ESM 203: Groundwater

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ESM 203 Groundwater

• Jeff Dozier Tom DunneFall 2007

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From lecture on Planetary HydrologyContinental
hydrology (largely subsurface)

• Storage and transmission of water below ground
  generates a resistance to evapotranspiration,
  allowing water to escape from the radiation load
  at Earths surface and remain liquid and
  available as a water supply in groundwater and
  streams

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Ground water storage and discharge Conceptual
model 1
Rnet
E

• Ddepth of root zone
• ?volume fraction of water
• V(t) volume of groundwater storage resulting
  from balance between drainage from soil and
  drainage to rivers Q(t)

Advection of sensible (H) heat
P
Quickflow R
Soilwater SM(t)D?(t)
Recharge when SM(t)gtSMmax
Ground water V(t)
Delayed flow Q(t)
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Ground water storage and discharge Conceptual
model 1
 
SM(t) transient soil-moisturecontent
(vol/area) SM(t) ?(t)D, where D root-zone
depth
? P E ? R?
Soil Storage SM(t)
Gravitational drainage occurs when ? gt ?fc, a
critical value called field capacity
    Ground water V(t)
V(t) volume of groundwater storage resulting
from balance between drainage from soil drainage
to rivers
Outflow to rivers
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Groundwater storage and discharge

• Groundwater discharge behaves approximately as a
  linear reservoir that is, the volume of
  outflow in some unit of time (?V / ?t) is some
  fixed proportion of the volume stored (V).

E.g the rate of outflow in m3/day is 1 per day
of the volume that is stored. So k 0.01 per
day. Since ?V is a decrease, we use a negative
sign in front of it
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Groundwater storage and discharge
Linear storage-outflow relationship
In differential form, taking limits as ?t ?0
Reorganizing
Integrating both sides
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Groundwater storage and discharge (cont.)
We know a boundary condition when t 0, V V0.
Therefore
ln V0 C
Substitute this result back into the equation
above
Taking antilogs and moving V0
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Exponential decline in volume stored
V0
V
t 0
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Implications

• If the groundwater is recharged by drainage from
  the soil during a wet season, a snowmelt season,
  or a rainstorm (i.e. if its volume is re-set to
  V0), the volume in storage will decline
  exponentially through time
• Since the volume of groundwater storage is
  reflected in the height of the water table, then
  the water table behaves in the same way

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Also

• Since river discharge in the absence of quickflow
  originates from groundwater drainage,
• The flow of streams will also decline
  exponentially through time after some sharp rise
  due to a pulse of recharge

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Conceptual model 1
Rnet
E

• Ddepth of root zone
• ?volume fraction of water
• V(t) volume of groundwater storage resulting
  from balance between drainage from soil and
  drainage to rivers Q(t)

Advection of sensible (H) heat
P
Quickflow R
Soilwater SM(t)D?(t)
Recharge when SM(t)gtSMmax
Ground water V(t)
Delayed flow Q(t)
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Physical model of groundwater

• If water drains down to some impermeable boundary
  by gravity, it will accumulate above that
  boundary and fill all voids (pores, fractures) in
  the rocks up to some height

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Groundwater Conceptual model 2
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Physical model of groundwater

• If water drains down to some impermeable boundary
  by gravity, it will accumulate above that
  boundary and fill all voids (pores, fractures) in
  the rocks up to some height
• Above that height, water will not fill all of the
  voids there will be some air-filled spaces
• The air-water interfaces in small openings (like
  capillary tubes) will result in concave air-water
  interfaces that develop negative pressure and
  hold up water within the voids against
  gravitational drainage

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Groundwater Conceptual model 2
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Physical model of groundwater

• If water drains down to some impermeable boundary
  by gravity, it will accumulate above that
  boundary and fill all voids (pores, fractures) in
  the rocks up to some height
• Above that height, water will not fill all of the
  voids there will be some air-filled spaces
• The air-water interfaces in small openings (like
  capillary tubes) will result in concave air-water
  interfaces that develop negative pressure and
  hold up water within the voids against
  gravitational drainage
• If we dig an unlined hole (a well) into the
  saturated soil or rock, water will flow into the
  hole and stand to some constant level, which is
  called the water table
• Water table the height at which the pressure in
  the fluid is at atmospheric pressure. If
  pressures are expressed relative to atmospheric
  pressure, then above the water table pressures
  are negative below they exceed atmospheric
  pressure

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Groundwater Conceptual model 2
Impermeable rock
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Some terms

• Aquifer - geologic formation that stores a large
  volume of groundwater and allows it to drain to
  streams, springs, or wells at rates that humans
  consider useful
• Aquiclude geologic formation that does not
  transmit water at rates useful to humans
• Confined aquifer bounded on its upper surface
  by an aquiclude, which precludes direct recharge
  from the overlying land surface, but only from
  some remote upstream zone of the surface
• Unconfined aquifer upper boundary of saturated
  zone is a water table atmospheric pressure and
  connected directly to the atmosphere

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Confined and unconfined aquifers
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Groundwater flows down gradients of potential
energy in the water

• A unit (mass, volume, or weight) of water has
  potential energy by virtue of
• its elevation above some datum
• Its pressure
• Energy per unit weight has dimensions of length,
  so we convert each energy component to this form
  and obtain a convenient measure of potential
  energy called head

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Measurement of head

• Insert solid-wall pipe (not a well), called a
  piezometer, into groundwater to measure pressure
  and head
• Measure elevation (relative to some datum, e.g.
  sea level) to which water rises in the pipe

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Darcys Law for flow through a porous medium

