Fluid Flow in Homogeneous and Heterogeneous Porous Media

Fluid Flow in Homogeneous and Heterogeneous Porous Media

Olusegun Olalekan Alabi Department of Mathematical & Physical Sciences (Solid Earth Physics Research Laboratory) College of Science, Engineering and Technology, Osun State University, P M B 4494, Osogbo, Nigeria e-mail: [email protected]

ABSTRACT The rate, magnitude and direction of seepage of fluid flow in porous media are vital tools in seepage and drainage control in soils. These can be determined by Darcy’s law of laminar flow of fluids in porous media. Three different types of heterogeneous media from five soil samples of different porosities were considered; with constant-head permeameter the saturated hydraulic conductivity for each medium was determined. The results show that fluid flows faster in mixed heterogeneous medium than layered heterogeneous medium. However, fluid flow in homogeneous porous media is generally faster than that of heterogeneous media of similar geometry and grain packing. Thus, layered heterogeneous medium is more appropriate as a protective filter for seepage control.

KEYWORDS: Permeability, Heterogeneous media, Hydraulic gradient, Porosity, Seepage.

INTRODUCTION Control of the movement of water and prevention of the damaged caused by the movement of water in soils are vital aspects of soil engineering (Leonard, 1962). The study of seepage patterns in cross section with soil with permeability more than one is one of the most worthwhile and rewarding applications, especially in selecting a protective filter or seepage control in man-made constructions(Casagrande, 1937; Elsayed and Lindly, 1966). Excessive seepage is caused by high permeability or short seepage path (Sower and Sower, 1970). Its permeability can be reduced by a proper selection of materials, for example, mixing a small amount of clay with the sand (protective filter) used for construction can reduce the permeability greatly (Sower and Sower, 1970). A filter or protective filter is any porous material whose opening is small enough to prevent movement of the soil into the drains and which is sufficiently pervious to offer little resistance to seepage (Jacob, 2001). Frequently a soil is employed as a filter, and in preparing a good filter, knowledge of permeability of homogeneous and heterogeneous media is very essential. A medium is homogeneous if the permeability is constant from point to point over medium while it is heterogeneous if permeability changes from point - 61 -

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to point in the medium. The permeability is the most important physical property of porous a medium, which is a measure of the ability of a material to transmit fluid through it. The application of Darcy’s law enables hydraulic conductivity to be determined. The permeability can be determined or computed from hydraulic conductivity (Domenico and Schwartz, 2008). An earlier experiment to study the effects of porosity on rate of fluid flow by using Darcy’s law in saturated porous media shows that there is a very strong statistical correlation between porosity and rate of fluid flow in porous media (Popoola et al., 2007a and Popoola et al., 2007b). These suggest that porosity of a material has a great influence on hydraulic conductivity and permeability. Also, the size distribution and percentage of gravel and coarse fraction present are some of the factors that are responsible for variability in hydraulic conductivity. In this study, riverbed sand samples were used as porous media for both homogeneous and heterogeneous media. The objectives of this study were to investigate the difference in the flow rate of fluid in homogeneous media and different types of heterogeneous media. These are necessary for proper selection of material for seepage control in man-made constructions.

THEORY When water flows from a soil of low permeability into a soil of higher permeability, less area is required to accommodate the same quantity of water and lower gradients are needed. If the flow is from high permeability into lower permeability, steeper (or higher) gradient are required and a relatively more area is needed to accommodate the flow (Cedergren, 1976). If layers of beds of porous media of different porosity are considered and it is assumed that each layer is homogeneous and isotropic, then each layer is however characterized by a different hydraulic conductivity rendering the sequence as a whole heterogeneous. It was found that for horizontal flow, the most permeable unit dominates the system. For vertical flow the least permeable unit dominates the system. Under the same hydraulic gradient, horizontal flow is of the order of six orders of magnitude faster than vertical flow (Domenico and Schwartz, 2000). Fluid flow through a porous material of permeability k, by Darcy is generally written as(Frick and Taylor, 1978; Olowofela and Adegoke, 2005):

