Page 160 - gas transport in porous media

Chapter 8: Gas Injection and Fingering in Porous Media
which should be compared with Eq. (8.17). If the time step T satisfies
T/( x y) ≤ C, then 153
8
δC ij T
=− −ˆn.∇C dl (8.19)
C x y C
faces ∂A ij
For the full problem (convection and dispersion) we can split the time evolution as

∂C ∂C ∂C
ˆ
φ = Q + φD L (8.20)
∂t ∂τ ∂T
In this equation ∂/∂t and ∂/∂T represent separate convection and dispersion pro-
cesses. Convective evolution is described by Eq. (8.18), while dispersion evolution
is represented by Eq. (8.20). Thus, we have a sequential finger evolution in which
convection initiates the growth, which is then modified or moderated by dispersion. If
we fix the time interval ∂t, then, ∂τ = (Q/ψ)δt, and ∂T = D L δt, which then define
ˆ
the Peclet number, Pe = δτ/δT. In practice, δτ is set by, δτ = τ = x y C,
implying that T = τ/Pe. In most of practical cases, Pe > 1, and Eq. (8.20) is
properly normalized. However, if Pe < 1, δT is subdivided into n D intervals to obtain
T = (δτ/Pe)n D , where n D > Pe −1 . King and Scher (1990) simulated miscible dis-
placements, with dispersion effect included, for various values of the viscosity ratio
and obtained reasonable agreement with the experimental data.
The second probabilistic model is due to Araktingi and Orr (1990). In their model
theporousmediumisrepresentedbyathree-ortwo-dimensionalgridofcubic(square)
blocks. At the beginning of each time step the pressure field is calculated, given
the distribution of the permeability and the current distribution of fluid viscosities.
Tracer particles that carry a finite concentration of displacing fluid are injected into
the system and are moved with velocities based on the pressure field. The velocities
are calculated at the midpoint between grid nodes. Velocities for particles that are
not on such nodes are obtained by linear interpolation. After moving the particles by
convection to their current positions, the effect of dispersion is simulated by random
perturbations of particle positions in the longitudinal and transverse directions. Since
for diffusive dispersion the standard deviations of the particles’ motion are given by,
√ √
σ x = 2D L t and σ y = 2D T t, the distribution of the particles about a mean position
can be simulated by multiplying these standard deviations by a number between
−6 and +6. This number is obtained by adding a sequence of 12 random numbers,
distributed normally with a zero mean and unit standard deviation, to −6. The values
+6 and −6 are used because, on a practical basis, the probability of a particle moving
beyond 6 standard deviations on either side of the mean is less than 1%. After the
particles arrive at their new position, the current time step is determined. To avoid
having particles travel a distance greater than a grid block, the time step is chosen to
allow movement equal to half of a grid block length (or width), traveled at the highest
existing velocity. The new pressure field is calculated, and a new position for each
particle is determined. This procedure is repeated many times.