Involves study of subsurface flow in saturated soil media (pressure greater than atmospheric); Groundwater (GW) constitutes ~30% of global total freshwater, ~99% of global liquid freshwater

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Groundwater

• Involves study of subsurface flow in saturated
  soil media (pressure greater than atmospheric)
  Groundwater (GW) constitutes 30 of global total
  freshwater, 99 of global liquid freshwater
• Basic definitions
• An aquifer is a geologic unit capable of
  storing/transmitting significant amounts of
  water Flow still governed by Darcy Law (P gt 0)
• Unconfined aquifers
• Pores saturate by pooling up on top of an
  impervious or low conductivity layer
• Aquifer upper boundary is the water table (w.t.)
  p0 at w.t.
• Aquifer supplied by recharge from above
• Elevation of w.t. changes as storage in aquifer
  changes (analogous to flow in streams)
• Piezometric surface (hP/?g z) corresponds to
  w.t.
• Confined aquifers
• Saturated soil that is bounded above and below
  by low conductivity layers
• Boundary of aquifer does not change in time
  (analogous to pipe flow)
• Major recharge occurs upstream or via leaky
  confining layers
• Water generally under high pressure piezometric
  surface is above top of aquifer

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Unconfined vs. Confined Aquifers
Figure 6.1.3 (p. 144)Types of aquifers (from
U.S. Bureau of Reclamation (1981)).
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Water Storage in Aquifers

• Unconfined aquifers
• storage changes correspond directly to change in
  water table level (w.t. increases water going
  into storage and vice versa)
• storage parameter specific yield or storage
  coeff. (Sy) volume of GW released per unit
  decline in water table (per unit area)
• Confined aquifers
• storage changes correspond mainly to compression
  of aquifer as weight of overlying material is
  transferred from liquid to solid grains (change
  in porosity) when water is removed (or vice
  versa)
• storage parameter specific storage (Ss)
• Also often use storativity (S)

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Confined aquifer
Unconfined aquifer
Figure 6.1.5 (p. 145)Illustrative sketches for
defining storage coefficient of (a) confined and
(b) unconfined aquifers (from Todd (1980)).
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Derivation of 2D GW Flow Equation

• From mass balance considerations the 3D GW flow
  equation is given by
• Provided the necessary boundary conditions (2 per
  spatial dimension) and an initial condition we
  could solve the above 2nd order PDE for
  h(x,y,z,t)
• However, GW flow in aquifers largely consists of
  horizontal flow (in the x-y plane)
• Dupuit approx. We can simplify the equation for
  these cases by integrating the above equation
  with respect to z, assuming no vertical flow
  inside aquifer (qz 0) and that the aquifer has a
  horizontal lower boundary (h h(x,y,t))
• We can integrate from z0 to zh, where h h
  (water table) for unconfined aquifers and hb
  (aquifer thickness) for confined aquifers
• This equation consists of four terms (one on LHS
  three on RHS). We will integrate them
  one-by-one. First (term 1)

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Next (term 2)
Similarly (term 3),
Finally (term 4),
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Putting the four terms back together to get 2D GW
Flow Equations Confined aquifers which is a
linear PDE R gt 0 only if leaky upper
boundary Unconfined aquifers which is a
nonlinear PDE in h. To make the equation linear
use Substituting Which is linear in the
variable h2.
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By expressing both equations in linear form, we
can write a unified governing linear 2D GW flow
equation for a homogeneous/isotropic aquifer
where For pumping problems (flow
toward a single pumping well) it is useful to
write the governing equation in cylindrical
coordinates NOTE We will often use this
equation as the starting point for solving
steady-state and/or 1D problems. In the case of
steady-1D problems, the above PDE reduces to an
Ordinary Differential Equation (ODE) which we can
generally easily integrate. The linearity of the
governing equation also allows for the
application of the principle of superposition.
Unconfined Confined



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Figure 6.4.4 (p. 159)Well hydraulics for a
confined aquifer.
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Figure 6.4.6 (p. 161)Well hydraulics for an
unconfined aquifer.
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Drawdown for Transient Pumping from a Confined
Aquifer
Tabulated Well Function
Table 6.5.1 (p. 163)Values of W(u) for Various
Values of u.
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Steady-state pumping (single well in infinite
confined aquifer)
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Steady-state pumping (single well in infinite
unconfined aquifer)
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Superposition of GW solutions

• The linearity of the governing equations in GW
  flow allow for the application of the principle
  of superposition of solutions. This means we can
  build-up the solution to a more complicated
  problem by summing up solutions to simpler
  problems (that when added together represent the
  actual GW problem). In the context of pumping
  wells, we have solutions for a single well in an
  infinite aquifer we would like to extend these
  solutions to more realistic problems.
• Cases where this idea is useful in pumping
  problems include
• An infinite aquifer with multiple wells
• Non-infinite aquifers with a no-flux BC
• Non-infinite aquifers with a fixed-head BC
• Combinations of the above cases