ν =−

k

μ

∇ ( p − ρgz )

(2)

which can be expressed as

ν =−

k  (dp) dz  − ρg   μ  ds ds 

(3)

where s = distance in the direction of flow, v = volume flux across a unit area of the porous medium in unit time along the flow path, z = vertical coordinate, considered downward, ρ = density of the fluid, g = acceleration of gravity, dp/ds = pressure gradient along s at the point to which v refers , μ = viscosity of the fluid, k = permeability of the medium, and dp/ds = sin q (it is the angle between z and horizontal)

Eq. 3 can be expressed further as follows:

k  dz dp   ρg −  μ  ds ds  dz dp νμ = ρg − k ds ds

ν=

then

νμ

dp k ds νμ dp − ρ g sin θ = ds k = ρ g sin θ −

 ds  = sin θ    ds  (4)

Integrating both sides

 νμ    k − ρg sin θ ds = − P dp 1 P2

 νμ  − ρg sin θ  s = p1 − p 2 = p   k  νμ p = + ρg sin θ k s For downward (vertical) flow when the driving head (difference in hydraulic head) is h;

(

)

dz = sin 90 0 θ = 90 0 which implies that sin 90 0 = 1 ds Then,

νμ k

=

p + ρg s

νμ therefore

 ρgh  = + ρg  k  s 

( P = ρgh)

νμ

h  = ρg  + 1 k s  ρg  h  ν= k  + 1 μ L 

(5)

where L = length of the sample (or medium) and it is equivalent to s in the general equation, h = head constant (or hydraulic head) Hydraulic conductivity as related with permeability by equation (Domenico and Schwartz, 2000 and Hubbert, 1940)

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K=

ρg μ

k (6)

where K = hydraulic conductivity (ms-1), k = permeability (m2), μ = viscosity of the fluid (Nsm-2), and ρ = density of the fluid, (kgm-3) By using equation (6) in equation (5), equation (5) becomes (Jacob and Arnold, 1990)

h  + 1 L  Q ν =q= A

ν = K

(7)

Then

h  q = K  + 1 L 

(8)

h  + 1 is the hydraulic gradient, i. This L 

where q is the volume flux (ms– 1) and 

expression of Darcy’s law(Equation 8) shall be used in this work to determine the hydraulic conductivities of the samples. The slope of the graph of volume flux against hydraulic gradient indicates hydraulic conductivity. The hydraulic conductivities obtained were later converted to permeabilities by using equation (6).

MATERIALS AND METHOD Sand samples were collected from the riverbed of two different rivers. The samples were washed, rinsed, sun dried and later placed in an oven. The samples were later sieved into five different grain sizes. Each of these samples was used as homogeneous medium. The permeability properties of porous media were both determined in the laboratory using a vertical form of Darcy’s equation. Five different homogeneous media and three different heterogeneous media were used during the experiment. The homogeneous samples were obtained by the use of a sieve of a known grade while the heterogeneous samples were obtained by mixing sieved sand on different grain sizes. Three different types of heterogeneous media were formulated. They were mixed heterogeneous medium (MHT), ascending layered heterogeneous medium (ALHT) and descending layered heterogeneous medium (DHLT). The first heterogeneous medium (MHT) is the mixture of five homogeneous media in the same proportion. The second medium (ALHT) is the layer of each of five heterogeneous medium of the same thickness in ascending order of porosity from the bottom. The third heterogeneous medium (DLHT) is the layer of each of the five homogeneous medium of the same thickness in descending order of their porosity from the bottom. From the experiment, the porosity of the media were determined using volumetric method, while the fluxes were determined

from vertical permeameter set up with constant head method. The transparent cylindrical tube of cross-sectional area 7.06 x 10-4 m2 was used as a permeameter (Fig. 2). A continuous steady supply of water was fed through the sand samples of length L packed under gravity and at height h a hole was drilled, this enable the height h to be maintained. The volume of water discharge, Q through each sample for a period of 60 seconds was measured by measuring cylinder at different hydraulic gradient, i and the volume flux was determined from respective volume discharged using eqn.(7). The hydraulic conductivity of each of the medium was obtained from Darcy’s equation (equation 8).

