Hydrogeology

Groundwater flow and Darcy’s law– Experiment by Henry Dracy:– Principles of groundwater flow

• (steady state flow)

Page 2

Set up principle of Darcy’s experiment (Bear &Verruijt’in (1987).

Conclusions of Darcy Indicated that steady state flow is a function of hydraulic head and

coefficient of hydraulic conductivity (characteristic of the soil)• Flow is maintained by differences in hydraulic head, not (only) pressure

differences.• Differences of hydraulic head can be used to assess the tendency water

to flow from one point to the other in soil

Potential energy of water can be described in terms of hydraulic head or fluid potential

Hydraulic head : potential energy/unit weight (physical dimension L length)

Potential = potential energy/unit mass (dimension L2/T2, L= length, T= time)

Page 3

Darcy’s law•  Darcy’s law expresses the relation between hydraulic head differences and

hydraulic conductivity•  Can be expressed as :

• Where Q=discharge (volumetric flow rate) [L3/T], h hydraulic head, l distance, A cross-sectional area

• n= porosity• q=Q/A= “specific discharge”= flow per unit rate• Flow is therefore driven by the negative gradient of hydraulic head• Average linear flow rate V

Page 4

l

hKq

l

hKAQ

l

h

n

KnqV

Q has been called also Darcy velocity which has been also a subject of confusion. It is not flow rate but unit discharge !

Darcy’s law

–  can be used to produce rough estimates and working hypotheses– In modelling:– Differential equations are formulated by assuming that equation

determines flow in each point in an aquifer (infinitesimally small unit volume)

– How long it takes when maximum impacts of groundwater pollution emerge around tailing ponds without sealing bottom structures.

Page 5

Example

–  An tailing pond has radius of 100 m, tailing are saturated to the surface (90 m asl). Gw-level is at level 80 m asl (perennial level of discharge in drainage ditches). Subsurface is till with K=10-7 m/s. Calculate average linear velocity of groundwater underneath the pond if porosity of till is 15 %

– How long would it take for contaminated water leaked into the till at the center of the pond to leave the area of the tailing pond area (horizontal travel time).

Page 6

+90 m

+80 m

• Hydraulic head is a sum of elevation head and pressure head

– One should not say “groundwater is moving from high pressure to low pressure”

– See the examples.• Ingebritsen, Sanford & Neuzil,

1999. Groundwater in Geologic Processes, 2nd ed.

Hydraulic conductivity of soil/ porous media• Hydraulic conductivity in Darcy’s law

– Physical dimension: L/T

• Hydraulic conductivity K– Properties of porous media– Properties of the fluid (g specific weight, m viscosity)

•   • ki is intrinsic permeability”/permeability [dimension L2]

– “basic parameter” in oil geology/engineering

Page 8

mg

m

ii kg

kK

Page 9

Hydraulic conductivity in subsurface• Unsaturated zone• Pores contain water and gas

Darcy’s law is valid but– Conductivity is a function of water

content– Kunsat< Ksat– Water pressure is influenced by

capillary suction (pressure of soil moisture is less than air pressure)

• Hydraulic conductivity can be can be applied for fractured rock mass (although should be done with care)– Equivalent porous media– Flow follows Darcy’s law

• K is not “strongly dependent “ on porosity

Note in next slices range of K values in nature!Compare to porosity range! Coarse gravel or sand can have similar porosity

but K-values are quite differentFractured rocks can have porosity of few

percent but high K-values

Hydraulic conductivity values• Freeze and Cherry (1979)• Note the range of K-values

from 10-11 to 1 m/s • Eurocode requirements:

– K must be measured in situ

Page 10

Porosity

• Effective porosity:

Interconnected pores

0-35%vol

Total porosity:

In unconsolidated clays up to 70%

More typically see the table

In clays extremely high porosity but extremely low K

Page 11

Soil moisture

• Unsaturate zone– Seasonal variation of moisture

content– In Nordic conditions recharge

takes mostly place after snow melt

• (in some humid areas also recharge in autum)

– Heavy raining rarely induces recharge by infiltration

• horisontal: water content (relative) vertical: z

Governing equations of groundwater flow

x

qy

qz

q dxdydzw x w y w z 0

In practice, the mathematical formulation the governing equation proceeds by assuming that

the volume of water leaving or entering a macroscopic elementary volume follows

Darcy's law.

