2.2.1 $\unicode[STIX]{x1D6FC}<1$

When $\unicode[STIX]{x1D6FC}<1$ , similar to the case of $\unicode[STIX]{x1D6FC}=0$ , equations (2.18a,b ) are valid before a critical drain-out time $T_{d}$ . By substituting (2.18b ) into (2.18a ), we obtain a nonlinear ODE for $L(T)$ :

With the initial condition $L(0)=0$ , equation (2.19) can be solved numerically. For the early-time period, i.e. $T\ll 1$ , the capillary forces dominate and the time dependence of the depth is $L\propto T^{1/2}$ . For the late-time period, i.e. $T\gg 1$ , gravity dominates and the time dependence of the depth is $L\propto T$ .

The critical drain-out time $T_{d}$ can be obtained by setting $\unicode[STIX]{x1D719}L(T_{d})=T_{d}^{\unicode[STIX]{x1D6FC}}$ , i.e. when all of the added liquid just drains into the porous substrate, and there is no liquid above ( $H=0$ ). After the drainage occurs, i.e. $T>T_{d}$ , the model (2.18a,b ) fails. We expect that the porous medium will be unsaturated with the added liquid, and the post-drain-out dynamics is beyond the scope of current work.

2.2.2 $\unicode[STIX]{x1D6FC}=1$

When $\unicode[STIX]{x1D6FC}=1$ , the liquid supply rate is constant, $Q_{supp}\equiv 1$ , which is smaller than the early-time liquid loss rate $Q_{loss}(T)\approx \unicode[STIX]{x1D719}(H+H^{\ast }+L)/L$ , because $L\ll 1$ . Hence, there is no liquid accumulation at the beginning, i.e. $H=0$ . As time proceeds, $L$ increases and hence $Q_{loss}$ decreases. After a critical time $T_{a}$ , the liquid loss rate becomes smaller than the liquid supply rate, i.e. $Q_{loss}(T_{a})=1$ , then the liquid starts to accumulate above the porous medium, i.e. $H>0$ . This criterion yields an implicit equation:

From global volume conservation (2.18b ), we obtain $L(T_{a})=T_{a}/\unicode[STIX]{x1D719}$ , which when used with (2.20) provides

After $T>T_{a}$ , the governing equation of $L(T)$ is also given by (2.19) with the initial condition $L(T_{a})=\unicode[STIX]{x1D719}H^{\ast }/(1-\unicode[STIX]{x1D719})$ , with which we can numerically solve for $L(T)$ . For the late-time asymptotic behaviours, i.e. $T\gg 1$ , it is safe to neglect the influence of the regime when $T<T_{a}$ . From the scaling arguments, we know that in the late-time period, i.e. $T\gg 1$ , there is a regime of constant speed invasion of the porous medium, where

As an example, the numerical solutions for the cases when $\unicode[STIX]{x1D6FC}=1$ , $\unicode[STIX]{x1D719}=0.37$ are shown in figure 2(b) when $H^{\ast }=0$ (solid blue curve), and $H^{\ast }=100$ (dashed orange curve). With respect to (2.21), the critical starting time and the corresponding starting depth for $H^{\ast }=0$ is $T_{a}=L(T_{a})=0$ , and those for $H^{\ast }=100$ are $T_{a}\approx 21.7$ and $L(T_{a})\approx 58.7$ . The regime when $T<T_{a}$ is where the current model fails, so we denote it with a shaded area. We find that the numerical solutions for $H^{\ast }=0$ collapse with the asymptotic solution (black dotted curve) given by (2.22b ) for the entire time period, while those for $H^{\ast }=100$ collapse with the asymptotic solution at late times, which means that neither the capillary effects nor the drain-out occurring in the early-time period influence the late-time behaviours.

2.2.3 $\unicode[STIX]{x1D6FC}>1$

When $\unicode[STIX]{x1D6FC}>1$ , the liquid supply rate $Q_{supp}(T)=\unicode[STIX]{x1D6FC}T^{\unicode[STIX]{x1D6FC}-1}$ is small at the beginning, while the liquid loss rate $Q_{loss}(T)$ is large, so (2.18a,b ) are invalid until a critical time $T_{a}$ , where the liquid supply rate starts to exceed the liquid loss rate, i.e. $Q_{supp}(T)\geqslant Q_{loss}(T)$ . Then, the liquid starts to accumulate above the porous medium, i.e. $H>0$ when $T>T_{a}$ . Thus, similar to the case when $\unicode[STIX]{x1D6FC}=1$ , we can estimate $T_{a}$ by setting $Q_{supp}(T_{a})=Q_{loss}(T_{a})$ , which yields an implicit equation:

We cannot obtain a simple expression for $L(T_{a})$ from the global volume conservation as that for $\unicode[STIX]{x1D6FC}=1$ , because when $\unicode[STIX]{x1D6FC}>1$ , the porous medium is unsaturated with the liquid in the early-time period, which is beyond the scope of current work. However, because the late-time asymptotic behaviours are not influenced by what happens at early times, we can still use (2.19) to predict that in the late-time period, i.e. $T\gg 1$ , gravity dominates, and the late-time time dependence of the depth is $L\propto T^{(\unicode[STIX]{x1D6FC}+1)/2}$ .

