Distribution of liquid flow in a pore network during evaporation

The variation of the distribution of the liquid flow in porous media during evaporation is still a puzzle. We resolve it with the pore network modeling approach. The distribution of the evaporation-induced liquid flow in a pore network composed of about 2.5 million pores is determined. The probability density function of the magnitude of the normalized liquid flow rate is obtained. For the low normalized liquid flow rate, the probability density function is power-lawlike. The power-law exponent depends on both the liquid saturation and the location of the moving meniscus in the main liquid cluster. The evaporation-induced liquid flow in the pores in the pore network can be correlated. Whether the liquid flow distributions in various zones in the pore network are similar or not relies significantly on the location of the moving meniscus in the main liquid cluster. The functions depicting the relation between the power-law exponent and the local liquid saturation for the zones adjacent to and away from the open side of the pore network are different. These findings from the pore scale studies provide insights into developing the accurate continuum model for evaporation in porous media.

Figure 1

(a) Size and configuration of the pore network used in the present study. The pore network is attached to a diffusion boundary layer and divided into four zones. Zone 1 is at the top side, while zone 4 is at the bottom side of the pore network. (b) A 2D schematic of the pore network composed of spherical pore bodies connected by cylindrical pore throats. The gas, solid, and liquid phases are shown in white, gray, and blue, respectively. The grids in the diffusion boundary layer near the top open side of the pore network are also shown. (c) A 3D schematic of a unit cell in the pore network and the adjacent nine grids in the diffusion boundary layer. The length of the side of the central grid in the in-plane direction is equal to the diameter of the pore throat attached.

Figure 2

Variation of the normalized evaporation rate and the location of the moving meniscus in the main liquid cluster in the course of evaporation. Here, $S_t$ is the total liquid saturation, $E$ the evaporation from the pore network, $E_i$ the initial evaporation rate at $S_t=1$, and $z^{*}=l_{z,i}/L_z$ is the fractional distance along the $z$ direction of the moving meniscus in the main liquid cluster, for which $l_{z,i}$ is the distance along the $z$ direction from the moving meniscus to the open side of the pore network, and $L_z$ the length of the pore network in the $z$ direction.

Figure 3

Distributions of the magnitude of the normalized liquid flow rate in the x-z plane of the pore network at the middle of the $y$ direction for (a) $S_t=0.76$ and $z^{*}=0.51$; (b) $S_t=0.75$ and $z^{*}=0.985$; (c) $S_t=0.7$ and $z^{*}=0.95$; and (d) $S_t=0.49$ and $z^{*}=0.955$. Here, $z^{*}=l_{z,i}/L_z$ is the fractional distance along the $z$ direction of the moving meniscus in the main liquid cluster. Only the liquid filled pore throats are presented; the solid, pore bodies, and empty pore throats are not shown for the sake of the clarity.

Figure 4

Variation of the probability density function (PDF), $f_q$, with the magnitude of the normalized liquid flow rate, $q$, in all the filled pore throats in the pore network for (a) $S_t=0.75$ ($z^{*}=0.985$) and $S_t=0.76$ ($z^{*}=0.51$) and (b) $S_t=0.49$ ($z^{*}=0.955$) and $S_t=0.70$ ($z^{*}=0.95$).

Figure 5

(a) Variation of β with $S_t$ and $z^{*}$. (b) Variation of β with $z^{*}$ for $z^{*}\le 0.875$. (c) Variation of β with $S_t$ for $z^{*}>0.875$. Here, β is the power-law exponent in the relation $f_q\sim q^{-\beta }, S_t$ the total liquid saturation, and $z^{*}=l_{z,i}/L_z$ the fractional distance along the $z$ direction of the moving meniscus in the main liquid cluster.

Figure 6

Variation of the PDF, $f_q$, with the magnitude of the normalized liquid flow rate, $q$, for various zones in the pore network for (a) $S_t=1$ and $z^{*}=0.003$ and (b) $S_t=0.75$ and $z^{*}=0.99$. Zone 1 is at the top open side of the pore network, and zone 4 is at the bottom side, as illustrated in Fig. 1. $f_q$ for a certain zone is obtained based on the liquid flow rates in the filled pore throats only in this zone. $z^{*}=l_{z,i}/L_z$ is the fractional distance along the $z$ direction of the moving meniscus in the main liquid cluster.

Figure 7

Variation of the spatial correlation function, $C_{qq,z}$, with $m$ ($R_z=ma$) for the main liquid cluster in the pore network. Here, $a$ is the distance between centers of two adjacent pore bodies, as illustrated in Fig. 1.

Figure 8

(a) Variation of ${\beta }_{\mathrm{zone},1}$ with $S_{\mathrm{zone},1}$ during the evaporation of the pore network. (b) Variation of ${\beta }_{\mathrm{zone},1}$ with $S_{\mathrm{zone},1}$ and $z^{*}$ for $S_{\mathrm{zone},1}>0.9$. (c) Variation of ${\beta }_{\mathrm{zone},2}$ with $z^{*}$ for $z^{*}\le 0.4$. (d) Variation of ${\beta }_{\mathrm{zone},2}$ with $S_{\mathrm{zone},2}$ for $z^{*}>0.4$. (e) Variation of ${\beta }_{\mathrm{zone},3}$ with $z^{*}$ for $0.25\le z^{*}\le 0.62$. (f) Variation of ${\beta }_{\mathrm{zone},3}$ with $S_{\mathrm{zone},3}$ for $z^{*}>0.62$. (g) Variation of ${\beta }_{\mathrm{zone},4}$ with $z^{*}$ for $0.5\le z^{*}\le 0.96$. (h) Variation of ${\beta }_{\mathrm{zone},4}$ with $S_{\mathrm{zone},4}$ for $z^{*}>0.96$. Here, ${\beta }_{\mathrm{zone},i}$ is the exponent of the power-law function depicting the relation between $f_q$ and $q$ for zone $i$. $S_{\mathrm{zone}i}$ is the local liquid saturation of zone $i$. $z^{*}=l_{z,i}/L_z$ is the fractional distance along the $z$ direction of the moving meniscus in the main liquid cluster.