Chapter 11 Modelling seawater-pore water exchange near seabeds

Oil Pollution and its Environmental Impact in the Arabian Gulf Region M. Al-Azab, W. El-Shorbagy and S. Al-Ghais, editors © 2005 Elsevier B.V. Allrightsreserved.

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Chapter 11

Modelling seawater-pore water exchange near seabeds Arzhang Khalili Max Planck Institute for Marine Microbiology,

Celsiusstr.l,

28359 Bremen,

Germany

Abstract When sediments are permeable, a direct link benveen the pore water and overlying water column is given. In such cases - compared to impermeable sediments - the transport of nutrients and chemicals from the seawater into the seabed (and vice versa) is enhanced. So far, the conventional approach to study the interfacial phenomena has been to collect sediment samples and study them in the lab. The purpose of this chapter is to present a mathematical method that is able to quantify interfacial fluxes near and through preamble interfaces, simultaneously, without using any hydrodynamic interface conditions. The results of this model have been verified in non-invasive experiments. The model may be applied for calculation of contaminant transport through permeable sediments.

1. Introduction and governing equations in a porous layer The transport of nutrients and particulate matter across permeable sediment-water interfaces plays an important role in many situations of marine biogeochemistry such as in rivers, oceans and lakes. The quantification of the amount of substances transported through the interface is, however, a difficult task. Common experimental techniques for in-situ quantification of the interfacial exchange are micro sensors (Glud et al., 1995, 1996), benthic chambers (Gust, 1990), sediment cut-off (Huettel and Gust, 1992) or landers (Witte et al., 2003). In many of these measurement methods, however, the real situation is perturbed due to the existence of devices intruded into the fluid or the porous bed, and additional unwanted flows are generated that may change the concentration field of the substance in question. Besides, the interfacial transport is governed by many different parameters such as the topography shape, the physical properties of the porous medium (permeabiHty K and porosity <^), and the velocity field. Therefore, an efficient mathematical representation for the interfacial exchange is a highly desired tool. The first mathematical attempt in this direction was the Darcy's law (1856) to account for the interaction of the flow density (v) with the pressure gradient (V/?) given in modem vector notation as Vp=-^v

(1) K

E-mail address: [email protected] (A. Khalili).

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with K being the permeabihty of the porous medium and /x the fluid density. However, it is well-known tat the linear Darcy law holds for flows at low Reynolds numbers in which the driving forces are small and balanced by the viscous forces, solely. The Darcy's law is also found to be valid for homogeneous isotropic and non-deformable porous media. When these assumptions do not hold, the applicability of the Darcy's law is limited, and extensions need to be made to the linear Darcy equation. First non-linear generalization of Darcy's law has been suggested by Forchheimer (1901), which is known as Darcy-Forchheimer equation given by V;,= - i i v - ^ l v l v

(2)

with Cf and pf denoting the form drag coefficient and the fluid density, respectively. The second extension to the Darcy's law was made by Brinkman (1947), and was intended to account for the viscous drag by adding fiV^v {fl being the effective viscosity) to the Darcy equation. With this term included in the Darcy equation, the Brinkman-extended Darcy equation can be written by Vp = ilV^v-

-V.

(3)

K

Considering both extensions simultaneously results in Vp = /lV\-

^v-

^Ivlv.

(4)

Finally, including the total acceleration following Wooding (1957), one obtains the most general form of the momentum equation for a saturated homogeneous porous medium as Pf[3tV + (V-V)V] = -V/7 + /iV'v - ^ v - ^

Iviv

(5)

with 8t denoting differentiation with respect to time and • being the scalar product symbol. It should be noted that the intrinsic velocity V (average of velocity over a volume that contain fluid only) and the Darcy velocity v (average of velocity over a volume that contain both fluid and solid matrix) are linked to each other by the so-called DupuitForchheimer relation (Kaviany, 1999) given as v = ipV

(6)

With this equation, the general momentum equation would read as Pf[(p~^atV +
Ivlv.

(7)

After non-dimensionalization with the characteristic length L, velocity U and time L/U, the flow model equations (conservation of momentum and mass) containing all extensions to Darcy's model in vector form are (see for example Prasad, 1991): [atV + (p'Vv-V)v] = -(pV/7 4-A — V ' ~ BJ v + FlvlvV Re \ Re Da / Vv = 0.

(8) (9)

Modelling seawater-pore water exchange near seabeds

149

The parameters appearing in Eq. (8), namely A, Re, Da and F are, respectively, the viscosity ratio, Reynolds number, Darcy number, and the Forchheimer number and are given by A=f^,

(10)

LU Re=—,

(11)

V

Da=-^,

(12)

F = - ^ .

(13)

The parameter B is a binary constant that may take the values zero or unity, and will be explained in the next section. Likewise, the concentration (c) equation in dimensionless form can be given as b,c + cp-\v'^)c = ^^V^c

(14)

Re Sc where F and Sc are the diffusivity ratio and the Schmidt number given by ^

or

r = - , a Sc=-, a with a and a denoting the diffusivities of c inside the porous and fluid layer.

