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 9:

2. 2 JB/SWICA/SW SEA WATER INTRUSION and VARIABLE DENSITY FLOW Sea water intrusion into coastal aquifers. Contaminant transport with density effects. Solute/brine transport. 26/1/01 Typical coastal aquifer. Recall: scales are highly distorted!

3. Natural Replenishment (N) Artificial Injection (R) Pumping (P) Excess Freshwater Flowing to the Sea Seawater Wedge Sharp Interface Transition Zone Recirculating Flow/Transport (May be thin or wide) 3 26/1/01 JB/SWICA/SW Typical cross-section of a coastal aquifer. Water balance components: Fresh water From inland

4. 4 26/1/01 JB/SWICA/SW Overpumping produces lowering of piezemetric head Reduction in Freshwater Flow to the Sea Landward Advance of Seawater Widening of Transition Zone Upconing to Wells Pumping Above Interface Contamination/salinization of Pumped Freshwater Consequence: Shutoff of Contaminated Wells As a result of pumping… L

5. 5 26/1/01 JB/SWICA/SW A FEW MORE SITUATIONS

6. 6 26/1/01 JB/SWICA/SW Sharp interface Approximation! Reality is 3-d!

7. 7 26/1/01 JB/SWICA/SW Multiple sub-aquifers, and semipervious layers. Mixed water

8. Obviously, the aquifer may be more complicated. Again: INTERFACE vs. TRANSITION ZONE Is this still a question nowadays? (Coastal plane in Israel) 8 26/1/01 JB/SWICA/SW

9. 9 26/1/01 JB/SWICA/SW And the case of a salt dome, with density effects. …enough of the bla-bla!… W H E R E I S the MATHEMATICS?

10. 10 26/1/01 JB/SWICA/SW The SHARP INTERFACE approximation vs. and (the more realistic) TRANSITION ZONE MODEL of sea water intrusion.

11. 11 26/1/01 JB/SWICA/SW THE SHARP INTERFACE APPROXIMATION Mathematical model: two regions, each with a different fluid. (…immiscible….sharp interface….capillary pressure?….) Fresh water head Sea water head

12. 12 26/1/01 JB/ SWICA /SW Equation describing the interface: F = F(x,y,z,t) For a point on the interface at an elevation:  (x,y,t) The interface can, thus, be described by: z =  (x,y,t) Or, by: F(x,y,z,t) = z  (x,y,t) = 0. The pressure at a point on the interface is the same when the interface is approached from either side: Or: 40! Once we have solved for h f and h s, we can find the shape of the interface from;

13. 13 26/1/01 JB/ SWICA /SW HOWEVER, it is a highly nonlinear problem and… we encounter the vicious circle, stemming from the interface being also an (a-priori-unknown) boundary, or a free surface. Over the years, efforts to solve: The sharp interface approximation. A single fluid phase, with variable density ---a solute transport problem. ---a transition zone.

14. 14 26/1/01 JB/ SWICA /SW ….a sharp interface approximation…..a nostalgic moment… USE OF HELE-SHAW MODELS. ~1960

15. 15 26/1/01 JB/ SWICA C/SW A multilayered coastal aquifer. Studying the effect of injection. ~1963

16. 16 26/1/01 JB/ SWICA /SW The slope of the (sharp) interface. Dynamic equilibrium at a stationary interface: For

17. 17 26/1/01 JB/ SWICA /SW For stationary sea water, when (q f ) r increases, so does . F-water head in F-water zone S-water head in S-water zone No flow of S-water

18. 18 26/1/01 JB/ SWICA /SW For stationary sea water, when (q f ) r increases, so does . The BADON-GHYBEN - HERZBERG APPROXIMATION: (1888)(1901) Equivalent to the case of a vertical equipotential, or horizontal flow, …….DUPUIT ASSUMPTION! Is this real/possible near the coast? Dutchman Belgian

19. 19 26/1/01 JB SWICA /SW In reality: Actual flow near the coast. Recall: still assuming a sharp interface. without accretion.

20. 20 26/1/01 JB/ SWICA /SW MASS BALANCE EQUATIONS BASED ON DUPUIT. In 3-d: Essentially horizontal flow

21. 21 26/1/01 JB/ SWICA /SW By integration along the vertical, taking into account the boundary conditions at the boundaries of integration: ………………… For steady state: sea water is stationary everywhere. h s = constant. And: Boundary conditions at the coast?

