Tímarit Verkfræðingafélags Íslands - 01.04.1981, Blaðsíða 16

Geothermal Reservoir Testing
based on Signals of Tidal Origin
by Gunnar Böðvarsson, School of Oceanography, Oregon State University,
and Jonathan M. Hanson, Lawrence Livermore Laboratory,
University of California
*
INTRODUCTION
The theory of pressure and water lev-
el osciallations of tidal origin in Darcy
type aquifers and petroleum reservoirs
has been discussed in a number of re-
cent publications (see for example,
Bredehoeft, 1967; Bodvarsson, 1970,
1977, 1978a, 1978b; and Arditty et al.,
1978). There is a general agreement that
observational data on the tidal pressure
phenomena may be applied to obtain
useful estimates of important reservoir
parameters such as the permeability.
A SIMPLE BASIC MODEL
In the simplest setting involving a
single open well connected by a small
spherical cavity to a large homogeneous
and isotropic reservoir, the mechanism
of the tidal well test is easily com-
prehended on the basis of the model il-
lustrated in Fig. 1 below. Let the
permeability of the porous medium be
k, the density of the fluid be p, its
kinematic viscosity be v and hence the
fluid conductivity of the medium be
c = k/v- Moreover, let s be the hydraulic
capacitivity or storage coefficient of the
medium and the diffusivity therefore
a = c/pis = k//ts where /r is the absolute
viscosity of the fluid. The skin depth of
the medium at an angular frequency u
is then d = (2a/u)'/2 (Bodvarsson, 1970).
For the present purpose, concentrating
first on cases where boundary effects
can be ignored, we assume that the skin
depth of the reservoir material at tidal
frequencies is smaller than the extent of
the reservoir including the depth of the
well. In other words, the reservoir can
be assumed to be infinite as viewed
* Paper given at the Fourth Workshop on
Geothermal Reservoir Engineering, December,
1978, Stanford University, Stanford, California.
from the well-cavity. Introducing a
spherical coordinate system with the
radial coordinate r and with the origin
placed at the center of the cavity, the
fluid pressure field p(r,t) in the porous
medium is governed by the diffusion
equation (Bodvarsson, 1970).
3 tp - a|arr + (2/r)ar]p = -( s/s)3tb
0)
where t is time, b(t) the tidal dilatation
of the medium and e is the formation
matrix coefficient. Let rQ be the radius
of the cavity, f the cross section of the
well and g the acceleration of gravity.
The boundary condition at r = rQ is then
(f/g)3tp - Fc3rp = O, (2)
where F = 47tr02 is the surface area of
the cavity.
The expression for the oscillations of
the water level in the well in response to
the dilatation is obtained by solving the
equation (1) with the boundary condi-
tion (2) and deriving the pressure at the
cavity which is equal to the pressure at
the well bottom. To simplify our results
without any appreciable loss of generali-
ty, we can in most cases assume that the
skin depth of the medium is much larger
than the dimensions of the cavity, that
is, d >> rQ. Assuming therefore an in-
finite medium and that b and p « exp
(iut), the solution of (1) in terms of
amplitudes is (Bodvarsson, 1970)
p = (B/r)exp|-(l+i)r/d| - ( Fb/s), (3)
where B is a constant to be determined
by the boundary condition (2). Inserting
(3) into (2), we finally obtain for the
amplitude of the water level in the well
h = -K eb/Pgs)T/(l+T), (4)
where b is the dilatation amplitude and
T is the tidal factor,
T = -4ingcr0/fu (5)
Gunnar Böðvarsson lauk f.h. prófi I
vélaverkfrœði frá TH í Miinchen 1936,
verkfrœðiprófi í stœrðfrceði, kraftfrœði
og skipavélfrœði frá TH í Berlín 1943.
PhD-próf frá California Inst. of
Technology í Bandaríkjunum 1957.
Verkfrœðingur hjá vélsmiðjunni Atlas
/4S 1 Khöfn 1943-45, hjá Rafmagnseftir-
liti ríkisins í Rvík 1945-47. Yfirverk-
frœðingur við Jarðboranir ríkisins og
jarðhitadeild Raforkumálaskrifstofunn-
ar 1947-61. Fór á vegum Sþ, til Santa
Lucia í Vestur-Indíum 1951, Mexíkó
1954, Costa Rica 1963, fjölmargar
ferðir til El Salvador, Guatemala og
Nicaragua 1965-76, Chile 1972, Islands
1972 og Kína 1981 til að athuga mögu-
leika á vinnslu jarðvarma. Námsdvöl
við Cal. Inst. og Technology 1955-57.
Meðstofnandi ráðgefandi verkfrœði-
fyrirtækisins Vermis sf. og starfaði við
það 1962-64. Prófessor í stærðfrœði og
jarðeðlisfræði við Oregon State Univer-
sity í Bandaríkjunum frá 1964.
An elementary potential theoretical
argument shows that the steady state
admittance or conductance of the cavity
A = 47tcr0, (6)
Figure 1. Single well model.
28 — TÍMARIT VFÍ 1981