Under steady state mass balance of in an infinitesimal volume is

w xq dydz

w x w xq dydzx

q dxdydz

x

q dxdydzw x

y

q dydxdzw y

z

q dzdxdyw z

x

qy

qz

q dxdydzw x w y w z 0

•Mass-flux entering in x-direction

• Mass flux leaving:

_________________________________________•difference

In y-direction:

In z-direction

•Assuming no-net change in storage and summing up the components gives

Which can be fulfilled only if

x

qyq

zqx y z 0

x

Kh

x yK

h

y zK

h

zx y z 0

K

x

K

y

K

zx y z 0

Kh

xK

h

yK

h

zx y z

2

2

2

2

2

2 0

2

2

2

2

2

2 0h

x

h

y

h

z

•Q is given by Darcy’s law

•If homogeneous

•And isotropic (Kx=Ky=Kz)

•We need only Laplace’s equation for solving flow

Unsteady flow

• Changes in water storage• During unsteady flow, changes of the volume of water in

a macroscopic elementary volume of an aquifer can be expressed in terms of the specific storage coefficient Ss.

• The volume of fluid that is released or retained by a unit volume of an aquifer per a unit change in hydraulic head is:

• where Va is the unit volume of the aquifer, and Vp is the volume of water in pores.

h

V

VSs

a p1

Storage and compressibility• Specific storage is actually a lumped parameter resulting from the analogy

between heat and groundwater flow (Theis, 1935). • In well testing of oil reservoirs and sedimentary rock aquifers, it is

commonly represented simply as:• where r is the porosity and ct

is the total compressibility. – However, different formulas representing specific storage as a function of the

compressibility of rock and water have been developed based on the vertical compaction of an aquifer (e.g. in Freeze and Cherry, 1979; Bear and Verruijt, 1994, ).

• Storativity is a dimensionless parameter that determines the volume of water that is released or retained by aquifer per unit surface area per unit change of head perpendicular to that surface. As transmissivity, storativity was initially developed for the analysis of confined porous aquifers.

tcgSs

SsbS Where b is the thickness of an aquifer

Specific yield

• Storativity in an unconfined aquifer is determined by specific yield– the volume of water which a unit volume of a saturated aquifer,

will yield by gravity; – Approximately same as effective porosity– Compression of/changes in porous media or compression of

water are substantially smaller:

• • where h’ is the thickness of saturated aquifer

– In sand and gravel Sy a magnitude to decade higher than Ssh’

• S typically 0,02-0.3

S Sy Ssh Sy

Measurement of hydraulic conductivity

1) Laboratory test• Representativity of the sample is a problem: small volume and

disturbance during sampling

2) Empirically based on grain-size distribution

Hazen method ( K in cm/s)

• Where d10 is defined from cumulative grain-size distribution (from sieve results)

• C is a cofficient • fine sands, sandy loams C=40-80

medium coarse, sorted sands and coarse poorly sorted sands, C= 80-120

• sorted coarse sand and gravels C= 120-150

3) In situ –measurements i.e. hydraulic tests

Page 19

210CdK

Hydraulic tests

Page 20

In situ –measurements byHydraulic tests

Several methods: choice depends on investigation objectives and

hydrogeology (soil, rock fracturing etc.)

Most important

• Test pumping by constant rate or pressure

• Spinner-tests• Slug- ja bail-tests

For poorly conductive soils and rocks

21Tekijä ja päiväys

Slug-tests

– Slug- ja bail-tests – An instantaneous change in hydraulic head and

measurement of recovery• Either increase of head (slug) or reduction (bail)

– Nowadays slug-test is used as a general term • Interpretation similar

– Several interpretation methods• Cooper-Papanpodoulus confined aquifer, fully penetrative well• Hvroslev-method unconfined aquifer, partial penetration• Bouwer-Rice

Hydraulic conductivities

• Zero pressure at the underground mine (at depth of about 600 m)

• Under extremely high gradient flow can be significant

• We must be able to measure extremely low hydraulic conductivites in situ!!!

