1

Permeability of soft porous media under one-dimensional compaction

D. H. Akaydin, A. Pierides, S. Weinbaum & Y. Andreopoulos

Department of Mechanical Engineering The City College of New York, CUNY

New York, NY 10031

Abstract

A novel experimental approach to measure permeability of porous material samples under variable longitudinal compression has been developed. The material has a non-linear behavior and exhibits a small hysteresis during loading and unloading experiments. The new permeameter includes a moving piston with controllable speed inside a plexiglas cylinder and a test section where the porous material sample is placed under compression by two grids with adjustable positions. Time-dependent pressure was recorded at four different locations along the sample together with the velocity of the piston. Experiments with two different sample lengths have been performed at three different Reynolds numbers based on the apparatus diameter. The results show that pressure gradient and permeability data do not depend on sample length. All experiments included measurements at various compression rates of the material followed by measurements during decompression/expansion of the material. No hysteresis was observed in the pressure gradient and permeability data during compression and expansion of the material. The effects of small velocity fluctuations due to variable friction of the moving piston with the cylinder’s wall were also considered. These velocity fluctuations result in pressure fluctuations within the first half of the sample which are damped completely towards the last part of the material sample. It was found that permeability, which is a material property, is drastically reduced with increased compression ratio of the material while its solid fraction changes substantially and its porosity remains practically unchanged. A comparison with the Carman-Kozeny expression for random porous media was also examined. The theory predicts qualitatively the reduction of permeability with compression. However, the predicted values of permeability are very sensitive to the initial value of porosity.

Introduction Permeability is a property of porous media and it is part of the proportionality constant in

Darcy’s law (Darcy, 1856) which relates the flow rate of discharge qi (i=1, 2, 3) relative to the pore motion (i.e. relative specific mass flux) and the fluid physical properties (i.e. viscosity μ) to a pressure gradient ∂p/∂xj (j=1, 2, 3) applied to the porous media

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂−=

jx

p

μijK

iq (1)

In this context Kij is the permeability tensor, a second order tensor, defined for the general case of anisotropic porous media. If it is isotropic then the diagonal values are equal K11=K22=K33>0 while the other components are 0.

Darcy’s law can be considered as a phenomenological constitutive relation (Dagan, 1986 & 1989) or can be derived from momentum balance considerations i.e. Navier-Stokes equations (Lage Krueger and Narasinhan, 2005). In either case and under the assumption that the pressure gradient is always aligned with the flow direction, Darcy’s law becomes

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−==−

ix

p

μiK

iqsiufiu )(φ (2)

2

where φ is the volumetric macroscopic porosity of the porous media given as s

/ρρ1φbulk

−= ,

εs= s/ρρφ-1

bulk= is the solid fraction with bulkρ being the bulk density of the medium with the

pores and the ufi is the fluid velocity inside the pores which are moving with velocity usi. Since the value of Ki originating from (2) represents a material property, it is expected that this value would not change with the size of the porous material under test i.e. its length and diameter or flow rate but only with changes in its internal structure. This paper is one of series of papers demonstrating various poroelastic phenomena associated with the dynamic compression of soft porous media. The theoretical investigation by Feng and Weinbaum (2000), hereafter referred to as F&W, demonstrated by using lubrication theory that periodic fiber arrays could transiently support very large loads beneath a planing surface for short duration if the trapped fluid within the fibrous medium could not escape on the time scale of the passage of the planning surface. This theory has been extended to random porous media in Mirbod et al. (2009) in which dramatically different behavior is observed as the permeability parameter α= h/√K increases (h layer thickness, K Darcy permeability) depending on whether the porous layer is attached to a moving inclined upper boundary or to a stationary horizontal lower boundary in which the inclined upper boundary is moving. The basic idea was first conceptualized in trying to explain the ‘pop out” phenomena that was observed in the motion of red blood cells (RBCs) in capillaries. At velocities < 20 µm/s the RBCs were observed to glide on a compressed endothelial surface layer (ESL) of sulfated proteoglycans and glycoproteins that line the endothelial cells (ECs) of our entire microcirculation and that when the motion of the RBC was arrested the ESL collapsed and the RBC filled the entire lumen of the capillary (see Weinbaum et al., 2007). F&W showed that there is a remarkable hydrodynamic similarity between the motion of the RBC and a human skier or snowboarder skiing on soft snow powder even though their difference in mass is of order 1015. The behavior of random fibrous materials undergoing rapid compression was investigated experimentally by Al-Chidiac et al (2009) in a piston-porous walled cylinder apparatus that was first utilized to examine the behavior of snow and goose down feathers (Wu et al. 2005; Wu et al. [7]). This work clearly shows the different time scales on which the air and solid fiber phases act during the compression of the porous layer and the deformation and stresses developed in the solid phase. These measurements clearly demonstrate that the excess pore pressure builds up inside the porous material and reaches its maximum before there is any significant rise in the solid phase force.

The possibility of supporting a train car on a giant ski using a material with permeability properties similar to feathers was first suggested in Wu et al. (2004). A recent paper by Mirbod et al. (2009) shows theoretically that it is quantitatively feasible to propel a 200 passenger train car weighing 70 mt at speeds approaching 700 km/hr on a nearly frictionless soft porous track using jet engines of only 10,000 lbf, less than 1/5 the thrust and fuel consumption of commercial jet planes with the same passenger load.