(15) (16)

2. Boundary and interface condition and equations in composite fluid-sediment layer While the boundary conditions at free fluid surfaces and solid walls can be formulated in a straight forward manner, a clear consensus on the mathematical form of the hydrodynamic condition that prevail at the interface between a fluid and a sediment layer, does not exist. Beavers and Joseph (1967) were the first who formulated a slip condition, later extended further by Saffman (1971). Since then, the Beavers and Joseph condition has opened a vital discussion on the interface condition problem. There are other suggestions that include the continuity of velocity, pressure and shear across the interface (Taylor, 1971; Ross, 1983; Vafai and Kim, 1990) or, more recently, the stress-jump condition of Ochoa-Tapia and Whitaker (1995). It should be noted that all these suggestions have been derived for one-dimensional problems, and that the formulation of the correct set of interface conditions for threedimensional flows is seldom obvious. However, it is possible to solve the momentum equation (Eq. 8) in the vicinity of an interface without applying any interface condition. By proper choice of parameter B, (p, A and F in Eq. (8), namely, the momentum equation in a porous layer as well

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as fluid region may be obtained when setting: ^fO, B

in fluid region ir

j l , irm porous region in fluid region

(o< ^ < 1, in porous region f 1, A =X

in fluid region

(^ 7^ 1, in porous region and

r- Ui,

in fluid region

in porous region Hence, Eqs (8), (9) and (14) may be treated as equations of a single domain in which different input parameters are taken, and therefore matching of variable values across the interface is inherent in the formulation itself, thus doing away with the need for separate interface conditions. Further details are given by Basu and Khalili (1999). 3. Solution method The Eqs (8), (9), and (14) need to be put into a proper coordinate system depending on the geometry of the problem in question. Using approximation methods such as finite differences, finite elements or finite volumes, discretized form of the above equations may be obtained. The resulting systems of algebraic equations can then be solved by standard numerical techniques. An effective solution method based on the MAC algorithm combined with the fractional step method (Kim and Moin, 1985) is given by Khalili et al. (1997). 4. Application and examples Figure 1 demonstrates the field of application of the mathematical method developed. As can be seen it contains of the flow and concentration distribution over impermeable as well as permeable beds. The flow may be driven by simple currents, waves or a combination of both. The effect of the activity of the living organisms, referred to as bio-irrigation, also may be included using the so-called irrigation parameter (Emerson et al., 1984; Aller, 1990). In order to verify the results of the method developed, first a cylindrical geometry with a rotating lid was constructed that was filled up to half height with a permeable medium and up to top with water. When the disk was set into rotation, a three-dimensional motion was generated that transported fluid, and with this, substances into the sediment layer underneath (see Fig. 2).

151

Modelling seawater-pore water exchange near seabeds

Iniptrmeable

BioiiTlgation

Permeable Sediment

Surface Topography

Figure 1. Different possible applications include effect of currents and waves on impermeable sediments and effect of advection on permeable seabeds. The flow and concentration field generated by the rotation of the disk was calculated (Basu and Khalili, 1999) and then compared with experimental results (Khalili et al, 1999) using different non-invasive techniques with good agreement between the simulations and the experiments as shown in Figure 3. Another situation is given when seabeds with periodic ripples are subject to horizontal oscillatory motion. The seabed may be solid (Fig. 4), porous but impermeable (Fig. 7) or permeable (Fig. 8).

Figure 2. Interfacial exchange of fluid due to the secondaryflowinduced by the disk rotation.

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A. Khalili Experiment

Calculation

Figure 3. Comparison of the concentration profile in the cylindrical container by numerical simulation (left image) and experiments (right image) with PET (only half of the container is shown here).

In the first case (solid ripples), the computational domain contains only fluid, and the flow equations are given by the Navier-Stokes equations. A standing wave given by an oscillatory function drives the flow over solid ripples of known wave number. The complete flow cycle is shown by the stream function distribution within one period (Fig. 5). As can be seen therein, the flow is confined to a narrow layer in the vicinity of the solid bed. The corresponding concentration distribution may be seen from Figure 6. An interesting feature of the program is demonstrated by Figures 7 and 8, where an impermeable seabed (Fig. 7) or permeable one (Fig. 8) replaced the solid bed of Figure 4. These programs developed may easily switch to any case desired. Figure 7 demonstrates the concentration distribution over an impermeable bed above the crest (left image) and

U(t) = A Sin (ft)

U(t) = A Sin (ft)

Water

Figure 4. A sinusoidal oscillation over a periodic seabed topography.

Modelling seawater-pore water exchange near seabeds

153

2.5

-

2.5 0.2T

0 2

1.5

-

1

-

1.5 h

1 V-

0.5

^^^Mll 0

^nillff/^^

^Si^l. L^^mii^r 0.5 1 1.5 2 X

2.5

2.5 h 0.5T

2h

1.5

0.8T 2h

i.5r

Figure 5. Stream function over solid ripples within one period.

the trough (right image). It is interesting to observe how the concentration of a given substance in the overlying water is transported into the impermeable bed underneath. Finally, when a permeable bed is considered, the situation changes thoroughly, because of the advection of flow through the bed. In this situation, namely, a pressure gradient is formed which is responsible for the transport of substances dissolved in the fluid. This is shown in Figure 8 by the vorticity transport (left image). The corresponding concentration

154

A. Khalili -0.5625-

-0.5-

1.5 \-0.4375-

-0.375-

-0.3215— 0.25"0.1875-0.125-

0.0625

0.5

Figure 6. Concentration distribution over solid ripples.

