principles of groundwater flow · 2019-03-27 · groundwater flow forms of energy that ground water...
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Principles of Groundwater Flow
Hsin-yu ShanDepartment of Civil EngineeringNational Chiao Tung University
Groundwater Flow
Forms of energy that ground water possesses
MechanicalThermalChemical
Ground water moves from one region to another to eliminate energy differentialsThe flow of ground water is controlled by the law of physics and thermodynamics
Outside Forces Acting on Ground Water
Gravity – pulls ground water downwardExternal pressure
Atmospheric pressure above the zone of saturation
Molecular attraction –Cause water to adhere to solid surfacesCreates surface tension in water when the water is exposed to airThis is the cause of the capillary phenomenon
Resistant ForcesForces resisting the fluid movement when ground water is flowing through a porous media
Shear stresses – acting tangentially to the surface of solidNormal stresses acting perpendicularly to the surface
These forces can be thought of as “friction”
Mechanical Energy
constant2
2
=++=g
Pzg
vhρ
Bernoulli equationh hydraulic head (L, J/N)First term – velocity head (ignored in ground water flow)Second term – elevation headThird term – pressure head
Ground surface
z
hp
hz
hp
h
datum
Force Potential and Hydraulic Head
)( pp hzg
ghgzPgz +=+=+=Φ
ρρ
ρ
gh=Φ
Heads in Water (Liquid) with Various Densities
ppghP ρ=1 ff ghP ρ=2
21 PP =
ffpp ghgh ρρ =
pf
pf hh
ρρ
=
Definition of point-water head and fresh-water head
Point-water head for a system of three aquifers, each containing water with a different density
hp2hp1
z2z1
datum
Darcy’s Law
)(dldhKAQ −=
Q flow rate (L3/T)K hydraulic conductivity (L/T)h head (L)dh/dl hydraulic gradientA cross-sectional area of porous media (L2)
The Application of Daycy’sLaw
Laminar flow – viscous forces dominatesReynolds Number
R Reynolds number, dimensionlessρ fluid densityv discharge velocityd diameter of passageway through which fluid movesµ viscosity (M/TL)
µρvdR =
Fig. 5.6
Laminar flow
Turbulent flow
Specific Recharge and Average Linear Velocity
dldhK
AQv −==
v is termed the specific discharge, or Darcy flux.It is the apparent velocity
dldh
nK
AnQv
eex −==
Seepage velocity, or average linear velocityne is the effective porosity
Equations of Ground-Water Flow
Control volume for flow through a confined aquifer
Representative Elementary Volume (REV)
Confined Aquifers(5.25)Net total accumulation of mass in the control volume
dxdydzqz
qy
qx zwywxw )( ρρρ
∂∂
+∂∂
+∂∂
−
Change in the mass of water in the control volume
) ( dxdydzntt
Mwρ∂
∂=
∂∂
Compressibility of water, β:
w
wddPρρβ =
Compressibility of aquifer, α: (only consider volume change in the vertical direction)
dzdzddP )(
=α
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
])1[(0 dxdydznddVs −==Differentiation of the above equation yields:
)()1( dzdndndz −=
and
dzdzdndn )()1( −
=
(eq. 5-31)gdhndn wαρ)1( −=
Change of mass with time in the control volume
dxdyt
ndztndz
tdzn
tM w
ww ])([∂∂
+∂∂
+∂
∂=
∂∂ ρρρ
thdxdydzgng
tM
www ∂∂
+=∂∂ ρβραρ )( (5.36)
Eq. (5.25) = Eq. (5.36)
thgng
zh
yh
xhK ww ∂
∂+=
∂∂
+∂∂
+∂∂ )()( 2
2
2
2
2
2
βραρ
)( gngbS ww βραρ +=
Two-dimensional flow with no vertical components:
th
TS
yh
xh
∂∂
=∂∂
+∂∂
2
2
2
2
(5-42)
Steady-state flow no change in head with time
Laplace equation: (three-dimensional flow)
02
2
2
2
2
2
=∂∂
+∂∂
+∂∂
zh
yh
xh (5.43)
Two-dimensional flow with leakage
th
TS
Te
yh
xh
∂∂
=+∂∂
+∂∂
2
2
2
2 (5.44)
Unconfined AquifersBoussinesq equation:
th
KS
yhh
yxhh
xy
∂∂
=∂∂
∂∂
+∂∂
∂∂ )()(
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
th
KbS
yh
xh y
∂∂
=∂∂
+∂∂
2
2
2
2
Solution of Flow EquationsIf aquifer is homogeneous and isotropic, and the boundaries can be described with algebraic equations
Analytical solutionsComplex conditions with boundaries that cannot be described with algebraic equations
Numerical solutions
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:
gh=Φ
To obtain the potential energy: measure the heads in an aquifer with piezometers and multiply the results by gIf 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)
Fig. 5.8
Equipotential lines in a three-dimensional flow field and the gradient of h
The diagram in the previous slide shows the equipotential surfaces of a two-dimensional uniform flow fieldUniform means the horizontal distance between each equipotential surface is the sameThe gradient of h: a vector roughly analogous to the maximum slope of the equipotential field.
