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