2.1. Governing equations and boundary conditions

Consider a fully saturated porous medium, which is initially unstrained in the absence of flow and gravitational stresses. We let $\phi$ denote the solid fraction of the porous medium, and $z$ denote the vertical axis; in the unstrained state, the medium has uniform stress-free solid fraction $\phi (z)=\phi _0$ and occupies depth $h_0$ (see figure 1 a). The medium is composed of solid phase with density $\rho _s$ and fluid phase with density $\rho _{\!f}$ such that its bulk density is $\rho = \rho _s \phi + \rho _{\!f} (1-\phi )$ . When $\rho _s\gt \rho _{\!f}$ , the medium differentially compacts under gravity to depth $h_g\lt h_0$ with solid fraction profile $\phi _g(z)$ ; we refer to this state as ‘gravity-slumped’ (see figure 1 b).

Figure 1.

The set-up for §§ 2–4. (a,b) No flow ( $\Delta p=0$ ) with (a) $\Delta \rho = \rho _s-\rho _{\!f}=0$ , stress-free, and (b) $\Delta \rho \gt 0$ , gravity-slumped. (c,d) The subsequent flow-induced compaction set-up, with $\Delta \rho \gt 0$ , for flow applied (c) downwards ( $Q\lt 0$ , driven by a negative pressure drop $\Delta p\lt 0$ ) and (d) upwards ( $Q\gt 0$ , driven by a positive pressure drop $\Delta p\gt 0$ ). The upper and lower walls are permeable to the fluid, but not to the solid.

Consider now that the gravity-slumped medium is lying between semi-permeable plates at $z=0$ and $z=h_g$ , which allow the fluid phase to freely pass through but do not allow passage of the solid phase. When an excess fluid pressure drop $\Delta p$ is applied across the sample, fluid flows through the medium with velocity $w_{\!f}(z)$ , which drives deformation of the medium itself with solid velocity $w_s(z)$ . We consider unidirectional flow, which may be generated in either vertical direction, depending on the sign of $\Delta p$ (see figures 1 c,d), and we choose to specify that a positive $\Delta p$ drives upward flow in the positive $z$ direction. As compaction occurs, the solid fraction profile $\phi (z)$ evolves, and the medium compacts to a depth $h=h_t-h_b$ , where $z = h_b, h_t$ are the positions of the bottom and top boundaries of the medium, respectively. The boundary out of which fluid flows remains fixed in position as the medium compacts against the relevant semi-permeable plate.

We consider both the solid and fluid phases to be individually incompressible, and enforce conservation of mass in each phase, Darcy’s law and total stress balance to give

(2.1) \begin{align}&\qquad - \frac {\partial \phi }{\partial t} + \frac {\partial }{\partial z} [(1-\phi )w_{\!f}] = 0, \end{align}

(2.2) \begin{align}&\qquad\qquad\, \frac {\partial \phi }{\partial t} + \frac {\partial }{\partial z}\left [\phi w_s\right ] = 0, \end{align}

(2.3) \begin{align}& (1-\phi )(w_{\!f}-w_s) = -\frac {k}{\mu }\left (\frac {\partial p}{\partial z}+\rho _{\!f} g\right )\!, \end{align}

(2.4) \begin{align}&\qquad\qquad\quad \frac {\partial \varPi }{\partial z} + \rho g = 0, \end{align}

where $k$ is the permeability of the medium, $\mu$ is the fluid viscosity, $p$ is the local pressure in the fluid, $g$ is the gravitational acceleration, and $\varPi$ is the total pressure of the two-phase system. In line with standard poromechanical modelling, we further define the effective pressure $\sigma = \varPi -p$ such that (2.3) and (2.4) become

(2.5) \begin{align} (1-\phi )(w_{\!f}-w_s) = \frac {k}{\mu } \left ( \frac {\partial \sigma }{\partial z}+\Delta \rho\, g\phi \right ) \!, \end{align}

where $\Delta \rho = \rho _s-\rho _{\!f}$ . The effective pressure $\sigma$ and permeability $k$ are material constitutive functions that encode the rheology of the medium. We refine our focus to general elastic media such that $\sigma$ depends only on the vertical strain, or equivalently, on the solid fraction $\phi$ (MacMinn, Dufresne & Wettlaufer Reference MacMinn, Dufresne and Wettlaufer2016). The stress-free solid fraction $\phi _0$ , introduced above, satisfies $\sigma (\phi _0)=0$ . We similarly assume that the permeability $k(\phi )$ depends only on the current solid fraction of the medium.

We must enforce that solid velocity is zero at the boundary out of which fluid flows. As such, $w_s(z=h_t) = 0$ in upward flow and $w_s(z=h_b) = 0$ in downward flow. The solid velocity at the opposite boundary is given simply by $w_s(h_{b,t}) = \partial h_{b,t}/\partial t$ . At the upper boundary $z=h_t$ , we scale the pore pressure $p$ to zero, thus at the lower boundary $p(h_b)=\rho _{\!f} g h+\Delta p$ , since $\Delta p$ is the applied pressure drop in excess of the background hydrostatic pressure.

The asymmetry induced by gravity requires the specification of different boundary conditions for $\varPi$ in upward and downward flows. We focus first on the case of downward flow ( $\Delta p \lt 0$ ); the boundary conditions for upward flow need more care and are presented in § 4.

In the absence of flow, the total pressure at the upper boundary is zero, $\varPi (h_t)=0$ . The pressure at the lower boundary follows from (2.4) and consists of the relative weight per unit area of the overlying solid and fluid. We neglect any additional resistance to flow coming from the entry conditions to the porous medium (e.g. Xu et al. Reference Xu, Zhang, Young, Yue and Feng2022) and as such, the total pressure at the upper boundary remains zero in the presence of downward flow. These pressure boundary conditions for downward flow can be expressed in terms of effective pressure $\sigma$ :

(2.6) \begin{align} &\sigma |_{z=h_t} = 0, \end{align}

(2.7) \begin{align} & \sigma |_{z= h_b} = \Delta \rho g h_0 \phi _0 + |\Delta p|, \end{align}

where the dependence on $\phi$ has been simplified by recognising that conservation of total solid mass enforces

