Applied Hydrogeology V

1. Applied Hydrogeology V Прикладная Гидрогеология Yoram Eckstein, Ph.D. Fulbright Professor 2013/2014 Tomsk Polytechnic University Tomsk, Russian Federation Spring Semester 2014

2. Useful links • http://www.onlineconversion.com/ • http://www.digitaldutch.com/unitconverter/ • http://water.usgs.gov/ogw/basics.html • http://water.usgs.gov/ogw/pubs.html • http://ga.water.usgs.gov/edu/earthgwaquifer.html • http://water.usgs.gov/ogw/techniques.html • http://water.usgs.gov/ogw/CRT/

3. Applied Hydrogeology V. Principles of Groundwater Flow

4. Forms of energy endowed in ground-water • Mechanical • Thermal • Chemical Ground water moves from one region to another to eliminate energy differentials The flow of ground water is controlled by the law of physics and thermodynamics

5. Forces Acting on Ground- Water • Gravity –pulls ground water downward • External pressure • Atmospheric pressure above the zone of saturation • Molecular attraction – • Cause water to adhere to solid surfaces • Creates surface tension in water when the water is exposed to air • This is the cause of the capillary phenomenon

6. Resistive Forces • Forces resisting the fluid movement when ground water is flowing through a porous media: • Shear stresses –acting tangentially to the surface of solid • Normal stresses acting perpendicularly to the surface • These forces can be thought of as “friction”

7. Mechanical Energy • Bernoulli equation • h - hydraulic head (L, J/N) • First term –velocity head (ignored in ground water flow) • Second term –elevation head • Third term –pressure head h = =

8. Heads in Water with Various Densities P1 = P2 =

10. Darcy’s Law

11. The limits of validity of Darcy’s Law • Laminar flow –viscous forces dominate • R - Reynolds number, dimensionless • ρ - fluid density • v - discharge velocity • d - diameter of passageway through which fluid moves • μ - viscosity (M/TL)

12. Specific DischargeAverage Linear Velocity v is termed the specific discharge, or Darcy flux; It is the apparent velocity = vx is seepage velocity, or average linear velocity ne is the effective porosity

13. Derivation of the Governing Equation R x y q z x y • Consider flux (q) through REV • OUT – IN = - Storage • Combine with: q = -K grad h

14. Confined Aquifers Net total accumulation of mass in the control volume: Change in the mass of water in the control volume:

15. Confined Aquifers Compressibility of water, β: Compressibility of aquifer, α: (only consider volume change in the vertical direction)

16. Confined Aquifers As the aquifer compresses or expands, n will change, but the volume of solids, Vs, will be constant. If the only deformation is in the z-direction, d(dx) and d(dy) will be equal to zero: Differentiation of the above equation yields: and

17. Confined Aquifers Change of mass with time in the control volume:

18. Confined Aquifers Two-dimensional flow with no vertical components: Steady-state flow: no change in head with time Laplace equation (three-dimensional flow): = 0

19. Two-dimensional flow with leakage: Confined Aquifers vertical leakage downward to the confined aquifer

20. Unconfined Aquifers Boussinesq equation: If the Δh in the aquifer is very small compared with the saturated thickness, h, can be replaced with an average thickness, b, that is assumed to be constant over the aquifer

21. Solution of Flow Equations If aquifer is homogeneous and isotropic, and the boundaries can be described with algebraic equations analytical solutions Complex conditions with boundaries that cannot be described with algebraic equations numerical solutions

22. Gradient of Hydraulic Head • The potential energy, or force potential of ground water consists of two parts: elevation and pressure (velocity related kinetic energy is neglected) • It is equal to the product of acceleration of gravity and the total head, and represents • mechanical energy per unit mass:

23. Gradient of Hydraulic Head • To obtain the potential energy: measure the heads in an aquifer with piezometers and multiply the results by g • If the value of h is variable in an aquifer, a contour map may be made showing the lines of equal value of h (equipotential surfaces)

24. Gradient of Hydraulic Head Equipotential lines in a three-dimensional flow field and the gradient of h • The diagram above shows the equipotential surfaces of a two-dimensional uniform flow field • Uniform the horizontal distance between each equipotential surface is the same • The gradient of h: a vector roughly analogous to the maximum slope of the equipotential field.

