Asymmetric invasion in anisotropic porous media

We report and discuss, by means of pore-scale numerical simulations, the possibility of achieving a directional-dependent two-phase flow behavior during the process of invasion of a viscous fluid into anisotropic porous media with controlled design. By customising the pore-scale morphology and heterogeneities with the adoption of anisotropic triangular pillars distributed with quenched disorder, we observe a substantially different invasion dynamics according to the direction of fluid injection relative to the medium orientation, that is depending if the triangular pillars have their apex oriented (flow aligned) or opposed (flow opposing) to the main flow direction. Three flow regimes can be observed: (i) for low values of the ratio between the macroscopic pressure drop and the characteristic pore-scale capillary threshold, i.e., for $\mathrm{\Delta }{p}_0/p_c\le 1$, the fluid invasion dynamics is strongly impeded and the viscous fluid is unable to reach the outlet of the medium, irrespective of the direction of injection; (ii) for intermediate values, $1<\mathrm{\Delta }{p}_0/p_c\le 2$, the viscous fluid reaches the outlet only when the triangular pillars are flow-opposing oriented; (iii) for larger values, i.e., for $\mathrm{\Delta }{p}_0/p_c>2$, the outlet is again reached irrespective of the direction of injection. The porous medium anisotropy induces a lower effective resistance when the pillars are flow-opposing oriented, suppressing front roughening and capillary fingering. We thus argue that the invasion process occurs as long as the pressure drop is larger then the macroscopic capillary pressure determined by the front roughness, which in the case of flow-opposing pillars is halved. We present a simple approximated model, based on Darcy's assumptions, that links the macroscopic effective permeability with the directional-dependent front roughening, to predict the asymmetric invasion dynamics. This peculiar behavior opens up the possibility of fabrication of porous capillary valves to control the flow along certain specific directions.

Figure 1

(a) A sketch of the porous capillary valve exhibiting asymmetric invasion dynamics. The viscous phase (invading fluid) is injected aligned (red) or opposing (blue) to the pillars, leading to a different front displacement, with the former case inducing capillary fingering and unstable displacement, see the sketch in (b). For certain values of the dimensionless forcing $\mathrm{\Delta }{p}_0/p_c$, with $\mathrm{\Delta }{p}_0$ the pressure drop and $p_c$ being the characteristic capillary threshold, the valve acts asymmetrically, i.e., conducting the flow primarily along one direction. Panel (c) depicts the average front position $m$ (dashed lines) and the maximum and minimum penetration depth (the shaded area) at breakthrough time, computed in number of invaded pore rows, for aligned (red) and opposing (blue) injections. As the forcing increases, the medium becomes penetrable in both directions and the front stands at a maximum position $max\left(m\right)>4$ which denotes that the viscous phase has somewhere invaded the fourth row and reached the outlet. In panel (d) the characteristic invasion rate $t_0/t_b$ is plotted for the different cases, with $t_b$ and $t_0={\mu }_1/\mathrm{\Delta }{p}_0$ being the breakthrough and characteristic times, respectively.

Figure 2

A sketch of the crystal-like anisotropic porous medium composed of triangular pillars with regularly distributed defects. All the characteristic lengths are given in dimensionless form, as functions of the characteristic pore size ${\ell }_p$. The porous domain is composed of $M=M_{\parallel }{M}_{\perp }$ pores. The representative elementary volume has the following length scales: the transversal length $1/\zeta$, the transversal distance between defects on contiguous rows ${\ell }_d$, and the periodicity of the system along the streamwise direction $1/\left(\zeta {\ell }_d\right)$. The characteristic pore throat size and the pillars size are $2r/{\ell }_p$ and ${\ell }_t/{\ell }_p$, respectively. Note that the same geometrical lengths and sizes are used for characterizing the random configurations investigated in Sec. 4, an example of which is given in Fig. 3.

Figure 3

An example of a random configuration of the anisotropic medium. The investigated system consists of a fluid invading a porous medium composed of $N$ solid pillars initially filled with a less viscous fluid. The pillars composing the medium are $(1-\zeta )N$ equilateral triangles uniformly distributed with interpillar distance ${\ell }_p$ and $\zeta N$ randomly distributed. In the upper panel (a), a porous medium composed of triangular-shaped pillars with a percentage of defects $\zeta =0.06$ randomly distributed is depicted. In the lower panels, the dynamics of the invading front with flow-aligned (b) and flow-opposing oriented (c) pillars is presented. The shading colors indicate the position of the interface at different invasion times $t^{*}=t/t_c$. It is interesting to notice that, when the pillars are flow-opposing oriented (c), successive invasion events often occurs along a serpentine-wise direction, with a streamwise invasion when the front reaches a defect and jumps to the successive pore row, followed by many transversal invasion events.

