1. 1 JB/SWICA 01-01-01 MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE by JACOB BEAR SHORT COURSE ON Copyright © 2002 by Jacob Bear, Haifa Israel. All Rights Reserved. To use, copy, modify, and distribute these documents for any purpose is prohibited, except by written permission from Jacob Bear. Lectures presented at the Instituto de Geologia, UNAM, Mexico City, Mexico, December 6--8, 2003 Professor Emeritus, Technion—Israel Institute of Technology, Haifa, Israel Part 7: 250-284

2. 250 01-01-01 JB/SWICA CONTAMINANT TRANSPORT (ISOTHERMAL) Examples of subsurface contamination and remediation: Water table Waste Clay cover Slurry wall Impervious

3. 251 01-01-01 JB/SWICA

4. 252 01-0-101 JB/SWICA OBJECTIVE: Develop complete (macroscopic) models that describe the transport of contaminants (as components in water and gas) in the subsurface. c = c(x,y,z,t) Recall that we have also the case of a contaminant in the form of a separate fluid phase, but we model the dissolved chemicals in water. We shall present this topic in four lectures: Flux of a dissolved chemical species (the present lecture). Mass balance equation for a single dissolved component. Complete Models. Multiple components.

5. 253 01-01-01 JB/SWICA MEASURES OF SOLUTE CONCENTRATION Mass concentration, c  a, expresses the mass of  -species, per unit volume of a fluid a-phase: m   = mass of the  -species within a REV, and U o  denotes the volume of the  -phase inside the REV. Common units: are g/l (= grams of  per liter of fluid, or mg/l (= milligrams of  per liter of fluid). Molar concentration, or molarity, n  , expresses the number of  -moles, N  a, per unit volume of the a-phase: M  = molecular mass of the  -species. Note that the units of M  should be consistent with the unit used for mass. One mole means the molecular mass of a compound in grams

6. 254 01-01-01 JB/SWICA Molar fraction, or mole fraction, X  a, is defined as the ratio between the number of moles of  and the total number of moles in the a-phase (dimensionless, or mole per mole): Molality, or molar concentration, m , of a solute is the number of moles of the latter per kg mass of solvent (especially water). The unit is mol/kg. For a solution composed of N  moles of a  -solute and N  a moles of a  -solvent, having a molar mass M , we have: Mass fraction,   , is the mass of  per unit mass of the phase (dimensionless, or mass per mass):

7. 255 01-01-01 JB/SWICA FLUXES OF CONTAMINANTS ADVECTIVE FLUX = the intrinsic phase average (mass weighted) velocity of the phase. = the intrinsic phase average concentration of the contaminant. The advective flux of a component: = mass of the component passing through a unit area of porous Medium, normal to V, per unit time. The advective flux can be obtained by averaging over an REV

8. 256 01-01-01 JB/SWICA Given: A 2-d horizontal flow domain, with streamlines, and a front between clean water and contaminated water at t. Question: Where will the contaminated water be after some time?

9. 257 01-01-01 JB/SWICA ANSWER: We consider the movement contaminated water in an aquifer. with n initial sharp front between contaminated and clean water. Use Darcy's law to determine velocities of water particles on the front Use to determine the position of the front after. Is this the new distribution of contaminated water in the aquifer ?

10. 258 01-01-01 JB/SWICA TWO CONCEPTUAL FIELD EXPERIMENTS: Longitudinal spreading of a contaminant. Uniform flow in a two-dimensional field. Initial sharp front between contaminated (c = 1) and clean water (c = 0). Observations taken after a period  t. How will the contaminant be distributed? WE OBSERVE: NOT A MOVEMENT OF A SAHRP FRONT! (as Described by Darcy’s law.

