Ground Water HydrologyIntroduction - 2005

Philip B. Bedient

Civil & Environmental Engineering

Rice University

GW Resources - Quantity

• Aquifer system parameters• Rate and direction of GW flow • Darcy’s Law - governing flow relation• Dupuit Eqn for unconfined flow• Recharge and discharge zones• Well mechanics- pumping for water supply,

hydraulic control, or injection of wastes

GW Resources - Quality

• Contamination sources

• Contaminant transport mechanims

• Rate and direction of GW migration

• Fate processes-chemical, biological

• Remediation Systems for cleanup

Trends in Ground Water Use

Ground Water: A Valuable Resource

• Ground water supplies 95% of the drinking water needs in rural areas.

• 75% of public water systems rely on groundwater.

• In the United States, ground water provides drinking water to approximately 140 million people.

• Supplies about 40% of Houston area

Regional Aquifer Issues

Typical Hydrocarbon Spill

Aquifer Characteristics

1. Matrix type2. Porosity (n)3. Confined or unconfined4. Vertical distribution (stratigraphy or layering)5. Hydraulic conductivity (K)6. Intrinsic permeability (k)7. Transmissivity (T)8. Storage coefficient or Storativity (S)

Vertical Distribution of Ground Water

Vertical Zones of Subsurface Water

• Soil water zone: extends from the ground surface down through the major root zone, varies with soil type and vegetation but is usually a few feet in thickness

• Vadose zone (unsaturated zone): extends from the surface to the water table through the root zone, intermediate zone, and the capillary zone

• Capillary zone: extends from the water table up to the limit of capillary rise, which varies inversely with the pore size of the soil and directly with the surface tension

Typical Soil-Moisture Relationship

Soil-Moisture Relationship

• The amount of moisture in the vadose zone generally decreases with vertical distance above the water table

• Soil moisture curves vary with soil type and with the wetting cycle

Vertical Zones of Subsurface Water Continued

• Water table: the level to which water will rise in a well drilled into the saturated zone

• Saturated zone: occurs beneath the water table where porosity is a direct measure of the water contained per unit volume

Porosity– Porosity averages about 25% to 35% for most

aquifer systems

– Expressed as the ratio of the volume of voids Vv to the total volume V:

n = Vv/V = 1- b/m

where:

b is the bulk density, and

m is the density of grains

Porosity

Water

Arrangement of Particles in a Subsurface Matrix

Porosity depends on:

• particle size

• particle packing

• Cubic packing of spheres with a theoretical porosity of 47.65%

• Rhombohedral packing of spheres with a theoretical porosity of 25.95%

Soil Classification Based on Particle Size

(after Morris and Johnson)

Material Particle Size, mm

Clay <0.004

Silt 0.004 - 0.062

Very fine sand 0.062 - 0.125

Fine sand 0.125 - 0.25

Medium sand 0.25 - 0.5

Coarse sand 0.5 - 1.0

Soil Classification…cont.

Material Particle Size, mm

Very coarse sand 1.0 - 2.0

Very fine gravel 2.0 - 4.0

Fine gravel 4.0 - 8.0

Medium gravel 8.0 - 16.0

Coarse gravel 16.0 - 32.0

Very coarse gravel 32.0 - 64.0

Particle Size Distribution Graph

Particle Size Distribution and Uniformity

• The uniformity coefficient U indicates the relative sorting of the material and is defined as D60/D10

U is a low value for fine sand compared to alluvium which is made up of a range of particle sizes

Cross Section of Unconfined and Confined Aquifers

Unconfined Aquifer Systems

• Unconfined aquifer: an aquifer where the water table exists under atmospheric pressure as defined by levels in shallow wells

• Water table: the level to which water will rise in a well drilled into the saturated zone

Confined Aquifer Systems

• Confined aquifer: an aquifer that is overlain by a relatively impermeable unit such that the aquifer is under pressure and the water level rises above the confined unit

• Potentiometric surface: in a confined aquifer, the hydrostatic pressure level of water in the aquifer, defined by the water level that occurs in a lined penetrating well

Special Aquifer Systems

• Leaky confined aquifer: represents a stratum that allows water to flow from above through a leaky confining zone into the underlying aquifer

• Perched aquifer: occurs when an unconfined water zone sits on top of a clay lens, separated from the main aquifer below

Ground Water FlowDarcy’s LawContinuity EquationDupuit Equation

Darcy’s Law• Darcy investigated the flow of water through beds of

permeable sand and found that the flow rate through porous media is proportional to the head loss and inversely proportional to the length of the flow path

• Darcy derived equation of governing ground water flow and defined hydraulic conductivity K:

V = Q/A where:

A is the cross-sectional area V -∆h, and V 1/∆L

Darcy’s LawV= - K dh/dl

Q = - KA dh/dl

Example of Darcy’s Law

• A confined aquifer has a source of recharge.

• K for the aquifer is 50 m/day, and n is 0.2.

• The piezometric head in two wells 1000 m apart is

55 m and 50 m respectively, from a common

datum.

• The average thickness of the aquifer is 30 m,

• The average width of flow is 5 km.