Page 22 Freeze and Cherry (1979)

Slug tests

• Sudden change in head+ monitoring of the change (recovery)

• Recovery period can be slow, overall testing can be very slow

• + in fractured rocks could be used also an interference test (USA) Hinkkanen, 2011, Posiva Working report 2011-40

• Difference flow method

• Flow rates that are slower than the growth of your hair!

Posiva working report WR2006-47

Thiem equation• h is hydraulic head• hs is hydraulic head at the

radius of influence• r0 radius at the well bore

• Ts transmissivity of the test section (typically 0,5- 2 m)

• Qs0 and Qs1 flows out of the borehole in natural and pumped conditions

• h0 an h1 are the measured heads

27Tekijä ja päiväys

Example: Slug test

– Hvroslev-method– unconfined system,

partial penetration (well or piezometer penetrates only a part of a groundwater formation)

– Semilogarithmic plot h/h0 (log) vs. t

– Straigline fitting– Define t0 when h/h0 = 0.37=

1/e

0

2

2

ln

tL

RLr

Ke

e

Le screen legths

R screen radius: (assumption Le/R >8, if <8 another forms of Hvroslev-equation must be used)

28Tekijä ja päiväys

Water injection tests

• Empirical injection test a.k.a. Lugeon-test,

• Pressure is maintained in a bore hole and the injection rate is measured– Repeated using several

pressure levels during the test • As a result an estimate of

hydraulic conductivity expressed in therms of a Lugeon unit (litre/minute*length of testes section* 100 kPa),

• Lugeon is about 1-2 x 10-7 m/s (de Marsily, 1986),

http://www.google.fi/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRw&url=http%3A

%2F%2Fwww.geotechdata.info%2Fgeotest%2FLugeon_test.html&ei=Hd56VNfxFur4ywPRkYCQAw&bvm=bv.80642063,d.bGQ&psig=AFQj

CNH8MvIfSbgaLG4j5HNAK7mmGwPUSg&ust=1417424796483853

Lugeon-tests are widely used because they provide pragmatic estimates for

groutingFor environmental studies not

sensitive enough!

29Tekijä ja päiväys

Objectives of test pumping

– 1. in situ- estimates of (large scale) hydraulic properties of soils and rock

– 2. assessment of long term impacts (over 1 month)– In the context of mining: planning of dewatering– Assuring water quality and yield for (drinking water/fresh water

supply)– 3. Calibration/verification of numerical flow models

– Measurements during test pumping• Mostly by pressure transducers and automatic systems

– Q (constant), drawdown vs time– Or Q vs time, in h constant– (derivative for log time)

– Interpretation using graphic and curve fitting methods

• Developed particularly for oil and gas testing,

– Utilizing analytical models• Solutions of differential equations

Page 30

Lo g -lo g p lo t c ha ra c te ristic

Line a r flo w Ra d ia l flo w Sp he ric a l flo w

10

10

10

10

100 10 1 10 2 103 104

-1

0

1

2

1

lo g tim e

log

dra

wd

ow

n 1/2-lo g -lo g -slo p e

lo g tim e

The is-c urve

lo g tim e

2

Sp e c ia lize d p lo t c ha ra c te ristic s

tim e

dra

wd

ow

lo g tim e

dra

wd

ow

dra

wd

ow

tim e1/

10

10

10

10

100 101 102 10 3 10 4

-1

0

1

2

10

10

10

10

100 10 1 10 2 103 10 4

-1

0

1

2

log

dra

wd

ow

n

log

dra

wd

ow

n

Groundwater, effective stress and total stress• Terzaghi (1928) • Compaction of soil takes

place as a response to a change in effective stress

• Shear strength depends effective stress

• Where is external load or total vertical stress, is a coefficient ranging from 0-1