11. INSTEAD: A transition from contaminated to clean water. As flow continues, the transition zone expands. We observe SPREADING of the contaminant LONGITUDINALLY. Can Darcy's law explain this spreading and the evolution of the transition zone? Another experiment: 01-01-01 JB/SWICA 259

12. Observe: Both LONGITUDINAL and TRANSVERSAL spreading. Of the contaminant. Uniform, two-dimensional flow. A finite volume of tracer labeled water is injected into a well at x = 0, y = 0, at some initial time t = 0. Can Darcy's law explain this spreading? CONCLUSION: We observe a SPREADING ( DISPERSION) That CANNOT BE EXPLAINED?PREDICTERD by DARCY’s LAW 26001-01-01 JB/SWICA

13. Another experiment: Displacement in a laboratory column. Conclusion: Displacement of one fluid by another DOES NOT Follow the movement of a sharp front determined by DARCY’s LAW. BREAKTHROUGH CURVE c = c(t). 26101-01-01 JB/SWICA

14. EXPLANATIONS: Water particles (carrying contaminants) move at the MICROSCOPIC velocity in the void space. This velocity varies in MAGNITUDE and DIRECTION from point to point within the void space. Darcy's law gives the AVERAGE VELOCITY of the water. Parabolic velocity distribution in a capillary tube. Later, one additional factor. Because of the shape of the interconnected pore space, the (microscopic) streamlines deviate from the mean direction of flow. 26201-01-01 JB/SWICA

15. MECHANICAL (fluid mechanics) DISPERSION of any initially close group of tracer particles: Reasons: FLOW THE PRESENCE OF A PORE SYSTEM Why is the spreading irreversible? Why is the plume growing wider and wider? These questions cannot be explained by mechanical dispersion alone. An additional phenomenon: MOLECULAR DIFFUSION 26301-01-01 JB/SWICA

16. MOLECULAR DIFFUSION. Molecular diffusion produces an additional flux of tracer particles (at the microscopic level) from regions of higher tracer concentrations to those of lower ones, relative to the advective flux. This phenomenon tends to equalize the concentration ALONG (minor effect) the stream tube, and BETWEEN ADJACENT streamlines(major effect). This phenomenon explains the observed IRREVERSIBILITY, and Ever-growing transversal dispersion. Molecular diffusion of a component in a fluid is caused by the random motion of the molecules in a fluid. This motion, when averaged (over the molecular level), produces a flux of the component's particles at the microscopic level from regions of higher concentrations to those of lower ones. This flux, called molecular diffusion, is relative to the advective one, produced by the velocity of the fluid phase. 26401-01-01 JB/SWICA

17. 265 Within the void space, we observe two diffusion phenomena (actually, the SAME molecular diffusion): As the tracer spreads out along each microscopic stream-tube, as a result of mechanical dispersion, a concentration gradient is produced. This gradient produces an additional flux of the component by molecular diffusion. The latter phenomenon tends to equalize the concentration along every microscopic stream-tube. Thus along the stream-tube, we have two transport mechanisms: advection by the prevailing fluid velocity, and molecular diffusion, due to concentration gradient at the advancing front. At the same time, a concentration gradient is also produced between adjacent stream-tubes, causing lateral molecular diffusion across streamlines. This molecular diffusion flux tends to equalize the concentration across pores. This phenomenon, combined with the randomness of the streamlines, explains the observed ever-growing transversal dispersion and the irreversibility. i.e., at the microscopic level 25801-01-01 JB/SWICA

18. 266 01-01-01 JB/SWICA The deviations in solute concentration within a fluid phase From the concentration distribution that would have been obtained by assuming advection only (at the average velocity) are due to two simultaneous phenomena: Variations in the microscopic velocity within the phase, with respect to the averaged velocity. Molecular diffusion. Molecular diffusion contributes to spreading of a solute in two ways: As an additional flux (just try no advection!) By continuously transporting solute particles from any stream-tube to the adjacent one More on molecular diffusion later. And, obviously, we’ll need fluxes at the MACROSCOPIC LEVEL!

19. 267 01-01-01 JB/SWICA DISPERSIVE FLUX Ideal tracer (concentration does not affect density and viscosity) Dispersive flux: A consequence of the averaging process. The advective flux of a solute (= mass of solute per unit area of fluid) at a (microscopic) point, x', within a fluid phase (f) that occupies part of the void space within a REV centered at a point x, is given by cV. The intrinsic phase average of this flux is. In order to express this flux in terms of the average values, and, the velocity, V (x', t; x), and the component concentration, c (x', t; x), are decomposed into two parts: an intrinsic phase average value and a deviation from that value, in the form:

20. 268 01-01-01 JB/SWICA By averaging, noting that: Advective flux (per unit area of fluid) Dispersive flux Conclusion: The macroscopic flux of a component = sum of two macroscopic fluxes: Advective flux Dispersive flux (results from velocity fluctuations)

21. 269 01-01-01 JB/SWICA The price for circumventing the lack of information about the microscopic details is the creation of a new COEFFICIENT. DISPERSIVE FLUX Fickian-type law: or In indicial notation, invoking Einstein's summation convention: D, a second rank symmetrical tensor, with components D ij, is the coefficient of advective (or mechanical) dispersion. Note that in the above equation, i.j = 1, 2, 3 (in three dimensions), or, equivalently, x 1, x 2, x 3, or x, y, z. COEFFICIENT OF DISPERSION

22. 270 01-01-01 JB/SWICA Effect of diffusion Effect of velocity Effect of pore space geometry. Pe = Peclet mumber. Expresses the ration between the rate of transport by advection and by diffusion.  f = hydraulic radius of the fluid occupied portion of the void space, serving as acharacteristic length. D  f = coefficient of molecular diffusion of . Usually: Henceforth, we’ll assume:

23. 271 01-01-01 JB/SWICA a  jkl = components of a fourth rank tensor, a, called DISPERSIVITY of the porous medium (dims. L). DISPERSIVITY is a property of the geometry of the void space. In multiphase flow, it is a property of the domain within the REV occupied by a phase, i.e., function of . A symmetric second rank tensor (principal directions, etc.). In the three principal directions: D x1 x1 is called coefficient of longitudinal dispersion, D x2 x2 and D x3 x3 are called coefficients of transversal dispersion.

24. 272 01-01-01 JB/SWICA The tensor D and this means also its principal directions) depends also on the (macroscopic) velocity field. At a point on a streamline: A unit vector, T, in the direction of the tangent to the streamline. A unit vector, , in the direction of  = d  /ds, where s is the distance measured along the streamline. A unit vector,  = T , which is normal to both T and . In an isotropic porous medium, the principal directions of D coincide with the directions of these three unit vectors.

25. 273 01-01-01 JB/SWICA DISPERSIVITY The dispersivity, a, is a FOURTH rank tensor that has 81 a iklj -components in a three-dimensional space. All but 36 components are zeros. In an ISOTROPIC porous medium, only 21 components are nonzero. remain. They, in turn, depend only on TWO parameters: The 21 components depend only on two parameters: a L = longitudinal dispersivity. a T = transversal dispersivity. Both have a length dimension that characterizes the microscopic configuration of the phase within the REV. Say, a L is of the order of magnitude of a pore size. a T is 8 to 24 times smaller than a L. (we often use a factor 10).

26. For an isotropic porous medium, we can express the components of the dispersivity tensor in terms of a L and a T : And: The Kronecker delta is defined by: In cartesian coordinates; 01-01-01 274 JB/SWICA

27. 01-01-01 In the special case of uniform flow, say V x =V 0, V y = 0, V z = 0, In multiphase flow, each of the dispersivity components, as well as the longitudinal and transversal dispersivities, depend on phase saturation. In an ANISOTROPIC porous medium, the number of dispersivity Coefficients is larger, depending on the kind of symmetry. For axial symmetry, we need 5 dispersivities. 275 JB/SWICA

28. 276 01-01-01 JB/SWICA DIFFUSIVE FLUX Consider a multicomponent fluid phase consisting of a number of components, each made up of a large number of identical molecules that are in constant random motion. When the fluid phase is in motion, the movement of the molecules is relative to the movement of the fluid. The continuum approach to molecular diffusion at the microscopic level. Macroscopic law is obtained by averaging molecular behavior. MICROSCOPIC flux Microscopic values. ?

