2.
Fundamentals of Groundwater Flow
Course Outline (8hrs)
2.1 Aquifer properties and groundwater flow: effective porosity,
storage coefficient, specific yield, permeability
2.2 Darcy’s experiment and empirical expression of Darcy’s law
and its extension with 3-d generalization, validity of Darcy’s law
2.3 Definition of Hydraulic conductivity (with their typical
values), aquifer transmissivity, aquifer homogeneity,
heterogeneity, isotropic and anisotropy.
2.4 Determination of Hydraulic conductivity
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2.1 Aquifer properties and groundwater flow
• Two propeties of an aquifer material related to its
storage function are its porosity and specific yield
• All the water stored in a water bearing stratum
cannot be drained out by gravity or by pumping,
because a portion of the water is rigidly held in the
voids of the aquifer by molecular and surface tension
forces
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Specific Yield:
• Volume of water expressed as a percentage of the total volume of
the saturated aquifer, that can be drained by gravity is called the
specific yield ‘Sy’ and is depend upon grain size, shape and
distribution of pores and compaction of the formation
• It is defined mathematically by the equation:
Sy = (Vw/V)x100% where, Vw is the volume of water in a unit
volume of earth materials (m3)
V is the unit volume of earth material,
including both voids and solids (m3)
Specific Retention:
• Volume of water retained by molecular and surface tension forces,
against the force of gravity, denoted as ‘Sr’
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Porosity
• The property of a rock possessing pores or voids.
• It is the ratio of the volume of voids of pores in a soil mass to
its total volume.
• Shape, size and packing of grain affect the porosity.
• The larger the pore space or the greater their number, the
higher the porosity and the larger the water holding capacity.
It is defined mathematically by the equation:
n = (Vv/V) x 100%
Where n is the porosity (percentage)
Vv is the volume of void space, m3
V is the unit volume including both voids and solids,m3
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Porosity contd…
• In sediments or sedimentary rocks, the porosity depends on
grain size, the shape of the grains, the degree of sorting and
the degree of cementation.
• In rocks, the porosity depends upon the extent, spacing and
pattern of cracks and fractures.
• The porosity of well-rounded sediments, (which have been
sorted so that they are all about the same size), is
independent of particle size, depending upon the packing.
• Well-rounded coarse-grained sediments (well sorted
sediments means grain size of similar size) usually have higher
porosity than fine-grained sediments, because the grains
don’t fit together well
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Porosity contd…
• Poorly sorted sediments (sediments contains a mixture of
grain sizes) usually have lower porosity because the fine-
grained fragments tend to fill the open space
• Porosity can range from zero to more than 60%. Recently
deposited sediments have higher porosity. Dense crystalline
rock or highly compacted soft rocks such as shale have lower
porosity.
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Storage coefficient (S)
• It is the volume of water released from storage or taken into
storage per unit of aquifer storage area per unit change in
head.
• The storage coefficient is also called storativity.
• It is dimensionless as it is the ratio of the volume of water
released from original unit volume.
• Storage coefficient of an confined aquifer ranges from
0.00005 to 0.005 and for water table aquifers 0.05 to 0.30
• The storage coefficient for unconfined aquifer corresponds to
its specific yield
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STORAGE COEFFICIENT (STORATIVITY)
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Transmissivity (T) and Permeability(K)
Transmissivity (T) or coefficient of transmissibility
• defined as the rate at which water of prevailing kinematic viscosity
is transmitted through a unit width of aquifer under a unit hydraulic
gradient.
• It is the product of field permeability (k) and saturated thickness of
aquifer (b)
i.e. T = Kb (m/day)x(m) = m2/day where b is the saturated
thickness of the aquifer.
Permeability (K)
• It is a quantitative measure of the ease of water movement through
aquifer materials. For example, sand is more permeable than clay
because the pore spaces between sand grains are larger than pore
spaces between clay particles.
• Permeability can be defined as the flow per unit cross sectional
area of the formation when subjected to a unit hydraulic head per
unit length of flow (per unit hydraulic gradient) and has the
dimension of velocity.
