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 6: 201-248

2. 01-01-01 JB/SWICA 201 MOTION EQUATIONS (for unsaturated flow). CONCEPTUAL MODEL: Each fluid (here, either air or water) occupies a certain portion of the void space. Water, the wetting phase, occupies primarily the smaller pores, while air, the non-wetting phase, occupies primarily the larger ones. The film, behaves as if it were an extended part of the solid in transmitting momentum from Each fluid to the solid. Hence, each fluid has a microscopic interface with the solid, in addition to an interface with the other fluid. The transfer of momentum across the fluid-fluid (microscopic interface) within the void space may be neglected. Since volume occupied by each phase within an REV varies with its saturation, the resistance to the flow of each fluid phase also depends on its saturation.

3. 01-01-01 JB/SWICA 201 CONCLUSION: Darcy's law, can be used also as a good approximation for the flow of a fluid phase in a multiphase system. Densities of water and air. Porosity Fluid saturations. EFFECTIVE PERMEABILITIES to water and to air. (Tensor)

4. 01-01-01 JB/SWICA 202 In vector notation, and noting that the permeability depends on saturation: Relative specific discharge: The two motion equations are not independent of each other. Linked by: Linked by dependence of effective permeabilities on saturations.

5. 01-01-01 JB/SWICA 203 For constant fluid densities, we may use the piezometric heads: EFFECTIVE PERMEABILITIES, k, and HYDRAULIC CONDUCTIVITY, K, to water and to air: Often, only the flow of water is of interest. Assumption of immobile air is unjustified (e.g., when air flow is produced by air injection and/or vacuuming). Assumption of `water flow only' neglects resistance to air flow. Better assumption: air pressure is constant (atmospheric).

6. 01-01-01 JB/SWICA 204 In terms of suction,  : (assumption of zero atmospheric pressure) In terms of saturation: MOISTURE DIFFUSIVITY: For horizontal flow: Or  w  Looks like Fick’s Law

7. 01-01-01 JB/SWICA 205 NONLINEARITY. The dependence of D w, or K w, or  on  w, introduces a non-linearity into the equations of motion. We usually obtain the retention curve form a static kind of test. In the motion equation, the relationship, say, p c = p c (  w ) is used under dynamic (= flow) conditions. Non-uniqueness of K w (S w ) and p c (S w ) relationships (hysteresis). DIFFICULTIES: Or: with

8. 01-01-01 JB/SWICA 206 EFFECTIVE AND RELATIVE PERMEABILITIES k w and k a depend the microscopic configuration of the portion of the void space occupied by each phase. Hence, dependence on phase saturation k w = k w (S w ), k a = k a (S a ) For an anisotropic porous medium, each of the ij components, of either the tensor k w = k w (S w ), or K w = K w (S w ) may have a DIFFERENT FUNCTIONAL RELATIONSHIP to saturation. RELATIVE PERMEABILITY ONLY for an isotropic porous medium, we define may define and use the tem RELATIVE PERMEABILITY

9. 01-01-01 JB/SWICA 207 TYPICAL RELATIVE PERMEABILITY CURVE FOR A WETTING PHASE (say, WATER). Note the point at  wr. Assumption: Each fluid establishes its own tortuous pathways through a certain network of channels within the porous medium.

10. 01-01-01 JB/SWICA 208 Typical relative permeability curves (a) without hysteresis, (b) with hysteresis  (a)(b) Determined experimentally Drainage from full water saturation. Rapid decline in k rw as the larger pores are drained first. At irreducible water saturation, S wr : water is discontinuous, in the form of isolated pendular rings and very thin films on solid. Residual air saturation, S wr

11. 01-01-01 JB/SWICA 209 ANALYTICAL EXPRESSIONS. Obtained by experiments and curve fitting. Irmay (1954) suggested 3 rd power parabola : Reduced (effective) saturation Corey (1957) finds that for many consolidated rocks, K wr is proportional to S e , while K nwr is proportional to (1-S e )  (1-S e  ). Gardner (1958) suggested: Suction