29. 277 01-01-01 JB/SWICA Advective flux: of the  -component carried by the mass-weighted velocity, V. Diffusive flux: of the  -component relative to the moving phase. This flux is the FLUX OF MOLECULAR DIFFUSION: per unit area of fluid phase Note: FICK'S LAW (binary system) cc The binary system assumption is valid only as long as the phase is sufficiently dilute

30. 278 01-01-01 JB/SWICA When, i.e., a homogeneous fluid, or when we may write Fick's law, in terms of the concentration, c  Typical values of D  diff at 25 o C, in a liquid phase, are in the range of 0.5-10x10 -5 cm 2 /s. For example, for Ca 2+, D  diff = 7.9x10 -6 cm 2 /s, for K +, D g diff = 19.6x10 -6 cm 2 /s, and for Cl -, D  diff = 20.3 x 10 -6 cm 2 /s. Values D  diff of for gas are much higher. Diffusive mass flux of a component is driven by nonequilibrium in the chemical potential,

31. 279 01-01-01 JB/SWICA Fick's law for molecular diffusion in a porous medium: By averaging, we obtain: Another form for a component in a multicomponent phase: (So far, we have discussed a binary system). = coefficient of molecular diffusion that expresses the Contribution to the flux of  by a unit gradient in the concentration of . = macroscopic mass fraction of . = (a second rank symmetric tensor) coefficient of molecular diffusion within a phase in a porous medium. Per unit area of fluid

32. 280 01-01-01 JB/SWICA POROUS MEDIUM TORTUOSITY is a second rank symmetric tensor. In an isotropic porous medium, the components are: Kronecker delta

33. Indicial notation, and constant density fluid: Einstein's summation convention is invoked. Millington and Quirk (1961): 281 01-01-01 JB/SWICA

34. Total flux = advection + diffusion + dispersion. TOTAL FLUX: The coefficient of hydrodynamic dispersion, D , is The concept of dispersion, is a consequence of the heterogeneity of the porous medium at the pore scale, i.e., due to the presence of distributed solid and void sub-domains within the REV This heterogeneity produces velocity variations at the microscopic Level. The macroscopic level of description is obtained by averaging over an REV The dispersive flux, is introduced as means of circumventing the need to know the details of the velocity distribution and of other transport features at the microscopic level. 281 01-01-01 JB/SWICA

35. MACRODISPERSION ALL porous media are heterogeneous, say, with respect to permeability. Because pressure propagates very fast, the effect of this inherent heterogeneity is felt less when considering fluid flow. However, its effect is significant when considering phenomena of transport of heat and of mass of a component, as these extensive quantities are carried advectively by the relatively slow movement of fluid phases As in the passage from the microscopic level to the macroscopic one, a new representative elementary volume is needed in order to perform the smoothing: Representative Macroscopic Volume (abbreviated as RMV) of the porous medium domain. The same approach of averaging may be applied also to heterogeneity at the macroscopic level. As a result of this second-order averaging, a new continuum is obtained, which describes the porous medium and phenomena occurring in it at a level called the megascopic level 282 01-01-01 JB/SWICA

36. d * = length characterizing the macroscopic heterogeneity that we wish to smooth out, L = length characterizing the considered porous medium domain. The characteristic size, l , of averaging volume MACRODISPERSIVE FLUX: = An advective flux,, expressing the flux carried by the fluid at the latter's average specific discharge,. = macrodispersive flux. It is a megascopic (= field scale) flux caused by the variations in the specific discharge and fluid concentration in the vicinity of a (megascopic) point in a porous medium domain. 283 01-01-01 JB/SWICA

37. Assumption: Macrodispersive flux can also be expressed in As a Fickian type law (like the flux of mechanical dispersion). The macrodispersive flux is proportional to the gradient of the averaged concentration. The coefficient of proportionality is the coefficient of macrodispersion. Summary: Dispersion and macrodispersion are analogous phenomena. Both are a consequence of velocity variations that are due to inhomogeneity, but at different scales. In pore size In the ratio k/  IN PRACTICE: Apart from the different magnitude of the dispersivity to be employed, the expressions for advective and dispersive fluxes presented earlier, remain valid also when we model field conditions. Scale of heterogeneity grows with the size of the considered domain. Field experiments and model calibrations indicate that the longitudinal dispersivity is of the order of magnitude of 1/10 of the size of the domain of interest, 284 01-01-01 JB/SWICA End of part 7

38. 250 01-01-01 JB/SWICA MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE by JACOB BEAR WORKSHOP I Part 7:slides 250-300 Copyright © 2000 by Jacob Bear, Haifa Israel. All Rights Reserved.