3.3.1 Regime I: $0\leqslant \unicode[STIX]{x1D6FC}<1/2$

Now, we present the self-similar solutions. When $0\leqslant \unicode[STIX]{x1D6FC}<1/2$ , the governing equations (3.11a–c ) reduce to

As is clear from (3.17a ), spreading dominates over drainage in the continuity equation of the liquid above the porous medium, and the global volume conservation is dominated by the liquid above the porous medium (see (3.17c )). Thus, the solutions for $H(X,T)$ and $X_{F}(T)$ recover Huppert’s (1982) solution, where a gravity current spreads over a rigid horizontal surface without drainage.

By defining a similarity variable $s\equiv X/(\unicode[STIX]{x1D709}_{f}T^{(3\unicode[STIX]{x1D6FC}+1)/5})$ , where $\unicode[STIX]{x1D709}_{f}$ is a stretching constant to be determined, the shape of the current can be written in a self-similar form, described as $X_{F}=\unicode[STIX]{x1D709}_{f}T^{(3\unicode[STIX]{x1D6FC}+1)/5}$ , $H=\unicode[STIX]{x1D709}_{f}^{2/3}T^{(2\unicode[STIX]{x1D6FC}-1)/5}f(s)$ , and $L=\unicode[STIX]{x1D719}^{-1/2}\unicode[STIX]{x1D709}_{f}^{1/3}T^{(\unicode[STIX]{x1D6FC}+2)/5}g(s)$ on the domain $s\in [0,1]$ . Here, $f(s)$ , $g(s)$ and $\unicode[STIX]{x1D709}_{f}$ can be determined from the equations:

subject to boundary conditions $f(1)=g(1)=0$ . The primes in (3.18a,b ) denote differentiation with respect to $s$ . We can also determine the local asymptotic behaviour at the front from (3.18a ), which yields

Equation (3.19) provides the values of $f(1-\unicode[STIX]{x1D716})$ and $f^{\prime }(1-\unicode[STIX]{x1D716})$ with $\unicode[STIX]{x1D716}\ll 1$ . Together with $g(1)=0$ , they can be used to numerically solve ODEs (3.18a,b ) by shooting from $s=1-\unicode[STIX]{x1D716}$ towards $s=0$ . The stretching constant $\unicode[STIX]{x1D709}_{f}$ can be obtained from (3.18c ). The asymptotic behaviour of $g(s)$ as $s\rightarrow 1^{-}$ is

Thus, we obtain the relation, $g(s)\propto (f(s))^{2}$ , near the location of the front.

The functions $f(s)$ and $\unicode[STIX]{x1D709}_{f}$ are given by Huppert (Reference Huppert1982b ). We also find that $H^{\ast }$ does not appear in (3.18a–c ), so $f(s)$ , $g(s)$ and $\unicode[STIX]{x1D709}_{f}$ are independent of $H^{\ast }$ . As an example, we set $\unicode[STIX]{x1D6FC}=0$ , for which we can find analytical solutions

The normalized self-similar solutions $f/f_{max}$ and $g/g_{max}$ for $\unicode[STIX]{x1D6FC}=0$ are shown in figure 4 as the red curve, where the subscript ‘max’ denotes the maximum values of the functions.

Figure 4. Normalized self-similar shapes given by analytical solutions in § 3.3 and appendix B under different addition exponents $\unicode[STIX]{x1D6FC}$ in the early-time period, i.e. $T\ll 1$ . For the regime boundaries $\unicode[STIX]{x1D6FC}=1/2,7/4$ , we choose the dimensionless capillary length $H^{\ast }=1$ and $\unicode[STIX]{x1D719}=0.37$ . For the special case $\unicode[STIX]{x1D6FC}=3$ , we choose $Q=1$ .

The current theory is based on the assumptions that $x_{f}(t)\gg h(0,t)$ , $x_{f}(t)\gg \ell (0,t)$ and $|\text{d}x_{f}/\text{d}t|\gg |\unicode[STIX]{x2202}\ell /\unicode[STIX]{x2202}t|$ (so that the spreading gravity current is always flowing over dry porous media). Using the analytical results in this section, we arrive at the self-consistent requirement:

As long as the left-hand side of (3.22) is much smaller than $1$ , and the right-hand side is of order $1$ or much larger than $1$ , the drainage theory is valid in the early-time period.

3.3.2 Regime II: $1/2<\unicode[STIX]{x1D6FC}<7/4$

When $1/2<\unicode[STIX]{x1D6FC}<7/4$ , the governing equations (3.11a–c ) reduce to

Given that (3.23a,c ) are the same as (3.17a,c ) in regime I and subject to the same boundary and initial conditions, we conclude that $H(X,T)$ and $X_{F}(T)$ are the same as that in regime I. The small difference between these two problems is in the dynamics for $L(X,T)$ , which includes $H^{\ast }$ : the time evolution of $L(X,T)$ is mainly driven by the capillary effects here while it is mainly driven by gravity in regime I.