22. 22 26/1/01 JB/ SWICA /SW Real vs. Dupuit shape of the interface near the coast. The OUTFLOW FACE. x Boundary conditions at x = 0 under the Dupuit assumption? Obviously, some kind of approximation. Where do we place this boundary?

23. 23 26/1/01 JB/ SWICA /SW Bear and Dagan (1964) compared exact and Dupuit solutions, using the hodograph technique.

24. 24 26/1/01 JB/ SWICA /SW G Consider boundary at x = 0: Along the outflow face (AB): For fresh water: R = resistance of ABC Similarly, for the sea water: Etc. x C B A x x z G f s

25. 25 26/1/01 JB/ SWICA /SW Glover (1959): Henry (1959): This term is deleted if we make the Dupuit assumption.

26. 26 26/1/01 JB/ SWICA /SW Some analytical solutions for a stationary interface, based on the DUPUIT ASSUMPTIONS: Confined Phreatic Paraola Parabola

27. 27 26/1/01 JB/ SWICA /SW Conclusion: There exists a relationship between the length of sea water Intrusion, L, the discharge to the sea, Q 0, and the piezometric head above the toe,  0. AS Q 0 INCREASES, L DECREASES! This is the basis for management of a coastal aquifer.

28. 28 26/1/01 JB/ SWICA SW UPCONING of the interface under a pumping well. 1 2 3 1.Seaward flow everywhere above the interface. 2.In the vicinity of the well, flow towards the well. 3. Above a critical point—an unstable cusp. …and in reservoir engineering.

29. 29 26/1/01 JB/ SWICA /SW USE OF THE HODOGRAPH TECHNIQUE:

30. 30 26/1/01 JB/ SWICA /SW

31. 31 26/1/01 JB/ SWICA /SW Upconing by the hodograph method.

32. 32 26/1/01 JB/ SWICA /SW Unsteady upconing by the method of small perturbations.

33. 33 26/1/01 JB/v/SW The transition zone in the coastal aquifer MATHEMATICAL MODEL Conceptual Model Mathematical Model (variable density flow)) –Governing equations –Constitutive equations –Parameters –Main features The Special Case for Seawater Intrusion –Governing equations –Constitutive equations –Parameters and main features –Initial conditions –Boundary conditions

34. 34 26/1/01 JB/ SWICA /SW Types of piezometric heads in a well in a variable density fluid. –Well is filled with the water present AT the measured POINT. –Well contains the SAME water as along the soil column adjacent to the well. –Well contains a reference fluid. Conceptual Model All processes are described and studied, at the continuum (macroscopic) level. The porous medium may be heterogeneous and anisotropic. The solid matrix is non-deformable, except for the effect of its compressibility on the specific storativity. The medium’s permeability is independent of salt concentration. The porous medium is saturated. A sharp phreatic surface is assumed to exist. Isothermal conditions are assumed to prevail. Freshwater and seawater are assumed to be a single fluid phase, with a variable salt concentration and density. A transition zone develops between freshwater and seawater.

35. 35 26/1/01 JB/ SWICA /SW Conceptual model (cont.) Fluid’s density and dynamic viscosity depend on salt concentration Darcy’s law is valid for fluid motion. Fick’s law describes the diffusive flux of salt at the macroscopic level. A Fickian-type law is used for the dispersive flux. No adsorption/desorption, and no chemical reactions occur. Mathematical Model (General Density-dependent Flow) Mass balance equation for the water: Since  (c,p)

36. 36 26/1/01 JB/ SWICA /SW The motion (Darcy) equation for a variable density flow: c m = Normalized salt concentration in terms of. Buoyancy force

37. 37 26/1/01 JB/ SWICA /SW Constitutive equation for water density: Not a too good approximation, as it does not consider changes in volume due to changes in concentration. Linear Effect of density variations on dynamic viscosity:

38. 38 26/1/01 JB/ SWICA C/SW Fluid’s density and dynamic viscosity as functions of mass fraction of concentrated salt solution (brine).

