We present a dynamic model of immiscible two-phase flow in a network representation of a porous medium. The model is based on the governing equations describing two-phase flow in porous media, and can handle both drainage, imbibition, and steady-state displacement. Dynamic wetting layers in corners of the pore space are incorporated, with focus on modeling resistivity measurements on saturated rocks at different capillary numbers. The flow simulations are performed on a realistic network of a sandpack which is perfectly water-wet. Our numerical results show saturation profiles for imbibition in agreement with experiments. For free spontaneous imbibition we find that the imbibition rate follows the Washburn relation, i.e., the water saturation increases proportionally to the square root of time. We also reproduce rate effects in the resistivity index for drainage and imbibition.

We study a nonlinear convective-diffusive equation, local in space and time, which has its background in the dynamics of the thickness of a wetting film. The presence of a nonlinear diffusion predicts the existence of fronts as well as shock fronts. Despite the absence of memory effects, solutions in the case of pure nonlinear diffusion exhibit an anomalous subdiffusive scaling. Due to a balance between nonlinear diffusion and convection we, in particular, show that solitary waves appear. For large times they merge into a single solitary wave exhibiting a topological stability. Even though our results concern a specific equation, numerical simulations support the view that anomalous diffusion and the solitary waves disclosed will be general features in such nonlinear convective-diffusive dynamics.

We study diffusion processes on clusters of non-wetting fluid dispersed in a wetting one flowing in a two-dimensional porous medium under steady-state conditions using a numerical model. At the critical saturation and capillary number of 10 −5 , where the cluster size distribution follows a power law, we find anomalous diffusion characterized by two critical exponents, d rw = 2.35 ± 0.05 in the average flow direction and d rw⊥ = 3.51 ± 0.05 in the perpendicular direction. We determine the conductivity exponents to be µ = 1.6 ± 0.2 and µ ⊥ = 1.25 ± 0.1, respectively. The high-frequency scaling exponents of the AC conductivity we find to be η = 0.37 ± 0.1 and η ⊥ = 0.36 ± 0.1, respectively.

We present numerical studies of electrical breakdown in disordered materials using a twodimensional thermal fuse model with heat diffusion. A conducting fuse is heated locally by a Joule heating term. Heat diffuses to neighbouring fuses by a diffusion term. When the temperature reaches a given threshold, the fuse breaks and turns into an insulator. The time dynamics is governed by the time scales related to the two terms, in the presence of quenched disorder in the conductances of the fuses. For the two limiting domains, when one time scale is much smaller than the other, we find that the global breakdown time tr follows tr ∼ I 2 and tr ∼ L 2 , where I is the applied current, and L is the system size. However, such power law does not apply in the intermediate domain where the competition between the two terms produces a subtle behaviour.

We present numerical results of electrical resistivity of two-phase flow in reservoir rocks using a dynamic network model. The model accounts for viscous and capillary forces, as well as wetting layers in the crevices of the pore space. It can be used as a unified model for drainage, imbibition and steady-state displacement. We use the model to study viscous effects on electrical resistivity for two-phase flow under strongly water-wet conditions. The pore network is extracted from a realistic pore space of a sandstone. For unsteady drainage and imbibition, our numerical results display capillary number dependent non-Archie behavior and hysteresis of the resistivity index. For steady-state displacement the resistivity index exhibits no significant hysteresis. For increasing capillary number we observe a higher degree of non-Archie (negative curvature) behavior. The simulated data are compared with relevant experimental data, and are in good agreement. Our conclusion is that the dynamic network model successfully reproduces viscous effects on the resistivity index in drainage, imbibition and steady-state displacement processes.