A similarity variable is defined as $s\equiv X/(\unicode[STIX]{x1D709}_{f}T^{(3\unicode[STIX]{x1D6FC}+1)/5})$ , so we can rewrite the functions as $X_{F}=\unicode[STIX]{x1D709}_{f}T^{(3\unicode[STIX]{x1D6FC}+1)/5}$ , $H=\unicode[STIX]{x1D709}_{f}^{2/3}T^{(2\unicode[STIX]{x1D6FC}-1)/5}f(s)$ and $L=(H^{\ast }/\unicode[STIX]{x1D719})^{1/2}T^{1/2}g(s)$ on the domain $s\in [0,1]$ . As mentioned above, $f(s)$ and $\unicode[STIX]{x1D709}_{f}$ are the same as the solutions in regime I, and so are independent of $H^{\ast }$ . The ODE and the boundary condition for $g(s)$ are

which yields

We note that $f(s)$ , $g(s)$ and $\unicode[STIX]{x1D709}_{f}$ are all independent of $H^{\ast }$ . As an example, the normalized self-similar shape for $\unicode[STIX]{x1D6FC}=1$ is shown in figure 4 as the black curve. The stretching constant is computed as $\unicode[STIX]{x1D709}_{f}\approx 1.00$ .

Again, we can a posteriori validate the approximations made in the derivation of the governing equations. In particular, the assumptions that $x_{f}(t)\gg h(0,t)$ , $x_{f}(t)\gg \ell (0,t)$ and $|\text{d}x_{f}/\text{d}t|\gg |\unicode[STIX]{x2202}\ell /\unicode[STIX]{x2202}t|$ provide the self-consistent requirements:

As long as the right-hand sides of (3.26a,b ) are much smaller than $1$ , the drainage theory is valid in the early-time period.

We can also solve for $L(X,T)$ directly from (3.23b ), which yields

where $T_{0}(X)$ is the time when the front reaches the location $X$ , i.e. $X_{F}(T_{0})=X$ , because the drainage starts at $X$ when $X_{F}(T_{0})=X$ . From the scaling of $X$ , we obtain

By combining equations (3.27) and (3.28), we can obtain an explicit expression for $L(X,T)$ :

We note that the derivation of (3.29) is equivalent to that of (3.25), but with more physical insights.

3.3.3 Regime III: $\unicode[STIX]{x1D6FC}>7/4$

When $\unicode[STIX]{x1D6FC}>7/4$ and $\unicode[STIX]{x1D6FC}\neq 3$ the governing equations (3.11a–c ) reduce to

We note that there is no time derivative term in (3.30a ), which means that the flow above the porous medium is quasi-steady. Also, equation (3.30c ) indicates that $H\ll L$ in this regime, which means that almost all of the liquid is in the porous medium.

In this case, a similarity variable is defined as $s\equiv (H^{\ast }\unicode[STIX]{x1D719})^{1/2}X/(\unicode[STIX]{x1D709}_{f}T^{\unicode[STIX]{x1D6FC}-1/2})$ , and we can rewrite the functions as $X_{F}=(H^{\ast }\unicode[STIX]{x1D719})^{-1/2}\unicode[STIX]{x1D709}_{f}T^{\unicode[STIX]{x1D6FC}-1/2}$ , $H=(H^{\ast }\unicode[STIX]{x1D719})^{-1/8}\unicode[STIX]{x1D709}_{f}^{1/2}T^{\unicode[STIX]{x1D6FC}/2-3/8}f(s)$ and $L=(H^{\ast }/\unicode[STIX]{x1D719})^{1/2}T^{1/2}g(s)$ on the domain $s\in [0,1]$ . Hence, we can rewrite the equations (3.30a–c ) as

The self-similar depth $g(s)$ can be solved analytically from (3.31b ) with boundary condition $g(1)=0$ , which yields

In addition, from (3.31c ), we obtain an analytical expression for the stretching constant

Also, an asymptotic analysis of (3.31a ) near $s=1$ , subject to $f(1)=0$ , provides

Equation (3.34) gives the values of $f(1-\unicode[STIX]{x1D716})$ and $f^{\prime }(1-\unicode[STIX]{x1D716})$ with $\unicode[STIX]{x1D716}\ll 1$ , which are used in a shooting procedure to numerically solve (3.31a ) from $s=1-\unicode[STIX]{x1D716}$ towards $s=0$ . It can be seen from (3.31a–c ) that $f(s)$ , $g(s)$ and $\unicode[STIX]{x1D709}_{f}$ are all independent of $H^{\ast }$ .

Once again the assumptions that $x_{f}(t)\gg h(0,t)$ , $x_{f}(t)\gg \ell (0,t)$ and $|\text{d}x_{f}/\text{d}t|\gg |\unicode[STIX]{x2202}\ell /\unicode[STIX]{x2202}t|$ provide the self-consistent requirements:

So long as the right-hand sides of (3.35a,b ) are much smaller than $1$ , the drainage theory is valid in the early-time period.