12. 01-01-01 JB/SWICA 210 Brooks and Corey (1964) suggested k w is in cm 2, p b (in dyne/cm 2 ) is the bubbling pressure, or air entry value, related to the largest pore size forming a continuous network of water occupied channels within the porous medium, and m is an index of the pore-size distribution of the porous medium. Van Genuchten (1980) suggested

13. 01-01-01 JB/SWICA 211 13 10/4/00 JB/MGFC

14. 01-01-01 JB/SWICA 212 HYSTERESIS and ENTRAPPED AIR in a CAPILLARY PRESSURE CURVE DRAINAGE AND IMBIBITION SCANNING CURVES:

15. 01-01-01 JB/SWICA 213 HYSTERESIS: Ink bottle effect. results from the shape of the pore space, with interchanging narrow (throats) and wide passages. We note the same meniscus curvature at different elevations., Raindrop effect is due to the fact that the contact angle at the advancing trace of a water--air interface on a solid surface is larger than at the receding one.

16. 01-01-01 JB/SWICA 214 HYSTERESIS IN EFFECTIVE PERMEABILITY CURVES: The difference between the retention curves for wetting and for drying stems from the difference in the configuration of the water-occupied portion of the void space during the processes of drainage and imbibition. The relationship k w (S w ) shows much more hysteresis than k w  probably due to the large hysteresis in the function  (S w ). Hysteresis in k w (S w ) is usually ignored because the function p c (S w ) usually exhibits far greater hysteretic effects We have discussed earlier: Similarly----hysteresis in hydraulic conductivity.

17. 01-01-01 JB/SWICA 215 DARCY'S LAW FOR THREE PHASE FLOW Wetting fluid (w) Nonwetting fluid (n), Intermediate wetting fluid, NAPL (o). Only two saturations may be independent. Three capillary pressure relations: MOTION EQUATIONS: Effective permeability to the wetting phase in the presence of the intermediate and nonwetting phases.

18. 01-01-01 JB/SWICA 216 EFFECTIVE PERMEABILITIES Underlying assumption: Recall: In a two-phase system, the effective permeabilities to the wetting phase and to the nonwetting one, are functions of their respective saturations only, i.e., In a three-phase system: The effective permeabilities to the wetting and nonwetting phases are the same functions of their respective saturations as in a two-phase system, i.e., The effective permeability to the intermediate wetting phase is a function of BOTH the wetting and nonwetting saturations: TENSOR

19. 01-01-01 JB/SWICA 217 MASS BALANCE EQUATIONS AND COMPLETE MODELS The balance equation for the mass of a phase. Two (motion) equations, but…….three or four (at least) variables. Valid for each phase.

20. 01-01-01 JB/SWICA 218 The balance equation for the mass of a phase. The mass flux of an  -phase is expressed by   q , where q  mass represents the specific discharge of that phase. The fluid phase mass contained in a unit volume of porous medium is expressed as     (check: = [mass of fluid per unit volume of fluid] x [volume of fluid per unit volume of porous medium]), or    S . The source of mass of the  -phase within the considered volume is due here only to the transfer of the mass of the a-phase from the  -one, across their common interface, per unit volume of porous medium. This rate of transfer is given by. Source/sink due to evaporation, condensation, dissolution..

21. 01-01-01 JB/SWICA 219 CAREFUL! Recall the difference between a PHASE and A COMPONENT of a phase. BALANCE OF WATER “Water” as a PHASE, or H 2 O as a component in a liquid? Leave out the sink term. BALANCE EQUATIONS could also be obtained by averaging the microscopic balance equations. IN MULTIPHASE FLOW a source of mass of a phase can be added, e.g., a point source/a well. Variables in balance equations: ++

22. 01-01-01 JB/SWICA 220 8 Variables in the two balance equations (water and air): q w, q a, p w, p a S w, S a,  w,  a, We started with two flux equations (Darcy's law), with 4 variables: q w, q a, p w, p a. Two constitutive relations:   w =  w (p w ),  a =  a (p a ). One definition: S w, + S a = 1.0. One relationship (actually, also a constitutive equation)—retention curve: p a - p w = p c (S w ), Now we have 8 equations and 8 variable. We say that we have a closed set of equations.