Figure 4

Dynamics of the cumulative number of pore invaded over the average number of pores per row, $m\left(t\right)$ (solid lines), and number of pore throats at the liquid-gas interface over the average number of pores per row, $\mathcal{C}\left(t\right)$ (dashed lines). The dynamics is represented for flow-aligned (a) and flow-opposing (b) oriented triangular pillars. In the insets, the cases with regularly placed defects are shown (crystal-like structures). The solid lines refer to the average value between different statistical realisations, the shaded areas represent the statistical uncertainty, estimated via the standard error (the standard deviation divided by the square root of the number of realization). The color gradient of the solid lines is related to different fractions of defects $\zeta$, with darker lines denoting a higher fraction. We notice that when the pillars are flow-opposing oriented, the roughness of the front, $\mathcal{C}$, is reduced while the number of pore invaded (saturation), $m\left(t\right)$, is enhanced at long times, for both the random and crystal-like configurations.

Figure 5

Sketch of an invading fluid structure after a burst event (the full invasion of a defect). The base of the invading fluid structure is located in correspondence of the defect (blue square). Each square of the sketch in the panel (a) corresponds to a unitary pore space composing the medium (b). The tips of the front are defined as the invaded pore at the two-phase interface that are connected with at least two not invaded pores (red squares). To each tip, the closest defect is identified and the Cartesian components of the base-tip distance $h_{\parallel }$ and $h_{\perp }$ are measured and used for the computation of the dimensionless pressure gradients defined in Eq. (7).

Figure 6

Probability distribution functions $\mathcal{P}$ of the pressure gradients established between bases and tips of the invading fluid structures, $-{\mathbf{\nabla }}_i^{*}{p}^{*}$, as defined in Eq. (7), with $i=\parallel ,\perp$ indicating a pressure gradient along the streamwise and transverse directions, respectively. We report the case of flow-aligned (a) and flow-opposing oriented (b) triangular pillars with random defect distributions. The probabilities are computed at the simulation times ${10}^2<t^{*}<{10}^4$. We observe a substantial difference between streamwise and transverse pressure gradients for the case with flow-opposing oriented pillars (b).

Figure 7

Pore invasion events depends on the isotropic or anisotropic distribution of capillary thresholds. In the case of flow-opposing oriented pillars (b), a marked anisotropic pore invasion mechanism is observed. The capillary threshold at the pore throat connecting two pores longitudinally or transversally is $p_{c,i}\sim \sigma /r_i$, with $i=\parallel ,\perp$ indicating the streamwise and transverse directions, respectively. Here $r_i$ represents the minimum radius of curvature of the moving meniscus at the pore throat. Since the medium is neutrally wetted, the two-phase interface must form a contact angle of ${90}^{\circ }$ at the three-phase contact point while invading a pore throat. Using simple geometrical consideration we obtain for the flow-aligned pillars (a) $r_{\parallel }\sim r_{\perp }\sim {\ell }_t$, marking an isotropic spatial distribution of percolation threshold. On the other hand, for the flow-opposing pillars (b) it results instead $r_{\parallel }\sim r_{\perp }/2\sim {\ell }_t/2$, which denotes a lower capillary threshold for invasion along the transverse direction.

Figure 8

Dynamics of the cumulative number of pore invaded transversally, ${\ell }_{\perp }\left(t\right)$ (solid lines), and along the streamwise direction, ${\ell }_{\parallel }\left(t\right)$ (dashed lines), over the average number of pores per row $M_{\perp }$. The case of flow-aligned (a) and flow-opposing oriented (b) pillars are represented. In the insets, the cases with regularly placed defects are shown. We observe a similar invasion rate $d{\ell }_i\left(t\right)/d{t}^{*}$, for $i=\parallel ,\perp$ in the case of flow-aligned pillars (a), since along both directions the capillary thresholds are similar, i.e., $p_{c,i}=\sigma /{\ell }_t$, as depicted in Fig. 7. On the contrary, when the pillars are flow-opposing oriented (b), the invasion rate is substantially higher along the transverse direction for which the capillary threshold is substantially lower, i.e., $p_{c,\perp }=\sigma /{\ell }_t$. In both cases, at the initial stage, for very short dimensionless times, the viscous fluid invades the first pore row along the streamwise direction, subjected to inertial forces.