In all these applications of skiing, snowboarding or the airborne jet train the dynamic compression of the solid and fluid phases of the soft porous media causes a mutual interaction between the two phases which changes the flow resistance through the pores of the material as is expressed by its permeability. These applications were the catalyst for the present work to determine experimentally the effects of compaction of soft porous materials on permeability.

Detailed information of permeability under compression is vital in several biomechanical applications particularly in articular cartilage (see Mansour and Mow, 1976), material processing using fiber reinforcement (Buntain and Bickerton, 2003), rock mechanics (Wang and Park, 2002; Oda et al. 2002) and geotechnical and structural engineering (Picandet et al., 2001). It is very interesting to mention that the bulk behavior of permeability under compressive stress is different in each application and it depends on the material structure. In biological tissues and material processing, for instance, permeability decreases with compression. In experiments with rocks and cement under compression, permeability has been found to increase with applied stress because of microfractures and microcracks developing in the materials. Thus, the behavior of the material used in the present investigation, a soft compressible fibrous material found in inexpensive pillows, under compression is not known a priori.

Most of the techniques to measure permeability under compression have been developed with a specific application in mind. In the present work, we designed an apparatus and

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developed a technique which provided detailed time-dependent information within several locations of the non-linear soft compressible porous material under compaction which was not available before.

In section 2 we review the basic transport equations in porous media, while the results of material characterization tests are presented in section 3. The novel experimental set up to measure permeability of porous material under compaction is described in section 4. The results are presented in section 5 and section 6 discusses the major conclusions.

Theoretical background The governing equations for momentum conservation in the fluid phase (Bear and

Bachmat, 1990) is

( ) ( )

( )siufiusiufiufFCiK

siufiu

jxij

ixpTC

ix

pTCfkufiuf

kxfiuft

−−−−

−

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂

φρ)(φμ

τφφρφρ

(3)

where ufk and usk are the velocity components of the fluid and solid phases respectively with k=1, 2, 3 and ρf and ρs being the corresponding densities. CT is the tortuosity constant for the flow path through the porous material and CF is the Forchheimer constant (Forchheimer, 1901). For a steady state flow with self similar velocity profiles, the inertia terms on the left hand side of this equation vanish or under the assumptions of lubrication theory they are smaller than the terms on the right hand side and they can be neglected. After substitution with the constitutive

equation ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂=

i

fj

j

fiij x

uxu

μτ , equation (3) for i=1 becomes

( )11111

1112

11sfsffF

sf

jj

fTT uuuuC

Kuu

xxu

xpC

xpC −−−

−−

∂∂

∂+

∂∂

=∂∂ φρ

φμφμφ )()(

(4)

The above equation illustrates some of the physics in this balance of forces: the applied

pressure gradient will be provided by the pressure built up due to non uniform porosity1x

pCT ∂∂φ ,

the viscous stress/viscous diffusion termjj

f

xxu∂∂

∂ )( 12 φ

μ , the Darcy viscous term 1

11

Kuu sf )( −

−φμ

and

the inertia drag term ( )1111 sfsffF uuuuC −−− φρ . One has to compare the definition of

permeability provided by Darcy’s law in equation (2) and the permeability obtained from equation

(4). Thus, in order to obtain K1 out of equation (4) with a known applied pressure gradient 1xp

∂∂ its

breakdown into the four terms of the right hand side of equation (4) should be known. In the present work the solid phase was steady and therefore us1=0 and CF=C/K1

1/2 (see Joseph et al., 1983) where C is the frictional drag. Then

( )112111

112

11fff

f

jj

fTT uu

KC

Ku

xxu

xpC

xpC φρ

φμφμφ

/

)()(−−

∂∂

∂+

∂∂

=∂∂ (5)

Material characterization

The material tested here is regular polyester pillow material obtained in local retail stores. It consists of 95% polyester and 5% silk and it is fabricated by Mountain Mist (Heritage Collection). The results presented here have been obtained in experiments with this material which had fibers of 10 μm in diameter that provided low flexural rigidity EI where E is Young’s modulus and I is its moment of inertia (Mirbod et al., 2009).

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The porosity of the fiber material was determined by measuring the solid volume of a sample consisting of eight circular layers after dipping it into water. The bulk volume of the sample was first measured when it is not compacted. The layers are then dipped and left one by one into a liquid in a graded glass container to measure the volume displacement the sample causes. This step is performed carefully in order to avoid any visible air bubbles trapped inside the fibers. In addition some dishwashing liquid is mixed in the water to reduce its surface tension and mitigate the bubble formation. The solid volume of the fiber sample measured was used to estimate its porosity as φ=1-Vs/V=0.9968. The weight of the sample was also determined, giving a mass density of ρ=1667kg/m3.

We have measured the quasi-steady force generated when the fiber material is subject to incrementally increasing compressive forces. The material is not confined in its radial direction and the compression is along its axial vertical axis (see figure 1). The load is added gradually to a light solid plate on top of the material so that the air in the pores could freely escape without elevating the pore pressure in contrast to the dynamic experiments in Al-Chidiac et al. (2009) and Wu et al. (2005) in which the air in the pores is temporarily trapped before it escapes. The compaction of the porous material sample, -Δh, was measured immediately after the load application. Thus, creep deformation was excluded. The initial height of the present sample is h0 =9.5 cm, its diameter D=4 cm and it consists of 9 layers of fibers with a total mass of 3 gr.

