Principles of Groundwater Flow. Hsin-yu Shan Department of Civil Engineering National Chiao Tung University

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1 Principles of Groundwater Flow Hsin-yu Shan Department of Civil Engineering National Chiao Tung University

2 Groundwater Flow Forms of energy that ground water possesses Mechanical Thermal Chemical

3 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

4 Outside 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

5 Resistant 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

6 Mechanical Energy h = v g + Bernoulli equation z + P ρ g h hydraulic head (L, J/N) = constant First term velocity head (ignored in ground water flow) Second term elevation head Third term pressure head

7 Ground surface h p h p h z h z datum

8 Force Potential and Hydraulic Head Φ = P ρghp gz + = gz + = g( z + hp) ρ ρ Φ = gh

9 Heads in Water (Liquid) with Various Densities P = ρ p gh p P = ρ f gh 1 f P 1 = P ρ gh = p p ρ gh f f ρ p h f = hp ρ f

10 Definition of point-water head and fresh-water head

11 Point-water head for a system of three aquifers, each containing water with a different density

12 h p1 h p z 1 z datum

13 Darcy s Law dh Q = KA( dl ) Q flow rate (L 3 /T) K hydraulic conductivity (L/T) h head (L) dh/dl hydraulic gradient A cross-sectional area of porous media (L )

14 The Application of Daycy s Law Laminar flow viscous forces dominates Reynolds Number R Reynolds number, dimensionless ρ fluid density v discharge velocity d diameter of passageway through which fluid moves µ viscosity (M/TL) R = ρvd µ

15 Fig. 5.6 Laminar flow Turbulent flow

16 Specific Recharge and Average Linear Velocity v = Q A = K dh dl v is termed the specific discharge, or Darcy flux. It is the apparent velocity

17 v x = n Q e A = K n e dh dl Seepage velocity, or average linear velocity n e is the effective porosity

18 Equations of Ground-Water Flow Control volume for flow through a confined aquifer

19 Representative Elementary Volume (REV)

20 Confined Aquifers Net total accumulation of mass in the control volume ( ρw qx + ρwqy + ρwqz ) dxdydz x y z (5.5) Change in the mass of water in the control volume M t = t ( n dxdydz) ρ w

21 Compressibility of water, β: β dp = dρw ρ w Compressibility of aquifer, α: (only consider volume change in the vertical direction) αdp = d( dz) dz

22 As the aquifer compresses or expands, n will change, but the volume of solids, V s, will be constant. If the only deformation is in the z-direction, d(dx) and d(dy) will be equal to zero dv s = 0 = d[(1 n) dxdydz] Differentiation of the above equation yields: dz dn = ( 1 n) d( dz) and dn = ( 1 n) d( dz) dz dn = ( 1 n)αρ gdh w (eq. 5-31)

23 Change of mass with time in the control volume dxdy t ndz t n dz t dz n t M w w w ] ) ( [ + + = ρ ρ ρ t h dxdydz g n g t M w w w + = ρ βρ αρ ) ( (5.36) Eq. (5.5) = Eq. (5.36) t h g n g z h y h x h K w w + = + + ) ( ) ( βρ αρ

24 S = b( αρ g + nβρ g) w w Two-dimensional flow with no vertical components: h x + h y = S T h t (5-4)

25 Steady-state flow no change in head with time Laplace equation: (three-dimensional flow) 0 = + + z h y h x h (5.43) Two-dimensional flow with leakage t h T S T e y h x h = + + (5.44)

26 Unconfined Aquifers Boussinesq equation: t h K S y h h y x h h x y = + ) ( ) ( If the drawdown 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 t h Kb S y h x h y = +

27 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

28 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: Φ = gh

29 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)

30 Fig. 5.8 Equipotential lines in a three-dimensional flow field and the gradient of h

31 The diagram in the previous slide shows the equipotential surfaces of a twodimensional uniform flow field Uniform means 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.

32 grad h = dh ds 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 groundwater flow

33 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

34 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

35 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

36 Effect of Anisotropy on Flow Net 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

37 Fig What if the direction of K max is perpendicular the grad h? Relationship of flow lines to equipotential field and grad h. A. Isotropic aquifer. B. Anisotropic aquifer

38 Flow Net 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

39 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

40 The soil and water are incompressible Flow is laminar, and Darcy s law is valid All boundary conditions are known

41 Boundary Conditions No-flow boundary: Ground water cannot pass a no-flow boundary Adjacent flow lines will be parallel to a noflow boundary Equipotential lines will intersect it at right angles Boundaries such as impermeable formation, engineering cut off structure

42 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

43 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

44 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

45 Procedure for Constructing a Flow Net 1. Identify the boundary conditions. Make a sketch of the boundaries to scale with the two axes of the drawing having the same scale 3. Identify the position of known equipotential and flow-line conditions

46

47 4. Draw a trial set of flow lines. A. The outer flow lines will be parallel to noflow boundaries. B. The distance between adjacent flow lines should be the same at all sections of the flow field

48

49 5. Draw a trial set of equipotential lines. A. The equipotential lines should be perpendicular to flow lines. B. They will be parallel to constant-head boundaries and at right angles to no-flow boundaries. C. 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 D. Should be spaced to form areas that are equidimensional, be as square as possible

50 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

51

52 Fig. 5.1 Flow net beneath an impermeable dam

53 Computing Flow Rate q = Kph f q is the total volume discharge per unit width of aquifer 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

54 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 in the two strata is the same (continuity)

55 Fig Streamtube crossing a hydraulic conductivity boundary

56 Q = K a 1 1 dh dl 1 1 Q = K c dh dl Q 1 = Q h 1 = h dh 1 K 1a = dl1 a K 1 = dl 1 K K c dh dl c dl

57 a = bcosσ 1 c = bcosσ b 1 = dl1 sinσ1 K K 1 = b 1 = dl sinσ tanσ1 tanσ B. From low to high conductivity. C. From high to low conductivity

58 Fig Low conductivity High conductivity

59 Steady Flow in a Confined Aquifer q = Kb dh dl q is the flow per unit width dh/dl is the slope of the potentiometric surface h = h 1 q Kb x x is the distance from h 1

60 Fig Steady flow through a confined aquifer of uniform thickness

61 Steady Flow in a Unconfined Aquifer Fig Steady flow through an unconfined aquifer resting on a horizontal impervious surface

62 q = Kh dh dx h is the saturate thickness of the aquifer Integrate both sides of the equation Dupuit Equation q = 1 K ( h L h 1 ) L is the flow length

63 Control volume for flow through a prism of an unconfined aquifer with the bottom resting on a horizontal impervious surface and the top coinciding with the water table

64 Unconfined flow, which is subjected to infiltration or evaporation

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