Finally, for $\unicode[STIX]{x1D6FC}=3$ the governing equations (3.13a–c ) reduce to

In this case, a similarity variable is defined as $s\equiv \unicode[STIX]{x1D719}^{1/2}X/(Q\unicode[STIX]{x1D709}_{f}T^{\unicode[STIX]{x1D6FC}-1/2})$ , and we can rewrite the functions as $H=Q^{1/2}\unicode[STIX]{x1D719}^{-1/8}\unicode[STIX]{x1D709}_{f}^{1/2}T^{\unicode[STIX]{x1D6FC}/2-3/8}f(s)$ and $L=\unicode[STIX]{x1D719}^{-1/2}T^{1/2}g(s)$ on the domain $s\in [0,1]$ . Hence, we can rewrite the (3.36a–c ) as (3.31a–c ), and use the same steps to solve for the solutions. For this case, $f(s)$ , $g(s)$ and $\unicode[STIX]{x1D709}_{f}$ are all independent of $Q$ . The normalized self-similar shape for $\unicode[STIX]{x1D6FC}=3$ is shown in figure 4 as the green curve. The stretching constant is calculated as $\unicode[STIX]{x1D709}_{f}\approx 1.44$ .

From figure 4, we can see that the self-similar shape of the current is thinner for larger addition exponents $\unicode[STIX]{x1D6FC}$ , which corresponds to smaller addition rates when $T\ll 1$ . This result is intuitive because the smaller is the addition rate, the less liquid is in the system, which is consistent with a thinner current shape.

3.4.1 Regime IV: $0<\unicode[STIX]{x1D6FC}<1$

When $0<\unicode[STIX]{x1D6FC}<1$ , as mentioned above, the front recedes after it reaches its maximum position $X_{max}$ , and the height $H$ decreases as time proceeds. The governing equation (3.11a ) reduces to

which means that the flow above the porous medium is quasi-steady. By integrating equation (3.39) from $X=0$ to $X=X_{F}(T)$ and recalling $H^{3}\unicode[STIX]{x2202}H/\unicode[STIX]{x2202}X=0$ at the front, we obtain

where we have used the boundary condition (3.38a ). We can also solve for $H(X,T)$ from (3.39) with boundary conditions (3.14a,b ), which yields

in which we have substituted the expression (3.40) for $X_{F}(T)$ . From (3.40) and (3.41), we know that even though the front recedes and the height decreases, the liquid will not drain out in a finite time. The normalized current shape can be obtained by combining equations (3.40) and (3.41), which yields

A representative example when $\unicode[STIX]{x1D6FC}=1/2$ is shown in figure 7. For this case, the late-time self-similar solutions for $H_{max}(T)$ and $X_{F}(T)$ are

which are calculated, respectively, from (3.41) and (3.40) by setting $\unicode[STIX]{x1D6FC}=1/2$ and $X=0$ . It can be seen that the PDE solutions for $X_{F}(T),H_{max}(T)$ and the normalized current shape approach their late-time self-similar solutions (3.42) and (3.43a,b ) in figure 7(a–c).

With a receding front at late times, the only assumption we need to justify the original governing equations is $x_{f}(t)\gg h(0,t)$ ; the other two assumptions, $x_{f}(t)\gg \ell (0,t)$ and $|\text{d}x_{f}/\text{d}t|\gg |\unicode[STIX]{x2202}\ell /\unicode[STIX]{x2202}t|$ , are only necessary for calculating the depth $\ell$ that the fluid penetrates into the porous medium, which does not play an important role in the governing equation (3.39) for this regime. Thus, we arrive at a self-consistent requirement:

As long as the right-hand side of (3.44) is much larger than $1$ , the drainage theory is valid in the late-time period.

3.4.2 Regime V: $1\leqslant \unicode[STIX]{x1D6FC}<3$

When $1\leqslant \unicode[STIX]{x1D6FC}<3$ , the governing equations (3.11a–c ) reduce to

By defining the similarity variable as $s\equiv X/(\unicode[STIX]{x1D709}_{f}T^{\unicode[STIX]{x1D6FC}-1})$ , we can rewrite the functions as $X_{F}=\unicode[STIX]{x1D709}_{f}T^{\unicode[STIX]{x1D6FC}-1}$ , $H=\unicode[STIX]{x1D709}_{f}^{1/3}T^{(\unicode[STIX]{x1D6FC}-1)/2}f(s)$ and $L=\unicode[STIX]{x1D719}^{-1}Tg(s)$ on the domain $s\in [0,1]$ , and then write equations (3.45a–c ) as

By solving equations (3.46a–c ) with boundary conditions $f(1)=g(1)=0$ , $f^{3}f^{\prime }|_{s=1}=0$ , we can obtain an analytical solution for $f(s)$ , $g(s)$ and $\unicode[STIX]{x1D709}_{f}$ , given by

Note that when $\unicode[STIX]{x1D6FC}=1$ , the self-similar depth $g(s)$ is a step function at leading order, which means that as time proceeds, the shape of the liquid–air interface in the porous medium approaches a shock solution. In addition, $X_{F}\rightarrow 1$ when $T\rightarrow \infty$ for $\unicode[STIX]{x1D6FC}=1$ , i.e. the current front reaches a steady position in the late-time period. The normalized self-similar shapes for $\unicode[STIX]{x1D6FC}=1$ (red curve), $1.1$ (orange curve) are shown in figure 6.

