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Stability and relaxation of flows in porous materials

Collaborator: B. Albers , K. Wilmanski  

Cooperation with: T. Wilhelm (Universität Innsbruck, Austria)

Description: Aims and results of the project

The main aim of the project is the linear stability analysis of various flow processes coupled with the deformation of the saturated skeleton made of a poroelastic material. During the last two years three problems have been investigated:

1.

Loss of stability of the 1-d steady state flow through a granular material due to liquefaction ([1]),

2.

Relaxation properties of a longitudinal disturbance of a 1-d steady state flow through a poroelastic material ([3]),

3.

Stability properties of a 1-d steady state flow through a poroelastic material with respect to a transversal disturbance.

It has already been reported that it is necessary to introduce a contribution that be nonlinear with respect to the relative velocity of components to the source of momentum in order to describe a threshold to the fluidized state of a granular material. This result has been well-confirmed by experiments performed at the University of Innsbruck (Austria).

The problem of stability of a 1-d flow through a saturated poroelastic material has been divided into two parts.

In the first part the steady state solution has been disturbed by a dynamical disturbance in the longitudinal direction. Such a flow through the immobile skeleton is described by the following field equations ([2])

$$\frac{\partial \rho ^{F}}{\partial t}+\frac{\partial \rho ^{... ...artial c}{\partial x}\right) =\left( 1-c\right) \hat{\rho}^{A},$$

$$\hat{\rho}^{A}:=-\rho _{ad}^{A}\frac{d\xi }{dt},\quad \frac{... ...}\left[ \frac{cp^{F}}{p_{0}}\left( 1-\xi \right) -\xi \right] ,$$

 

$$\rho ^{F}\left( \frac{\partial v^{F}}{\partial t}+v^{F}\frac... ...\quad p^{F}=\stackrel{0}{p}^{F}+\kappa \stackrel{1}{\rho }^{F}.$$ (1)

The first two equations are, of course, mass balance equations for the fluid component and for the adsorbate in the fluid. The concentration of the adsorbate is denoted by c. $\xi$ is the microstructural variable called the number of occupied sites. $\kappa ,\pi ,p_{0},\tau _{ad},\rho _{ad}^{A}$are constants. We consider two cases. The first one is the longitudinal disturbance without mass exchange (i.e. $\hat{\rho}^{A}\equiv 0$). In the second one the adsorption in the nonsteady disturbance is different from zero.

The base steady state flow in a pipe is described for both cases by the following relations:

1.

Skeleton is deformed and does not move any more,

2.

Longitudinal velocity and pressure in the fluid satisfy the following relations

 

$$\stackrel{0}{v}^{F} f\;\frac{n_{E}\left( p_{l}-p_{r}\right) }{\pi l}=const.,$$ =

$$\stackrel{0}{p}^{F} n_{E}p_{l}-f\;n_{E}\left( p_{l}-p_{r}\right) \frac{x}{l}-\frac{n_{E}\left( p_{l}-p_{r}\right) }{2+\frac{\alpha \pi l}{\rho _{0}^{F}}},$$ = (2)

$$=\frac{\frac{\alpha \pi l}{\rho _{0}^{F}}}{2+\frac{\alpha \pi l}{\rho _{0}^{F}}},$$ f :

where we use the following notation. p_l,p_r denote the external pressure at the left and right ends of the pipe, respectively. n_E is the constant porosity of the skeleton. $\rho _{0}^{F}$ denotes the initial mass density of the fluid component, l is the length of the pipe, and $\pi ,\alpha$ are material constants. The first one describes the bulk permeability, and the second one the surface permeability of the material. The above solution satisfies the boundary conditions of the third type describing the flow through a permeable boundary. In the special case $\alpha \rightarrow \infty$, this boundary condition becomes: $\left. p^{F}\right\vert _{x=0}=p_{l},\quad \left. p^{F}\right\vert _{x=l}=p_{r},$ which is commonly used in Darcy-like models of flows.

The longitudinal disturbance may or may not lead to the exchange of mass between the fluid as a carrier of an adsorbate and the skeleton.

The main result of this work is that the flow (2) is stable with respect to any such disturbance. However, the analysis leads as well to some results on the relaxation properties of longitudinal disturbances related to different values of permeability coefficients $\pi$ and $\alpha$. We demonstrate these results in the two figures below.

 

Fig. 1: Relaxation without adsorption

 

Fig. 2: Relaxation with adsorption

In Fig. 1 we show the inverse of the relaxation time of disturbance (in seconds) as a function of the bulk permeability $\pi$ for the case of disturbance without mass exchange. It is seen that the relaxation is fastest for medium values of the permeability. The value appearing in the most common applications in soil mechanics is $\pi =10^{6}-10^{8}\quad \left[ \frac{kg}{m^{3}s}\right]$. Simultaneously there exists an essential influence of the surface permeability $\alpha$. The relaxation time becomes longer for smaller values of this permeability. It should be mentioned that in the range of smaller values of the bulk permeability the dynamical disturbance yields not only damping but also vibrations.

In Fig. 2 we show similar results under the presence of mass exchange. Qualitatively the changes in comparison to the previous case are not very big even though there appear additional plateaus. However, quantitatively the relaxation becomes much slower and it depends on the relaxation time of adsorption $\tau _{ad}$.

The results for transversal disturbance of the same steady state flow are still preliminary. We consider the 2-d flow through a channel of the width 2b. The boundary conditions in the direction of the channel are the same as before. The boundary conditions in the transversal direction are: $\left. v_{z}^{F}\right\vert _{z=\pm b}=0$, where v_z^F is the transversal component of the fluid velocity.

The flow becomes unstable in the range of high values of the bulk permeability $\pi$. The exchange of stability ($\mbox{Re }\omega =0$, where $\omega$ is the complex frequency of the transversal disturbance) appears for the values of $\pi$ which grow with the growing surface permeability $\alpha$. This problem shall be further investigated in the next year.

References:

1.   T. WILHELM, K. WILMANSKI, On the onset of flow instabilities in granular media due to porosity inhomogeneities, to appear in: Internat. J. Multiphase Flows.
2.   B. ALBERS, Makroskopische Beschreibung von Adsorptions-Diffusions-Vorgängen in porösen Körpern, PhD thesis, Technische Universität Berlin, Logos-Verlag, Berlin, 2000.
3.   B. ALBERS, K. WILMANSKI Relaxation properties of a 1-d flow through a porous material without and with adsorption, WIAS Preprint no. 707, 2001.

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