– Experiment by Henry Dracy:
– Principles of groundwater flow
•
(steady state flow)
Set up principle of Darcy’s experiment
(Bear & Verruijt’in (1987).
Page 2

 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 L 2 /T 2, L= length, T= time)
Page 3

• Darcy’s law expresses the relation between hydraulic head differences and hydraulic conductivity
• Can be expressed as :
Q
 
KA

 l h q
 
K
 h
 l
• Where Q=discharge (volumetric flow rate) [L 3 /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
V
  q n
 
K n
 h
 l
Q has been called also Darcy velocity which has been also a subject of confusion. It is not flow rate but unit discharge !
Page 4

– 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

–
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).
+90 m
+80 m
Page 6

• 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 in Darcy’s law
– Physical dimension: L/T
•
• Hydraulic conductivity K
– Properties of porous media
– Properties of the fluid ( g specific weight, m viscosity)
K
 k i


 g m


 k i

 g m


• k i is intrinsic permeability”/permeability [dimension L
– “basic parameter” in oil geology/engineering
2 ]
Page 8

• Unsaturated zone
• Pores contain water and gas
Darcy’s law is valid but
– Conductivity is a function of water content
– K unsat
< K sat
– 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 different
Fractured rocks can have porosity of few percent but high K-values
Page 9

• 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

•
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

• 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

Under steady state mass balance of in an infinitesimal volume is

 
 x
 w q x


 y
 w q y


 z
 w



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.

• Mass-flux entering in x-direction
• Mass flux leaving:
 q dydz w x
 w q dydz


 x

 w

•
_________________________________________ difference
In y-direction:


 x

 q dxdydz w x



 y

 w q y
 dydxdz
In z-direction
•Assuming no-net change in storage and summing up the components gives
Which can be fulfilled only if


 z

 q dzdxdy w z


 
 x
 q w x


 y
 q w y


 z
 q dxdydz w z



0

 
 x q x


 y q y


 z q z


0

•Q is given by Darcy’s law




 x


K x

 h x



 y



K y
 h
 y



 z


K z
 h
 z




0
• If homogeneous
•And isotropic (Kx=Ky=Kz)
•We need only Laplace’s equation for solving flow

K
 x x


K y
 y


K
 z z

0


K x

 x
2 h
2

K y

 y
2 h
2

K z

 z
2 h
2


0
 2 h
 x
2

 2 h
 y
2


 z
2
2 h

0

• 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:
Ss

1
V a



V p
 h

 where Va is the unit volume of the aquifer, and Vp is the volume of water in pores.

• 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: the total compressibility. Ss
 where
 is the porosity and c t is
 g
 c t
– 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.
S

Ssb Where b is the thickness of an aquifer

•
• 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:
S

Sy

Ssh
 
Sy
• 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

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) K
 2
Cd
10
•
•
•
• Where d
10 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

In situ –measurements by
Hydraulic 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
Page 20

– 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
Tekijä ja päiväys 21

• 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)

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
• r
0 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

– 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 K
 r
2 ln



L e
2 L e t
0
– Define t
0
1/e when h/h
0
= 0.37=
Le screen legths
R 


R screen radius: (assumption Le/R >8, if <8 another forms of Hvroslev-equation must be used)
Tekijä ja päiväys 27

• 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
(de Marsily, 1986),
-7 m/s
Lugeon-tests are widely used because they provide pragmatic estimates for grouting
For environmental studies not sensitive enough!
http://www.google.fi/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CA cQjRw&url=http%3A%2F%2Fwww.geotechdata.info%2Fgeotest%2FLugeon_test.html&ei=Hd56
VNfxFur4ywPRkYCQAw&bvm=bv.80642063,d.bGQ&psig=AFQjCNH8MvIfSbgaLG4j5HNAK7mm
GwPUSg&ust=1417424796483853
Tekijä ja päiväys 28

– 1. in situestimates 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
Tekijä ja päiväys 29

– 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)
Linear flow
– Interpretation using graphic and curve fitting methods
• Developed particularly for oil and gas testing,
Log-log plot characteristic
– Utilizing analytical models
• Solutions of differential equations
10
2
10
1
1/2-log-log-slope
1
10
2
10
1
Radial flow
Theis-curve
10
0
10
0
10
-1
10
0
10
1
10
2 log tim e
10
3
10
4
10
-1
10
0
10
1
10
2
10
3 log time
10
4
Spheric al flow
10
2
10
1
10
0
2
10
-1
10
0
10
1
10
2
10
3 log tim e
10
4
Specialized plot characteristics tim e log time 1/ tim e
Page 30

• Terzaghi (1928)
• Compaction of soil takes place as a response to a change in effective stress
• Shear strength depends effective stress
• 𝜎 ′ = 𝜎 − 𝛼𝑝 𝑣
• Where 𝜎 is external load or total vertical stress, 𝛼 is a coefficient ranging from 0-1
• Equation is valid for changes of stresses and pressure as well s ’ effective stress p v fluid pressure

•
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 𝜎 ′ = 𝜎 − 𝛼𝑝 𝑣 s ’ grain pressure p v pore water pressure

𝜎 = 𝜎 ′ + 𝛼𝑝 𝑣 𝜎 ′ = 𝜎 − 𝛼𝑝 𝑣
• 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

the stress state to conditions where shear failure takes place
– In soil
– Existing fracture
– Fracturing s
'
 s 
P

• Vajont dam catastrophy
• Huokosveden paineen nousun aiheuttama maanvyöry ja hyökyaalto
• http://www.landslideblog.org/2008/12/vaiont-vajontlandslide-of-1963.html

Some aspects of

• 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

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.

• 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
GTK’s image database
Maanmittauslaitoksen numeerinen korkeusmalli: © National
Survey lupanro 13/MML/09 ja Logica Suomi Oy

• Intensively fractured planar zones of intensive fracturing
• Commonly products of multiple deformations

• Fault gouge
• Ductile deformation
• Alteration
• Weathering
Strong weathering along fractures to clay in test tunnel of
Otaniemi
Ultramylonitic fault

• 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
T f

K f w

 gw
3
12 m
• 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

Engineering geological investigations of fracture zones
Regional interpretation
aerogeophysics
aerial photos
satellite data
Electrical tomography
Detailed interpretation
seismic refraction
electrical tomography
other methods
5 km fracture zones
Aeromagnetic map

• Intensity of fracturing typically reduced after 100 m in depth
• Conductive fracture zones continue to much greater depths