Figure 6. Normalized self-similar shapes of the current under different addition exponents $\unicode[STIX]{x1D6FC}$ in the late-time period, i.e. $T\gg 1$ . For the regime boundary $\unicode[STIX]{x1D6FC}=3$ , we choose $Q=1$ and $\unicode[STIX]{x1D719}=0.37$ . The dimensionless capillary length $H^{\ast }$ does not influence the late-time normalized self-similar shapes.

Figure 7. The late-time asymptotic behaviours, i.e. $T\gg 1$ , when $\unicode[STIX]{x1D6FC}=1/2$ , $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ . (a) The front location $X_{F}$ as a function of time $T$ , where the triangles represent the PDE numerical results, and the dashed blue curve represents the late-time self-similar solution (3.40). (b) The maximum height $H_{max}$ as a function of time $T$ , where the circles represent the PDE numerical results and the dashed blue curve represents the self-similar solution (3.41). (c) The normalized shape of the current above the porous medium, where the PDE numerical solutions (red curve) in the direction of the arrow are at $T=\{1,2,3,6\}$ , respectively, and the self-similar shape is given by (3.42) (blue curve). The dimensionless capillary length $H^{\ast }$ is only used to numerically solve PDEs (3.11a–c ), without influencing the self-similar solutions.

The assumptions that $x_{f}(t)\gg h(0,t)$ , $x_{f}(t)\gg \ell (0,t)$ and $|\text{d}x_{f}/\text{d}t|\gg |\unicode[STIX]{x2202}\ell /\unicode[STIX]{x2202}t|$ provide the self-consistent requirements

Thus, we recognize that as long as (3.48b ) is satisfied, the drainage theory is valid in the early-time period.

3.4.3 Regime VI: $\unicode[STIX]{x1D6FC}>3$

When $\unicode[STIX]{x1D6FC}>3$ , the simplified governing equations are the same as those in regime I (§ 3.3.1) in the early-time period, so the solutions of $H$ , $L$ and $X_{F}$ should also be the same. As an example, the normalized self-similar shape for $\unicode[STIX]{x1D6FC}=5$ is shown in figure 6 as the blue curve with the stretching constant $\unicode[STIX]{x1D709}_{f}\approx 0.75$ . From figure 6, we can see that for larger addition exponents $\unicode[STIX]{x1D6FC}$ , i.e. larger addition rates when $T\gg 1$ , the shape above the porous medium is fatter, while the shape inside the porous medium changes from a step function to a thinner shape and then becomes fatter again.

Once again we recall that the current theory is based on the assumptions that $x_{f}(t)\gg h(0,t)$ , $x_{f}(t)\gg \ell (0,t)$ and $|\text{d}x_{f}/\text{d}t|\gg |\unicode[STIX]{x2202}\ell /\unicode[STIX]{x2202}t|$ , so that we arrive at the self-consistent requirements:

As long as the right-hand sides of (3.49) are of order $1$ or smaller, the drainage theory is valid in the late-time period.

3.4.4 Special case: $\unicode[STIX]{x1D6FC}=0$

For the fixed-volume case, i.e. $\unicode[STIX]{x1D6FC}=0$ , as mentioned above, the front will recede after it reaches its maximum position $X_{max}$ and the height $H$ decreases as time proceeds. Meanwhile, there exists a time when all of the liquid has drained into the porous medium, which is denoted as $T_{d}$ . The difference between the current case and regime IV (§ 3.4.1) is that the time derivative term dominates in the governing equation (3.11a ).

Figure 8. The asymptotic behaviours near the drain-out time, i.e. $\unicode[STIX]{x1D70F}\rightarrow 0$ , when $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ , $H_{0}=10$ . (a) The normalized shape of the current above the porous medium, where PDE numerical solutions (red curve) in the direction of the arrow are at $\unicode[STIX]{x1D70F}=\{0.595,0.195,0.095,0.045,0.005\}$ , respectively. In all the figures, the blue arrows denote the evolution direction as time proceeds, i.e. $\unicode[STIX]{x1D70F}\rightarrow 0$ . (b) The front location $X_{F}$ as a function of time $\unicode[STIX]{x1D70F}$ , where the triangles represent the PDE numerical results and the dashed blue curve represents the fitting curve $X_{F}=2\unicode[STIX]{x1D70F}^{1/2}$ . (c) The maximum height $H_{max}$ as a function of time $\unicode[STIX]{x1D70F}$ , where the circles represent the PDE numerical results, and the dashed blue curve represents the self-similar solution, $H_{max}=\unicode[STIX]{x1D70F}$ . The dimensionless capillary length $H^{\ast }$ is only used to numerically solve PDEs (3.11a–c ), without influencing the self-similar solutions.