25. Gradient of Hydraulic Head • s is the distance parallel to grad h • Grad h has a direction perpendicular to the equipotential lines • If the potential is the same everywhere in an aquifer, there will be no ground-water flow

26. Relationship of Ground-Water-Flow Direction to Grad h • The direction of ground-water flow is a function of the potential field and the degree of anisotropy of the hydraulic conductivity and the orientation of axes of permeability with respect to grad h • In isotropic aquifers, the direction of fluid flow will be parallel to grad h and will also be perpendicular to the equipotential lines

27. Relationship of Ground-Water-Flow Direction to Grad h • For anisotropic aquifers, the direction of ground-water flow will be dependent upon the relative directions of grad h and principal axes of hydraulic conductivity • The direction of flow will incline towards the direction with larger K

28. Flow-Lines and Flow-Nets • A flow line is an imaginary line that traces the path that a particle of ground water would follow as it flows through an aquifer • In an isotropic aquifer, flow lines will cross equipotential lines at right angles • If there is anisotropy in the plane of flow, then the flow lines will cross the equipotential lines at an angle dictated by: • the degree of anisotropy and; • the orientation of grad h to the hydraulic conductivity tensor ellipsoid

29. Flow-Lines and Flow-Nets Relationship of flow lines to equipotential field and grad h. A. Isotropic aquifer. B. Anisotropic aquifer

30. Flow-Nets • The two-dimensional Laplace equation for steady-flow conditions may be solved by graphical construction of a flow net • Flow net is a network of equipotential lines and associated flow lines • A flow net is especially useful in isotropic media

31. Assumptions for Constructing Flow Nets • The aquifer is homogeneous • The aquifer is fully saturated • The aquifer is isotropic (or else it needs transformation) • There is no change in the potential field with time • The soil and water are incompressible • Flow is laminar, and Darcy’s law is valid • All boundary conditions are known

32. Boundary Conditions • No-flow boundary: • Ground water cannot pass a no-flow boundary • Adjacent flow lines will be parallel to a no-flow boundary • Equipotentiallines will intersect it at right angles • Boundaries such as impermeable formation, engineering cut off structures etc.

33. Boundary Conditions • Constant-head boundary: • The head is the same everywhere on the boundary • It represents an equipotential line • Flow lines will intersect it at right angles • Adjacent equipotential lines will be parallel • Recharging or discharge surface water body

34. Boundary Conditions • Water-table boundary: • In unconfined aquifers • The water table is neither a flow line nor an equipotential line; rather it is line where head is known • If there is recharge or discharge across the water table, flow lines will be at an oblique angle to the water table • If there is no recharge across the water table, flow lines can be parallel to it

35. Flow Net • A flow net is a family of equipotential lines with sufficient orthogonal flow lines drawn so that a pattern of “squares” figures results • Except in cases of the most simple geometry, the figures will not truly be squares

36. Procedure for Constructing a Flow Net Identify the boundary conditions Make a sketch of the boundaries to scale with the two axes of the drawing having the same scale Identify the position of known equipotential and flow-line conditions

37. Procedure for Constructing a Flow Net

38. Procedure for Constructing a Flow Net • 4. Draw a trial set of flow lines. • The outer flow lines will be parallel to no-flow boundaries. • The distance between adjacent flow lines should be the same at all sections of the flow field

39. Procedure for Constructing a Flow Net

40. Procedure for Constructing a Flow Net 5. Draw a trial set of equipotential lines. The equipotential lines should be perpendicular to flow lines. They will be parallel to constant-head boundaries and at right angles to no-flow boundaries. If there is a water-table boundary, the position of the equipotential line at the water table is base on the elevation of the water table They should be spaced to form areas that are equidimensional, be as square as possible

41. Procedure for Constructing a Flow Net 6. Erase and redraw the trial flow lines and equipotential lines until the desired flow net of orthogonal equipotential lines and flow lines is obtained, e.g.

42. Procedure for Constructing a Flow Net Flow net beneath an impermeable dam

43. Example of Flow Net Between Two Streams • The water table is neither a flow line nor an equipotential line; rather it is line where head is known

44. Computing Flow Rate from a Flow Net • q’ is the total volume discharge per unit width of aquifer (e.g. between adjacent flow lines = flow path) • p is the number of flow paths • h is the total head loss • F is the number of squares bounded by any two adjacent pairs of flow lines and covering the entire length of flow

45. Refraction of Flow Lines • When water passes from one stratum to another stratum with a different hydraulic conductivity, the direction of the flow path will change • The flow rate through each stream tube (flow path) in the two strata is the same (continuity)

46. Refraction of Flow Lines Stream tube crossing a hydraulic conductivity boundary

47. Refraction of Flow Lines = h1 = h2

48. Refraction of Flow Lines

49. Refraction of Flow Lines B. From low to high conductivity. C. From high to low conductivity

50. Refraction of Flow Lines