Figure 2a shows a typical incremental loading curve of the present sample and its comparison with the results of Al-Chidiac et al. (2009) obtained in a loading/unloading experiment with a sample of different dimensions of the same material with an initial porosity of 0.998 (experiment ALC1). The material has a non-linear behavior which is characterized by large deformations under small loads in the beginning of compression followed by small deformations under large loads. The data of Al-Chidiac et al. indicate a small hysteresis as a result of the non-linear behavior of the material. In addition these data have shown a substantial radial deformation during its axial compression with a Poisson ratio ν=-ΔR/R/Δh/h0=0.125. The present axial compression data shown in figure 2a is not expected to agree with the data of Al-Chidiac et al. because the sample geometries are different, with aspect ratios h0/2R=0.75 and 0.426, respectively. However, when these two data sets are normalized with the corresponding maximum deformations and loadings they agree very well. Figure 2b shows this comparison. In addition, this figure shows the data of a loading/unloading experiment of the same material in a confined cylinder first shown in Al-Chidiac et al. (experiment ALC2). A significant hysteresis can be observed during the static unloading of the compressed layer inside the confined cylindrical apparatus. The data also show that for a given compressive load σ1/σ1,max much greater deformation takes place in the unconfined compression than in the confined one. The difference in the behavior of the material between confined and unconfined compression is mainly attributed to the friction forces that develop at the cylinder surface as a result of the confined radial deformation of the material. In order to establish the effect of friction between the present material and the wall of the plexiglass tube a series of experiments was performed in the apparatus shown in figure 3. The material was placed inside a plexiglas tube of 4’’ diameter (D=2R=100mm) with an initial length of the layer h0=9.5 cm. The fiber was constrained by two rigid discs of slightly smaller diameter than the diameter of the cylinder and minimum thickness made out of light weight compressed paper, cut out from the back of a paper pad. Three pieces of string were used to keep the fiber and the two discs together under a fixed compaction. Knots at fixed locations were tied on the strings in order to provide the desired compression ratio Δh/ho. The fiber, strings and disc assembly was set to a prescribed length and weights in small increments were used to start moving the fiber assembly. The minimum weight required to set the motion of fiber was assumed to be the total friction force at the corresponding h i.e. Ff (h) = W. Figure 4a shows the frictional shear stress Rh2Fff πτ /= at various compression ratios Δh/h0. Values of shear stress obtained in the present experiment Exp2 are compared with the values of Al-Chidiac et al (2009) indicated as ALC2. The fact that the two data sets agree rather well in figure 4b is quite accidental since the measured frictional forces are very different as are the two inner cylindrical surfaces. What is very interesting to observe here is the fact that both data sets suggest that frictional stress increases with the compression in a non-linear way. An almost four plus fold compression results in a twelve fold increase in shear stress in Exp2 and a three plus

5

compression in ALC2 results in a five plus frictional stress. The friction coefficient μ, defined through the relation Ff=μFR FR is the radial force, has also been computed by using the relation

1

1

2

1

σ

τ

ν

)ν(

ν

)ν(μ f

sFfF

h

R −=

−= (6)

which has been derived in Al-Chidiac et al. by assuming that Poisson’s ratio is the same in both experiments Exp2 and ALC2. The values of μ in Exp2 are shown in figure 4b and they range from 1.3 to 1.15 while values of μ in ALC2 range from 2.4 at small compression ratios to 2.5 at large compression ratios. Both experiments showed that μ is more or less independent of compression ratio Δh/h0. The reason for this behavior is that Poisson’s ratio ν is reasonably constant during most of the compression except at the beginning and that τf and σ1 vary similarly with Δh/h0. Thus, the friction coefficient μ depends mostly on the surface roughness and porosity and the fiber material rather than on the values of Δh/h0 or the values of τf and σ1 as is usually the case. Friction coefficients obtained in Exp2 under compressive tension inside the smooth plexiglass tube inner surface has values roughly half of the values measured inside the cylinder apparatus of Al-Chidiac et al. (2009) because the large values of μ in ALC2 are due first to the roughness of the inner cylindrical surface and second to the pores in the Rigimesh wall which tangle and grab the fibers of the material during its motion. Experimental set-up

A new permeameter was designed, fabricated and tested with the objective to obtain permeability measurements of the material under various compression ratios/deformation -Δh/h0. Figure 5a shows a schematic of the permeameter. It consists of a heavy piston tethered on an electrical motor which slides vertically with a controlled constant velocity that is monitored by a rotary encoder of a computer mouse located on top of the cylinder and connected to the piston via a long string. The piston was fitted with o-rings to seal air from leaking through its gap with the plexiglass pipe. Thus the total mass of the air displaced by the moving piston passed through the porous sample located at the lower open end of the pipe. An array of 13 check-valves pressure ports placed in small intervals from each other was fabricated and attached along the test section of the apparatus holding the sample to measure the pressure as shown in figures 5b and d. Four pressure transducers were used to measure the pressure within the porous material at four different locations and therefore estimate the pressure gradient within the material sample. Each of these MKS pressure transducers were connected to one of the available pressure ports. This setup allowed for monitoring of the pressure variation along the porous material sample and therefore it was possible to estimate the pressure drop at several locations. The output voltage of the pressure transducers was digitized by a data acquisition board.

The diameter of the plexiglas tube was D=135 mm and the height of the permeameter was 1800 mm. A picture of the apparatus is shown in figure 5b.