(2.8) \begin{align} \int _{h_b}^{h_t} \phi\, {\rm d} z= h_0\phi _0. \end{align}

2.2. Non-dimensionalisation

This system of equations is closed by the specification of constitutive laws for the permeability $k$ and the effective pressure $\sigma$ . Given such laws, we can extract characteristic dimensional scales for the elastic modulus $\sigma ^*$ and permeability $k^*$ . To remove dimensions from the problem, we scale lengths with the stress-free depth $h_0$ , pressures with $\sigma ^*$ , velocity with $k^*\sigma ^*/h_0 \mu$ , and time with $h_0^2\mu /k^* \sigma ^*$ , the poroelastic time scale. Non-dimensionalisation introduces two dimensionless groups,

(2.9) \begin{align} \mathcal{G}=\frac {\Delta \rho g h_0}{\sigma ^*}, \quad \mathcal{P}=\frac {\Delta p}{{\sigma ^* }}, \end{align}

where $\mathcal{G}$ is a dimensionless measure of the relative importance of gravitational stress to elastic stresses, and $\mathcal{P}$ is the dimensionless pressure drop. We can also frame the relative gravity term $\mathcal{G}$ in the context of the compaction length scale $L= {\sigma ^* } /\Delta \rho g$ (Head & Wilson Reference Head and Wilson1992; Wilson & Head Reference Wilson and Head1994; Mergny & Schmidt Reference Mergny and Schmidt2024), such that $\mathcal{G} = h_0/L$ is a ratio of the medium’s depth to the depth over which gravitational compaction occurs; when $L\gg h_0$ , little gravitational compaction is expected.

The material flux $Q$ is defined by

(2.10) \begin{align} Q\equiv \phi w_s +(1-\phi )w_{\!f}, \end{align}

which, by combining (2.1)–(2.2), is constant in space. Equation (2.3) with all variables now dimensionless is

(2.11) \begin{align} (1-\phi )(w_{\!f}-w_s) &= k(\phi )\left (\sigma '\frac {\partial \phi }{\partial z}+\mathcal{G}\phi \right )\!, \end{align}

using the notation $\sigma ' = \partial \sigma /\partial \phi$ . The flux can therefore be identified as

(2.12) \begin{align} Q=w_s+ k(\phi )\left (\sigma '\frac {\partial \phi }{\partial z}+ \mathcal{G}\phi \right )\!, \end{align}

and the boundary conditions (2.6)–(2.7) become

(2.13) \begin{align} &\sigma |_{z=h_t} = \sigma (\varPhi _t) = 0, \end{align}

(2.14) \begin{align} & \sigma |_{z=h_b}= \sigma (\varPhi _b) = \mathcal{G} \phi _0+\left | \mathcal{P}\right | \!, \\[12pt] \nonumber \end{align}

where we have introduced notation for the solid fraction at the respective boundaries $\varPhi _t=\phi (z=h_t)$ and $\varPhi _b=\phi (z=h_b)$ . By assuming that the constitutive law $\sigma (\phi )$ is invertible, we can simply transform (2.13)–(2.14) into conditions on $\varPhi _{t,b}$ . Note simply that in a downward flow, the upper boundary is stress-free and $\varPhi _t=\phi _0$ .

Our primary concern here is to explore the steady solutions of compaction, but a brief summary of transient solutions is presented in Appendix A, which also serves as a numerical validation of the following steady solutions. In a steady state, it is most convenient to transform the mass conservation equation (2.8) into an integral over $\phi$ using (2.12) with $w_s\equiv 0$ , to yield an implicit equation for the unknown flux $Q$ :

(2.15) \begin{align} \int _{\varPhi _b}^{\varPhi _t} \frac {k\sigma '\phi }{Q-k\mathcal{G}\phi }\,{\rm d}\phi = \phi _0. \end{align}

Having determined $Q$ , the depth $h$ follows from direct integration of (2.12),

(2.16) \begin{align} \int _{\varPhi _b}^{\varPhi _t} \frac {k\sigma '}{Q-k\mathcal{G}\phi }\,{\rm d}\phi &= h, \end{align}

while the solid fraction profile $\phi (z)$ follows implicitly from integrating only up to $z$ ,

(2.17) \begin{align} \int _{\varPhi _b}^\phi \frac {k\sigma '}{Q -k\mathcal{G} u}\,{\rm d} u&=z(\phi ) -h_b, \end{align}

where $h_b\equiv 0$ in downward flow.

We can identify the grouping $D=k\sigma '\phi$ in (2.15) as the dimensionless form of the poroelastic diffusivity $ D^*=k\sigma '\phi /\mu$ , which is well known to control the essential behaviour of poroelastic compaction, and is related to the poroelastic time scale $h_0^2\mu /k^* \sigma ^*$ (Landman et al. Reference Landman, Sirakoff and White1991; Audet & Fowler Reference Audet and Fowler1992; Worster, Peppin & Wettlaufer Reference Worster, Peppin and Wettlaufer2021). As we will see, the dependence on the combination $k\sigma '$ , which is the product of the permeability and the effective stiffness of the medium, reflects the intrinsic interplay of these features in the mechanics of this problem, and dictates the qualitative behaviour of the deformable medium under compaction.

2.3. Constitutive laws

The model outlined above is defined for any constitutive laws for permeability $k(\phi )$ and any invertible effective pressure $\sigma (\phi )$ that are functions of solid fraction $\phi$ alone. When considering solutions to the model, we will focus our attention on specific choices for these laws. Generally, these laws must follow some physically intuitive restrictions. In particular, the medium cannot be compacted beyond a maximum solid fraction, $\phi _m \leqslant 1$ ; this identifies a corresponding minimum solid depth $h_m= \phi _0 h_0/\phi _m$ at which the medium is uniformly held at the maximum solid fraction. We choose the permeability to vanish at the maximum solid fraction $\phi _m$ but to diverge as the solid fraction tends to zero. Similarly, we require the effective pressure to vanish at the stress-free solid fraction $\phi _0$ , and we also expect it to diverge as $\phi \rightarrow \phi _m$ . These effects are captured by simple laws of the general form

(2.18) \begin{align} k = \frac {(\phi _m-\phi )^\delta }{\phi ^\eta }, \quad \sigma = \frac {(\phi -\phi _0)^\gamma }{(\phi _m-\phi )^\lambda }, \end{align}

for $\delta , \eta , \gamma ,\lambda \gt 0$ .