Crest

D-0.005 D-0.01 - - D-0.05 D-0.1 D-1

I I I I I'

0

0.1

I I I I ' I I I I ' I I I I I ' I I

0.2

0.3 C

0.4

0.5

0

0.1

0.2

Figure 7. Concentration distribution over solid ripples (impermeable sediment).

0.3 C

0.4

0.5

155

Modelling seawater-pore water exchange near seabeds Vorticity Numerical simulation time = 150.

Concentration Numerical simulation time = 150.

HoJi Mai

Figure 8. Vorticity transport (left image) and concentration distribution (right image) over a topography with two permeable ripples.

equation, demonstrating the penetration depth of the substances into the pore water is shown in the right image of Figure 8.

5. Conclusions A numerical model and laboratory experiments were developed to study the flow field and concentration exchange at permeable water-sediment layers. With the help of such a method, percolation of pollutions into permeable seabeds may be traced and quantified. The application of such techniques may be extended to many other situations in science and nature where composite fluid-sediment layer occurs. It has been shown that the model developed (Khalili et al., 1997; Basu and Khalili, 1999) is capable of providing reliable results on exchange of matter and fluid over solid as well as porous beds with very low (almost impermeable) and high permeability.

References Aller, R.C., 1990. Bioturbation and manganese cycHng in hemipelagic sediments. Philos. Trans. R. Soc. London, 5 1 - 6 8 . Basu, A.J., Khalili, A., 1999. Computation of flow through a fluid-sediment interface in a benthic chamber. Phys. Fluids 11, 1395-1405. Beavers, G.S., Joseph, D.D., 1967. Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197-207. Brinkman, H.C., 1947. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27-34. Darcy, H., 1856. Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris. Emerson, S., Jahnke, R., Heggie, D., 1984. Sediment-water exchange in shallow water estuarine sediments. J. Mar. Res. 42, 709-730. Forchheimer, P., 1901. Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45, 1782-1788.

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Glud, R.N., Gundersen, J.K., Revsbech, N.P., Jorgensen, B.B., Huettel, M., 1995. Calibration and performance of the stirred flux chamber from the benthic lander Elinor. Deep-Sea Res. I 42, 1029-1042. Glud, R.N., Forster, S., Huettel, M., 1996. Influence of radial pressure gradients on solute exchange in stirred benthic chambers. Mar. Ecol. Prog. Ser. 141, 303-311. Gust, G., 1990. Method of Generating Precisely Defined Wall Shearing Stresses, U.S. Patent, 4, 973,165. Huettel, M., Gust, G., 1992. Solute release mechanisms from confined sediment cores in stirred benthic chambers and flume flows. Mar. Ecol. Prog. Ser. 82, 187-197. Kaviany, M., 1999. Principles of Heat Transfer in Porous Media, 2nd ed. Springer, New York. Khalili, A., Basu, A.J., Mathew, J., 1997. A non-Darcy model for recirculating flow a fluid-sediment interface in a cylindrical container. Acta Mech. 123, 75-87. Khalili, A., Basu, J., Pietrzyk, U., RafFel, M., 1999. An experimental study of recirculating flow through fluid-sediment interfaces. J. Fluid Mech. 383, 229-247. Kim, J., Moin, P., 1985. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59, 308-323. Ochoa-Tapia, J., Whitaker, S., 1995. Momentum transfer at the boundary between a porous medium and a homogeneous fluid I: theoretical development. Int. J. Heat Mass Transfer 38, 2635-2646. Prasad, V., 1991. Convective heat and mass transfer in porous media. In: Kaka9, S. (Ed.), Convective Flow Interaction and Heat Transfer between Fluid and Porous Layer, NATO ASI Series E: Applied Science, Vol. 196, pp. 563-615. Ross, S.M., 1983. Theoretical model of the boundary condition at a fluid-porous interface. AIChE J. 29, 840-846. Safifman, P.G., 1971. On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50, 93-101. Taylor, G.I., 1971. A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49, 319-326. Vafai, K., Kim, S., 1990. Fluid mechanics of the interface region between a porous medium and a fluid layer - an exact solution. Int. J. Heat Fluid Flow 11, 254-256. Witte, U., Wenzhofer, F., Sommer, S., Boetius, A., Heinz, P., Aberle, A., Sand, M., Cremer, A., Abraham, W.R., J0rgensen, B.B., Pfannkuche, O., 2003. In situ experimental evidence of the fate of a phytodetritus pulse at the abyssal sea floor. Nature 424, 763-766. Wooding, R.A., 1957. Steady state free thermal convection of liquid in a saturated permeable medium. J. Fluid Mech. 2, 273-285.