We performed many numerical simulations, which suggest that a self-similar solution exists for the height $H(X,T)$ in the limit when $T\rightarrow T_{d}$ (see figure 8(a) for typical solutions). We define an approach time $\unicode[STIX]{x1D70F}\equiv T_{d}-T$ , and also assume that $L\gg H$ when $T\rightarrow T_{d}$ . Then, the governing equation (3.11a ) reduces to

The two boundary conditions for $\unicode[STIX]{x1D6FC}=0$ are

where $X_{F}(\unicode[STIX]{x1D70F})$ is the front location. Now, we aim to solve equation (3.50) with boundary conditions (3.51a,b ) for a self-similar solution for $H(X,\unicode[STIX]{x1D70F})$ in the limit when $\unicode[STIX]{x1D70F}\rightarrow 0$ .

By balancing the first term on the left-hand side with the right-hand side of (3.50), we obtain the time dependence of the height is $H\propto \unicode[STIX]{x1D70F}$ (see figure 8 b). The numerical solutions of the PDEs (3.11a–c ) when $\unicode[STIX]{x1D6FC}=0$ show $X_{F}\propto \unicode[STIX]{x1D70F}^{1/2}$ (see figure 8 c). Hence, the second term on the left-hand side of (3.50) scales with $\unicode[STIX]{x1D70F}^{3}$ , which is negligible compared with the other two terms in the equation when $\unicode[STIX]{x1D70F}\ll 1$ . Now, in the limit when $\unicode[STIX]{x1D70F}\rightarrow 0$ , equation (3.50) reduces to

which yields

where $\unicode[STIX]{x1D70F}_{0}(X)$ is the time when the front reaches the location $X$ , i.e. $X_{F}(\unicode[STIX]{x1D70F}_{0})=X$ . Based on the numerical solutions for $X_{F}$ , we can write $X_{F}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D709}_{f}\unicode[STIX]{x1D70F}^{1/2}$ , which yields

By combining equations (3.52) and (3.54), we obtain the expression for the height:

By substituting $H_{max}=\unicode[STIX]{x1D70F}$ , $X_{F}=\unicode[STIX]{x1D709}_{f}\unicode[STIX]{x1D70F}^{1/2}$ into (3.55), we arrive at the self-similar shape of the current:

We solve the PDEs (3.11a–c ) with $\unicode[STIX]{x1D6FC}=0$ , and find that near the drain-out time $T_{d}$ , i.e. $\unicode[STIX]{x1D70F}\rightarrow 0$ , the maximum height $H_{max}$ approaches the solution $H_{max}=\unicode[STIX]{x1D70F}$ (see figure 8 b). In addition, the normalized current shape above the porous medium approaches the self-similar shape described by (3.56) as $\unicode[STIX]{x1D70F}\rightarrow 0$ (see figure 8 a). We note the solution (3.56) describes our numerical data but we do not have an a priori argument for the functional form of $X_{F}(\unicode[STIX]{x1D70F})$ .

Similar to regime IV, with receding front at late times, the only assumption we need is $x_{f}(t)\gg h(0,t)$ since we are only interested in the fluid above the porous medium, which provides a self-consistent requirement:

As long as the right-hand side of (3.57) is of order of $1$ or larger, the drainage theory is valid in the late-time period.

3.5.1 Fixed volume: $\unicode[STIX]{x1D6FC}=0$

For the case when $\unicode[STIX]{x1D6FC}=0$ , i.e. a fixed volume, there are self-similar solutions in two asymptotic time limits: $T\ll 1$ and $T\rightarrow T_{d}$ . The good agreement between the PDE numerical solutions and the self-similar solutions when $T\rightarrow T_{d}$ has already been discussed in § 3.4.4. Hence, we only focus on comparing the PDE numerical solutions with the early-time self-similar solutions, i.e. $T\ll 1$ , and the transition between the early-time to the late-time asymptotic behaviours.

Figure 9. The PDE numerical solutions and the early-time self-similar solutions (3.58a–c ) when $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ for $X_{F}(T)$ , $H_{max}(T)$ and $L_{max}(T)$ with initial conditions described by (3.16) with: (a,c,e) $H_{0}=10$ ; (b,d,f):  $H_{0}=3$ . In all the legends, ‘num soln’ $=$ numerical solution.

Figure 10. The early-time evolution of the normalized shape of the current, when $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D719}=0.37$ , and $H^{\ast }=1$ , with initial conditions (3.16) and $H_{0}=10$ . The PDE numerical solutions for $H$ shown in the direction of the arrows are at $T=\{10^{-7},10^{-6},10^{-5},10^{-4},10^{-3},0.01,0,1,0.2,0.3\}$ , respectively. The PDE numerical solutions for $L$ shown in the direction of the arrow are at $T=\{10^{-7},10^{-6},10^{-5},10^{-4},10^{-3}\}$ , respectively. In the legend, ‘soln’ $=$ solution.