For the purposes of illustrating model results, we set $\delta = 3$ , $\eta = 2$ , $\gamma =1$ , and leave $\lambda$ as a free parameter. The choices of $\delta$ and $\eta$ are motivated by the widely used Kozeny–Carman permeability law for the modelling of uniform packed spheres (Carman Reference Carman1997), while the choice of $\gamma$ is made to specify a linear relation between $\sigma$ and $\phi$ for small deformations. As such, all solutions presented here are governed by the laws

(2.19) \begin{align} k = \frac {(\phi _m-\phi )^3}{\phi ^2}, \quad \sigma = \frac {(\phi -\phi _0)}{(\phi _m-\phi )^\lambda }, \end{align}

where results will be sensitive to choice of $\lambda$ . For simplicity, for the rest of this paper we will set $\phi _m=1$ and $\phi _0=0.6$ .

Note that in the chosen flow set-up (figure 1), the system can never be in a state of negative effective pressure (i.e. $\phi \lt \phi _0$ ). In such a state, the constitutive behaviour would vary significantly depending on whether the medium was cohesive (like a sponge) or comprised a collection of deformable particles (like a pack of hydrogel beads). For example, flow upwards through a collection of particles without confinement can lead to fluidisation (Van Zessen et al. Reference Van Zessen, Tramper, Rinzema and Beeftink2005), which is not considered here.

Figure 2.

Steady-state solutions for a medium compacted by a downward flow induced by pressure drop $\mathcal{P}\lt 0$ , showing (a,d) the flux $Q$ and (b,e) the depth $h$ as functions of pressure drop. (c,f) Graphs of the scaled solid fraction profiles $\phi (z/h)$ . Solutions are presented for media governed by two different effective pressure laws given by (2.19), with (a–c) $\lambda =4$ and (d–f) $\lambda =1$ .

3.1. Observations when $\mathcal{G}=0$

We first consider solutions when $\mathcal{G}=0$ , as would be the case if, for example, the porous medium comprised solid and fluid phases with comparable densities. While this case has been the focus of previous work (Parker et al. Reference Parker, Mehta and Caro1987; Paterson et al. Reference Paterson, Eaves, Hewitt, Balmforth and Martinez2019), it will be helpful to interrogate this limit a little further here to set the scene for the remainder of this work. In this situation, upward and downward flows are identical. Figure 2 illustrates flux $Q$ , depth $h$ , and profiles of solid fraction $\phi$ as a downward flow is induced with increasing pressure drop $|\mathcal{P}|$ across the medium. Flow through the porous medium induces differential compaction that results in denser packing at the lower boundary, where the fluid exits, while the medium remains stress-free at the upper boundary, where fluid enters.

Figure 2 highlights two possible behaviours for compaction as $|\mathcal{P}|$ increases for two media with different constitutive laws. The first medium (figure 2 a–c) shows a continual increase in $Q$ with $\mathcal{P}$ ; this is the standard behaviour for a rigid porous medium, but extends to deformable media if they remain sufficiently permeable under increasing compaction. The depth $h$ decreases towards the minimum solid depth $h_m=0.6$ , and $\phi$ correspondingly compacts towards the maximum solid fraction $\phi _m=1$ , although this limit is not attained at finite $\mathcal{P}$ . The second medium (figure 2 d–f) illustrates the existence of a maximal value of $Q$ as $\left |\mathcal{P}\right |$ increases. As $Q$ approaches this maximum, $h$ approaches a minimum that is larger than $h_m$ , and $\phi$ tends towards an asymptotic profile with a curvature different to that of the profiles in figure 2(c). All of these measures ( $Q$ , $h$ , $\phi$ ) effectively attain their asymptotic value or profile for $\mathcal{P}\approx \textit {O}(1)$ , and become insensitive to $\mathcal{P}$ for larger pressure drops, in clear contrast to the results in figure 2(a)–2(c). The difference between the media in this figure is that the second medium has a stiffness with a weaker dependence on $\phi$ than the first; that is, it stiffens less significantly under compaction. Due to its lower stiffness, the permeability – or more specifically, the loss of permeability – plays a more dominant role in determining the response of the medium: further compaction or increase in $Q$ is more strongly opposed by the resistance to flow.

As already noted, there is a marked difference in the curvature of the compacted profiles $\phi (z)$ between the two media as the pressure drop increases. This difference is encapsulated in the contrasting gradients of solid fraction $\partial \phi /\partial z$ at the lower boundary. The profiles in figure 2(c) show $\partial \phi /\partial z$ tending towards zero at the lower boundary, while the solid fraction steadily compacts across the depth of the medium as $\mathcal{P}$ increases. In contrast, figure 2(f) illustrates a limit towards an infinite gradient at the lower boundary as the solid fraction quickly reduces away from the lower boundary value.

In the next subsection, we will classify how these observed behaviours can be predicted from the constitutive laws. Some of these ideas were briefly discussed in the appendix of Hewitt et al. (Reference Hewitt, Nijjer, Worster and Neufeld2016a ), and we expand on and formalise those ideas here.

3.2. ‘Types’ of behaviour

3.2.1. Limiting flux

In a steady state and for $\mathcal{G}=0$ , the flux is given explicitly from (2.15) by

(3.1) \begin{align} Q=\frac 1 {\phi _0} \int _{\varPhi _b}^{\varPhi _t} k\sigma '\phi\, { \rm d} \phi =\frac 1 {\phi _0} \int _{\varPhi _b}^{\varPhi _t} D\, {\rm d} \phi . \end{align}

We consider a downward flow in which a negative pressure drop $\mathcal{P}$ induces negative flux $Q$ and has boundary conditions $\varPhi _b = \sigma ^{-1}({\left |\mathcal{P}\right |})$ and $\varPhi _t = \phi _0$ . The derivative of $|Q|$ with respect to $\left |\mathcal{P}\right |$ immediately shows

(3.2) \begin{align} \frac {\partial |Q|}{\partial {\left |\mathcal{P}\right |}} = \left .\frac {k\sigma '\phi }{\phi _0}\right |_{\varPhi _{b}}\frac {\partial \varPhi _{b}}{\partial \mathcal{P}} = \left .\frac {k \phi }{\phi _0}\right |_{\varPhi _{b}}\gt 0, \end{align}

since $\partial \varPhi _{b}/\partial {\left |\mathcal{P}\right |} = \sigma '(\varPhi _{b})^{-1}$ . As such, the magnitude of flux must increase with increasing pressure drop across the medium. This confirms that either $|Q|$ increases without bound, or $|Q|$ increases to an asymptotic limit as $\left |\mathcal{P}\right |$ increases; the two behaviours of flux observed in figure 2 are the only two possibilities.

