1. 41 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 2:slides 41-80

2. 42 JB/SWICA 01-01-01 PIEZOMETRIC HEAD = Elevation of point = Specific weight of water = Pressure in the water Piezometric head is measured with respect to a DATUM LEVEL.

3. 43 JB/SWICA 01-01-01 CONFINED AQUIFER : bounded from above and below by impervious formations. Piezometric surface above ceiling of aquifer. CLASSIFICATION OF AQUIFERS ACCORDING TO THE PIEZOMETRIC HEAD SYSTEM

4. 44 JB/SWICA 01-01-01 ARTESIAN AQUIFER Portion of a confined aquifer in which the piezometric head is above ground surface.

5. 45 JB/SWICA 01-01-01 PHREATIC AQUIFER: bounded from above by a phreatic surface. PERCHED AQUIFER: local phreatic aquifer above the phreatic surface of a major one.

6. 46 JB/SWICA 01-01-01 LEAKY (PHREATIC OR CONFINED) AQUIFER: bounded from above and/or below by an aquitard.

7. 47 JB/SWICA 01-01-01 MULTIPLE AQUIFERS Distorted scale!!

8. 48 AQUIFERS ARE VERY THIN DOMAINS, RELATIVE TO HORIZONTAL DISTANCES OF INTEREST CONFINED AQUIFER Very small vertical flow component. CONFINED AQUIFER WITH PARTIALLY PENETRATING WELLS 01-01-01 JB/SWICA

9. 49 JB/SWICA 01-01-01 Leaky-confined aquifer Phreatic aquifer River Note the streamlines Recall: distorted scale! LOCAL vs. REGIONAL PROBLEMS.

10. 50 JB/SWICA 01-01-01 Flow in a vertical cross-section, with horizontal water table Flow near inflow And outflow boundaries.

11. 51 JB/SWICA 01-01-01 MOTION EQUATIONS FO SINGLE PHASE FLOW: DARCY's LAW Two ways for presenting Darcy’s law The MOTION EQUATION is an APPROXIMATE FORM of the AVERAGED MOMENTUM BALANCE EQUATION FOR A FLUID ---the NAVIER STOKES EQUATIONS (neglecting inertial effects and terms that express energy dissipation as a result of momentum transfer within the fluid). Or….the French engineer in charge of the water system in the city of Dijon, Henry Darcy, 1856,..…conducted experiments on sand packed filter columns, reaching EMPIRICAL CONCLUSION

12. 52 JB/SWICA 01-01-01 K = coefficient of proportionality called hydraulic conductivity. Q = volume of fluid per unit time passing through a column of constant cross-sectional area, A and length L. h 1, h 2 = elevations of inflow and exit reservoirs of the column. z = elevation of the point at which the piezometric head is measured, above a datum level. p,  = fluid's pressure and mass density. z = elevation of the point at which the piezometric head is measured. p,  = fluid's pressure and mass density

13. 53 JB/SWICA 01-01-01 In a compressible fluid,  =  (p), we define HUBBERT's POTENTIAL (Hubbert, 1940): In Darcy’s law, h 1 - h 2 =  h = Energy loss across the column due to friction at the microscopic solid-fluid interface. HYDRAULIC GRADIENT: Flow x x+  x x

14. 54 JB/SWICA 01-01-01 SPECIFIC DISCHARGE: (…NOT “apparent velocity”, “Darcy velocity”, etc). So far... flow through a finite length, L. What happens AT A POINT? ss s+  s s-  s s s

15. 55 JB/SWICA 01-01-01 ss s+  s s-  s s q s Consider flow in a column aligned in the direction of the unit vector s. Along the column, h = h(s). Use Darcy's law for the segment: In the limit, as  s 0, we obtain:

16. 56 JB/SWICA 01-01-01 Thus, AT A POINT: with q s considered positive when it coincides with the positive direction of the s-axis. FLOW TAKES PLACE FROM HIGH PIEZOMETRIC HEAD (ENERGY) TO LOW PIEZOMETRIC HEAD. NOT NECESSARILY FROM HIGH TO LOW PRESSURE. Do not use the piezometric head when the density varies (by temperature and/or concentration changes).

17. 57 JB/SWICA 01-01-01 VELOCITY, V, is the distance traveled per unit time. (Omit averaging symbol) EFFECTIVE POROSITY. Part of the void space is not available to fluid flow, due to dead-end pores (immobile Fluid), zone with fine grained material.

18. HYDRAULIC CONDUCTIVITY, K (dims. L/T). Can be defined as: Specific discharge per unit gradient (in 1-d flow in an isotropic porous medium). The hydraulic conductivity depends of fluid properties and void space configuration (width of passages and tortuosity). PERMEABILITY, k (dims. L 2 ), depends only on void space Configuration. 58 JB/SWICA 01-01-01  = dynamic viscosiy = kinematic viscosity UNITS for HYDRAULIC CONDUCTIVITY: m/d, cm/s, ft/d, gal/d-ft 2, (SI: m/s) UNITS for PERMEABILITY: m 2, cm 2, ft 2, (SI: m 2 )

19. 59 JB/SWICA 01-01-01 In petroleum Engineering: FORMULAE FOR PERMEABILITY: Empirical and semi-empirical formulae: C = dimensionless coefficient. d = effective grain diameter, say d 10. (and many other formulae)

20. Changes of permeability with time, due to Compaction by external load. Dissolution of solid. Precipitation. Clogging by fines, Biological activities... Shrinkage of clay soil. RANGE OF VALIDITY OF DARCY’s LAW 60 JB/SWICA 01-01-01 Experiments:

21. 61 JB/SWICA 01-01-01 In pressurized conduits, and channels, the (dimensionless) REYNOLDS NUMBER, Re, expresses the ratio of inertial to viscous forces acting on a fluid. Helps to distinguish between LAMINAR FLOW, at low Re and turbulent flow at higher Re. By analogy, for flow through porous media : d = some representative (microscopic) length characterizing void space, e.g. d 10. = kinematic viscosity of fluid (e.g., Darcy's law is valid as long as the Re, that indicates the magnitude of the inertial forces relative to the viscous drag ones, does not exceed a value of about 1 (but sometimes as high as 10):

22. 62 JB/SWICA 01-01-01 EXTENSIONS OF DARCY'S LAW: To three dimensions. To compressible fluids. To inhomogeneous porous medium. To anisotropic porous medium. Remember: Darcy's law is a simplified form of the averaged momentum balance equation. THREE-DIMENSIONAL FLOW For a homogeneous isotropic porous medium, K = a constant SCALAR Since the specific discharge is a VECTOR:

23. 63 JB/SWICA 01-01-01 Are components of the VECTORS q,

24. 64 JB/SWICA 01-01-01 Recall: The hydraulic gradient is a vector equal to the negative of the gradient vector.

25. 65 JB/SWICA 01-01-01 COMPRESSIBLE FLUID: Here,  =  (p), and we use the motion equation (Darcy's law) written in terms of h* (= Hubbert’s potential): INHOMOGENEOUS POROUS MEDIUM: We use K = K(x,y,z) in Darcy’s Law. ANISOTROPIC POROUS MEDIUM If the permeability at a given point depends on direction, the porous medium at that point is said to be ANISOTROPIC. Reasons for anisotropy: layering, shape of grains, vertical stress.

26. 66 JB/SWICA 01-01-01 Darcy's law for an anisotropic porous medium: This is a linear relationship between the components of q and the components of.

27. 67 JB/SWICA 01-01-01 The permeability is represented by NINE COEFFICIENTS: K xx, K xy, K xz,…., etc. K ij may be interpreted as the contribution to q i by a unit of. Components of the SECOND RANK TENSOR OF. Symmetric tensor, i.e., K ij = K ji Six distinct components. The coefficients K ij are non-negative. HYDRAULIC CONDUCTIVITY TENSOR

28. 68 JB/SWICA 01-01-01 In matrix forms, in 3-d domain : In 2-d domain:

29. 69 JB/SWICA 01-01-01 69 Other (compact) forms of Darcy"s law: Vector form: Indicial form: subscripts i,j indicate x i, x j. Einstein's summation convention: subscript (or superscript) repeated twice and only twice in any product or quotient of terms is summed over the entire range of values of that subscript (or superscript).

30. 70 JB/SWICA 01-01-01 PRINCIPAL DIRECTIONS OF A SECOND RANK TENSOR IN AN ANISOTROPIC POROUS MEDIUM, FLOW AND GRADIENT ARE NOT CO-LINEAR! When principal directions are used as a coordinate system:

31. 71 JB/SWICA 01-01-01 GENERALIZED DARCY LAW By averaging the momentum balance equation for a Newtonian fluid, and introducing the simplifying assumptions: Inertial effects are negligible relative to viscous ones. Shear stress WITHIN the fluid is negligible in comparison with the drag at the fluid-solid interface. we obtain for saturated flow: V, p, ,  = (average) velocity, pressure, density, and viscosity of the fluid, respectively. V s = (average) velocity of the solid. z = elevation. k(x,y,z) = permeability tensor.

32. 72 JB/SWICA 01-01-01 For Stationary, non-deformable porous medium, V s = 0: When x,y,z are PRINCIPLE DIRECTIONS:

33. 73 JB/SWICA 01-01-01 What about 2-d flow in an aquifer? MODELING FLOW IN AN AQUIFER---- ESSENTIALLY HORIZONTAL FLOW USE DARCY'S LAW

34. 74 JB/SWICA 01-01-01 ….and in vector form when K and T are tensors: : Q’= Discharge per unit width through entire aquifer thickness. h=h(x,y,t) = Average head along aquifer's thickness, B. The product KB is called TRANSMISSIVITY. When K=K(x,y,z), T=T(x,y) = K(x,y,z) dz,

35. 75 JB/SWICA 01-01-01 FLOW IN A CONFINED AQUIFER. Recall: on phreatic surface, p = 0, h = z. h = h(x,y,z,t) = Piezometric head. H=H(x,y,t) = Elevation of water table above datum level.

36. 76 JB/SWICA 01-01-01 Introduce DUPUIT (1863) ASSUMPTION(s) In vector notation: When bottom is at elevation  =  (x,y): ESSENTIALLY HORIZONTAL FLOW IN THE AQUIFER. Equivalently: Equipotentials are vertical, Pressure distribution is hydrostatic along theoretical, Velocities are uniform along the vertical. K(h -  ) plays the role of transmissivity of the phreatic aquifer. It is a tensor in an anisotropic medium.

37. 77 JB/SWICA 01-01-01 Boundary conditions. What about the seepage face? When employing the Dupuit assumption, we neglect the seepage face. By integration, we get the DUPUIT-FORCHHEIMER DISCHARGE FORMULA: Use Dupuit assumption:

38. 78 JB/SWICA 01-01-01 HOWEVER….non-horizontal flow: End of part 2

39. 41A JB/SWICA 01-01-01 MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE by JACOB BEAR WORKSHOP I Part 2:slides 41-80 Copyright © 2000 by Jacob Bear, Haifa Israel. All Rights Reserved. The Second International conference on Salt Water Intrusion and Coastal Aquifers --Monitoring, Modeling and Management