1. Groundwater withdrawal and recharge (steady state situation)

2. Groundwater withdrawal and recharge (steady state situation)
Prepared by:
Fikadu Woldemariam, MSc, Hydrogeologist, Dilla University

3. Outline of presentation
Steady-state groundwater flow in confined aquifer
2. Steady-state groundwater flow in unconfined aquifer

4. Steady-state groundwater flow in confined aquifer
The equation describing time-independent (steady state) groundwater flow toward a well in a confined aquifer, also known as Theim equation
Source: Kresic, 1997

5. The velocity of the flow at distance r is given as
Deriving the differential equation of groundwater flow toward a fully penetrating well in a confined aquifer.
The cross-sectional area, A at distance r from the pumping well is the side of the cylinder with the radius r and the thickness b (which is the thickness of the confined aquifer):
The velocity of the flow at distance r is given as
Giving:
Source: Kresic, 1997

6. To solve this, differential equation the variables h (hydraulic head) and r (radial distance from the well) first need to be separated:
Integration of this equation with boundary conditions:
at distance rw (which is the well radius) the hydraulic head is hw (head in the well),
at distance r from the well the hydraulic head is hr:
The integrals on both sides of the equation are readily solved:
Finally, the hydraulic head at any distance from the pumping well is given as:

7. This equation can be integrated with the following boundary conditions:
at distance rw (which is the well radius) the hydraulic head is hw (head in the well),
at distance R from the well (which is the radius of well influence) the hydraulic head is H (which is the undisturbed head, equal to the initial head before the pumping started).
Giving:
The pumping rate of the well is then:

8. Steady state Flow in a Unconfined Aquifer
Scheme for deriving the differential equation of groundwater flow toward a fully penetrating well in an unconfined aquifer
llustration of Dupuit’s hypothesis (left) and the actual flow (right) toward a fully penetrating well in an unconfined aquifer.
Note that the actual equipotential line is curvilinear, and that the velocity vector has both horizontal and vertical components (Kresic, 1997). h1, h2 is hydraulic head in the observation well 1and 2 respectively; hw = hydraulic head in the well; r1, r2 = the distance from the well to the observation well 1 and 2; Q = the discharge amount (the pumping rate).

9. The cross-sectional area of flow at distance r from the pumping well is the side of the cylinder with radius r and thickness h (which is the thickness of the saturated zone):
The velocity of flow at distance r is given as the product of the aquifer's hydraulic conductivity (K) and the hydraulic gradient (i) which is an Infinitesimally small drop in hydraulic head (dh) over an infinitesimally small distance (dr):
The flow rate is then:

10. and then integrated for known boundary conditions
To solve this differential equation the variables h (hydraulic head) and r (radial distance from the well) have to be separated:
and then integrated for known boundary conditions
at distance rw (which is the well radius) the hydraulic head is hw (head in the well)
at distance r from the well the hydraulic head is h:
which gives:
pumping rate of fully penetrating well in an unconfined aquifer

11. Radial flow from recharge wells penetrating
The same theoretical considerations apply to wells that either extract water or inject water.
Radial flow from recharge wells penetrating
(a) confined and (b) unconfined aquifers
For a confined aquifer with water being recharged into a well completely open to the aquifer at a rate Qr, the following equation is applicable:

12. For a recharging well penetrating an unconfined aquifer, the following equation is applicable:
Where
Q = rate of injection, in m3/day
K = Hydraulic conductivity, in m/day
H = head above the bottom of aquifer while recharging, in m.
Ho = Head above the bottom of aquifer when no pumping is
taking place, in m.
ro = radius of influence, in m
rw = radius of injection well, in m.

13. Bibliography
Driscoll, F.G. (1986). Groundwater and wells. Johnson Filtration Systems Inc., Minnesota, USA.
Fetter, C.W Applied Hydrogeology. Fourth edition. Prentice-Hall, Inc. New Jersy.
Kresic., N Quantitative solutions in Hydrogeology and groundwater modeling. Lewis publisher. New York.
Mostafa, M. S., Philip, E.L., Bashir, A. M., Fakhry, A.A., & James, W.L Environmental Hydrogeology.
Suresh, R., (2005). Watershed Hydrology. A.K. Jain, Delhi.
Tenalem Ayenew &Tamiru Alemayehu Principles of Hydrogeology. Addis Ababa University Printing Press. Addis Ababa, Ethiopia.

14. Thank you!