Phase coexistence in consolidating porous media

The appearance of the fluid-rich phase in saturated porous media under the effect of an external pressure is investigated. For this purpose we introduce a two field second gradient model allowing the complete description of the phenomenon. We study the coexistence profile between poor and rich fluid phases and we show that for a suitable choice of the parameters nonmonotonic interfaces show up at coexistence.

Figure 1

From the bottom to the top the graphs of ${\epsilon }_{\text{f}}\left(p\right)$, $m_{\text{s}}\left(p\right)=b{\epsilon }_{\text{s}}\left(p\right)$, and $m_{\text{f}}\left(p\right)$ for $a=0.5$, $b=1$, and $\alpha =100$.

Figure 2

Graph of the overall potential energy ${\Psi }_{\text{s}}\left(p\right)$ and ${\Psi }_{\text{f}}\left(p\right)$ for $a=0.5$, $b=1$, and $\alpha =100$.

Figure 3

Graph of the constraint curve in the plane $m-\epsilon$. The three disks represent the fluid-rich phase, the standard phase and the saddle (gray disk) of the potential energy $\Psi$. Parameters: $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=1$, and $p=p_{\text{co}}=0.24218$.

Figure 4

Graph of the constraint curve in the plane $x-y$. The three disks represent the fluid-rich phase, the standard phase and the saddle (gray disk) of the potential energy $\Psi$. Parameters: $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=1$, and $p=p_{\text{co}}=0.24218$.

Figure 5

Function $V\left[x,y\left(x\right)\right]$ in the case $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=1$, and $p=p_{\text{co}}=0.24218$.

Figure 6

Heteroclinic (coexistence profile) in the case $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=1$, and $p=p_{\text{co}}=0.24218$. Time (space in the original model) on the horizontal axis and $x$, $y$, $\epsilon$, and $m$ on the vertical axis, respectively, in (a), (b), (c), and (d).

Figure 7

Graph of the function $U\left(m,\epsilon \right)$ and the heteroclinic connecting the two maxima. Parameters: $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=1$, and $p=p_{\text{co}}=0.24218$.

Figure 8

The constraint curves and the points representing the fluid-rich phase, the standard phase and the saddle (gray disk) of the potential energy $\Psi$. Parameters: $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=0.7,1.9$ (solid, dashed), and $p=p_{\text{co}}=0.24218$.

Figure 9

The constraint curves and the points representing the fluid-rich phase, the standard phase and the saddle (gray disk) of the potential energy $\Psi$. Parameters: $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=-0.3,-0.4,-1.0$ (solid, dashed, dotted), and $p=p_{\text{co}}=0.24218$.

Figure 10

Homoclinic profile (critical nucleus) in the case $a=0.5$, $b=1$, $\alpha =100$, $k_3=1$, $k=1$, and $p=p_{\text{co}}-{10}^{-2}{p}_{\text{co}}$ (left) and $p=p_{\text{co}}+{10}^{-2}{p}_{\text{co}}$ (right) with $p_{\text{co}}=0.24218$. Time (space in the original model) on the horizontal axis and $x$ [(a) and (b)] and $\epsilon$ [(c) and (d)] on the vertical one.