Compaction of the specimen was provided by adjusting the distance between the two grids which were used to confine the material in place. Figure 5c shows a picture of the porous material layers used to form the sample together with the two plastic grids used to hold the material under compressive deformation while fig. 5d shows the sample under compression before it is installed in the test section of the apparatus.

Results

The Reynolds number based on the fiber diameter d (=10 μm) used in the present work was very small so that no boundary layers were developed around the fibers. Thus the viscous term ∂τ1j/∂xj=∂τ11/∂x1+∂τ12/∂x2 appearing in equations (3) and (4) within the fibrous region away from the pipe wall was zero. This is also evident if one considers the fiber interaction layer thickness which is negligible. Viscous diffusion from the pipe wall was also negligible because the Reynolds number based on the pipe diameter was small. Alternatively it can be argued that ∂τ11/∂x1=0 because a similarity in the velocity profile is assumed, and since no pressure gradient was detected during experiments without the porous material in place ∂τ11/∂x1 is also zero. Alternatively the theory and methodology of Lage and Krueger (2005) can be invoked to obtained estimates of the pressure gradient due to viscous diffusion from the pipe boundary. According to

6

this work the pressure gradient due to this effect can be estimated as 1

1,

μf

ubx

p=

∂

∂where

b=Dh2/2Po where Dh is the hydraulic diameter and Po is the Poiseuille number Po=2τwDh/μu1,f. In

the present case it appears that b=D2/32 and for the highest velocity of 0.08m/s the pressure drop along a 0.1 m length appears to be 0.001 Pa which is quite small in comparison to the measured pressure of 3 to 30 Pa and therefore this effect is insignificant in the present application and it can be ignored.

The tortuosity constant CT which characterizes the average non-dimensional length of the fluid particle path through the pores of the material was assumed to be 1 because of the high porosity of the material.

The material sample tested in the permeameter consisted of several superimposed layers. This layout may have introduced periodic variation of porosity φ when the material is uncompressed which was not characterized. However, an average porosity φ over the sample’s length can be determined and therefore its variation with x1 is zero. When compression is applied to the sample by moving the two material confining grids one may expect that the radial tension of the material due to Poisson’s ratio effects will cause some variation in porosity along the length of the sample. However, the local pressure gradients measured within the sample did not indicate any significant variation so that it is very reasonable to assume ∂φ/∂x1=0 even in the cases of compressed material samples. Thus equation (5) becomes

( )f1uf1ufφρ1/21K

C

1K

)f1φμ(u

1x

p−−=

∂

∂ (7)

The ratio of these two terms on the right hand side appears to be proportional to the Reynolds number based on K1

1/2 i.e.

KCKfu

CR Reμ

/ρ

Darcy

rForchheime===

2111 (8)

Values of ReK obtained in the present work are in the range of 0.12 to 1.5 and for an average frictional drag coefficient C=0.1 (see Joseph et al., 1982) the ratio of the two forces appears to be between 0.012 and 0.15. Thus inertia forces may be marginally negligible at the large ReK flows of the present work i.e. in the cases with large K1 and high velocity. The data shown below have been first analyzed under the assumption that the inertia term is negligible. Evidence will be provided showing that this assumption is justified.

The volumetric flow rate is calculated from the speed of the piston which falls along the cylinder with a controlled speed ranging from 0.02 to 0.08 m/s. The bulk speed of the airflow across the pipe can be assumed to be equal to the piston speed because the pressure developing in the cylinder is small enough (<200 Pa) to neglect compressibility and no leakage in the cylinder was detected. Figure 6a shows a typical output pulsating signal from the encoder with range from 0 to 3.5 v. This signal is subsequently converted to pulses in the 0 to 1 v range by applying a threshold to the original signal. This digital boxcar signal was then used to obtain the linear speed of the piston by considering the known number of openings and diameter of the encoder wheel. The corresponding signal of the speed of the piston is shown in figure 6b. It indicates an unsteady piston motion which is most probably due to the unevenness of the inner surface of the cylindrical plexiglass pipe due to rubber O-ringsand the resulting variation of friction developed between the piston and this surface. This signal, however, appears to be stationary with mean value 0.021 m/s and standard deviation 0.00043 m/s.

The output signals of the pressure transducers were low-pass filtered digitally by using a Butterworth filter. The four pressure signals corresponding to the experiment shown in figure 6b are shown in figure 6c. All four signals exhibit some fluctuations which are due to the unsteady motion of the piston. These fluctuations are highly correlated to each other and seem to be higher in the two locations closer to the piston. It appears that this unsteady motion is considerably dampened as the air flows towards the end of the sample i.e. closer to the open exhaust into the atmosphere. This quenching of fluctuations appears to be a characteristic of porous materials. The pressure signals are stationary and their time-averaged statistical values can be calculated.

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Typical mean values of gauge pressure are plotted in figure 6d along the sample (case ε1=-Δh/h0=0). One observes that the gauge pressure is considerably reduced along the length of the sample from its initial value upstream of the piston to the atmospheric value at its exit. This figure also includes an additional mean pressure profile of the same sample under moderate compression (case ε1=0.55). A comparison between the pressure distributions along the sample indicates that higher pressures are developed along the sample when it is under compression than without compression.

A typical signal of permeability obtained in the case of ε1=-Δh/h0=0 is shown in figure 6e. The signal exhibits some fluctuations which are mostly correlated with the velocity fluctuations. If one decomposes velocity and pressure into the corresponding mean fluctuating parts i.e.