In order to deduce for which laws we expect flux to be unbounded or limited with increasing $\left |\mathcal{P}\right |$ , we consider the limit $\left |\mathcal{P}\right |\rightarrow \infty$ , following Hewitt et al. (Reference Hewitt, Nijjer, Worster and Neufeld2016a ). In this limit, we expect $\phi \rightarrow \phi _m$ at the base, so we can generically express permeability and effective pressure as power laws of the form

(3.3) \begin{align} k \approx (\phi _m-\phi )^\alpha ,\quad \sigma \approx (\phi _m-\phi )^{-\beta }, \end{align}

for some $\alpha , \beta \geqslant 0$ . Note that in our chosen constitutive laws, given by (2.19), $\alpha = 3$ and $\beta = \lambda$ . Considering the effective pressure at the compacted boundary $\sigma (\varPhi _b)={\left |\mathcal{P}\right |}$ , we identify $(\phi _m-\varPhi _b)\approx {\left |\mathcal{P}\right |}^{-{1}/ \beta }$ , therefore it follows from integration of (3.2) that the flux scales like

(3.4) \begin{align} |Q| \approx {\left |\mathcal{P}\right |} ^{|1-\alpha/ \beta |} \quad & \text{for } \alpha \lt \beta , \end{align}

(3.5) \begin{align} |Q| \approx \ln ({\left |\mathcal{P}\right |}) \quad & \text{for } \alpha = \beta , \end{align}

(3.6) \begin{align} |Q| \approx Q_\infty + {\left |\mathcal{P}\right |} ^{-|1-\alpha/ \beta |} \quad & \text{for } \alpha \gt \beta , \end{align}

as ${\left |\mathcal{P}\right |}\rightarrow \infty$ , where $Q_\infty$ is a constant. Two types of behaviour can be identified: type 1 ( $\alpha \leqslant \beta$ ), for which $Q$ is unbounded with increasing $\left |\mathcal{P}\right |$ ; and type 2 ( $\alpha \gt \beta$ ), for which $Q$ is bounded. This classification can be recast in terms of the diffusivity $D = k\sigma '\phi$ , with type 1 corresponding to $D$ diverging faster than a simple pole, $(\phi _m-\phi )D\rightarrow \infty$ as $\phi \rightarrow \phi _m$ .

In summary, for type 1 media, $k$ approaches zero more slowly than $\sigma$ diverges as $\left |\mathcal{P}\right |$ is increased. Physically, this means that as a type 1 medium is steadily compacted towards the maximum solid fraction $\phi _m$ everywhere, the medium maintains sufficient permeability for flow. Therefore, like a rigid porous medium, the flux can be increased without bound simply by increasing the pressure drop. The insensitivity to loss in permeability allows for further continual compaction towards $\phi _m$ . For type 2 media, in contrast, $k$ tends to zero quicker than $\sigma$ diverges as $\left |\mathcal{P}\right |$ is increased, and consequently the mechanics of compaction close to the maximum solid fraction are controlled primarily by the loss of permeability. Microstructural effects enforce that at solid fractions less than the theoretical maximum $\phi _m$ , the medium becomes increasingly impermeable, continually impeding flow such that the magnitude of flux $|Q|$ increases asymptotically towards a finite limiting value $Q_{\infty }$ . Since further compaction is limited by the decreased permeability, the depth tends towards a finite compaction depth $h_{\infty }$ that is larger than the theoretical minimum solid depth $h_m$ .

In general, it is difficult to predict the ‘type’ of a given deformable medium due to the intrinsic dependence on the combination of permeability and stiffness. However, it is possible to find examples of constitutive laws corresponding to both ‘types’ in previous literature. For example, constitutive laws giving type 1 media have been reported for nylon fibres in glycerine (Hewitt et al. Reference Hewitt, Paterson, Balmforth and Martinez2016b ) and intervertebral discs (Riches et al. Reference Riches, Dhillon, Lotz, Woods and McNally2002). Reported type 2 media include hydrogel beads (MacMinn et al. Reference MacMinn, Dufresne and Wettlaufer2015), cellulose fibres (Hewitt et al. Reference Hewitt, Paterson, Balmforth and Martinez2016b ), compacting shale layers (Audet & Fowler Reference Audet and Fowler1992) and waste water treatment sludge (Stickland & Buscall Reference Stickland and Buscall2009).

3.2.2. Limiting compaction profiles

In a downward flow, the gradient of solid fraction $\partial \phi / \partial z$ at the lower compacted boundary is found from rearrangement of (2.12) and is indicative of the overall steady compaction profile $\phi (z)$ . With $\mathcal{G} = 0$ , the gradient in the limit $\phi \rightarrow \phi _m$ is simply $\partial \phi / \partial z\rightarrow Q(\phi _m-\phi )^{1+\beta -\alpha }$ , which is dependent on the powers $\alpha,\beta$ and the limiting form of $Q$ as defined in (3.4)–(3.6). As before, we can express $\mathcal{P}$ in terms of the compacted boundary solid fraction $\varPhi _b$ through the effective pressure boundary condition; $(\phi _m-\varPhi _b)\approx {\left |\mathcal{P}\right |}^{-{1}/ \beta }$ .

For type 1 media ( $\alpha \leqslant \beta$ ), $Q$ is unbounded for increasing $\left |\mathcal{P}\right |$ , with precise form given by (3.4)–(3.5). Combined with the limiting expression for $\left |\mathcal{P}\right |$ , we deduce that at the lower boundary, $\partial \phi / \partial z \rightarrow 0$ as ${\left |\mathcal{P}\right |} \rightarrow \infty$ .

For type 2 media ( $\alpha \gt \beta$ ), $Q$ tends to a limiting value $Q_{\infty }$ , so the limit of the gradient is determined explicitly by the sign of $1+\beta -\alpha$ , giving the follow options for the gradient at the lower boundary:

1. (i) type $2i$ ( $\beta \lt \alpha \lt \beta +1$ ), $D \rightarrow \infty$ , ${\partial \phi }/{\partial z}\rightarrow 0$

2. (ii) type $2\textit{ii}$ ( $\alpha =\beta +1$ ), $D \rightarrow D_\infty$ (constant), ${\partial \phi }/{\partial z}\approx\text{constant}$

3. (iii) type $2\textit{iii}$ ( $\alpha \gt \beta +1$ ), $D \rightarrow 0$ , ${\partial \phi }/{\partial z}\rightarrow \infty$ .