39. 39 26/1/01 JB/ SWICA SW Altogether two equations in two variables: p,c m Replace the pressure, p, by a piezometric head, h f, based on a reference density,  0f, as the variable Then, Darcy’s law: More convenient:

40. 40 26/1/01 JB/ SWICA /SW Flow Equation (mass balance + motion equations) Two coupling terms: Natural advection along the vertical, produced by density difference. Change in mass due to change in concentration. Mass balance equation for the dissolved salt:

41. 41 26/1/01 JB/ SWICA SW Another form: SUMMARY: Two coupled density-dependent (1) flow equation, and (2) dissolved salt balance equation, in terms of the two primary variables: h f and c m. And V, or q:

42. 42 26/1/01 JB/ SWICA /SW Main Features Rotational flow due to the body-force term. Density-dependent flow Nonlinear flow equation Nonlinear transport equation Coupling between flow and transport

43. 43 26/1/01 JB/ SWICA /SW INITIAL AND BOUNDARY CONDITIONS for SEA-WATER INTRUSION: Domain:ABCDEFA Initial Conditions F

44. 44 26/1/01 JB/ SWICA /SW Boundary Conditions Impervious Bottom (AB) Flow Transport Right-side Boundary (BC) Flow Or Transport Or

45. 45 26/1/01 JB/ SWICA /SW Phreatic Surface (CD) Shape: Condition (flow) : To be expressed by Darcy’s law. Condition (transport) : Seepage Face (DE) Flow (points on seepage face) Transport SySy

46. 46 26/1/01 JB/ SWICA /SW Sea bottom: Flow: Pressure is dictated by sea level: Transport: Inflow: Outflow: Assume that next to the bottom, The concentration is that of the outflowing water. Hence: Point M is obtained by iterations. (V=0) M n Sea side:

47. 47 26/1/01 JB/MGFC/SW Solutions can be obtained by a transforming the mathematical model into a numerical one, and solving the latter by a computer program (code). And more, nowadays … Also: Sorek and pinder (1999) SWICA

48. Models: 2-D Horizontal +2-D Vertical +2-D Radial Symmetric, and 3-D Heterogeneous, anisotropic aquifers Multilayered aquifers Pumping and recharge wells Dynamic phreatic surface, and varying conditions along sea bottom Temporally variable inflow, outflow and sources/sinks Nonlinear density-dependent flow Nonlinear solute transport Coupled flow and transport Memory is allocated/released dynamically. Data and operations are encapsulated in each class, calls among different classes are made by class object pointer. Easy to extended to include new capabilities. 48 26/1/01 JB/ SWICA /SW Main Features of the Numerical Model Developed by Dr. Quanlin Zhou and Dr. J. Bensabat.

49. 49 26/1/01 JB/ SWICA /SW The solution is obtained by several steps: The flow equation is solved by the Galerkin FEM. The Lagrangian concentration is obtained by an adaptive pathline-based particle tracking algorithm. The remaining dispersion equation is solved by the Galerkin FEM. No more….. Some results on sea water intrusion:

50. 50 26/1/01 JB/ SWICA /SW Upconing Above a Saltwater Layer Initial and Boundary conditions for the upconing problem. Freshwater Seawater 20m 80m 20m Initial transition zone (2m) x z seawater freshwater

51. 51 26/1/01 JB/ SWICA /SW Numerical Analysis –Dispersivity-diffusivity parameters: Case A: Case B: Case C: –Mesh 2-D Radial Symmetric domain 2-D vertical rectangular mesh Mesh for the upconing problem (Np=114x61, Ne=113x60).

52. 52 26/1/01 JB/ SWICA /SW Upconing and Decay (Case B) – Upconing (0-3.95 years) Salt transports to the well by advection and dispersion. A salt cone is produced, and extends upward. The reference head within the cone increases, under the accumulative fluid density effect in the vertical. Vertical velocity and advection decreases. Horizontal velocity decreases in the lower portion of the cone. A recirculating or stagnant cell is produced gradually in [0, 200]x[0,22]m. The cone extends horizontally by the varying velocity field. High salt concentration (0.8 isochlor) never goes upward. Transition zone extends vertically. The well is shutoff at 3.95 years, when concentration in the mixed pumped water is 0.02 (2%). – Decay period (3.95-50 years) A large counterclockwise recirculating flow is produced. Velocity is relatively large within the salt cone. Salt transports downward to the bottom by advection. The salt cone recovers and disappears gradually. Transition zone becomes wider. The recovery proceeds very fast just after the shutoff of the well, but it needs a long time to recover. Completely.

53. 53 26/1/01 JB/ SWICA /SW Transient distributions of salt concentration in Case B. The left side indicates results under well pumpage, while the right one indicates results in the period of decay after well shutoff.

