Tímarit Verkfræðingafélags Íslands - 01.10.1981, Blaðsíða 6

potential of the elastomechanical
methods in practical reservoir engineer-
ing and related areas. Assuming simple
relevant situations, the strength of the
field signals will be estimated and com-
pared to other ground surface data such
as gravity and D.C. electrical signals
that are also of interest in reservoir
monitoring. Because of greater difficul-
ty in observing surface strain, we will
limit our discusion to vertical ground
displacement and tilt signals.
BASIC RELATIONS
For the sake of brevity we will con-
sider only liquid dominated reservoirs
embedded in porous/permeable half-
spaces that are ultrasimple in the sense
that the formation can be taken to res-
pond in bulk as a homogeneous and
isotropic Hookean solid. Because of the
two-phase situation, all elastic
parameters are composite and will con-
sequently have to be defined properly.
Moreover, Hooke’s law must be
generalized to include the effects of the
pore or fracture fluid (see for example,
Nur and Byerlee, 1971). In the present
note, where only orders of magnitude
are of interest, we will circumvent a
more detailed discussion of these
aspects by assuming that the saturated
formation has empirically well defined
effective elastic parameters. Armed with
this set of quite strong but useful
assumptions, we are now in the position
of considering the effects listed in the
introduction above.
Let the (x, y) plane of our coordinate
system be placed in the surface E of the
half-space with the z-axis vertically
down. The general field point is P = (x,
y, z). We assume that the reservoir is in
equilibrium at the onset of production
at time t = 0. Having at a later time pro-
duced a certain mass of fluid, we can
assume that the subsurface temperature
field has been perturbed by an amount
T(P) and the subsurface fluid pressure
field by p(P). Clearly, since the fluid
pressure vanishes in the drained forma-
tion there is a discontinuity in the
pressure field at the groundwater level.
Let the formation displacement vec-
tor resulting from both perturbations be
u(P) = (u,v, w) (P). Moreover, let A
and n be the effective Lamé parameters,
k the effective bulk modulus, v the tions
Poisson ratio a the effective thermal ex-
pansivity of the fluid/rock systems.
Since the half-space is assumed to be
Hookean, the elastomechanical equa-
tions for the displacement are
[in u—(A+//)V V-u = b , (1)
where n = —_V2 is the Laplacian
operator and b is the body force den-
sity field resulting from the temperature
and pressure perturbations. The boun-
dary condition at the ground surface E
is that of no stress. In the present case,
b is a sum of two terms, one of ther-
moelastic origin associated with the per-
turbation temperature field T and a se-
cond one that results from the perturba-
tion p of the pore pressure. Because of
the discontinuity at the groundwater
surface, it is convenient to split the se-
cond contribution into two parts, one
associated with the perturbation
pressure field in the wet formation and
the other one resulting from the drain-
ing of the formations by the subsidence
of the groundwater level. We have thus
b = —Vf, (2)
where
f=fx + fp + fg (3)
where the f’s are scalars and the
subscript T refers to temperature, p to
fluid pressure, and g to groundwater
level.
On the basis of the theory of ther-
moelasticity (Boley and Weiner, 1960)
we obtain the first factor
fT = akT (4)
There is some uncertainty as to the
proper form of fp . This factor depends
quite heavily on pore/fracture geometry
and connectivity. For the present pur-
pose we will adapt the classical pro-
cedure of Biot (1941) by assuming
fp=öp (5)
where 0 is a positive dimensionless fac-
tor quantizing the effects of the pore
pressure p on the rock matrix. Clearly,
this factor is less than unity, but actual
values may vary within rather wide
limits. Few experimental results are
available. We can only quote
elastomechanical data collected by Rice
and Cleary (1976) from which values of
0 =0.2 to 0.8 can be inferred. The
lower values apply to samples of marble
and granites whereas the higher values
are obtained for sandstones. Along the
same lines the third factor on the right
of (3) can also be estimated on the basis
of (5) where p is then the negative
hydrostatic pore pressure in the drained
rock above the groundwater level.
Given p(P) and T(P), the problem of
solving (1) for the displacement vector
Tí (P) is now well defined.
A simple procedure of solving (1) at
the above defined conditions has been
presented by Bodvarsson (1976). We
will refrain from discussing details of
the method and quote only the result of
main interest, that is, the expression for
the vertical displacement component w
at the ground surface E. Let S = (x,y,
o) be a field point on S and Q =
(x’, y’, z’) be the source point; then
w (S) = / gw(S, Q) f (Q) dvQ (6)
where dvQ = dx’dy’dz’ and gw (S,P) is
the appropriate Green’s function
gw(S,Q) = [(1—v)/n(A + 2fi)] z’/r3SQ,
r3SQ=(x-xT + (y-y’)3 + (z’)3 (8)
is the distance from Q to S. The tilt vec-
tor T(S) is obtained from (6) by T =
- ViiW where Vh=( 3x, 3x,°)> the
horizontal gradient in E. This vector
can thus be expressed in the same form
as (6) by
T(S) = /h* (S,Q)f(Q)dvQ (9)
where h = -V^g.
SIMPLE SITUATIONS
To present an overview of relevant
field amplitudes, we will consider
ground surface displacement and tilt
fields generated by temperature or
pressure perturbations within bounded
compact source regions. Moreover, let
the average depth of the source region
be rather larger than its greatest linear
dimension. Obviously, this situation is
best portrayed by a spherically sym-
metric source region of radius R placed
such that the depth d of the center is a
few times larger than R. In the case of
the first two factors in (3) the integrals
in (6) and (9) can then be quite well ap-
proximated by simple expressions. With
regard to the ground surface displace-
ment we are most interested in the max-
imum value wm that is obtained at the
point 0 vertically above the center of the
region. Assuming the Poisson relation
A = ,/i that applies quite well to com-
mon rock and taking that z’“rOQ—d,
we obtain on the basis of (6) and (7) that
wm = (15/36tt) AV/d2 = 0.13 AV/d2
(10)
66 — TÍMARIT VFÍ 1981