Figure 3.

(a) Steady-state profiles for the solid fraction $\phi$ for different pressure drops $\mathcal{P}$ given in the legend (black lines). All solutions have $\mathcal{G} =0$ . Constitutive laws are given by (2.19) such that for type 1, $\lambda = 4$ , for type $2i$ , $\lambda = 5/ 2$ , for type $2\textit{ii}$ , $\lambda = 2$ , and for type $2\textit{iii}$ , $\lambda = 1$ . The large $\mathcal{P}$ limit for each set of laws is illustrated with red dashed lines. (b) The flux $|Q|$ for each type of medium plotted against applied pressure drop $\mathcal{P}$ , with the limiting flux $Q_{\infty }$ shown by dashed lines for each type 2 medium. (c) The depth $h$ for each type, with the limiting compaction depth $h_{\infty }$ and minimum depth $h_m$ shown by dashed lines.

Figure 3 presents solutions for the solid fraction profile $\phi$ , flux $Q$ and depth $h$ as $\left |\mathcal{P}\right |$ increases, for each type of medium identified above. Type 1 media are shown to characteristically compact towards $\phi _m$ everywhere, with $\partial \phi / \partial z\rightarrow 0$ at the lower boundary where the fluid leaves the domain. Type $2i$ media have the same zero gradient of $\phi$ at the base, but they instead approach a non-uniform asymptotic $\phi (z)$ profile. Similarly, type $2\textit{ii}$ and type $2\textit{iii}$ media approach non-uniform asymptotic $\phi (z)$ profiles, but are differentiated by the finite gradient of $\phi$ at the base in type $2\textit{ii}$ media, and an infinite gradient of $\phi$ at the base in type $2\textit{iii}$ . Figures 3(b) and 3(c) illustrate the existence of limiting flux $Q_{\infty }$ and limiting depth $h_{\infty }$ in type 2 media. While the depth of type 1 media is of course also limited by the minimum solid depth $h_m = \phi _0/\phi _m$ , in type 2 media the limiting depth $h_\infty$ is distinct and larger than this value. We note that the order of $\left |\mathcal{P}\right |$ at which $Q$ and $h$ approach these limiting values increases significantly as the relative power of $k$ to $\sigma$ is decreased in the constitutive laws.

Note that these observations can be linked to the analysis of Hewitt et al. (Reference Hewitt, Paterson, Balmforth and Martinez2016b ) for the formation of boundary layers under a quick moving piston for the dewatering of soft porous media. They found that in the limit $\phi \rightarrow \phi _m$ , the boundary layer solutions are determined by whether $D$ remains finite or diverges. The boundary condition for solid fraction at the medium-piston interface was found as an output of the model with finite $D$ (type $2\textit{ii}$ and $2\textit{iii}$ media) corresponding to the solid fraction at this boundary attaining its maximum value and the pressure on the piston diverging, whereas $D\rightarrow \infty$ (type $1$ and $2i$ media) corresponds to solid fraction approaching but not reaching the maximum.

3.3. Gravity and type interaction

In the absence of flow and with $\mathcal{G}\gt 0$ , the medium non-uniformly compacts due to gravitational stresses, with denser packing at the lower boundary, and stress-free solid fraction at the upper boundary. Compaction due to gravitational stresses alone, which we call gravity-slumping, is unaffected by the permeability, unlike flow-induced compaction, so inevitably, the gravity-slumped profiles of solid fraction are different to those for flow-induced compaction (figure 4 a). Figure 4(b) shows the no-flow gravity-slumped depth $h_g$ for an example medium for increasing $\mathcal{G}$ , illustrating how the medium can slump under gravity all the way down to the minimum depth $h_m$ for sufficiently large $\mathcal{G}$ . This behaviour is independent of the ‘type’ of the medium.

Natural and industrial porous media give rise to a wide range of relative gravity values. Even supposedly stiff porous media can have large $\mathcal{G}$ values if the relevant length scale $h_0$ is large enough. For example, the gravitational compaction of ice over several hundreds of metres results in $\mathcal{G} = \textit {O}(10)$ (Mergny & Schmidt Reference Mergny and Schmidt2024); notably softer porous media, such as fibreglass insulation or wool stuffing, may have similar or larger relative gravity $\mathcal{G}$ on much shorter length scales ( $h_0\approx 1$ m). It is clear from figure 4(b) that such values of $\mathcal{G}$ can result in significant gravity-slumping, although the precise depth is highly dependent on the constitutive laws for the media.

Figure 4.

The gravity-slumped solutions for (a) profiles of solid fraction $\phi _g$ for different relative gravity $\mathcal{G}$ as given in the legend, and (b) depths $h_g$ against increasing $\mathcal{G}$ . The blue dashed line in (b) represents the minimum depth $h_m$ . The medium is governed by (2.19) with $\lambda =1$ , although the illustrated behaviour is independent of medium ‘type’.

The extension of the analysis in § 3.2 to the regime $\mathcal{G}\gt 0$ is more mathematically involved but draws the same conclusion of classification of types, as outlined in Appendix B. The plots in figure 5 illustrate how the interaction between gravitational and flow-induced compaction varies for type 1 and type 2 media. Under a downward flow, type 1 media can compact everywhere towards the maximum solid fraction $\phi _m$ as ${\left |\mathcal{P}\right |}\rightarrow \infty$ . As such, it is unsurprising that for large enough $\mathcal{P}$ , the value of $\mathcal{G}$ has a negligible impact on the profiles of $\phi$ (figure 5 a). In contrast, the limiting profiles of $\phi$ for type 2 media in a downward flow depend sensitively on the value of $\mathcal{G}$ (figure 5 b). Indeed, for large enough $\mathcal{G}$ , there is little difference between the gravity-slumped profile for $\mathcal{P} =0$ (left-hand plot) and the asymptotic profile as ${\left |\mathcal{P}\right |}\rightarrow \infty$ (right-hand plot); we deduce here that gravity plays the dominant role in compaction, and little further deformation by fluid flow is possible.