dsdhh = grad
s is the distance parallel to grad hGrad h has a direction perpendicular to the equipotential linesIf the potential is the same everywhere in an aquifer, there will be no ground-water flow
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 hIn isotropic aquifers, the direction of fluid flow will be parallel to grad h and will also be perpendicular to the equipotential lines
For anisotropic aquifers, the direction of ground-water flow will be dependent upon the relative directions of grad hand principal axes of hydraulic conductivityThe direction of flow will incline towards the direction with larger K
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 aquiferIn an isotropic aquifer, flow lines will cross equipotential lines at right angles
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
Fig. 5.10
What if the direction of Kmax is perpendicular the grad h?
Relationship of flow lines to equipotential field and grad h. A. Isotropic aquifer. B. Anisotropic aquifer
Flow NetThe two-dimensional Laplace equation for steady-flow conditions may be solved by graphical construction of a flow netFlow net is a network of equipotentiallines and associated flow linesA flow net is especially useful in isotropic media
Assumptions for Constructing Flow Nets
The aquifer is homogeneousThe aquifer is fully saturatedThe aquifer is isotropic (or else it needs transformation)There is no change in the potential field with time
The soil and water are incompressibleFlow is laminar, and Darcy’s law is validAll boundary conditions are known
Boundary ConditionsNo-flow boundary:
Ground water cannot pass a no-flow boundaryAdjacent flow lines will be parallel to a no-flow boundaryEquipotential lines will intersect it at right angles
Boundaries such as impermeable formation, engineering cut off structure
Constant-head boundary:The head is the same everywhere on the boundaryIt represents an equipotential lineFlow lines will intersect it at right anglesAdjacent equipotential lines will be parallel
Recharging or discharge surface water body
Water-table boundary:In unconfined aquifersThe water table is neither a flow line nor an equipotential line; rather it is line where head is knownIf there is recharge or discharge across the water table, flow lines will be at an oblique angle to the water tableIf there is no recharge across the water table, flow lines can be parallel to it
Flow Net
A flow net is a family of equipotentiallines with sufficient orthogonal flow lines drawn so that a pattern of “squares” figures resultsExcept in cases of the most simple geometry, the figures will not truly be squares
Procedure for Constructing a Flow Net
1. Identify the boundary conditions2. 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
4. Draw a trial set of flow lines. A. The outer flow lines will be parallel to no-
flow boundaries. B. The distance between adjacent flow lines
should be the same at all sections of the flow field
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
6. Erase and redraw the trial flow lines and equipotential lines until the desired flow net of orthogonal equipotential lines and flow linesis obtained
Fig. 5.12
Flow net beneath an impermeable dam
Computing Flow Rate
fKphq =′
q’ is the total volume discharge per unit width of aquiferp is the number of flow pathsh is the total head lossf is the number of squares bounded by any two adjacent pairs of flow lines and covering the entire length of flow
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 changeThe flow rate through each stream tube in the two strata is the same (continuity)
Fig. 5.13
Streamtube crossing a hydraulic conductivity boundary
1
111 dl
dhaKQ =2
222 dl
dhcKQ =
21 QQ =
2
22
1
11 dl
dhcKdldhaK =
21 hh =
22
11 dl
cKdlaK =
1cosσba = 2cosσbc =
11 sin1σ
=dlb
22 sin1σ
=dlb
2
1
2
1
tantan
σσ
=KK
B. From low to high conductivity. C. From high to low conductivity
Fig. 5.15
Low conductivity
High conductivity
Steady Flow in a Confined Aquifer
dldhKbq =′
q’ is the flow per unit widthdh/dl is the slope of the potentiometric surface
xKbqhh′
−= 1
x is the distance from h1
Fig. 5.16
Steady flow through a confined aquifer of uniform thickness
Steady Flow in a Unconfined Aquifer
Fig. 5.17
Steady flow through an unconfined aquifer resting on a horizontal impervious surface
dxdhKhq −=′
h is the saturate thickness of the aquifer
Integrate both sides of the equation Dupuit Equation
LhhKq )(
21 2
122 −=′
L is the flow length
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
Unconfined flow, which is subjected to infiltration or evaporation