Hysteresis in relative permeabilities suffices for propagation of saturation overshoot: A quantitative comparison with experiment

Traditional Darcy theory for two-phase flow in porous media is shown to predict the propagation of nonmonotone saturation profiles, also known as saturation overshoot. The phenomenon depends sensitively on the constitutive parameters, on initial conditions, and on boundary conditions. Hysteresis in relative permeabilities is needed to observe the effect. Two hysteresis models are discussed and compared. The shape of overshoot solutions can change as a function of time or remain fixed and time independent. Traveling-wave-like overshoot profiles of fixed width exist in experimentally accessible regions of parameter space. They are compared quantitatively against experiment.

Figure 1

Schematic illustration of a scanning loop (left subfigure) and a reversible scanning curve (right subfigure) for $k_{\mathbb{W}}^r\left(S\right)$. Scanning curves (solid lines) are labeled $\mathrm{sc}$. Main branches (dashed lines) are labeled $\mathrm{im}$ (imbibition) or $\mathrm{dr}$ (drainage).

Figure 2

Illustration of ${\mathcal{G}}_{\mathbb{W}}$ for the $\delta$-model with $\delta =0.3$ and $A=0.66$. For $S\in [0,A-\delta /2]$ the solid line is the (equilibrated) secondary drainage main branch, for $S\in (A-\delta /2,A+\delta /2)$ it is the reversible scanning curve, and for $S\in [A+\delta /2,1]$ it is the (equilibrated) primary imbibition main branch.

Figure 3

Shown on the left are the fractional flow curves $f_{\mathcal{G}}\left(S\right)$ for imbibition $\mathcal{G}={\mathcal{G}}^{\mathrm{im}}$ (solid line) and drainage $\mathcal{G}={\mathcal{G}}^{\mathrm{dr}}$ (dashed line) for the jump-type hysteresis model with parameters from Table 1. Straight line secants represent the imbibition front (solid line connecting $◯1$ and $▵$) and the drainage front (dashed line connecting $◯2$ and $\square$). On the right are fractional flow curves $f_{\mathcal{G}}\left(S\right)$ for imbibition $\mathcal{G}={\mathcal{G}}^{\mathrm{im}}$ (solid line) and drainage $\mathcal{G}={\mathcal{G}}^{\mathrm{dr}}\cup {\mathcal{G}}^{\mathrm{sc}}$ (dashed line) for the $\delta$-hysteresis model for the parameters from Table 1. Secants represent the imbibition front (solid line connecting $◯1$ and $▵$) and the drainage front (dashed line connecting $◯1$ and $\square$). In both plots the slope of a secant equals the front propagation speed from Eq. (19).

Figure 4

Fractional flow curves $f_{\mathcal{G}}\left(S\right)$ for $\mathcal{G}={\mathcal{G}}_{\mathrm{im}}$ (solid lines) and $\mathcal{G}={\mathcal{G}}^{\mathrm{dr}}\cup {\mathcal{G}}^{\mathrm{sc}}$ (dashed lines) for problem A (left) and problem B (right). The parameters are given in Tables 1 and 3. The secants connecting the points marked as $▵$, $\square$, and $◯$ represent the imbibition and drainage fronts.

Figure 5

Shown on the left is the numerical solution $S(z,t)$ of problem A using $\mathrm{Open}\nabla \mathrm{FOAM}$ for times $0\le t\le 500\mathrm{s}$. For better visibility a mesh with $\mathrm{\Delta }z=0.05\mathrm{m}$ and $\mathrm{\Delta }t=50\text{s}$ is shown as an overlay. The constant and identical propagation velocities of both fronts and the constant plateau width of both fronts for $t>t^{\mathrm{B}}$ are clearly visible. On the right are the experimental (dashed line) and theoretical (solid line) saturation profiles $S(z,420\text{s})$ at time $t=420\mathrm{s}$. The experimental profile is from [10] and the model parameters are from Tables 1 and 3.

Figure 6

Shown on the left is the numerical solution $S(z,t)$ of problem B for times $0\le t\le 3000\mathrm{s}$. For better visibility a mesh with $\mathrm{\Delta }z=0.05\mathrm{m}$ and $\mathrm{\Delta }t=250\text{s}$ is shown as an overlay. On the right are the experimental (dashed line) and theoretical (solid line) saturation profiles $S(z,2640\text{s})$ at time $t=2640\mathrm{s}$. The experimental profile is from [10] and the parameters are from Tables 1 and 3.

Figure 7

Shown on the left is the numerical solution $S(z,t)$ of problem C for $0\le t\le 500\mathrm{s}$. The solution exhibits overshoot saturation $S^{\mathrm{P}}=0.81$ and increasing overshoot width. For better visibility a mesh with $\mathrm{\Delta }z=0.05$ m and $\mathrm{\Delta }t=50\text{s}$ is shown. On the right is the numerical solution $S(z,t)$ of problem D for $0\le t\le 2000\mathrm{s}$. A mesh with $\mathrm{\Delta }z=0.1$ m and $\mathrm{\Delta }t=200\text{s}$ is shown. For $0\le t<t^{\mathrm{B}}=155\mathrm{s}$ the solution is a step function with a single increasing step height. For $155\text{s}\le t<200\mathrm{s}$ the profile is nonmonotone with decreasing overshoot width. For $200\text{s}\le t\lesssim 610\mathrm{s}$ the overshoot width and height decrease until $S^{\mathrm{P}}$ reaches $S^{\mathrm{P}}\approx 0.6903$ around $t\approx 610\mathrm{s}$. For $610\text{s}\le t\lesssim 1100\mathrm{s}$ the overshoot saturation decreases further to $S^{\mathrm{P}}\approx 0.674$, but the overshoot width starts to increase, because $c_{{\mathcal{G}}^{\mathrm{im}}}(0.01,S^{\mathrm{P}}\le 0.6903)\ge c_{{\mathcal{G}}^{\mathrm{dr}}\cup {\mathcal{G}}^{\mathrm{sc}}}(0.25,S^{\mathrm{P}}\le 0.6903)$. For $t\ge 1100\mathrm{s}$ the profile is nonmonotone with constant overshoot $S^{\mathrm{P}}\approx 0.674$ and increasing overshoot width.