地下水运移机理

2 2 2
(5.43)
Two-dimensional flow with leakage
∂ h ∂ h e S ∂h + 2+ = 2 ∂x ∂y T T ∂t
2 2
(5.44)
Unconfined Aquifers
Boussinesq equation:
S y ∂h ∂ ∂h ∂ ∂h (h ) + (h ) = ∂x ∂x ∂y ∂y K ∂t
∂ h ∂ h S y ∂h + 2 = 2 ∂x ∂y Kb ∂t
2 2
Solution of Flow Equations
If aquifer is homogeneous and isotropic, and the boundaries can be described with algebraic equations
Control volume for flow through a confined aquifer
Representative Elementary Volume (REV)
Confined Aquifers
Net total accumulation of mass in the control volume (5.25)
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
moves µ viscosity (M/TL)
ρvd R= µ
Fig. 5.6 Laminar flow
Turbulent flow
Specific Recharge and Average Linear Velocity
Q dh v = = −K A dl
Darcy flux. It is the apparent velocity
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
hp1
hp2
z1
z2 datum
Darcy’s Law
dh Q = − KA( ) dl
Q flow rate (L3/T) K hydraulic conductivity (L/T) h head (L) dh/dl hydraulic gradient A cross-sectional area of porous media (L2)
P = ρ p gh p 1
P = P2 1
P2 = ρ f gh f
ρ p gh p = ρ f gh f
ρp hf = hp ρf
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
2
Ground surface
hp
hp
h
z
h
z datum
Force Potential and Hydraulic Head
ρgh p Φ = gz + = gz + = g ( z + hp ) ρ ρ
P
Φ = gh
Heads in Water (Liquid) with Various Densities
Eq. (5.25) = Eq. (5.36) (5.36)
∂ 2h ∂ 2h ∂ 2h ∂h K ( 2 + 2 + 2 ) = (αρ w g + nβρ w g ) ∂x ∂y ∂z ∂t
S = b(αρ w g + nβρ w g )
Two-dimensional flow with no vertical components:
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:
These forces can be thought of as “friction”
Mechanical Energy
v P h= +z+ = constant ρg 2g
Bernoulli J/N)
First term – velocity head (ignored in ground water flow) Second term – elevation head Third term – pressure head
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
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
Change of mass with time in the control volume
∂ ( dz ) ∂ρ w ∂M ∂n = [ ρ wn + ρ wdz + ndz ]dxdy ∂t ∂t ∂t ∂t ∂h ∂M = (αρ w g + nβρ w g ) ρ wdxdydz ∂t ∂t
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 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.
Analytical solutions
Complex conditions with boundaries that cannot be described with algebraic equations
Numerical solutions
Gradient of Hydraulic Head
∂ 2 h ∂ 2h S ∂h + 2 = 2 ∂x ∂y T ∂t
(5-42)
Steady-state flow no change in head with time Laplace equation: (three-dimensional flow)
∂h ∂h ∂h + 2 + 2 =0 2 ∂x ∂y ∂z
Outside Forces Acting on Ground Water
Gravity – pulls ground water downward External pressure
Atmospheric pressure above the zone of saturation
Molecular attraction –
∂ ∂ ∂ ρ w q y + ρ w qz )dxdydz − ( ρ w qx + ∂x ∂y ∂z
Change in the mass of water in the control volume
∂M ∂ = ( ρ wn dxdydz ) ∂t ∂t
Compressibility of water, β:
dVs = 0 = d [(1 − n )dxdydz ]
Differentiation of the above equation yields:
dz dn = (1 − n )d ( dz )
and
(1 − n )d ( dz ) dn = dz
dn = (1 − n )αρ w gdh
(eq. 5-31)