d

h

L

OVERFLOWING ARRANGEMENT WATER

SAND SAMPLE

d = 1.85 x 10-2 m Cross sectional Area = 2.69 x 10-4 m2

BEAKER

FIG. 2: Sand Model for Vertical Flow under h [5]

RESULTS Table 1 is the values of volume flux for five different homogenous media tagged A, B, C, D, and E; and heterogeneous media The slopes of the plot of volume flux, q against hydraulic gradient i for each of the medium indicate the hydraulic conductivities for each medium (Figs. 3-8). The slope of the line of volume flux q against hydraulic gradients i indicates the hydraulic conductivity for each medium. The permeabilities of both homogeneous and heterogeneous media were computed from hydraulic conductivities using Hubert King relation and are presented in Table 2. This shows the hydraulic conductivities and permeabilities of the porous media in order of increase.

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Table 1: Values of volume flux for homogeneous and heterogeneous media at different hydraulic gradients Hydraulic

A

B

Gradient

q x 10

i

(ms )

(ms )

(ms )

(ms )

(ms )

(ms )

(ms )

(ms )

2.64

3.19±0.01

5.96±0.03

11.82±0.02

18.90±0.06

36.88±0.02

4.18±0.01

4.06±0.01

3.39±0.02

2.90

3.51±0.02

6.45±0.04

12.85±0.02

20.56±0.06

39.47±0.03

4.46±0.01

4.31±0.02

3.59±0.03

3.22

3.91±0.02

7.08±0.05

14.41±0.03

23.30±0.07

42.45±0.04

4.81±0.02

4.63±0.03

3.82±0.03

3.63

4.43±0.03

7.77±0.06

16.56±0.04

25.55±0.05

48.36±0.04

5.25±0.04

5.03±0.03

4.13±0.04

4.14

5.07±0.01

9.82±0.06

19.20±0.05

29.31±0.06

55.44±0.09

5.80±0.03

5.54±0.02

4.51±0.04

4.83

5.93±0.04

11.18±0.12

20.62±0.05

33.75±0.04

63.11±0.09

6.54±0.02

6.22±0.04

5.03±0.02

5.80

7.14±0.02

13.53±0.14

27.06±0.11

36.78±0.10

75.37±0.11

7.59±0.02

7.18±0.04

5.76±0.04

7.25

8.95±0.02

16.61±0.10

33.02±0.12

40.19±0.11

101.11±0.14

9.16±0.04

8.62±0.04

6.84±0.04

9.67

11.98±0.02

25.06±0.14

42.70±0.12

62.28±0.14

127.69±0.14

11.76±0.05

11.01±0.05

8.65±0.02

14.50

18.01±0.02

28.92±0.15

59.50±0.13

103.36±0.14

168.62±0.15

16.98±0.05

15.78±0.05

12.27±0.03

-4

Volume Flux, q (m/s)

-1

C

q x 10

-4

D

q x 10

-1

-4

-1

E

q x 10

-4

-1

MHT

q x 10

-4

-1

q x 10

ALHT -4

-1

q = 1.2498i - 0.1093 R2 = 1

Hydraulic Gradient, i

Figure 3: Graph of Volume Flux against Hydraulic Gradient for Homogeneous Sample A

q x 10

DLHT -4

-1

q x 10

-4

-1

Volume flux, q (m/s)

q = 2.1014i + 0.9281 R2 = 0.9609

Hydraulic gradient, i


Figure 4: Volume flux against Hydraulic gradient for Homogeneous Sample B

q = 4.0908i + 1.8103 R2 = 0.9944



Figure 5: Volume flux against Hydraulic gradient for Homogeneous Sample C

q = 6.8101i - 0.4953 R2 = 0.9784


Figure 6: Graph of Volume flux against Hydraulic gradient for Homogeneous Sample D

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q = 11.61i + 7.8407 R2 = 0.9875


Figure 7: Graph of Volume flux against Hydraulic gradient for homogeneous Sample E


q = 1.079i + 1.3322 R2 = 1 q = 0.9888i + 1.4458 R2 = 1 q = 0.7486i + 1.4135 R2 = 1


Figure 8: Graph of Volume flux against Hydraulic gradient for Heterogeneous Samples MHT, ALHT, and DLHT respectively.

