Hydrogeologic Principles

1

Jae K. (Jim) Park, ProfessorDept. of Civil and Environmental Engineering

University of Wisconsin-Madison

Darcy’s Law Empirical law developed in 1856 for flow through

porous media for saturated and unsaturated flow The flow of a fluid in a porous medium is equal to the

product of a constant multiplied by the gradient of the force driving the fluid through the system divided by the porosity of the medium.

where K = hydraulic conductivity (L/T), anddh/dl = fluid gradient.

Q = -KiA

where Q = flow rate (L3/T), i = fluid gradient (dh/dl) (L/L), and A = cross-sectional area (L2).

q Kdh

dl v

q

n

2

Darcy’s Law - continued

k = specific or intrinsic permeability (L2); = mass density of the fluid (M/L3); = dynamic viscosity (M/L/T); andg = acceleration due to gravity (L/T2).Typical intrinsic permeability in a landfill

kv = 10-11 to 10-12 m2; kh = 10-10 m2

At very high flow rates, the flow regime changes from laminar to turbulent conditions, and Darcy’s law becomes invalid.

The upper limit is defined by Reynold’s number:

where q = specific discharge and d = length (mean pore dimension, or mean grain diameter).

K = d2kg

Re = = Inertial forces qdViscous forces

3

Fluid Conductivity Laboratory measurement method

q (rate of flow) = K h/l A

Distilled water is generally used. Due to a decrease in permeability with distilled water when compared with those where pore fluid is used as the permeant, 0.1 N CaSO4 solution should be used to properly simulate leachate or other waste liquids.

l

Soil

Drainage hInfluent liquid

4

Field measurement methods¨ Slug tests: instantaneous displacement of a known

volume of water and recording the response with respect to time at the tested well - rising head (volume extracted) and falling head (volume added)

¨ Injection tests and pumping tests: addition or removal of water, respectively, for an extended period of time (1 hr to several days) and recording of response (water level measurement) with respect to time at the tested well and at monitoring wells

Alternative methodsK (m/sec) = 0.01 d10

2 - Hazen methodwhere d10 = effective soil particle size (mm) (10% of the

particles in a sample are of smaller size).

Fluid Conductivity - continued

5

Two-Stage Borehole PermeameterThe rate of flow of water into soil through the bottom of a sealed, cased borehole is measured in each of two stages, normally with a standpipe in the falling head procedure. The standpipe can be refilled as necessary. In Stage 1, the bottom of the borehole is flush with the bottom of the casing for maximum effect of Kv. The test is continued until the flow rate becomes quasi-steady. For Stage 2, the borehole is extended below the bottom of the casing for maximum effect of Kh.This stage of the test is also continued until the flow rate becomes quasi-steady. The direct results of the test are apparent hydraulic conductivities K1 and K2. The actual hydraulic conductivities Kv and Kh can be calculated from these values.

http://homepages.cae.wisc.edu/~wang1/Fieldlist.html 6

Hydraulic Gradient Determined through measurements of hydraulic head at

different locations in the subsurface Saturated zone: measured using single and nested

standpipe piezometers or pressure transducers Unsaturated zone: measured with hydraulic tensiometers

Elevation differenceof the water table, h

Separation distance, l

Piezometer Ground surface

i = hl

7

Piezometer For measuring pressure inside a vessel

or pipe in which liquid is there, a tube may be attached to the walls of the container (or pipe) in which the liquid resides so liquid can rise in the tube. By determining the height to which liquid rises and using the relation P1 = ρgh, gauge pressure of the liquid can be determined. Such a device is known as piezometer. To avoid capillary effects, a piezometer's tube should be about 1/2 inch or greater.

It is important that the opening of the device to be tangential to any fluid motion, otherwise an erroneous reading will result.

8

Piezometer - continued

Measure the pressure head of the liquid

9

Installation of Minipiezometer and Stilling Well

10

1. Driver mechanism consisting of solid steel driver rod (C) and steel outer casing with flange (A) hammered into sediment to suitable depth using a cap fitting (B)

2. Driver rod (C) removed with the steel outer casing retained 3. Minipiezometer inserted into the outer steel casing 4. Outer steel casing removed with minipiezometer held in position and sediment

was manually tamped around the minipiezometer. Bentonite clay can also be used to seal the annulus between minipiezometer and hole above the inlet.