• Negative sign because flow direction is down the
  gradient (i.e. in opposite direction to it)
• Yet another example of a diffusion process

(m)
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Hydraulic conductivity (K)

• For a given fluid and temperature (i.e. viscosity
  and density) the hydraulic conductivity K
  reflects the properties of the soil or rock
  containing the groundwater
• Hydraulic conductivity correlates roughly with
  the 2nd-3rd power of the radius of the largest
  pores or fractures in the medium, though it is
  not easy to specify exactly which fraction of the
  largest of these conduits

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Some values of hydraulic conductivity K

• Gravel 10,000-100 m/day
• Sands 100-1 m/day
• Silts, glacial till 1-0.001 m/day
• Clays lt 0.001 m/day

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Formation of a groundwater body, early stage (t
1)
Water table has not yet risen to stream channels,
so no outflow
Water table at t 1
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Formation of a groundwater body (t 2)
Water table rises to stream channels, which drain
water away, but still at a rate lower than areal
sum of (P-E) because the water table gradients
are low
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Formation of a groundwater body, t 3
Water table (fixed at stream channels) has
continued to rise due to recharge until the
gradient is sufficient to increase the outflow to
equal the areal sum of (P-E)
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(in words)

• Equilibrium state occurs when outflow of
  groundwater to streams (Darcys law) balances P-E
  on the land surface
• Water table is a diffuse mimic of the topography
• The lower the value of K, the steeper the water
  table must be to convey the water
• If (P E) varies seasonally, the water-table
  gradient and therefore height and outflow rate
  will also change

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Effects of lithologic heterogeneity

• Rocks/soils have very heterogeneous K values (14
  orders of magnitude)
• Therefore, the arrangement and orientation of the
  rocks also affect the volume, direction, and
  speed of groundwater flow

R
Unconfined Aquifers
P-E
Water table
Aquiclude (low K)
Aquiclude
Confined Aquifer (high K)
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Shape and steepness of water table

• Rock properties (mainly K) affect groundwater
  flow. The lower the K-value, the steeper is the
  equilibrium gradient required to convey the (PE)
  that is draining from the ridges.  
• Why are water tables convex-upward in homogeneous
  (i.e. constant-K) aquifers?

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Equilibrium water table profile (1)

• Strip of aquifer, 1 meter wide, parallel to the
  page
• At equilibrium, after a sufficiently long time of
  stable recharge, outflow (Q) to the stream, or
  any other distance, x, equals input from recharge
  on the upstream drainage area, which begins at x
  0 (half way between the channels)

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Equilibrium water table profile (2)
Darcys law

At equilibrium
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Equilibrium water table profile (3)

Boundary condition at x xL h hs
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Equilibrium water table profile (4)
(P E)

(P E)
Stream Channel
h
hS
The result is a convex-upward water table, the
height and steepness of which depend on the
height of the stream surface and the ratio of
recharge to conductivity
X
X 0
X XL
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Coastal aquifers
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Ghyben-Herzberg static model of the fresh water
lens above salt water at a coast
From Dingman, Physical Hydrology
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Calculation of height-depth ratio of freshwater
lens
?f density of fresh water ?s density of sea
water h height of water table above SL Z
depth of salt wedge below SL
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Coastal aquifer
Ghyben-Herzberg static model of the fresh water
lens above salt water at the coast

• Remember that h depends on (P-E)x
• If h is lowered ?h by pumping, Z will diminish by
  ?Z 40?h

h
Z
Z
From Dingman, Physical Hydrology
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Coastal aquifersReal flow fields are somewhat
more complex and diffuse than the G-H
idealization, but it captures the big picture
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Velocity of a parcel of groundwater flow

• Darcys law gives an apparent velocity, but
  strictly speaking this is a discharge per unit
  cross-sectional area of aquifer
• But if we are interested in the velocity of the
  water molecules themselves, or a solute, we have
  to consider the porosity p of the aquifer (which
  is the cross-sectional area of the pores)

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Examples of the range of flow velocity values
(and therefore time of travel of the groundwater
itself)

• Steep (20) forested hillslope after a rainstorm,
  K1m hr1 and p 0.4
• Velocity 0.85 m hr1 so a 100 m hillslope
  responds in days
• Gently sloping sandstone aquifer 1000 km long,
  slope 103, K 0.1m day1, and p 0.1
• Velocity 0.4m yr1, so water would take 2.5
  million years to travel through the aquifer
• Implication large groundwater bodies respond
  very slowly to changes in PE
• Aquifers in mid-western U.S. and Sahara
  originally supplied in wetter, cooler climate
  (upto 100,000s of years ago)
• Todays recharge rates are much less than
  exploitation

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QuestionIf groundwater systems are at
equilibrium, how can we extract any water without
depleting the volume stored?

• To a first approximation, we cant. If we pump
  (Q, volume of water per unit area of aquifer),
  then RP-E-Q
• The maximum value of Q (per year over the long
  term) cannot exceed (P-E) without reducing the
  volume in storage and the runoff, which must
  ultimately ? 0
• But, we can choose to reduce R by pumping and
  live with the new equilibrium R in the rivers
• In other cases, there is a feedback of Q reducing
  E. For example by drawing down water tables, we
  can reduce water loss by E and absorb the
  ecological changes. Demise of some riparian
  forests in Central California. 1960s UN proposal
  for the Sahara.
• In other cases, pumping can lower the water table
  and induce recharge from rivers
• In many cases, we pump at Q gt (P-E) and mine
  water emplaced in a wetter or cooler climate
  (upto 100,000s of years ago)

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Water Balance, including pumping, with feedback