Table 2: Hydraulic conductivities and Permeabilities of homogeneous and heterogeneous media Medium

Hydraulic conductivity -4

-1

Permeability -11

2

K x 10 (ms )

k x 10 (m )

DLHT

0.75

0.07

ALHT

0.99

1.01

MHT

1.08

1.1

A

1.25

1.28

B

2.1

2.14

C

4.09

4.17

D

6.81

7.04

E

11.61

11.9

DISCUSSION The results as presented in Table 2 show that except for sample A, the permeabilities of the three heterogeneous media are lower than the permeabilities of all the homogeneous media. Among the three heterogeneous media, the mixed heterogeneous has the highest permeability. This indicates that fluid flow faster in mixed heterogeneous than layered heterogeneous media. This is as a result of non-uniform grain distribution with large pore size interconnectivity, which enhanced high permeability. DLHT has the lowest permeability. This is as a result of re-arrangement of grain distribution. The finest grains of the least permeable sample at the topmost migrate downward under the influence of water weight and thereby fill in the pores in the subsequence more permeable and leads to reduction in permeability of the entire medium. Thus, the hydraulic conductivity of fluid through the medium becomes low. However, this could not occur easily in ascending layered heterogeneous medium because the grains at the top most are bigger in size relatively to the subsequence layer.It was observed that the least permeable sample dominates in all the three types of heterogeneous media considered. This is true because the permeability of the sample A is relatively closer to that of the three heterogeneous media than any other homogeneous medium (Table 2).

CONCLUSION At the end of the study, it was found that [1] under the same hydraulic gradient, fluid flow faster in homogeneous porous media than in heterogeneous porous medium; [2] fluid flows faster in a mixed heterogeneous porous media than layered heterogeneous porous media; [3] and the least permeable medium dominates in the permeability of both bed layers and mixed heterogeneous media. Thus, in selecting a suitable material of lower permeability for seepage control, a medium of layered heterogeneous which is made up of sand sample of different porosities arranged in descending order from bottom (DHLT) will serves better. The choice of the least permeable layer and its placement should be considered in the preparation of a protective filter.

REFERENCES 1. Casagrande, A. (1937) Seepage through dams (Harvard University Publication 209).Reprinted from Journal of the new England water Association. 2. Cedergren H.R. (1976) Seepage, drainage and flow nets (New York: Willey Interscience Publication). 3. Domenico P.A. and Schwartz F.W. (2000) Physical and chemical hydrogeology (New York: John Willey and Sons). 4. Elsayed A.A. and Lindly K.A. (1996) Estimating permeability of untreated roadway in Washington presented at International conference. 5. Frick T. C and Taylor R.W. (1978) Petroleum production handbook 23.

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6. Hubert M.K. (1940) Journal of Geology48. 7. Jacob B. (2001) Modelling groundwater flow and contaminant transport. (Israel: Technion-Israel Institute of Technology). 8. Jacob B. and Arnold V. (1990) Modeling groundwater flow and pollution (Reidel Publishing). 9. Leonards, G.A. (1962) Foundation Engineering (New York: McGraw-Hill) 107139. 10. Olowofela J.A. and Adegoke J.A. (2005) Indian Journal of Physics 79(8): 893897. 11. Popoola O.I. Adegoke J.A. and Sciences 2(5): 642-645.

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13. Sower G.B.and Sower G.F. (1970) Introductory soil mechanics and foundation (New York: Macmillan Publishing) 85-87.