Geoscience Reference
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K f = 1
1
s w
K o
+ s w
1.5
K w ,
4 49
Fast P -wave
1.0
where K o and K w denote the bulk moduli of NAPL and
water, respectively. The shear modulus is independent
of the saturation because neither of the fluids sustains
shear stresses. The bulk modulus of the fluid is given
by Equation (4.49), and the density and viscosity of the
fluid are given by
Slow P-wave
0.5
0.0
-0.5
-1.0
Analytical
Numerical
ρ f =1
s w ρ o + s w ρ w ,
4 50
-1.5
s w
η o η w
η o
-2.0
η f =
,
4 51
0
0.2
0.4
0.6
0.8
1
Time (s)
where ρ o and ρ w denote the density of the NAPL and
water, respectively, and η o and η w represent the dynamic
viscosity of oil and water, respectively. The difference of
fluid pressure between the two phases is controlled by
the capillary pressure curve which is given by the same
capillary pressure curve used to simulate the two-phase
flow problem (e.g., Karaoulis et al . , 2012).
The geometry of themodel used for the computation of
the seismic waves is shown in Figure 4.5. The seismic
sources is an explosive-like source located at position
So inWell A (Figure 4.5). The receivers comprise 28 pairs
of seismic stations and electrodes (noted as E1 - E28),
which are located in Well B. The separation between
these receivers is equal to 4 m.
First,we solve the poroelastodynamicwave equations in
the frequency domain, taking into account the variable
saturation of the water phase. We use the ( u , p )formula-
tion (see Jardani et al., 2010). The multiphysics modeling
package COMSOL Multiphysics 4.2a and the stationary
parametric solver PARDISO were used to solve the result-
ing partial differential equations (Schenk & Gärtner, 2004,
2006; Schenk et al., 2007, 2008). The problem is solved as
follows: first (i), we compute for the poroelastic and electric
properties distribution for the given porosity, fluid perme-
ability, and saturation distribution of the NAPL and water
phases, and then (ii) we solve for the displacement of the
solid phase, u , and the pore fluid pressure, p , in the fre-
quency domain. The solution in the time domain is com-
puted by using an inverse Fourier transform of the
solution in the frequency - wave number domain.
In the frequency domain, we use the frequency
range 8 - 800 Hz since the appropriate seismic wave and
associated electric field in this setting operate in this
range. Then, using this frequency range, we compute
Figure 4.2 Comparison between analytical and numerical
solutions of the seismic problem in a homogeneous medium
(benchmark test) with properties summarized in Table 4.1.
The figure shows a snapshot of the vertical component of the
macroscopic solid displacement at t =0.58 s. The two P -waves
can be observed. The synthetic numerical seismograms
(solid black line) and the analytical solution calculated by
Dai et al. (1995) are in close agreement as they cannot
be distinguished from each other.
(surface tension) of the oil can change because bacteria
produce biopolymers bridging the oil molecules to the
surface of the grains. This effect is not accounted for here.
We display six time step snapshots (T1 - T6) of the oil and
water saturations determined by the numerical simula-
tion (Figure 4.4).
4.2.2 Simulation of the seismoelectric
problem
For each of the snapshots shown in Figure 4.4,we
simulated a seismoelectric acquisition between the two
wells where the simulation of the seismoelectric problem
is done in two steps as described in the succeeding text.
Because this problem is formally different from the
unsaturated case discussed earlier (two-phase flow
problem vs. unsaturated problem), somehow we need to
accommodate this issue as discussed in the following text.
Step 1. This step models the propagation of the seismic
waves between the two wells. We use the material
property values given in Table 4.2 to compute the seismic
properties. The bulk modulus of the fluid is related to the
NAPL saturation through the use of the Wood formula,
as discussed in Section 4.3.1. From the Wood formula,
we have
 
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