1. Aquifer and Non Equilibrium
equation for unsteady radial
flow
Presented by: AbhiShek Gupta
2. Aquifer
• A saturated, permeable, geologic unit that can transmit a
significant amount of groundwater under an ordinary
gradient.
3. Unsteady flow in confined aquifer
• Assumptions
The aquifer is confined
The aquifer has infinite aerial extent
The aquifer is homogeneous, isotropic and of uniform thickness
The piezometric surface is horizontal prior to pumping
The aquifer is pumped at a constant discharge rate
The well penetrates the full thickness of the aquifer and thus
receives water by horizontal flow
5. • Change in volume,
S (Storage coefficient) is the volume of water released per unit surface area per
unit change in head normal to the surface.
• In this equation, h is head, r is radial distance from the well, S is storage
coefficient, T is transmissivity, and t is the time since the beginning of pumping.
6. The Theis Method (Curve Matching Method)
• Theis assumed that the well is replaced by a mathematical sink of constant
strength and imposing the boundary conditions h = h0 for t = 0, and h → h0
as r →∞ for t ≥0, the solution,
Where W(u) is the well function and u is given by,
7. After taking log on both equations,
therefore, a graph of log s against log t should be the same shape as a graph of log
(W(u)) against log (1/u)
constants
13. Unsteady Radial Flow in an Unconfined Aquifer
Equation of flow of water (Neuman’s equation),
where,
h is the saturated thickness of the aquifer (m)
r is radial distance from the pumping well (m)
z is elevation above the base of the aquifer (m)
is specific storage (1/m)
is radial hydraulic conductivity (m/day)
is vertical hydraulic conductivity (m/day)
T is time (day)
14. Three phases of drawdown
First phase:
• pressure drops
• specific storage as a major
contribution behaves as an
artesian aquifer
•flow is horizontal
• time-drawdown follows
Theis curve S - the elastic
storativity.
15. Second phase Third phase
• water table declines
• specific yield as a major
contribution
• flow is both horizontal and
vertical
• time-drawdown is a function
of Kv/Kh r, b
• rate of drawdown decreases
• flow is again horizontal
• time-drawdown again follows
Theis curve S - the specific
yield.
16. Neuman’ assumptions
• The aquifer is unconfined.
• The vadose zone has no influence on the drawdown.
• Water initially pumped comes from the instantaneous release of water from elastic
storage.
• Eventually water comes from storage due to gravity drainage of interconnected
pores.
• The drawdown is negligible compared with the saturated aquifer thickness.
• The specific yield is at least 10 times the elastic storativity.
• The aquifer may be- but does not have to be- anisotropic with the radial hydraulic
conductivity different than the vertical hydraulic conductivity.
17. Neuman’s solution,
Where,
is the well function of water-table aquifer
For early time,
and
For late time, and
and
Parameters can be
found by Penman
method
18. Penmen method to find parameters
• Two sets of type curves are used and plotted on log-log paper (Theoretical curve
vs 1/u).
• Superpose the early (t − s) data on Type-A curve.
• The data analysis is done by matching the observed data to the type curve.
• From the match point of Type-A curve, determine the values for
and the value of
• Use the previous equations to determine T and S
• The latest (s − t) data are then superposed on Type-B Curve for the Γ - values of
previously matched Type-A curve, from the match point of Type-B curve, determine
the values for
• By using the previous equations, the T and S can be determined.
.
20. Unsteady Radial Flow in a Leaky Aquifer
Equation for Unsteady
radial flow for leaky aquifer,
Where,
r is the radial distance
from a pumping well (m)
e is the rate of vertical
leakage (m/day)
21. Hantush-Jacob Method
Assumptions:
• The aquifer is leaky and has an "apparent" infinite extent,
• The aquifer and the confining layer are homogeneous, isotropic, and of uniform
thickness, over the area influenced by pumping,
• The potentiometric surface was horizontal prior to pumping,
• The well is pumped at a constant rate,
• The well is fully penetrating,
• Water removed from storage is discharged instantaneously with decline in head,
• The well diameter is small so that well storage is negligible,
• Leakage through the aquitard layer is vertical.
22. Hantush and Jacob solution for leaky aquifer,
Where,
where,
is the well function for leaky confined aquifer
B is the leakage factor given as
where,
b' is thickness of the aquitard (m)
K' is hydraulic conductivity of the aquitard (m/day)
24. Procedure
• Field data are plotted on drawdown vs. time on full logarithmic
scale.
• Field data should match one of the type curves for r/B
(interpolation if between two lines)
• From a match point, the following are known values
• Substitute in Hantush-Jacob equation:
(From match) r = distance between pumping well and
observation well
B = leakage factor