Tracer Dispersion in Porous Media with Spatial Correlations

We analyze the transport properties of a neutral tracer in a carrier fluid flowing through percolation-like porous media with spatial correlations. We model convection in the mass transport process using the velocity field obtained by the numerical solution of the Navier-Stokes and continuity equations in the pore space. We show that the resulting statistical properties of the tracer can be approximated by a Levy walk model, which is a consequence of the broad distribution of velocities plus the existence of spatial correlations in the porous medium [ H. A. Makse, J. S. Andrade Jr., and H. Eugene Stanley, Tracer Dispersion in Porous Media with Spatial Correlations, Phys. Rev. Lett. (submitted)].

The phenomenon of hydrodynamic dispersion---the unsteady transport of a neutral tracer in a carrier fluid flowing through a porous material ---has been widely investigated in the fields of petroleum and chemical engineering. One can identify different regimes of tracer dispersion according to the Peclet number Pe= v l /D_m, which is the ratio between the typical time for diffusion l^2/D_m and the typical time for convection l/v. Here v is the velocity of the carrier fluid, l a characteristic length scale of the porous media, and D_m the molecular diffusivity of the tracer.

In the small Peclet number regime, molecular diffusion dominates the way in which the tracer samples the flow field. In the large Peclet number regime, also called mechanical dispersion, convection effects are significant; the tracer velocity is approximately equal to the carrier fluid velocity, and molecular diffusion plays little role. The tracer samples the disordered medium by following the velocity streamlines. In a random walk picture, we may think of a tracer particle following the direction of the velocity field, and taking steps of length l and duration l/v.

Here we discuss the interesting physics that arises when the tracer moves in a flow field with a very broad velocity distribution. Consider, e.g., fluid flow in percolation clusters near the percolation threshold---a model system relevant to a porous medium with stagnant small-velocity zones that are linked with large-velocity zones. In this case the typical time for convection l/v is without bound since the velocity can be arbitrarily small in some fluid elements of the void space. Saffman showed that the mean square duration of a tracer step is not finite but diverges logarithmically unless an upper cut-off is introduced into the typical time step. This upper cut-off is imposed by the mass transport mechanism of molecular diffusion.

Here we propose a model of tracer dispersion in a porous medium. The porous medium is composed of blocks of impermeable material that occupy, with a given probability p, a square lattice. We consider a lattice at the site percolation threshold, so an incipient spanning cluster is formed that connects the two ends of the lattice. Previous studies modeled the convective local ``bias'' for the movement of the neutral tracer in the porous media assuming Stokes flow. Even at macroscopically small Reynolds conditions, this assumption might be violated in real flow through porous media, specially in the case of heterogeneous materials (e.g., percolation-like structures) where a broad distribution of pore sizes can lead to a broad distribution of local fluxes. As a consequence, inertial effects might be locally relevant. To avoid this problem, we use the steady-state velocity field obtained by solving the full set of Navier-Stokes in the percolation geometry. Then we study the transport properties of a dynamically-neutral tracer moving in the flow field.

We treat the competition between the effects of convection and diffusion. The velocity field presents a broad scale-invariant power-law distribution of magnitude values, and we find that there are regions of very small velocity in which the tracer can be trapped. If convection is important, the tracer follows the stream-lines of the fluid. When a very small velocity region is reached, molecular diffusion effects cannot be neglected, since by diffusion the tracer may access the stagnant zones---where it then spends a long time. We shall see that due to the existence of these stagnant zones, the statistical properties of the tracer--- e.g., the first-passage time and the root mean square displacement--- can be understood using a Levy walk model for the tracer motion. The existence of Levy statistics is also related to the geometrical properties of the medium---whether it is correlated or uncorrelated in the occupancy variables of the percolation cluster.

Tracer diffusion in the porous medium consisting of a correlated percolation cluster at p_c for large Peclet number, Pe=1.7. We release five walkers and the black dots indicate the sites visited at least one time by the walkers.

Inertial Effects on Fluid Flow through Porous Media

We investigate the origin of the deviations from the classical Darcy's law by numerical simulation of the Navier-Stokes equations in two dimensional disordered porous media. We show that the classical Forchheimer equation provides a valid phenomenological model to correlate the variations of the friction factor for different porosities and over a wide range of Reynolds conditions. At sufficiently high Reynolds numbers, when the contribution of inertia to the transport of momentum at the pore scale becomes relevant, we observe a distinctive transition from linear to non-linear behavior which is typical of experimental observations. We find that such a transition can be understood and statistically characterized in terms of the spatial distribution of kinetic energy in the flowing system [J. S. Andrade Jr., U. M. S. Costa, M. P. Almeida, H. A. Makse, and H. E. Stanley Inertial Effects on Fluid Flow through Porous Media, Phys. Rev. Lett. (submitted); U. M. S. Costa, J. S. Andrade Jr., H. A. Makse, and H. E. Stanley, [Proc. Int. Conf. on Percolation and Disordered Systems, Giessen], Physica A (1998)].

In Fig. 1, we show the results of our flow simulations in terms of Forchheimer variables f and Re' for three different lattice porosities (epsilon=0.7, 0.8 and 0.9). After computing and averaging the overall pressure drops for all realizations at different values of epsilon and Re numbers, we fit the results with the Forchheimer to estimate the coefficients alpha and beta and calculate the modified variables f and Re'. In agreement with real flow experiments, we observe a transition from linear (Darcy's law) to nonlinear flow. Moreover, all results of our computational simulations collapse onto a single curve for which the Forchheimer provides a satisfactory fit. The point of departure from linear to nonlinear behavior in the range [10^-1, 10^-2] is also consistent with previous experimental observations.

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