'uUU += and 'ppp += then time dependent permeability is

⎥⎦

⎤⎢⎣

⎡−

+≈

+

+=

dxpd

dxdp

dxpd

uU

dxdpdxpd

uUK

/

/'

/

)'(μ

)/'/(

)'(μ11 which suggests that

21)/(

)/'('μ

)/(

μ

dxpd

dxdpu

dxpd

UK −= (9)

and 21 )/()/'(μ

)/('μ'

dxpddxdpU

dxpduk −= (10)

The second term on the right hand side of equation (9) represents second order effects

and it can be neglected. In fact its ratio to the first term )/()/'('

dxpdUdxdpu

is estimated to be about

0.0113 at most. Then

)/(μ

dxpdUK =1 (11)

and the relative magnitude of permeability fluctuations is

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛=

)/()/'(''

dxpddxdp

Uu

Kk

1

1 (12)

In this relation u’ represents the velocity fluctuations upstream of the sample and dp’/dx is the fluctuating pressure gradient within the sample. Thus permeability fluctuations depend on the upstream velocity fluctuations and the fluctuating pressure gradient which can not be ignored because pressure fluctuations are suppressed along the sample. In that respect pressure gradient fluctuations reduce the fluctuations in permeability. Although the mean pressure gradient

dxpd / is practically the same throughout the length of the sample the fluctuating one is mostly significant in the upper part of the sample where typical values of dp’/dx can be up to 0.5 of

dxpd / while typical values of u’/U =0.1. One issue which needs to be discussed is the way the mean pressure gradient in (11)

has been determined. There are several possibilities to obtain this pressure gradient from the measured pressure data. In the present work, three methods were employed to estimate

dxpd / and therefore permeability. In the first method, called segment-wise method, the pressures between two successive pressure taps were used to estimate the pressure gradient along the segment i of the sample i.e. )/()(/ , iiiiii xxppdxpd −−=>< +++ 111 . The second method involves the difference between the pressures at each of the ports and the exit i.e. the atmospheric pressure )/()(/ , iaiaai xxppdxpd −−=>< . This method is called port-wise and includes the most commonly used practice to obtain the pressure gradient from p1 and the pa i.e. aidxpd ,/ >< . In the third method a linear best fitting was applied to the five pressure

values available which were subsequently used to determine the pressure gradient dxpd / . This

8

method includes the detailed pressure distribution along the sample which is expected to represent better the characteristic of the fibrous material sample.

Figure 7 shows values of permeability K1 obtained by the first two methods together with their mean values and the value of K1 obtained by best fitting the pressure distribution. The three mean values agree within about 7 percent from each other while the port wise method exhibits the most scatter in their individual values. An error analysis indicated that the third method appeared to be more accurate and was adopted in the rest of the data reduction.

Values of the pressure gradient obtained at Re=208 under various values of compressive deformations ε1 are shown in figure 8a. The first observation is that dp/dx increases with compression in a strongly non-linear fashion. The static load – deformation curve shown in figure 2a have indicated a small hysteresis between compression and decompression of the material. In order to investigate if there is any hysteresis effect in the present measurements a series of experiments with decompression of material was carried out. The results are also shown in figure 8a. They clearly show that there is no hysteresis in the pressure gradient curves between compression and decompression of the material.

The variation of permeability with compression is shown in fig. 8b. Mean K1 decreases in a slightly non-linear fashion with respect to the applied deformation. A compression of the material by 87% results in a reduction of permeability by more than an order of magnitude from a value of 1 x10-8 m2 in the case of undeformed material to a value of 0.08 x10-8 m2

under ε1=0.87. The decompression tests were performed in small steps to avoid lag in material restoration which is observed in Mirbod et al. (under review) in the cases of large decompression ratios that followed long compression periods. The K1 data during decompression/expansion of the material clearly show that there is no hysteresis in the permeability curves between compression and decompression of the material. The same conclusion is reached with the data obtained at higher Re=420 and 810, which are not shown here.

In order to demonstrate if the sample length has an effect on the data two additional experiments were carried one with 8 layers and a second with 24 layers of the porous material. The pressure gradient data shown in figure 9a indicate no difference in the values between the two cases. The corresponding permeability values are shown in figure 9b. It can be observed that the effect of initial geometry of the sample material on permeability seems to be negligible. Values of K1 are identical in compressions with ε1>0.3 and differ about 8% at ε1=0.0. Similar results are obtained for Re=420 and Re=800. Thus the present permeability values appear to be a material property independent of its geometry.

The combined effects of Reynolds number and deformation on permeability the measured values of K1 are plotted in figure 10 as a function of Re and ε1. The data suggest no effect of Re on K1 in the range of compression ratios and Re investigated here. This justifies the assumption made earlier that the effect of inertia appears to be negligible. Alternatively, one can consider the experimental data of friction factor plotted against ReK in figure 11. The friction factor

fK, typically defined as 2

211

u

p

h

KKf

ρ

/ Δ= , and relations (7) and (8) can be written as

CK

Kf +=Re

1 (13)

In addition to the experimental data the theoretical values of fK obtained from (13) are plotted for various values of C. It can be seen that the experimental data agree with the theory for C=0 which indicates that the effects of inertia are not important.