We can further interrogate type 2 media’s dependency on $\mathcal{G}$ by observing the minimum flow-induced compaction depth $h_\infty$ for increasing $\mathcal{G}$ . Figure 5(c) overlays $h_\infty$ onto the results of figure 4(b); increasing $\mathcal{G}$ decreases both the initial flow-free depth $h_g$ and the limiting depth $h_\infty$ . However, if we instead inspect the relative change in depth $(h_g-h_\infty )/h_g$ in figure 5(d), which can be considered as the actual flow-induced strain, then we observe that this induced strain is decreasing as $\mathcal{G}$ increases. This is not surprising: since the initial gravity-slumped state is compacting towards the maximum solid fraction $\phi _m$ , the medium is under higher initial pressure and is therefore harder to further compact by flow. As such, we can conclude that increasing relative gravity results in less flow-induced strain in type 2 media, despite the fact that lower depths are reached.

Figure 5.

Steady-state solid fraction $\phi$ for porous media governed by constitutive laws (2.19) with (a) $\lambda =4$ , a type 1 medium, and (b) $\lambda =1$ , a type 2 medium. Solutions are shown for different pressure drops $\mathcal{P}$ inducing downward flow and different relative gravity $\mathcal{G}$ , as marked. (c) Plots of both gravity-slumped depth $h_g$ (black) and minimum flow-induced compaction depth $h_\infty$ (orange) against relative gravity $\mathcal{G}$ for the type 2 medium in (b). The dashed blue line marks the minimum depth $h_m$ . (d) Plots of the same data as in (c), but as relative strain $(h_g-h_{\infty })/h_g$ .

4.1. Inclusion in the model

We previously established that for downward flow, the total pressure $\varPi$ and fluid pressure $p$ disappear at the upper boundary, which is therefore stress-free. Upward flow, however, enters through the stressed lower boundary, so the equivalent boundary conditions are not so straightforward. As discussed above, flow reduces the entry boundary pressure by an amount that is a priori unknown, since it relies on how the medium rearranges internally. Given (2.4), we can determine the effective pressure boundary conditions up to a constant $\varSigma$ :

(4.1) \begin{align} \sigma (\varPhi _t) &= \mathcal{P}-\varSigma (\mathcal{P}), \end{align}

(4.2) \begin{align} \sigma (\varPhi _b) &= \mathcal{G}\phi _0-\varSigma (\mathcal{P}), \end{align}

where the relieved pressure $\varSigma$ evolves with applied pressure drop $\mathcal{P}$ and is bounded by $0\leqslant \varSigma \leqslant \mathcal{G}\phi _0$ , enforcing $\sigma \gt 0$ everywhere. Note here that the pressure felt at the upper boundary due to $\mathcal{P}$ is equally reduced as pressure is taken up by the internal rearrangement of the solid fraction as gravitational compaction is ‘undone’.

While the entry pressure has not been fully relieved ( $\sigma (\varPhi _b)\gt 0$ ), the medium is able to simultaneously compact and decompact, maintaining a fixed depth $h_g$ . Once $\sigma (\varPhi _b)=0$ , or equivalently $\varSigma = \mathcal{G}\phi _0$ , the medium can no longer decompact, and further increase in $\mathcal{P}$ will drive bulk compaction ( $h\lt h_g$ ). Having non-dimensionalised as in § 2.2, this mechanism is included in the steady equations (2.15)–(2.16) with the adapted boundary conditions (4.1)–(4.2), as follows:

(4.3) \begin{align} &\int _{\varPhi _b}^{\varPhi _t} \frac {k\sigma '\phi }{Q-k\mathcal{G}\phi }\,{\rm d}\phi = \phi _0, \end{align}

(4.4) \begin{align} & \int _{\varPhi _b}^{\varPhi _t} \frac {k\sigma '}{Q-k\mathcal{G}\phi }\,{\rm d}\phi = \begin{cases} h_g &\text{for } \sigma (\varPhi _b)\gt 0, \\h &\text{for } \sigma (\varPhi _b)=0. \end{cases} \\[6pt] \nonumber \end{align}

If $\sigma (\varPhi _b)\gt 0$ , then the unknowns in (4.3)–(4.4) are $\varSigma$ and $Q$ , with $h=h_g$ known. If, instead, $\sigma (\varPhi _b)=0$ , then the two unknowns are $h$ and $Q$ . We denote the value of $\mathcal{P}$ that separates these two cases to be the ‘lift-off’ pressure $\mathcal{P}_{\textit{lift}}$ , found by setting both $h=h_g$ and $\sigma (\varPhi _b)=0$ in (4.3)–(4.4).

4.2. General solutions

Gravity immediately introduces an asymmetry between upward and downward flow. We present in figure 7 a simple comparison of this asymmetry in both type 1 and type 2 example media. Figures 7(a) and 7(b) compare the profiles of solid fraction under compaction by downward and upward flow, respectively, in a type 1 medium, highlighting three key features. First, upward flow acts to ‘undo’ compaction due to gravity: inducing a downward flow can only compact the medium more, whereas upward flow allows for some initial decompaction of the medium. Second, in upward flow, there is a range of low pressure drop in which the depth remains fixed at depth $h_g$ . Finally, during this fixed depth regime, flow-induced compaction can rearrange the internal packing of the medium to a uniform profile of solid fraction when the applied pressure drop $\mathcal{P}$ exactly balances the gravitational pressure $\mathcal{G}\phi _0$ on the lower boundary. This homogenisation of the medium must occur in the regime of fixed depth, as it requires both boundaries to be somewhat compacted. Figures 7(e) and 7(f) illustrate the same ideas in a type 2 medium.

In considering the evolution of depth $h$ and flux $Q$ (figure 7 c,g,d,h), we note that upward flow consistently corresponds to a larger depth (less compacted) and higher flux compared to downward flow. Upward flow must first ‘undo’ gravitational compaction, and as such, the pressure is taken up by the rearrangement of internal packing, resulting in a reduced effective pressure at the boundary out of which flow exits. For any given pressure drop $\mathcal{P}$ , we can expect downward flow to result in greater compaction and more dense packing, thus lowering the bulk permeability. We consider that the resistivity of the medium to flow is higher in a downward flow, and as such, flux is impeded. Consequently, the limiting depths $h_\infty$ and flux $|Q_\infty |$ in type 2 media will always be lower for a downward flow.

