Tímarit Verkfræðingafélags Íslands - 01.12.1983, Síða 11

(1) Wet rock compressibility
cw = i/kw = (i/v)aav|q=0 (38)
where V is wet rock volume and q is the
pore fluid flow density
(2) Pore volume elasticity
ep * 0/♦)»„♦„ (39)
where <t> is the porosity and the subscript
o references to constant external stress
(3) Formation capacitivity
s = (cf + epH = 4>Cf+ ap*0 (40)
where Cf is the fluid compressibility
(4) Pore volume dilaticity
dp = n/4.)3a<t>p (41)
(5) Pore pressure reactance
- -3oPlq=0 (42)
Below, we will consider these quan-
tities in the case of plausible rock
models.
One of the main difficulties with
regard to the present subject results
from the uncertainties as to the pore
geometry and connectivity. To obtain a
semi-quantitative picture of possible
parameter ranges, we will therefore con-
sider the two extreme pore models il-
lustrated in Figure 2 below.
The model shown in Figure 2a in-
volves a porosity due to a fluid-filled
three-dimensional network of channels
of a uniform width h. The number of
channels per unit length is n and the
porosity is thus <t>=3nh. Let the solid
rock compressibility be cr. On the
parallel channel model there is an
approximately complete rock/fluid
pressure contact such that the total
compressibility
c * (l-4>)cr + <fcf= (l-f)(3/5y) + fcf (43)
Obviously, in this particular case the
increase in fluid volume/unit volume
= d<þ resulting from an increase in pore
pressure dp is simply
df = c^dp (44)
and the formation capacitivity s is thus
equal to cw. Moreover, the pore
pressure is equal to the negative impos-
ed isotropic stress such that the pore
pressure reactance y = 1.
For later development we use the
symbol
S0 = (l-*)cr + *cf (45)
for the capacitance that applies in the
case of the parallel channel model. In
general, the expression for the
capacitance s will depend on the pore
geometry. It is to be noted that because
of the completely open matrix, ep and dp
cannot be properly defined on this par-
ticular model.
Turning to the other extreme, the
spherical pore model sketched in Figure
2b, we assume that the porosity is quite
small, that is, a few % at most. We can
then neglect pore-pore interaction. Let
the number of pores per unit volume,
that is, the pore density be n and the
pore radius be R. The porosity is thus
<þ = (4tt/3)nR5.
Based on the assumptions of small
porosity, pore non-interaction and the
development in section (12) we find that
approximately cw = cr= l/kr where kr is
the rock bulk modulus, and using the
notation in Table I, the pore volume
elasticity
ep = nE^ = nnR3/y = 3*/4 (46)
and the capacitivity thus
s = [ (3/4 y) + cfH (47)
For most practical purposes, the first
term in the brackets of (47) can be
neglected.
o ( ) + 2r
o
- »'■ o O
T o
Figure 2a. Parallel channel model. Figure 2b. Spherical pore model.
To obtain the pore pressure reactance
at non-flow conditions, we turn to
equations (19) and (20) and find that for
spherical pores containing water with
Cf=5x 10-10 Pa-1 and p = 2x 1010 Pa
Y = (1.35/y)/[(0.75/v) + cfl = 0.125 (48)
Turning to general cases, we observe
on the basis of (19) and (39) to (42) that
Y = *dp/s (49)
and using the relation o = kwb where b
is the volume strain, equation (42) leads
to
P = -(♦dpkw/s)b (50)
which we abbreviate
P = -(«/s)b (51)
where
s - *dpkw (52)
is the dilaticity number.
On the parallel channel model
cw=l/kw = s and therefore y=l and
p = -a. Also, the dilaticity number <5=1.
On the other extreme, the spherical
pore model with a small porosity and
where kr = kw, we obtain from Table I
and on the basis of Poisson’s relations
that
6 = *dpkw = 1.8nR3/y= 0.72f, (53)
In some cases it is convenient to ex-
press the pore pressure
P ■ -(rs0kw/s0)b (54)
where sQ is the formation capacitivity as
defined by (45). This can then be ab-
breviated
p = -(e/sQ)b (55)
where
c = Ys0kw (56)
is the matrix coefficient that is an im-
portant dimensionless factor. We
observe that in the case of the parallel
channel model s = (l/kw), y=l and
hence £ =1. But in the case of the
spherical pore model we have in the very
low porosity limit kw=kr, s = (1 /kr)
and y = 0.125. Hence e =0.125 for this
limiting case.
The relevance of this factor results
from the fact that it is at times conve-
TÍMARIT VFÍ 1983 — 91