Comparison with Carman-Kozeny In the work by Mirbod et al. (2009) an expression to estimate permeability of random

porous media has been derived which is based on an empirical Carman-Kozeny relation described in Happel and Brenner (1983). This relation is given by

Bfr

K)φ( −

=18

231 (14)

9

where rf is the fiber diameter and B is an experimentally determined constant which is a function of φ

⎭⎬⎫

⎩⎨⎧

−+−−−−−+

−−−+−−−=

])φ(/[])φ([)φln()φ()φ()φln()φ( 2112111

22114312

1B (15)

where 1-φ is the solid fraction. The first term of right hand side of the above relation represents contributions from the flow parallel to the fibers while the last one represents contributions from the two flow directions normal to the fibers.

Since the fibrous material is assumed homogeneous in the longitudinal direction during compaction the porosity φ is obtained trough

)ε(

)φ()φ(φ

1101

10101−

−−=−−=

h

h (16a)

where h0 and φ0 are the initial height and porosity respectively and ε1 is the applied deformation. In terms of solid fraction 16a becomes

)ε(

εεε

110

00

−== s

sh

hs (16b)

Our estimates of φ0 based on the fiber material density indicated that φ0=0.996 while the measured value was 0.9968.

Values of K1 obtained through the above relations are plotted in figure 12a. It appears that the theoretical values are about 1/3 of those experimentally determined. In order to investigate more the source of this disagreement a sensitivity analysis of equation 13 was carried out in a way similar to an uncertainty analysis by assuming uncertainties δrf, δφ and δB in the measurements of rf, φ and B. Then the partial relative errors can be computed

212

12

12

1

1

1

/

δδφφ

δδ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂+

∂

∂= B

B

KKfr

fr

K

K

Kand therefore the following relation for

δK1/K1 can be derived: 21

22

1

22

1

1

/

δ

)φ(

δφδδ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−+=

B

B

frfr

K

K (17).

The relative uncertainty δB/B can be estimated by using the approximate relation shown in

Mirbod et al. (2009) to be )φln()φ(

δφδ

−−−=

11B

Band therefore equation 17 becomes

212

11

2

1

22

1

1

/

)φln()φ(

δφ

)φ(

δφδδ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+

−+=

frfr

K

K(18).

If we assume a relative error of 1% in rf and φ i.e. δrf/rf=0.01 and δφ/φ=0.01 then for the cases of φ approaching 1 then the 2nd partial uncertainty (non-squared term) in the right hand side of 17, δφ/(1-φ), dominates the uncertainty since the last one is about 5.5 times smaller than

the second and the first one is even smaller. Thus s

sK

Kε

δε

)φ(

δφδ−=

−≈

11

1 for φ → 1 and typically

for δφ/φ=0.01 it appears that δK1/K1≈2.5. This demonstrates that small uncertainties in the measurement of porosity can result in huge uncertainties in the estimates of permeability. If the initial porosity φ0 is adjusted to fit the measured K1 which requires φ0=0.9978 then the above

10

equations predict the experimental data of permeability under compaction of the porous material in figure 12a reasonably well. In addition, the values of porosity φ at various compression ratios are also plotted in figure 12a. It can be seen that porosity is practically constant with compression while permeability changes by almost an order of magnitude.

The data in figure 12a clearly indicate a strong reduction in K with increasing compression ratio for the synthetic material used in the present investigation and feathers which have been also tested. The theory also predicts a reduction of K1 with compression. Figure 12b shows the relative change in K, ΔK/K0=1-K1/K0 at various compression ratios. It appears that ΔK/K0 increases with compression ratio Δh/h0. Feathers have the lowest permeability while their relative change is higher than that of the fiber material by about 30 percent. The relative change in porosity Δϕ/ϕ0 =1-ϕ/ϕ0 appears to be at most 2.4 percent at the highest compression while there is a ten fold change in solid fraction Δεs/εs0 =1-εs/εs0 at the maximum compression. Conclusions In this work we have developed a novel experimental approach to measure permeability of porous material samples under variable longitudinal compression. The material of interest is a soft compressible material found in pillows. This inexpensive material generates an excess pore pressure that builds up inside during a dynamic compression which can be used to lift heavy objects. The material has a non-linear behavior which is characterized by large deformations under small loads in the beginning of compression followed by small deformations under large loads. A small hysteresis has been observed during loading and unloading static experiments in which the samples are compressed by a known load that allows a radially unrestricted expansion. The present work provides data on permeability that can be used to fully characterize these materials. The design of the new permeameter includes a moving piston with controllable speed inside a plexiglas cylinder and a test section where the porous material sample is placed under compression by two grids with adjustable positions. Time-dependent pressure was recorded at four different locations along the sample together with the velocity of the piston. Experiments with two different sample lengths have been carry out at three different Reynolds numbers based on the apparatus diameter. The results show that pressure gradient and permeability data do not depend on sample length supporting the notion that permeability is a material property which independent of the geometry of the sample. All experiments included measurements at various compression rates of the material followed by measurements during decompression/expansion of the material. No hysteresis was observed in the pressure gradient and permeability data during compression and expansion of the material. The effects of small velocity fluctuations due to variable friction of the moving piston with the cylinder’s wall were also considered. These velocity fluctuations result in pressure fluctuations within the sample particularly in its first half which are damped completely towards the last part of the material sample. The pressure gradient measured on the wall of the test section along the sample increases in an exponential way with the applied strain ε1. It was found that permeability, which is a material property, reduces drastically with increased compression ratio of the material while its porosity remains practically unchanged. A comparison with the Carman-Kozeny theory for random fibrous materials was also attempted. The theory predicts qualitatively the reduction of permeability with compression. However, the predicted values of permeability are very sensitive to the initial value of porosity. Acknowledgment

The financial support provided by the National Science Foundation under Grant# 0432229 is greatly appreciated.