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2.2 Darcy’s Law
• Velocity of groundwater flow which is entirely laminar is given
by Darcy’s law which states that ” the velocity of flow in a
porous medium is proportional to the hydraulic gradient”.
V = K i where K = coefficient of permeability and
i = hydraulic gradient = h/L
(when head h is lost in length L)
• Darcy discovered that the discharge Q of water through a
column of sand is proportional to the cross sectional area A of
the sand column, and to the difference in piezometric head
between the ends of the column, h1– h2, and inversely
proportional to the length of the column L. That is:
Q = KA (h1-h2)/L = KiA
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Groundwater flows slowly through the voids between grains or the
cracks in solid rock. Much of our knowledge depends on field and
laboratory observations. Here, for example, is an experiment to
measure head loss in an aquifer.
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Darcy’s Experiment 1. Velocities small, V ~ 0, so:
Piezometers before and
after sand. Pipe is full, so
flow rate is constant
2. Head difference doesn’t change with inclination of the sand filter
3. Again, Darcy related reduced flow rate to head loss and length of column
through a constant of proportionality K,
V = Q/A = -K dh / dL
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Groundwater Flow
• Ground water flows from levels of higher energy to levels of
lower energy. Energy is the result of elevation and pressure,
the velocity head is neglected since the flow is laminar.
• The term hydraulic head, which is the sum of elevation and
water pressure divided by the weight density of water, is used
to describe potential energy in ground-water flow systems
• The quantity of ground-water discharge (flux) to and from
surface-water bodies can be determined for a known cross
section of aquifer by multiplying the hydraulic gradient, (which
is determined from the hydraulic-head measurements in wells
and piezometers), by the permeability of the aquifer materials.
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Validity of Darcy’s Law
• Darcy’s law applies to laminar flow in saturated granular
media, under steady-state flow conditions, considering the
fluid homogenous, isotherm and incompressible, and
neglecting the kinetic energy. (Laminar flow means that the
water moves in a sheet like fashion)
• Darcy’s law is valid for Reynold numbers less than or in
between 1 and 10
• May not be valid for very slow water flow through dense clay
and very steep hydraulic gradient (rock aquifers,
unconsolidated aquifers.)
• May not be applicable in the immediate vicinity of wells which
have steep hydraulic gradient.
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Darcy’s Law contd…
• The most restrictive hypothesis of Darcy's law is the one that
considers the flow laminar, and the fluid movement as dominated
by viscous forces. This occurs when the fluids are moving slowly,
and the water molecules move along parallel streamlines. When
the velocity of flow increases (for instance in the vicinity of a
pumping well), the water particles move chaotically and the
streamlines are no longer parallel. The flow is turbulent, and the
inertial forces are more influential than the viscous forces (Fetter,
2001).
• The ratio between the inertial forces and the viscous forces driving
the flow is computed by the Reynolds number, which is used as a
criterion to distinguish between the laminar flow, the turbulent
flow and the transition zone.
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Turbulence and Reynolds Number
• The path a water molecule is called a streamline. In
laminar flow, streamlines do not cross, and the viscous
forces due to hydrogen bonds are important.
• In turbulent flow acceleration and large scale motion
away from a smooth path is important (this is the familiar
inertial force F = ma) and streamlines cross.
• We could take the ratio of inertial to viscous forces.
When this number is “large,” inertial forces are more
important, and flows are turbulent.
• This ratio is known as the Reynolds number Re:
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• Where: Re is dimensionless
• ρ the fluid density (M/L3 ; kg/m3)
• q specific discharge (L/T ; m/s), Darcy velocity
• d usually, the mean grain diameter or the mean pore
dimension (L; m)
• μ dynamic viscosity (M/T.L ; kg/s.m)
• ν kinetic viscosity (L2/T ; m2/s)
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Viscosity
• Viscosity is a fluid’s resistance to flow.
• Dynamic viscosity m, units Pa·s = N·s/m2, or kg/(m·s) is
determined experimentally. If a fluid with a viscosity of
one Pa·s is placed between two plates, and one plate is
pushed sideways with a shear stress of one Pascal, it
moves a distance equal to the thickness of the layer
between the plates in one second.