Calculate:• the Darcy and seepage velocity in the aquifer

• the average time of travel from the head of the aquifer to a point 4 km downstream

• assume no dispersion or diffusion

The solution• Cross-Sectional area

30(5)(1000) = 15 x 104 m2

• Hydraulic gradient (55-50)/1000 = 5 x 10-3

• Rate of Flow through aquifer Q = (50 m/day) (75 x 101 m2) = 37,500 m3/day

• Darcy Velocity: V = Q/A = (37,500m3/day) / (15 x 104 m2) = 0.25m/day

Therefore:• Seepage Velocity:

Vs = V/n = 0.25 / 0.2 = 1.25 m/day (about 4.1 ft/day)

• Time to travel 4 km downstream: T = 4(1000m) / (1.25m/day) = 3200 days or 8.77 years

• This example shows that water moves very slowly underground.

Ground Water Hydraulics

• Hydraulic conductivity, K, is an indication of an aquifer’s ability to transmit water

– Typical values:

10-2 to 10-3 cm/sec for Sands

10-4 to 10-5 cm/sec for Silts

10-7 to 10-9 cm/sec for Clays

Ground Water Hydraulics

Transmissivity (T) of Confined Aquifer

-The product of K and the saturated thickness of the aquifer T = Kb

- Expressed in m2/day or ft2/day

- Major parameter of concern

- Measured thru a number of tests - pump, slug, tracer

Ground Water HydraulicsIntrinsic permeability (k)

Property of the medium only, independent of fluid properties

Can be related to K by:K = k(g/µ)where: µ = dynamic

viscosity = fluid densityg = gravitational

constant

Storage CoefficientRelates to the water-yielding capacity of an aquifer

S = Vol/ (AsH)

– It is defined as the volume of water that an aquifer releases from or takes into storage per unit surface area per unit change in piezometric head - used extensively in pump tests.• For confined aquifers, S values range between

0.00005 to 0.005• For unconfined aquifers, S values range

between 0.07 and 0.25, roughly equal to the specific yield

Regional Aquifer Flows are Affected by Pump Centers

Streamlines and Equipotential lines

Derivation of the Dupuit Equation - Unconfined Flow

Dupuit Assumptions

For unconfined ground water flow Dupuit developed a theory that allows for a simple solution based off the following assumptions:

1) The water table or free surface is only

slightly inclined

2) Streamlines may be considered horizontal

and equipotential lines, vertical

3) Slopes of the free surface and hydraulic

gradient are equal

Derivation of the Dupuit Equation

Darcy’s law gives one-dimensional flow per unit width as:

q = -Kh dh/dx

At steady state, the rate of change of q with distance is zero, or

d/dx(-Kh dh/dx) = 0

OR (-K/2) d2h2/dx2 = 0

Which implies that,

d2h2/dx2 = 0

Dupuit Equation

Integration of d2h2/dx2 = 0 yieldsh2 = ax + b

Where a and b are constants. Setting the boundary

     condition h = ho at x = 0, we can solve for b

b = ho2

Differentiation of h2 = ax + b allows us to solve for a,a = 2h dh/dx

And from Darcy’s law,hdh/dx = -q/K

Dupuit Equation

So, by substitution

h2 = h02 – 2qx/K

Setting h = hL2 = h0

2 – 2qL/KRearrangement gives

q = K/2L (h02- hL

2) Dupuit Equation

Then the general equation for the shape of the parabola is

h2 = h02 – x/L(h0

2- hL2) Dupuit Parabola

However, this example does not consider recharge to the aquifer.

Cross Section of Flow

q

Adding Recharge W - Causes a Mound to Form

Divide

Dupuit Example

Example:

2 rivers 1000 m apart

K is 0.5 m/day

average rainfall is 15 cm/yr

evaporation is 10 cm/yr

water elevation in river 1 is 20 m

water elevation in river 2 is 18 m

Determine the daily discharge per meter width into each

River.

ExampleDupuit equation with recharge becomes

h2 = h02 + (hL

2 - h02) + W(x - L/2)

If W = 0, this equation will reduce to the parabolicEquation found in the previous example, and

q = K/2L (h02- hL

2) + W(x-L/2)Given:

L = 1000 m K = 0.5 m/day

h0 = 20 m

hL= 28 m

W = 5 cm/yr = 1.369 x 10-4 m/day

Example

For discharge into River 1, set x = 0 m

q = K/2L (h02- hL

2) + W(0-L/2)

= [(0.5 m/day)/(2)(1000 m)] (202 m2 – 18 m2 ) +

(1.369 x 10-4 m/day)(-1000 m / 2)

q = – 0.05 m2 /dayThe negative sign indicates that flow is in the opposite direction

From the x direction. Therefore,

q = 0.05 m2 /day into river 1

Example

For discharge into River 2, set x = L = 1000 m:

q = K/2L (h02- hL

2) + W(L-L/2)

= [(0.5 m/day)/(2)(1000 m)] (202 m2 – 18 m2 ) +

(1.369 x 10-4 m/day)(1000 m –(1000 m / 2))

q = 0.087 m2/day into River 2

By setting q = 0 at the divide and solving for xd, the

water divide is located 361.2 m from the edge of

River 1 and is 20.9 m high

Flow Nets - Graphical Flow Tool

Q = KmH / n

n = # head dropsm= # streamtubesK = hyd condH = total head drop

Flow Net in Isotropic Soil

Portion of a flow net is shown below

Stream tube

Curvilinear Squares

Flow Net Theory1. Streamlines and Equip. lines are .2. Streamlines are parallel to no flow

boundaries.3. Grids are curvilinear squares, where

diagonals cross at right angles.4. Each stream tube carries the same

flow.

Seepage Flow under a Dam