References

M. Al-chidiac, P. Mirbod, Y. Andreopoulos and S. Weinbaum, J. Porous Media. (2009) In press.

11

Bear J and Bachmat Y 1990 Introduction to Modeling Transport Phenomena in Porous MediaKlumer Academic Publishers, Dordrecht

Brinkman, H. C., A calculation of the viscous force exerted by a flowing fluid in a dense swarm of particles. Appl. Sci. Res. A 1, 27, 1947. Buntain, M.J. and Bickerton S. 2003 Compression flow permeability measurement: a continuous technique, Composites: Part A 34, 445–457. Dagan G. Theory of Solute Transport by Groundwater Annual Review of Fluid Mechanics, Vol. 19: 183-213, 1987. Dagan G. 1989. Flow and Transport in Porous Formations. Berlin: Springer-Verlag Darcy, H. P. G. Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris, 1856. Feng, J. & Weinbaum, S., Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans, J. Fluid Mech. 422, 282-317, 2000. Forchheimer, P. ‘‘Wasserbewegung durch Boden,’’ Z. Ver. Deutsch. Ing. 45, 1782 1788, 1901. Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media. Springer. Joseph, D. D., Nield, D.A. & Papanicolaou, G., “Nonlinear equation governing flow in a saturated porous medium,” Water Resources Research. 18(4), 1982, pp.1049-1052. Lage J.L., Krueger P. S. and Narasimhan A. (2005) Protocol for measuring permeability and form coefficient of porous media Physics of Fluids 17, 088101, 2005. Mansour JM and Mow VC (1976) The permeability of articular cartilage under compressive strain and at high pressures J Bone Joint Surg Am. 58:509-516. Martin, A.R., Saltiel, C. and Shyy, W. ‘‘Frictional losses and convective heat transfer in sparse, periodic cylinder arrays in cross flow,’’ Int. J. Heat Mass Transf. 41, 2383 ~1998. Mirbod P., Andreopoulos Y. and Weinbaum S., On the generation of lift forces in a random, soft porous media, J. Fluid Mech. 619, 147–166, 2009. doi:10.1017/S0022112008004552.

Mirbod P., Andreopoulos Y. and Weinbaum S. An airborne jet-ski train that flies on a soft porous track,” J. Porous Media (under review), 2009. Oda M, Takemura T. and Aoki T. 2002 Damage growth and permeability change in triaxial compression tests of Inada granite, Mechanics of Materials 34 313–331. Papathanasiou, T. D., Markicevic, B., Dendy, E. D., “B A computational evaluation of the Ergun and Forchheimer equations for fibrous porous media” Physics of Fluids Vol. 13, No. 10, Oct. 2001, pp. 2795-2804 Picandet V., Khelidj, A. and Bastian G. 2001 Effect of axial compressive damage on gas permeability of ordinary and high-performance concrete, Cement and Concrete Research 31 (2001) 1525–1532 Wang J-A and Park H.D. 2002 Fluid permeability of sedimentary rocks in a complete stress–strain process Engineering Geology 63, 291– 300

12

Weinbaum, S., Tarbell, M. J., Damiano, E. R., “The Structure and Function of the Endothelial Glycocalyx Layer,” Annu. Rev. Biomed. Eng., 9, 121-167, (2007). Wu, Q., Andreopoulos, Y., Xanthos, S. & Weinbaum, S., Dynamic compression of highly compressible porous media with application to snow compaction, J. of Fluid Mechanics 542, 281-304, 2005a. Wu, Q., Weinbaum, S., & Andreopoulos, Y., Stagnation-point flows in porous medium, Chem. Eng. Sci. 60 , 123-134, 2005b.

Wu, Q., Andreopoulos, Y. & Weinbaum, S., “From red cells to snowboarding to a new concept for a train track” Physical Review Letters vol. 93(19), pp. 194501-1 to 194501-4, 2004.

Wu, Q., Igci, Y., Andreopoulos, Y. & Weinbaum, S. “Lift mechanics of downhill skiing and snowboarding” Medicine & Science in Sports & Exercise, 38 (6): 1132-1146 June, 2006.

Experiments

Type

Diameter D (cm)

Initial height h0 (cm)

Maximum force, Fmax

(kg) Present compression experiment: Exp1

Un-confined-by-walls compression

10 7.5 1.2

Present friction/compression experiment: Exp2

confined-by-walls compression (plexiglass tube)

10 9.5 NA

Al-Chidiac et al. (2009) compression experiment: ALC1

Un-confined-by-walls compression

40 18.5 15.75

Al-Chidiac et al. (2009) friction/compression experiment: ALC2

confined-by-walls compression (Rigimesh cylinder)

40 14.5 9.1

Table 1: List of experiments and corresponding parameters.