• Kinematic viscosity n, is the dynamic viscosity divided by
the density. The SI unit of ν is m2/s.
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Reynolds: Inertial/Viscous forces
Recall the ratio of Kinetic/Potential Energy
(KE/PE) is the Froude Number Fr
Fr = V / sqrt( g L)
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Non Darcy Flow in Sub-surface
• Non Darcy behaviour is important for describing fluid flow in porous
media in situations where high velocity occurs. Two types of
criterion, the Reynolds number and Forchheimer number have
been used in the past for identifying the beginning of non Darcy
flow.
• Experiments show that the critical Reynolds number for non-Darcy
flow to become significant is in the range of 40-80
• The inception of turbulent flow can be located at Reynolds number
greater than 60.
• Between the laminar and the turbulent flow, there is a transition
zone where the flow is laminar but non-linear.
• In a general way, Darcy's law can be written:
➢ When the index m = 1, the flow law is linear, being valid for the
laminar flow; it is the case of Darcy's law: q = -K.i.
➢ For m ≠ 1 and Re > 100, the law is non-linear and the flow is
turbulent
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The actual velocity (Va) at which the water is moving through an
aquifer i.e the velocity at which a tracer would move through a
permeable medium is given by
where v = Darcy velocity and
n = porosity
Type of soil or rock Porosity n (%)
Unconsolidated deposits
It is important to notice that, as Gravel 20 – 35
Sand 25 – 50
the particle sizes of a soil are Silt 35 – 50
smaller, the porosity and the void Clay 40 – 70
Rocks
ratio increase. Amazing or not, Fractured basalt 5 – 50
gravels for example have porosity Karst limestone 5 – 50
5 – 30
values much smaller than those Sandstone
0 – 20
of clays, in spite of the size of the Limestone, dolomite
0 – 10
pores. Shale
Fracture cristalline rock 0 – 10
Dense cristalline rock 0–5
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2.3 Hydraulic Conductivity
• A medium has a unit hydraulic conductivity if it will transmit in
unit time a unit volume of groundwater at the prevailing
kinematic viscosity through a cross section of unit area
measured at right angles to the direction of flow, under a unit
hydraulic gradient.
• The unit is K = v/(dh/dl) = m/day
• The hydraulic conductivity of a soil or rock depends on a
variety of physical factors including porosity, particle size and
distribution, shape of particles and arrangement of particles.
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Ranges of Hydraulic Conductivities
Unconsolidated Hydraulic conductivity Hydraulic conductivity
Rocks
deposits (m/s) (m/s)
Dense clay 10-13 ……10-8 Dense sandstone 10-9 ……10-7
Weathered clay 10-8 ……10-6 Karstic sandstone 10-7 ……10-5
Silt 10-7 ……10-5 Dense limestone 10-9……10-7
Alluvial deposits 10-5 ……10-3 Karstic limestone 10-5 ……10-3
Fine sand 10-5 ……10-4 Dolomite 10-10 ……10-8
Medium sand 5x10-4 ……5x10-3 Dense crystalline rocks 10-13 ……10-12
Fractured crystalline
Coarse sand 10-4 ……10-3 10-10 ……10-6
rocks
Fine gravel 10-3 ……5x10-1 Dense basalt 10-13 ……10-10
Medium gravel 5x10-2 ……10-1 Fractured basalt 10-7 ……10-4
Coarse gravel 10-2 ……5x10-1 Claystone 10-13 ……10-9
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Homogeneity and isotropy
• Homogeneous aquifer: K is same all over
• Isotropic aquifer: K is the same in all directions
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Average Hydraulic Conductivities
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2.4 Determination of Hydraulic Conductivity
• Calculation from formulas
• Laboratory methods : Permeameter in which flow is
maintained through a small sample of material while
measurement of flow rate and head loss are made
(constant head and falling head types of
permeameter)
• Tracer tests
• Auger hole tests
• Pumping tests of wells
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