From the analysis in § 3.3.1, we know that when $T\ll 1$ , the self-similar solutions for the front location $X_{F}(T)$ , the maximum height $H_{max}(T)$ and the maximum depth $L_{max}(T)$ are

The numerical solutions for $X_{F}(T)$ , $H_{max}(T)$ and $L_{max}(T)$ from PDEs (3.11a–c ) for $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ are compared with the self-similar solutions described by PDEs (3.58a–c ) in figure 9 with two initial conditions: (a,c,e) a step function given by (3.16) with $H_{0}=10$ ; (b,d,f) a step function given by equations (3.16) with $H_{0}=3$ . We find that for both cases with different initial conditions, the PDE solutions need some time to approach the early-time self-similar solutions, then collapse with the self-similar solutions and finally depart from them. Also, the case with $H_{0}=10$ exhibits a longer time period when there is a good agreement between the PDE solutions and the self-similar solutions than the case with $H_{0}=3$ , because only when $H\gg L$ , are the early-time self-similar solutions valid. A larger $H_{0}$ guarantees longer time for the assumption $H\gg L$ to hold, which elongates the time span for validity of the early-time self-similar solutions.

In addition, we observed a maximum value of the front location, $X_{F,max}$ in figure 9(a,b), as denoted by the horizontal blue dashed line. After the front location reaches $X_{F,max}$ , it starts to recede. Also, from figure 9(c,d), the maximum height $H_{max}\rightarrow 0$ as time proceeds, which predicts a drain-out time $T_{d}$ , as denoted by the black arrows. The receding front and the drain-out behaviour predicted by the PDE solutions are consistent with the discussion in § 3.4.4.

We also show the time evolution of the normalized shape of the current in figure 10 when $\unicode[STIX]{x1D6FC}=0$ , $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ with initial conditions (3.16) and $H_{0}=10$ . The PDE solutions for $H/H_{max}$ (blue curve above the grey horizontal line) first approach, then collapse with, and finally depart from the self-similar shape (red curve above the grey horizontal line), as time proceeds. However, even though the PDE solutions for $L/L_{max}$ (blue curve under the grey horizontal line) approach the self-similar shape (red curve under the grey horizontal line), as time proceeds, there is no perfect collapse as for $H/H_{max}$ . This difference is because when $T\ll 1$ , $L$ is determined by $H$ (see (3.17b )), which means only when $H$ begins to follow its self-similar solution, can $L$ start approaching its self-similar solution. In this example, the time when $H$ agrees with its self-similar solution is too short for $L$ to fully develop its early-time self-similar shape.

3.5.2 Constant addition: $\unicode[STIX]{x1D6FC}=1$

For constant addition, i.e. $\unicode[STIX]{x1D6FC}=1$ , the PDE numerical results of the current shape are shown in figure 11(a) at early times and 11(b) at late times with $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ . We find that at early times $H$ , $L$ and $X_{F}$ all increase as time increases. However, at late times, $H$ and $X_{F}$ stay almost unchanged, while $L$ increases, which means that because of the insufficient addition rate, almost all the liquid added above the porous medium is lost by drainage. We also notice that at very late times, e.g. $T=100$ , $L$ approaches a step function, which is consistent with the self-similar solution described by (3.47c ).

Figure 11. Time evolution of the shape of the current under constant addition rate, i.e. $\unicode[STIX]{x1D6FC}=1$ , with $\unicode[STIX]{x1D719}=0.37$ , $H^{\ast }=1$ : (a) at early times; (b) at late times. The results are calculated numerically from PDEs (3.11a–c ).

Figure 12. (a) Time evolution of the front location when $\unicode[STIX]{x1D6FC}=1$ , $\unicode[STIX]{x1D719}=0.37$ and $H^{\ast }=1$ . The numerical results of PDEs (3.11a–c ) (black triangles) are compared with the self-similar solutions in the early-time (red dashed curve) and late-time (blue dotted curve) periods. (b) Time evolution of the normalized shape of the current. The numerical results of PDEs (3.11a–c ) at different times $T$ are compared with the normalized self-similar solutions in the early-time (thick red line) and late-time (thick blue line) periods. In both cases, the shapes of the current approach the normalized self-similar shapes.

The asymptotic analyses in §§ 3.3.2 and 3.4.2 give the asymptotic solutions for the front location for $\unicode[STIX]{x1D6FC}=1$ :

The front location first advances and finally arrests.

The PDE numerical results of the time evolution of the front location $X_{F}(T)$ and the self-similar solutions for $\unicode[STIX]{x1D6FC}=1$ in the asymptotic time limits (see (3.59)) are shown in figure 12(a), when $\unicode[STIX]{x1D719}=0.37$ and $H^{\ast }=1$ . There is good agreement between the PDE solutions (black triangles) and the predictions from the self-similar solutions in both early-time (red dashed curve) and late-time (blue dotted curve) periods. The transition between the early-time and late-time asymptotic behaviours occurs near $T=\mathit{O}(1)$ .