Figure 7.

Steady solutions for initially gravity-slumped media (black) with $\mathcal{G}=5$ governed by constitutive laws (2.19), with (a–d) $\lambda = 3$ , type 1 medium, and (e–h) $\lambda = 1$ , type 2 medium. (a,b,e,f) Solid fraction profiles $\phi$ for media compacted by (a,e) a downward flow (red), and (b,f) an upward flow (blue). The corresponding depth $h$ and flux $|Q|$ evolutions are in (c,g) and (d,h), respectively.

Figure 8 presents the lift-off pressure $\mathcal{P}_{\textit{lift}}$ for changing relative gravity $\mathcal{G}$ in both type 1 and type 2 media. Increasing $\mathcal{G}$ will increase gravitational compaction, so must increase $\mathcal{P}_{\textit{lift}}$ . While this increase is continual in the type 1 medium, in a type 2 medium it becomes unbounded at a finite value of $\mathcal{G}$ (figure 8 b). Beyond this critical $\mathcal{G}$ , no such $\mathcal{P}_{\textit{lift}}$ exists: no pressure drop is able to induce bulk compaction in an upward flow, and the depth remains fixed at $h_g$ . The example illustrated in figure 7(g) falls in this range, and $h$ will remain fixed for any $\mathcal{P}$ .

The interaction of type 2 media with gravity is further illustrated in figure 9, which shows the limiting compaction depth $h_\infty$ for upward and downward flow. We have already observed that increasing $\mathcal{G}$ lowers $h_\infty$ for downward flow (figure 5 c); here, we see the opposite trend for upward flow for $\mathcal{G}$ sufficiently small as $h_\infty$ directly increases with $\mathcal{G}$ . However, when the critical $\mathcal{G}$ at which ${\mathcal{P}_{\textit{lift}}}\rightarrow \infty$ is exceeded, $h_\infty$ becomes fixed at the gravity-slumped depth $h_g$ and thus decreases with further increase in $\mathcal{G}$ . In this range, flow-induced bulk compaction is no longer possible in upward flow, and the depth is controlled solely by gravity-slumping.

Figure 8.

Lift-off pressure $\mathcal{P}_{\textit{lift}}$ against relative gravity $\mathcal{G}$ for media governed by constitutive laws (2.19) with (a) $\lambda =3$ , a type 1 medium, and (b) $\lambda =1$ , a type 2 medium. The dashed line in (b) illustrates the asymptote of $\mathcal{P}_{\textit{lift}}$ .

Figure 9.

The limiting compaction depth $h_\infty$ against relative gravity $\mathcal{G}$ for a type 2 medium governed by (2.19) with $\lambda =1$ . Solutions are presented for upward (blue) and downward (red) flow, together with the no-flow gravity-slumped depth $h_g$ (black). The dashed line shows the critical $\mathcal{G}$ above which the lift-off pressure for upward flow is unbounded.

5.1. Inclusion in the model

As with upward flow in § 4, when the entry boundary is stressed, the internal packing of the solid fraction can be rearranged, leaving the overall depth fixed at $h_i$ (figure 10 c). We denote this entry boundary pressure as $\sigma (\varPhi _e)$ , where $e=t,b$ in downward and upward flow, respectively. Once the entry pressure is relieved ( ${\sigma (\varPhi _e)}=0$ ), bulk compaction occurs as $\mathcal{P}$ is increased further, such that the depth of the medium decreases ( $h\leq h_i$ ; figure 10 d).

We will again focus on the steady-state solutions; pre-strain is incorporated into the steady equations (2.15)–(2.16), having appropriately non-dimensionalised as in § 2.2, as follows:

(5.1) \begin{align} &\int _{\varPhi _b}^{\varPhi _t} \frac {k\sigma '\phi }{Q-k\mathcal{G}\phi }\,{\rm d}\phi = \phi _0, \end{align}

(5.2) \begin{align} & \int _{\varPhi _b}^{\varPhi _t} \frac {k\sigma '}{Q-k\mathcal{G}\phi }\,{\rm d}\phi = \begin{cases} h_i &\text{for } {\sigma (\varPhi _e)} \gt 0, \\h &\text{for } {\sigma (\varPhi _e)}=0, \end{cases} \\[10pt] \nonumber \end{align}

where $e = b$ or $t$ . The boundary conditions are adapted from (2.13)–(2.14), but now incorporate the pre-strain pressure:

(5.3) \begin{align} \sigma (\varPhi _t) &= \begin{cases} \mathcal{P}+{\mathcal{S}}_0-\varSigma &\text{for }\mathcal{P} \geqslant 0, \\{\mathcal{S}}_0-\varSigma &\text{for } \mathcal{P} \leqslant 0, \end{cases} \end{align}

(5.4) \begin{align} \sigma (\varPhi _b) &= \begin{cases}\mathcal{G}\phi _0+{\mathcal{S}}_0-\varSigma &\text{for } \mathcal{P}\geqslant 0 , \\{\left |\mathcal{P}\right |} +\mathcal{G}\phi _0+{\mathcal{S}}_0-\varSigma &\text{for }\mathcal{P} \leqslant 0, \end{cases} \\[6pt] \nonumber \end{align}

where $\varSigma (\mathcal{P})$ is the relieved pressure. Note that for upward flow, both the pre-strain pressure and gravitational pressure must be relieved in order to induce bulk compaction ( $\varSigma =\mathcal{G}\phi _0+{\mathcal{S}}_0$ ), while downward flow need only relieve the pre-strain pressure ( $\varSigma ={\mathcal{S}}_0$ ). If ${\sigma (\varPhi _e)}\gt 0$ , then the two unknown variables in (5.1)–(5.2) are $Q$ and $\varSigma$ , with $h = h_i$ known. Alternatively, if ${\sigma (\varPhi _e)} = 0$ , then the two unknowns are $Q$ and $h$ , and the system coincides with the compaction of an unpre-strained medium. We again denote the applied excess pressure at which the entry pressure is fully relieved to be the lift-off pressure $\mathcal{P}_{\textit{lift}}$ , which is found by solving (5.1)–(5.2) with ${\sigma (\varPhi _e)}=0$ and $h=h_i$ .