23. 01-01-01 JB/SWICA 221 Note that we have not counted k w (S w ) and k a (S a ) as variables, but regard them as known coefficients. Also, the above set of equations means that the relationship p c = p c (S w ) is known. With point sources and sinks: Wells at points denoted by m=1, 2, 3,…, M, pumping water at rates P w (m) (dims. L 3 /T). Wells at points n=1, 2, 3,…., N, injecting water at rates Q w (m).

24. 01-01-01 JB/SWICA 222 FLOW EQUATIONS for AIR and WATER: Solid matrix is stationary and nondeformable, or that |  V s |<<|q w |. Written in terms of pressure For constant densities: qwqw qaqa Actually, these equations are written in terms of both pressure and saturation

25. 01-01-01 JB/SWICA 223 In terms of piezometric head (and saturation):qwqw qaqa For non-deformable porous media,.

26. 01-01-01 JB/SWICA 224 By inserting the appropriate flux expressions into the balance equations, we obtain flow equations. A set of 12 VARIABLES in the complete model for (two-phase) flow of water and air: in a nondeformable porous medium: i=1,2,3. A set of 12 EQUATIONS: Mass balance equations for the fluids (2 x 3). Momentum balance equations for the fluids (= Darcy's law) (2). Constitutive equations for the fluids (2). Capillary pressure relationship (1). Sum of saturations (1). A complete set of equations. We still need initial and boundary conditions. q wi, q ai, p w, p a S w, S a,  w,  a

27. 01-01-01 JB/SWICA 225 Summary: rigid porous matrix, no interphase transfer, no external sources: Mass balance for the water: Mass balance for air: Darcy law for the water: Capillary pressure relationship:

28. 01-01-01 JB/SWICA 226 Constitutive relations: Sum of saturations: Together: Eight equations in eight variables: PRIMARY VARIABLES: NP = 2 fluid phases, NC = 2 components, NF = 2 primary variables. For example, What are primary variables? How many?

29. 01-01-01 JB/SWICA 227 In two-phase isothermal flow in a nondeformable porous medium, the model consists of the following equations: 8 equations. 8 variables: q w, q a, p w, p a S w, S a,   w,  a, 2 phases 2 components, Hence: NF = 2 select S w and p a..

30. 01-01-01 JB/SWICA 228 PRIMARY VARIABLES: NP = 2 fluid phases, NC = 2 components, NF = 2 primary variables. For example, …and we have to solve only 2 partial differential balance equations. WATER FLOW ONLY. In terms of suction for a non-deformable porous medium. No sources. Moisture capacity

31. 01-01-01 JB/SWICA 229 In terms of moisture content: Equation (a) and (b) are known as RICHARD’S EQUATION. z Especially, in 1-d vertical homogeneous isotropic medium: Or:

32. 01-01-01 JB/SWICA 230 Another form of Richard’s equation: Saturated--unsaturated three-dimensional (or 2-d vertical) flow with a phreatic surface. The advantage of coupled saturated--unsaturated flow model: The phreatic surface is removed as a boundary. SINGLE DOMAIN IN BOTH CASES….it is the SAME MASS BALANCE EQUATION, and the same kinds of boundary conditions.

33. 01-01-01 JB/SWICA 231 The differences between the two zones are In the unsaturated zone, k = k(S w ). p w = p w (S w ). We could write the single mass balance equation in the form: with Variables: p w and S w. We also have the relationship p w = p w (S w ). Obviously, we need to know k w = k w (S w ). Locate phreatic surface from: p w (x,t) = 0.