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Superposition of Multiple Wells
Single well solutions are only strictly
applicable for the idealized conditions they were
derived under. However linearity of the governing
equation allows for superposition of single well
solutions. For the case of multiple wells in a
confined aquifer, the actual drawdown resulting
from all wells can be obtained simply by summing
up the drawdown from each single well. Note
Summation of drawdowns not strictly applicable
for unconfined aquifers since equations not
linear in h (and therefore not linear in drawdown
s).
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Method of Images No-flux Boundaries
s(x,y) ?
In realistic problems we are often dealing with a
non-infinite aquifer (i.e. there is some sort of
boundary within the radius of influence of the
pumping well). In the case above there is an
impermeable boundary meaning there is no flux at
that boundary. We are interested in solving for
the head (or drawdown) profile under these
conditions. The differences between the actual
head (drawdown) and that predicted by a single
pumping well (in an infinite aquifer) solution
are i) The gradient in head must be zero at
the boundary, and ii) the presence of the
boundary will reduce the flow to the well from
that portion of the domain, thereby reducing the
head (increasing the drawdown) everywhere as the
flow must be replaced from other portions of the
domain. We could attempt to solve this more
complicated problem numerically, or we can use
superposition to our advantage to determine the
actual solution.
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Method of Images No-flux Boundaries
To model the realistic problem, we can use the
concept of an image well to model the effects
of the impermeable boundary. Namely, if we place
an image well pumping at the same rate as the
real well at a symmetric distance on the other
side of the boundary (i.e. as a mirror image) and
add up the drawdowns (which we can do due to
superposition) we will ensure that i) The
gradient in head is zero at the boundary, and
ii) all flow that would have been captured by
the real well if it were in an infinite aquifer
will instead be captured by the image well,
thereby increasing the drawdown everywhere in
the real domain. Hence the drawdown anywhere in
the domain for the real problem is Where the
drawdown solutions on the right-hand-side are
those for an single well in an infinite aquifer.
Note The image well must use a different
coordinate since it is as a different location.
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Method of Images No-flux Boundaries
We can illustrate this with an example. As weve
seen, for an infinite aquifer with a single well,
the drawdown field would look like the following
Symmetric cone of depression
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Method of Images No-flux Boundaries
If we have a non-infinite aquifer, i.e. with a
no-flux boundary, the single well solution
obviously no longer holds
Head gradient not equal to zero at boundary
However, we can conceptualize the boundary
condition as a pumping image well (pumping at the
same rate)
Image well drawdown (assuming an infinite aquifer)
Single well drawdown (assuming an infinite
aquifer)
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Method of Images No-flux Boundaries
If the drawdown from the two single well
solutions are summed up, we get the actual
resultant drawdown
Note -- Head gradient is zero at boundary --
Drawdown is increased compared to single well
solution.
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Method of Images Fixed-head Boundaries
The other type of non-infinite aquifer condition
we often must deal with is that of a stream
boundary. For simplicity, the stream is often
modeled as a fixed-head boundary condition. We
are interested in solving for the head (or
drawdown) profile under these conditions. The
differences between the actual head (drawdown)
and that predicted by a single pumping well (in
an infinite aquifer) solution are i) The head
must equal the fixed-head at the boundary, and
ii) the presence of the boundary will increase
the flow to the well from that portion of the
domain, thereby increasing the head (decreasing
the drawdown) everywhere as the flow is
augmented by the fixed-head boundary. We could
attempt to solve this more complicated problem
numerically, or we can again use superposition to
our advantage to determine the actual solution.

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Method of Images Fixed-head Boundaries
To model the realistic problem, we can again use
the concept of an image well to model the
effects of the fixed head boundary. Namely, if
we place an image well recharging (negative
pumping rate) at the same rate as the real well
at a symmetric distance on the other side of the
boundary and add up the drawdowns (which we can
do due to superposition) we will ensure that i)
The head is equal to the fixed value at the
boundary, and ii) all flow that would have been
unavailable to the real well if it were in an
infinite aquifer (due to drawdown) will instead
be supplied by the image well, thereby
decreasing the drawdown everywhere in the real
domain. Hence the drawdown anyhwere in the domain
for the real problem is Where the drawdown
solutions on the right-hand-side are those for an
single well in an infinite aquifer. Note The
image well must use a different coordinate since
it is as a different location.
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Method of Images Fixed-head Boundaries
Using the same example as before , but now with a
recharge image well, if the drawdown from the two
single well solutions are summed up, we get the
actual resultant drawdown
Note -- Drawdown is zero at the boundary --
Drawdown is decreased compared to single well
solution.
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Numerical Example

• The governing equation for 2D steady-state GW
  flow in a homogeneous/isotropic confined aquifer
  (with recharge/ pumping) is

Assume island is square, with a head h0 meters
along the boundary
h0
h(x,y)?

• Spatially varying recharge and non-infinite
  aquifer make analytical solutions (even with
  superposition) difficult if not impossible
• Need to solve numerically

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Case 1

• Spatially variable recharge more recharge away
  from (x,y) (0,0)

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Case 2

• Spatially variable recharge with pumping (at 5
  wells)

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In general, 2D/3D GW problems must be solved
numerically, i.e. using MODFLOW
Figure 6.9.2 (p. 184)Cell map used for the
digital computer model of the Edwards (Balcones
fault zone) aquifer (after Klemt et al. (1979)).
Figure 6.9.4b (cont.)Water levels for Barton
SpringsEdwards Aquifer. (b) Perspective block
diagram of 1981 water levels viewed from the east
side of the aquifer (after Wanakule (1985)).