54. 54 26/1/01 JB/ SWICA /SW Transient fields of the reference head (on the left side) and velocity (on the right side) in case B under well pumping and after well shutoff. Case B

55. 55 26/1/01 JB/ SWICA /SW Transient distributions of salt concentration in Case A. The left side indicates results under well pumpage, while the right one indicates results in the period of decay after well shutoff. Case A

56. 56 26/1/01 JB/ SWICA /SW A recirculating flow cell beneath the pumping well in Case B in the region of (Region A).

57. 57 26/1/01 JB/ SWICA SW x z y 500m 5m 100m N Sea level 500m Domain and boundary conditions in the upconing problem in a phreatic coastal aquifer. Upconing in a Phreatic CoastalAquifer

58. Steady-state solutions in Stage A for the case of upconing in a phreatic coastal aquifer: (a) salt concentration distribution, and (b) velcoity field. The dashed line in (a) indicates the Ghyben-Herzburg interface. 58 26/1/01 JB/ SWICA /SW

59. 59 26/1/01 JB/ SWICA /SW The transient process of interface upconing under well pumping in Stage B.

60. 60 JB/SWICA-K 03-23-01 The management problem (B). Management of a water resources system means making decisions on pumping and recharge, e.g., Location of extraction and injection wells. Rates of pumping and injection. Quality of pumped and injected Water. to achieve specified goals (including sustainable (safe) yield). subject to specified constraints (technical, hydrological, legal,economic, political, environmental, ….reliability level,…, etc. And subject to the constraints embodied in the flow and transport model!..and pumping sea water The flow and transport model constitutes a constraint to be satisfied at every instant.

61. 61 JB/SWICA-K 03-23-01 The management problem (cont) In a coastal aquifer add to this general statement, the relationship between the rates and locations of withdrawal and injection wells, and the quality of pumped water...and pumping sea water Management under (almost) virgin conditions. Management once sea water intrusion has taken place beyond desirable (?) extent. Unfortunately, in many places. A distinction between:

62. 62 JB/SWICA-K 03-23-01 Important to note: the transition from one state of the ground water system to another one TAKES (sometime long) TIME! The one time reserve. “Mining” of fresh water. Pumping above the “interface zone” The coastal collector. Initial stages of development of a coastal aquifer: Evaluate sustainable yield. Optimal sea water encroachment (L). Corresponding extraction and injection (rates, locations, etc.) Combination with artificial recharge (for storage purposes). Pumping sea water (cooling, Swimming pools, fish ponds…) L

63. 63 JB/SWICA-K 03-23-01 Management under conditions of excessive sea water intrusion. Objective: Restoration. Back to sustainable yield. Means: Reduction of fresh water extraction. (!) Redistribution of pumping. (?) Shallow wells to avoid upconing (?) Artificial recharge by imported water (possibly with reclaimed sewage). Extraction of sea/saline water. (Restoration of fresh water levels may be fast, but restoration of transition zone and water quality is a L O N G process.)

64. 64 JB/SWICA-K 03-23-01 Other (including engineered) means: (mostly means to prevent/stem further sea water) Creating a fresh-water (piezometric head) mound/barrier close to and parallel to the sea: by injection into a phreatic aquifer by injection into a confined aquifer. Creating (by pumping) a trough (parallel to the coast) as a barrier. Creating a physical (impervious.semipervious) barrier (practical in a shallow thin aquifer, and even then…??)

65. 65 JB/SWICA-K 03-23-01 …and in each case,….MODELING….. We focused only on a coastal aquifer. We have not mentioned: Variable density flow. Contamination of a coastal aquifer from surface sources. Salinization from saline water bodies not (presently) associated with the sea). Uncertainty….in practically everything. and many other subjects associated with sea water intrusion into coastal aquifers.

66. 66 JB/SWICA-K 03-23-01 Do we know all we need to know in order to properly manage a coastal aquifer? What are the current gaps in knowledge? DATA: MODELING Finally: UNCERTAINTY: Identifying objectives. Including environmental aspects. Incorporating the geohydrological model into the management model. Effect of salinity variations on permeability. Improved numerical methods. Incorporating contamination aspects. Monitoring techniques. Identification of seawater intrusion by geophysical techniques. How to cope with uncertainty in the data (and in the model). OPTIMAL MANAGEMENT Etc.