34. 01-01-01 JB/SWICA 232 INITIAL and BOUNDARY CONDITIONS We have a closed set of equations. We need initial and boundary conditions to obtain a unique, stable solution. INITIAL CONDITIONS. Example: p w (x,y,z,0) = f(x,y,z), or S w (x,y,z,0) = f(x,y,z). f(x,y,z) = a known function. BOUNDARY CONDITIONS. DIRICHLET BOUNDARY CONDITION: e.g., boundary of prescribed saturation, S w (x,y,z,) = f(x,y,z) on the boundary, or prescribed pressure: p w (x,y,z,) = f(x,y,z) on the boundary, What about ground surface?

35. 01-01-01 JB/SWICA 233 For air: prescribed pressure: p a (x,y,z,) = p atm or S a = 1.0. For water: prescribed pressure: p w (x,y,z,) ????----no information on p w, except under a pond, where we know both pressure, or head, and S w = 1.0. HOWEVER, we know the flux normal to the boundary, including “no flux”. BOUNDARY OF PRESCRIBED WATER FLUX On such a surface (V s - u). n =0. Hence, with no infiltration, the conditions is: q wr. n =0.

36. 01-01-01 JB/SWICA 234 IMPERVIOUS BOUNDARY: q wr. n =0. q ar. n =0. BOUNDARY WITH (known) INFILTRATION or WITH PRESCRIBED FLUX. N = prescribed flux on the external side of a boundary described by F=F(x,t). n F=F(x,t). Equality of fluxes NORMAL to the boundary (here the stationary ground surface) means:  w q w.n =  *N.n. For a constant water density: q w.n = N.n, For vertical downward infiltration, N = -N.n, For a horizontal ground surface,

37. 01-01-01 JB/SWICA 235 The condition for a horizontal ground surface, isotropic soil and vertical infiltration:: Or in terms of  w,. It is possible to specify the rate of evaporation, E, as a prescribed flux. This is a BOUNDARY CONDITION OF THE THIRD KIND, or a CAUCHY BOUNDARY CONDITION.

38. 01-01-01 JB/SWICA 236 Let us now make a distinction between the known rate of precipitation reaching ground surface, R(t), and the unknown rate of actual infiltration, I(t), into the subsurface. Usually, the condition on the values of pressure, suction, or saturation on the boundary are a-priori unknown. They are known on the bottom of a pond with a specified depth. What is the (instantaneous) rate of INFILTRATION at ground surface, given rate of precipitation, HOW DO WE ESTIMATE INFILTRATION IN HYDROLOGY?

39. 01-01-01 JB/SWICA 237 INFILTRATION Information required in order to specify boundary Condition at ground surface. Water from precipitation or irrigation, or from pond, INFILTRATES Through ground surface. Then it PERCOLATES downward. Rate of infiltration depends on: Rate of application. Type of soil. Vegetation. Antecedent moisture conditions. OBSERVATION: At first, high rate of infiltration. Then, the Actual rate decreases to a maximum possible under the Prevailing conditions------INFILTRATION CAPACITY, I c.

40. 01-01-01 JB/SWICA 238 Under what conditions may we assume: To state a flux boundary conditions, we have to know I. What do we do when we know R, but not I?. I ? Depending on soil properties, on the prevailing moisture content and on the gradient in the latter, the soil just below ground surface can transmit a certain flux of water, IF such flux is applied at ground surface. A higher rate will result in surface runoff. How do we state this condition in the model, as part of the boundary condition?

41. 01-01-01 JB/SWICA 239 Simplify: assume that both are vertically downward, and that the ground surface is horizontal. The case of negative infiltration, i.e., evaporation is a simple extension. The simplest statement of the flux boundary condition is: Or: But I(t) IS NOT KNOWN!! We usually add the verbal condition that `NO PONDING OR RUNOFF IS ALLOWED ABOVE GROUND SURFACE', as a constraint on I(t). I

42. 01-01-01 JB/SWICA 240 As the infiltration continues, the soil just below ground surface approaches saturation,, and we obtain: in an anisotropic Soil. TWO CASES, with R = const., and initially relatively dry soil (field capacity). Deep water table. CASE A. At first, the soil absorbs all incoming water at a very high rate. Very high gradient in moisture content is produced. For a certain period, I = R, i.e., all incoming water infiltrates. We use I = R in the flux boundary condition (shown above). The advancing wetting front produces two phenomena at ground surface:

43. 01-01-01 JB/SWICA 241 The advancing wetting front produces two phenomena at ground surface: A gradual increase in saturation up to full saturation. Consequently, an increase in K w. up to The saturation gradient decreases, to the limit of zero. CONCLUSION: The only remaining driving force is gravity. Gradually, the soil at ground surface can no longer transmit the water at the incoming rate, R. The water pressure in the pore space reaches p w = p a | atm and also S w = 1. Actually, 1.0 - S ar, because of entrapped air. Until such conditions are reached, we have I = R. Initial period: lasts as long as S w < 1.0, or, equivalently, p w < p a | atm Once full saturation is reached, the condition switches to p w = p a | atm.

44. 01-01-01 JB/SWICA 242 As the model is being solved, we check the evolving rate of infiltration, I(t) (<R), as the model runs. We observe a gradual reduction in I(t), approaching the limiting value of We can follow the same steps for time-dependent R = R(t). We check whether the resulting I(t) < R. As soon as, we switch back from the pressure condition to the condition with I = R. To summarize : In our model, the initial period continues as long as S w < 1.0, or, equivalently, as long as p w < p a | atm. During this period, the condition is: I

45. 01-01-01 JB/SWICA 243 Once full saturation is reached, we have to replace the flux condition by the first type boundary condition p w = p a | atm. (where we often assume p a | atm = 0 as the datum) corresponding to a condition just below a state of zero ponding depth. Capillarity plays a role in the part of the wetting front that is ahead of the (practically) saturated zone. The same phenomena occur for but the saturation at ground surface will vary, without leveling off.

46. 01-01-01 JB/SWICA 244 Next we calculate the rate of infiltration, I(t)(<R), by substituting the solutions for  w (t) into I We would then observe a gradual reduction in I(t), approaching the limiting value of K z | sat. : With the resulting value of I(t), we should keep track of whether the calculated rate of infiltration, I(t), is less than that of application, R(t). As soon as we reach a situation of we should switch back to the flux condition with I = R: I, R I

47. 01-01-01 JB/SWICA 245 It is important to emphasize again, that initially, and for some time (which may be significant when considering irrigation, or an individual storm event), the rate of infiltration will exceed the limiting value of K z | sat. The latter is approached from above, provided the rate of application of water to the ground surface remains larger than the rate of infiltration. I, R

48. 01-01-01 JB/SWICA 246 CASE B. Precipitation (or irrigation) is applied at a constant rate R < K z | sat. Initially, saturation gradient at ground surface will be very high, and the soil will absorb all the incoming water. However, rather rapidly, as infiltration takes place, and saturation at ground surface increases, the saturation gradient there decreases. Eventually, asymptotically, a saturation level is reached with a zero saturation gradient, so that the rate of infiltration (which is equal to the rate of application) becomes equal to the effective hydraulic conductivity at the prevailing saturation. Under such conditions, the only force driving infiltration at ground surface is gravity. Capillarity still plays a role in the part of the wetting front that is ahead of the (practically) saturated zone.

49. 01-01-01 JB/SWICA 247 The same phenomena occur with,but Saturation at ground surface will vary, without leveling off. SUMMARY: ….and back to other boundary conditions.

50. 01-01-01 JB/SWICA 248 BOUNDARY BETWEEN TWO POROUS MEDIA. Two conditions (why two?): No-jump in fluid pressures of both air and water (which means a jump in saturation!). No jump in fluxes of both air and water normal to the boundary. PHREATIC SURFACE. The phreatic surface serves as the lower boundary of the unsaturated zone. We have discussed it in connection with the saturated zone. The CONDITION: continuity of the normal flux of water across the phreatic surface boundary. Often we assume (in problems with a sharp boundary): End of part 6.

51. 200 01-01-01 JB/SWICA MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE by JACOB BEAR WORKSHOP I Part 6:slides 200-249 Copyright © 2000 by Jacob Bear, Haifa Israel. All Rights Reserved.