The initial pre-strain depth $h_i$ and the corresponding initial pre-strain pressure ${\mathcal{S}}_0$ are related by the solution of (5.2)–(5.4) with $Q=0$ , such that (5.2) is simply

(5.5) \begin{align} h_i=\int _{\varPhi _b}^{\varPhi _t} \frac {-\sigma '}{\mathcal{G}\phi }\,{\rm d}\phi , \end{align}

with $\varPhi _t = \sigma ^{-1}({\mathcal{S}}_0)$ and $\varPhi _b = \sigma ^{-1}({\mathcal{S}}_0+\mathcal{G}\phi _0)$ .

Figure 11.

Steady solutions for (a) entry boundary pressure $\sigma (\varPhi_e)$ , (b) depth $h$ , and (c) flux $|Q|$ , against applied pressure drop $|\mathcal{P}|$ for a medium with $\mathcal{G}=5$ , pre-strained to depth $h_i=0.95$ , such that the pre-strain coefficient is $\varDelta = 0.008$ , for upward (blue) or downward (red) flow. The profiles of solid fraction $\phi$ are presented in (d) and (e) for upward and downward flow, respectively, at given values of $\mathcal{P}$ denoted by dots in (a–c). Dashed grey lines in (a–c) illustrate the respective lift-off pressures $|\mathcal{P}_{\textit{lift}}|$ , and dashed coloured lines in (b,c) illustrate the unpre-strained solutions for depth, which coincide with the pre-strained solutions once $\sigma (\varPhi_e )=0$ . Dotted lines in (d) and (e) illustrate the boundaries of the medium $h_b$ and $h_t$ . The medium is governed by constitutive laws (2.19) with $\lambda =4$ .

5.2. Solutions

Figure 11 illustrates the behaviour of flow-induced compaction for a pre-strained medium. If the applied pressure drop is less than the lift-off pressure, ${\left |\mathcal{P}\right |}\leqslant {\mathcal{P}_{\textit{lift}}}$ , then the entry pressure $\sigma (\varPhi _e)$ is reduced but not relieved, the depth $h$ is unchanged from $h_i$ , and the profile of solid fraction $\phi$ rearranges, simultaneously compacting and decompacting in different regions. Alternatively, if ${\left |\mathcal{P}\right |}\geqslant {\mathcal{P}_{\textit{lift}}}$ , then the entry boundary is stress-free, $h$ is reduced from $h_i$ , and the medium can no longer decompact with $\phi$ strictly increasing. Figure 11(a) highlights the difference in lift-off pressure due to the orientation of flow: downward flow has a much lower lift-off pressure $\mathcal{P}_{\textit{lift}}$ . As before, upward flow admits the possibility of homogenising the medium as increasing applied pressure drop passes through $\mathcal{P} = \mathcal{G}\phi _0$ and the gradient of solid fraction changes sign (figure 11 d).

Figure 12.

Lift-off pressure $\mathcal{P}_{\textit{lift}}$ against the amount of pre-strain $\varDelta = (h_g-h_i)/h_g$ for compaction induced by a downward flow (red) and upward flow (blue) with $\mathcal{G}=0.5$ . The media are governed by constitutive laws (2.19) with (a) $\lambda =3$ , a type 1 medium, and (b) $\lambda =1$ , a type 2 medium. Black dotted asymptotes in (b) correspond to $\varDelta = (h_g-h_{\infty })/h_g$ , where $h_{\infty }$ is the limiting compaction depth for an unpre-strained system.

Figure 11(b) compares the pre-strained evolution of $h$ as $\left | \mathcal{P}\right |$ increases with the unpre-strained evolution of $h$ , and illustrates how, once $\sigma (\varPhi _e)$ is relieved, the pre-strained depth exactly coincides with the unpre-strained depth. The steady solutions of the pre-strained medium see no history of being mechanically compressed once ${\left |\mathcal{P}\right |}\geqslant {\mathcal{P}_{\textit{lift}}}$ . By corollary, we deduce that $\mathcal{P}_{\textit{lift}}$ is exactly the applied pressure drop required to compact the unpre-strained medium to the pre-strain depth $h_i$ . Similarly, $Q$ in figure 11(c) is identical to the unpre-strained solution once ${\left |\mathcal{P}\right |}\geqslant {\mathcal{P}_{\textit{lift}}}$ , and as before, flux is higher in the upward flow than the downward flow.

In order to measure the relative strain from the no-flow state, we introduce the parameter $\varDelta =(h_g-h_i)/h_g$ , which quantifies the amount of pre-strain relative to the gravity-slumped depth $h_g$ . Figure 12 presents the lift-off pressure $\mathcal{P}_{\textit{lift}}$ for increasing $\varDelta$ . The solutions contrast upward and downward flow, and type 1 and type 2 media. These graphs have some analogy with figure 8, which illustrated how increasing $\mathcal{G}$ directly increased $\mathcal{P}_{\textit{lift}}$ . Much like gravitational compaction, mechanical pre-strain increases the solid fraction throughout the medium towards the maximum solid fraction $\phi _m$ , increasing the effective pressure and decreasing the permeability; consequently, $\mathcal{P}_{\textit{lift}}$ increases with $\varDelta$ . We note that in both types of media in figure 12, $\mathcal{P}_{\textit{lift}}$ is non-zero at zero strain $\varDelta$ for upward flow corresponding to the gravitational lift-off pressure that is characteristic of upward flows.

The behaviour discussed thus far is independent of the ‘type’ of the medium. However, there is some difference between the two types as we increase $\varDelta$ . Since $\mathcal{P}_{\textit{lift}}$ is exactly the pressure required to compact an unpre-strained medium to the pre-strained depth $h_i$ , there is no lift-off pressure in type 2 media if they are sufficiently pre-strained. More specifically, this occurs if the medium is mechanically compacted beyond the limiting flow-induced compaction depth $h_{\infty }$ . Since flow-induced compaction cannot induce sufficient force to reach depths $h_i\leqslant h_{\infty }$ in the unpre-strained regime, it also cannot relieve the associated pre-strain pressure exerted by mechanical compaction of the medium to this depth. This feature is illustrated in figure 12(b) by the asymptotes for $\mathcal{P}_{\textit{lift}}$ at values of the pre-strain coefficient $\varDelta (h_i = h_\infty )$ . In contrast, in a type 1 medium, $\mathcal{P}_{\textit{lift}}$ exists for all attainable pre-strain depths $h_i\gt h_m$ (figure 12 a).