• Equation is valid for changes of stresses and pressure as well

s’ effective stress

pv fluid pressure

Apparent cohesion• In unsaturated zone air and water• Pore water pressure less than air pressure• Capillary forces squeeze grains to each

others• Effective stress increases compared to

saturated situation• Soil is apparently more stable (excavations

will not collide as easily etc.)• Apparent cohesion disappears when

saturation is reached

s’ grain pressure

pv pore water pressure

𝜎 ′=𝜎−𝛼𝑝𝑣

Effective stress and hydraulic failure

• If effective stress is 0, no friction between grains (at least at that point)

• sediment liquefies/looses its consistency– Piping under tailing dams– Erosion– Slope failures– Debris flow

𝜎=𝜎 ′+𝛼𝑝𝑣

𝜎 ′=𝜎−𝛼𝑝𝑣

Pressure change• Increasing water pressure may change

the stress state to conditions where shear failure takes place – In soil– Existing fracture– Fracturing

P '

Dewatering of mines can induce compaction similar to underground construction and civil engineering projects

the extent can be huge!

Example of changes if groundwater pressure to rock stability• Vajont dam catastrophy • Huokosveden paineen nousun aiheuttama maanvyöry ja

hyökyaalto

• http://www.landslideblog.org/2008/12/vaiont-vajont-landslide-of-1963.html

HARD ROCK HYDROGEOLOGYSome aspects of

Characteristics of hard rocks• Hydraulic conductivity of large open fractures and

fracture zones can be high (>1e-5 m/s) – same as in sand or gravel

• Flow takes place in interconnected open fractures– Comprise only a small fraction of the fracture network!

• Tracer test, NWM studies, models, theoretical studies of percolation theory

• Fracture porosity is typically 1-2 %– Evidence NWM-studies, models, applied geophysics

• Compressibility of rocks extremely small• Storage-properties extremely low!• Impacts of dewatering or pumping spread fast and

extensively

• In sedimentary rocks faults commonly act as seams– Gouge and differential stress

• In hard rocks the situation is opposite!– Multiple tectonic events– Brittle deformation but less

gouge– Fracture zones that may be

highly conductive

Fracture mechanisms and Fracture zones

Caine and Forster, 1999, Fault zone architecture and Fluid Flow.

• In Finland, bedrock comprises almost entirely of Precambrian igneous and metamorphic rocks.

• The surface topography of bedrock is strongly controlled by rock fracturing.

• Long and narrow valleys commonly trace faults, shear zones and lithological boundaries along which bedrock is commonly more intensively fractured than in the surrounding outcropping areas.

Rock fracturing• Precambrian bedrock

– Caledonian orogeny and youger tectonic phenomena,ice ages

• Bedrock has mostly low hydraulic conductivity , relatively solid and unbroken

• Mosaic-like block structure • Poorly fractured bedrock blocks split

by fracture zones– Commonly follown old zones of

ductile deformation and bedrock contacts

– Can be distiguished as linear structures on topographic and aerogeophysical maps

• lineament

• Directions of lineaments and fractures from outcrops are statistically similar

Maanmittauslaitoksen numeerinen korkeusmalli: © National Survey lupanro 13/MML/09 ja Logica Suomi Oy

GTK’s image database

Fracture zones

• Intensively fractured planar zones of intensive fracturing

• Commonly products of multiple deformations

Processes reducing conductivity

• Fault gouge• Ductile deformation• Alteration • Weathering

Ultramylonitic fault

Strong weathering along fractures to clay

in test tunnel of Otaniemi

Conceptualization of fracture flow

• Fractured rocks can be considered as “porousmedia” only over very large dimensions– Block sizes exceeding 500 m in dimensions

• Single fractures can be considered as conduits with transmissivity/conductivity determined by the (average) aperture

• Between these extreme cases flow in fractured rock volumes must be considered as discrete networks comprising fracture conduits and matrix blocks (possibly with K and storage).

Lähde: Posiva

m12

3gwwT ff K

Engineering geological investigations of fracture zones

Aeromagnetic map

fracture zones

Electrical tomography

5 km

Regional interpretation

- aerogeophysics- aerial photos- satellite data

Detailed interpretation - seismic refraction

- electrical tomography

- other methods

Fracturing

• Intensity of fracturing typically reduced after 100 m in depth

• Conductive fracture zones continue to much greater depths