13

Figure 1: Schematic of unconfined compression experiments.

h

D=2R

W

x2

x1

Figure 2a: Typical static load – deformation curves for fiber materials with 95% polyester fibers and 5% silk in different experiments (Table 1).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ε1=−Δh/h0

0

500

1000

1500

σ1

(Pa)

Experiment: ALC1Experiment: Exp1

14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2−Δh/hmax

0.00.10.20.30.40.50.60.70.80.91.01.11.2

F/F m

ax

Experiment: ALC1Experiment: Exp1Experiment: ALC2

Figure 2b: Normalized static load – deformation data comparison of various experiments (table 1).

Figure 3: Experimental setup to estimate friction under compression.

D

Ff

FS

W

h

15

Figure 4: (a) Measured total frictional stress as a function of compression/deformation Δh/h0; (b) Friction coefficient as a function of compression/deformation Δh/h0.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ε1=−Δh/h0

0

100

200

300

400

τ f (P

a)Experiment: Exp2Experiment: ALC2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ε1=−Δh/h0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

μ

Experiment: Exp2Experiment: ALC2

16

Figure 5: (a) Schematic of apparatus to measure permeability under compression; (b) actual picture of apparatus; (c) pictures of fiber layers and holding grids; (d) compressed porous material.

Air flows out to atmosphere

Grid to compress fibers

Piston with O-ring sealing

Rotary encoder

Piston

Pressure is measured by a series of transducers

Electric motor to control piston

D A Q

Porous Sample

1P

4P3P2P

b

c

d

17

Figure 6a: Typical rotary encoder signal.

15 16 17 18 19 200

0.5

1

1.5

2

2.5

3

3.5

Time [s]

Enco

der V

olta

ge [V

]

Figure 6b: Speed of the piston obtained from rotary encoder signal in figure 6a.

15 16 17 18 19 200

0.01

0.02

0.03

0.04

Time [s]

Pist

on S

peed

[m/s

]

18

0 0.02 0.04 0.06 0.08 0.1 0.12-1

0

1

2

3

4

5

6

7

H [m]

Gag

e Pr

essu

re [P

a]

ε1=0.55ε1=0

Figure 6c: Time-averaged gage pressures along the sample. Slope of the linear fit for 1 0ε = is 42.5 [Pa/m]. Slope of the linear

fit for 1 0.55ε = is 143.7 [Pa/m].

Figure 6c: Time-dependent variation of the gauge pressures at various locations along the sample.

15 16 17 18 19 200

1

2

3

4

5

6

7

Time [s]

Gag

e Pr

essu

re [P

a]P1P2P3P4

19

Figure 6e: Typical permeability signal of the sample tested. Time averaged (from 5 to 45 seconds) value of permeability for this case is K1=8.7 x 10 -9 with a standard deviation of -10 2 =3.56 10 mσ × .

15 16 17 18 19 200

0.2

0.4

0.6

0.8

1

1.2

x 10-8

Time [s]

K1 [m

2 ]

Figure 7: Permeability values along the sample. pK =8.7 x 10 -9 m2 from best fit is (solid

line). Port-wise average is K1=9.5 x 10 -9 (dashed line) with standard deviation σ=7.0 x 10 -9. Segment-wise average is K1=9.2 x 10 -9 (dashed-dot line) with standard deviation σ=2.0 x 10 -9.

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

x 10-8

H [m]

K1 [m

2 ]

SegmentwisePortwise

20

0 0.2 0.4 0.6 0.8 10

200

400

600

800

ε1

dp/d

x [P

a/m

]CompressionExpansion

Figure 8.a: Variation of /dp dx vs 1ε at Re 208 for comparing compression and expansion of

8 layers of fibers. Curve fit equation 1/ 2.177exp(6.428 )+53.05dp dx ε= .

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x 10-8

ε1

K1 [m

2 ]

CompressionExpansion

Figure 8.b: Variation of K1 vs 1ε at Re 208 for comparing compression and expansion of 8 layers of

fibers. Curve fit equation -9 2 -8 -91 16.306 10 -1.568 10 +9.394 10pK ε ε= × × ×

21

Figure 9a: Variation of /dp dx vs 1ε at Re 208 for comparing compression of 8 layers and 24 layers of fibers.

Curve fit equation 1/ 1.901exp(6.731 )+50.5dp dx ε= .

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

700

800

ε1

dp/d

x [P

a/m

]8 layers24 layers

Figure 9b: Variation of K1 vs 1ε at Re 208 for comparing compression and expansion of 8 layers of fibers.

Curve fit equation K1=7.518x10-9ε12-1.704x10-8ε1+9.709x10-9.

.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x 10-8

ε1

K1 [m

2 ]

8 layers24 layers

22

Figure 10: Variation of pK vs Re for different 1ε compression ratios.

200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1x 10-8

ReD

K1 [m

2 ]

Figure 11: Variation of fK (friction factor based on K11/2) vs ReK (Reynolds number based on

K11/2). The solid lines represent values of fK according to eq. (12) for various values of the

Forchheimer constant C.

10-2 10-1 100100

101

102

ReK1

f K1

8 layers compressionF=0 (Darcy's law)F=1

23

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ε1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

K1 x

108 ,

ΦExperimental dataTheoretical dataTheoretical data (adjusted)Porosity

Figure 12a: Porosity and theoretical permeability values at various compression ratios and comparison with experimental data.

Figure 12b: Permeability and porosity change from as a function of compression.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ε1

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ΔK

1/K0,

Δε s

/εs0

x 1

0-1

Experimental dataTheoretical dataTheoretical data (adjusted)FeathersSolid fraction