds)] (68)
where As is the steady state admittance
for the spherical cavity as given in Table
II. Let S = 47tR2 be the surface area of
the cavity. The cavity admittance given
by (68) can then be expressed
92 — TÍMARIT VFI 1983