Figure 13. Time evolution of the shape of the current under continuous addition when $\unicode[STIX]{x1D6FC}=3$ , with $\unicode[STIX]{x1D719}=0.37$ , $Q=1$ : (a) at early times; (b) at late times. The results are calculated numerically from PDEs (3.11a–c ).

Figure 14. (a) Time evolution of the front location when $\unicode[STIX]{x1D6FC}=3$ , $\unicode[STIX]{x1D719}=0.37$ and $Q\,=1$ . The numerical results of PDEs (3.11a–c ) (black triangles) are compared with the self-similar solutions in early-time (red dashed curve) and late-time (blue dotted curve) periods. (b) Time evolution of the normalized shape of the current, when $\unicode[STIX]{x1D6FC}=3$ $\unicode[STIX]{x1D719}=0.37$ , and $Q=1$ . The numerical results of PDEs (3.11a–c ) at different times $T$ are compared with the self-similar solutions in early-time (thick red line) and late-time (thick blue line) periods. In both cases, the shapes of the current approach the self-similar shapes in asymptotic limits.

The asymptotic analyses in § 3.3.2 and 3.4.2 also provide the normalized self-similar shapes of the current. For the early-time period, the self-similar depth is given by

while the corresponding self-similar height $f(s)$ can be solved numerically. For the late times, the self-similar height $f(s)$ and depth $g(s)$ are given by (3.47a,b ), respectively. The comparisons of the normalized current shape, i.e. $H/H_{max}$ , $L/L_{max}$ versus $X/X_{F}$ , at different times $T$ with the normalized self-similar solutions $f/f_{max}$ , $g/g_{max}$ versus $s$ when $\unicode[STIX]{x1D6FC}=1$ , $\unicode[STIX]{x1D719}=0.37$ , and $H^{\ast }=1$ are shown in figure 12(b). The shapes of the current approach the normalized self-similar shapes in the asymptotic limits, $T\ll 1$ or $T\gg 1$ .

3.5.3 Continuous addition: $\unicode[STIX]{x1D6FC}=3$

For continuous addition when $\unicode[STIX]{x1D6FC}=3$ , the PDE numerical results of the current shape at different times are shown in figure 13(a) at early times and (b) at late times, with $\unicode[STIX]{x1D719}=0.37$ , $Q=1$ . The three dimensionless variables, $H$ , $L$ and $X_{F}$ increase during the entire time period. This result is because with a higher addition exponent $\unicode[STIX]{x1D6FC}$ , the liquid can still accumulate above the porous medium while draining into the porous medium.

The asymptotic analyses in § 3.3.3 and appendix C provide the early-time and late-time solutions for the front location for $\unicode[STIX]{x1D6FC}=3$ . When $\unicode[STIX]{x1D719}=0.37$ , $Q=1$ , the asymptotic solutions of the front location are

We compare the PDE results of the time evolution of the front location with the asymptotic solutions for $\unicode[STIX]{x1D6FC}=3$ , $\unicode[STIX]{x1D719}=0.37$ and $Q=1$ (see (3.61)) in figure 14(a). There is good agreement between the PDE solutions (black triangles) and the predictions from the asymptotic analysis in both early-time (red dashed line) and late-time (blue dotted line) periods.

In addition, the normalized current shape, i.e. $H/H_{max}$ , $L/L_{max}$ versus $X/X_{F}$ , at different times $T$ are compared with the normalized self-similar solutions $f/f_{max}$ , $g/g_{max}$ versus $s$ when $\unicode[STIX]{x1D6FC}=3$ , $\unicode[STIX]{x1D719}=0.37$ and $Q=1$ in figure 14(b). For the early times, the self-similar depth (see § 3.3.3) is given by

while the corresponding self-similar height $f(s)$ can be solved numerically. The late-time self-similar shape is solved numerically by the method described in appendix C. In both asymptotic limits, i.e. $T\ll 1$ or $T\gg 1$ , the normalized current shapes from the numerical results approach the normalized self-similar shapes.

By comparing figures 11(b) and 13(b), we find that as time goes on, for $\unicode[STIX]{x1D6FC}=1$ , the current above the porous medium becomes thinner and that in the porous medium becomes fatter, while for $\unicode[STIX]{x1D6FC}=3$ , the current above the porous medium stays almost unchanged and that in the porous medium becomes thinner. This difference is because for small $\unicode[STIX]{x1D6FC}$ , i.e. a slow addition in the late-time period, spreading of the current is slower than the drainage, which leads to the thinner shape of the liquid above and a fatter shape inside the porous medium, as time goes on. However, for large $\unicode[STIX]{x1D6FC}$ , i.e. a rapid addition in the late-time period, spreading of the current is comparable or faster than the drainage, which makes the shape of the current inside the porous medium thinner.