Is there a formulation for (self-)accelerating fluid flow through permeable medium?

Question

I have a permeable system where is an accelerating fluid flow. Imagine a sponge that is squeezed. The fluid starts at rest, accelerates and flows out from the sponge. How to calculate the fluid speed?

Is it possible to derive an equation where the fluid flow has a feedback that increases fluid flow? Perhaps something similar than percolating water through a sand wall or dam when it starts to breach?

A quasi-static solution could be for example, Bernoulli + Kozeny-Carman. In the below equations P is pressure, u is fluid velocity along x and t is time.

Bernoulli's equation (one u -> dt on the right):

$\frac{dP}{dx}\propto\frac{du}{dt}$

Kozeny-Carman (flow through permeable medium):

$\frac{dP}{dx}\propto u$

Now the idea is that fluid is pushed through the sponge. According to Bernoulli this creates a pressure gradient as the fluid speed accelerates from rest to the output velocity. This velocity also defines a pressure according to Kozeny-Carman. Note that the external work is not included here (fluid just accelerates and is resisted by medium, is this a problem? maybe P has external and internal components?). If it is possible to assume that these pressure gradients are locally the same, you can write a relation:

$ \frac{du}{dt} \propto u $

which has an exponential solution in time. The above is just an example what I had in mind and I was wondering if there exists a more rigorous treatments for accelerating fluids through resistive mediums. Both equations can be derived from Navier-Stokes, so there might be already a version where both of these components are.

Answer 1

For the cases you are describing, you will generally need to account for evolution of porosity and permeability. For example in the sponge case, by squeezing the sponge you are compacting the pores - the pressure is likely slaved to the compaction rate (i.e. you would "squeeze harder" as the sponge stiffens due to strain hardening and the permeability decreases due to smaller pore throats; technically your rate of squeezing would be accelerating, assuming an elastic loading response.) In the sand-dam case, the permeability and porosity increase as sand is removed. In this system mass is not conserved (i.e. sand is removed by the flow). While the sand-dam system is self-accelerating, I would say the sponge system is driven more by outside forces. (For a sort of combination of the two, check out "injectites".)

For a physics perspective, you might check out "Biot Consolidation" (some references here; the original paper is also quite readable). For a geology perspective, you might check out soft-sediment deformation.

Edit: Note that classical Biot consolidation assumes a linear-elastic porous medium. This means 1) infinitesimal strains, 2) linear "Hooke Law" for stress vs. strain, and 3) all strains are elastic, i.e. reversible. For your sponge case, at the least you would need to consider finite strain (if you like differential geometry, this may interest you; only 3d so simpler than general relativity I guess?). However with finite strain the model "constant coefficients", such as permeability & elastic moduli, will also most likely change (e.g. permeability reduction, nonlinear elasticity). Finally, when the porosity reduction is irreversible, the term compaction is used (i.e. consolidation vs. compaction = elastic vs. plastic strain).

Answer 2

Here is the Biot Consolidation GeoMatt22 is referring in his answer.

The differential equations presented by Biot to represent the isotropic consolidation of such saturated media, assuming incompressible grains and fluid are shown below:

$$\nabla^2 p(X, t) = \kappa \left[\frac{\partial p(X, t)}{\partial t} - \frac{\partial\sigma_{vol}(X,t)}{\partial t}\right]$$

http://www.sciencedirect.com/science/article/pii/S0955799702000929 https://www.rocscience.com/help/phase2/webhelp9/pdf_files/theory/Coupled_Consolidation.pdf