5. Stilling well fitted and secured using a star picket

Typical Piezometer Arrangements

11

Unconfined aquifer installation

Confined aquifer installation

Nested installation

Capillarity Between two fluids in contact with each other, or a fluid in contact

with a solid, there is a free interfacial energy created by the difference between the forces that attract the molecules toward the interior of each phase and those that attract them to the contact surface. As these forces increase, fluids such as pore water are retained in the porous medium above the elevation of the water table.

h1

d1 d2

h2

d1 > d2

12

h1 < h2

Total Porosity Space that is occupied by fluids

n = =Volume of the voids Vv

Total unit volume VT

13

Clay 0.45~0.55Sand 0.35~0.4Gravel 0.3~0.4Sand and gravel 0.2~0.35Peat 0.85~0.9Municipal waste 0.3~0.4

Effective Porosity

14

Effective Porosity -continued

There exist nonconnected, dead-end pores through which advective migration cannot occur.

Advective flow velocity (effective)

That proportion of the total pore space in a rock which is capable of releasing its contained water. Clay, for example, may have a total porosity of 50% or more, but little if any of the water contained in these pores may be released, because of the retentive forces (e.g. surface tension) that hold it within the rock.

In unsaturated flow through low-permeability soil (clay with porosity of 0.4~0.7), ne 0 15

ea n

Kiv

Moisture Content In the unsaturated zone, the pore spaces are partially

filled with water and partially filled with air. The volumetric moisture content,

For saturate flow, = n and for unsaturated flow, < n. Degree of saturation in the unsaturated zone , then,

hydraulic conductivity Hydraulic conductivity (municipal refuse)

1 10-2 to 1 10-5 cm/sec (typical 1 10-3 cm/sec)

= =Volume of water Vw

Total volume of pore space Vv

16

Hydraulic Conductivity

Soil type K, cm/sec Capillary head, cm

17

Gravel GPGW

10-1

10-2-6

Sand SPSW

5ⅹ10-4

10-3-

60

Clay CLCH

3ⅹ10-4

10-9180

200~400Refuse as placedShredded refuse

10-3

10-2~10-4--

Silt 10-5

10-7180

-

Chemical Migration - continued Retardation: due to interaction of chemicals with the

solid phase of the porous medium

where d = bulk density of the porous medium (g/m3); and Kd = Freundlich sorption coefficient.

Kd varies from zero to 103 L/kg; > 10 L/kg - immobile

Ex. q = 1.6 10-8 m/sec; d = 1.6 g/m3; n = 0.4; KdSr+2 = 10

L/kg. R and vc?

R = 1 + 1.6/0.4 10 = 41; v = q/n = 1.6 10-8/0.4 = 4.0 10-8 m/sec; vc

= 4.0 10-8/41 = 9.8 10-10 m/sec

= = R = 1 + Kdv Average velocity of groundwater d

vc Average velocity of a chemical n

18

Chemical Migration Advection: due to the bulk motion of the GW

Dispersion: due to spreading out of the solute, molecular diffusion and mechanical mixing

Dl = l v + D* D* = D’

where l = dispersivity (L);D* = apparent or effective diffusion coefficient (L2/T); = tortuosity factor (0.01 to 0.5); andD’ = diffusion coefficient in free solution (L2/T).

19

dl

dh

n

K

n

qv

Mass Transport Equations

tD2

tvzerfc

D

zvexp

tD2

tvzerfc

2

1

C

C*L

*

*L

*

*L

*

0

z

C

R

v

z

C

R

D

t

C L

2

2

tD2

tvzerfc

2

1

C

C*L

*

0

**L

L vR

v and D

R

D

Initial conditions: C(0,t) = C0 at t ≥ 0C(z,0) = 0 at z ≥ 0

Boundary condition: C(,t) = 0 at t ≥ 0

Governing Equation

Solution

Solution when v* 0

20