\ I /
 United States
 Environmental Protection
 Agency
 Robert S. Kerr Environmental
 Research Laboratory
 Ada OK 74820
 Research and Development
EPA/600/S2-85/066 Aug. 1985
 Project  Summary
 Upconing  of a  Salt/Fresh-
 Water  Interface  Below a
 Pumping  Well
 Jan Wagner and Douglas Kent
  Analytical solutions for the upconing
 of an abrupt salt-water/fresh-water in-
 terface beneath a pumping well and for
 the concentration profile across a mov-
 ing interface are developed for two
 types of upconing problems. The first
 considers the position of the interface
 and the salinity of the pumped water
 for a specified pumping rate. The sec-
 ond  type of problem  addresses the
 pumping schedules to prevent saliniza-
 tion  of a well or to  reach a predeter-
 mined salinity in the pumped water.
  An interactive Fortran computer code
 has been developed to obtain solutions
 to both types of problems. The user is
 provided with options to modify the
 definition of a given  problem, and,
 therefore, can gain some insight into
 the effects of geometry and physical
 properties on the rate and extent of up-
 coning and the salinization of a well.
  This Project Summary was devel-
 oped by EPA's Robert S. Kerr Environ-
 mental Research  Laboratory, Ada, OK,
 to announce key findings  of the  re-
 search project that is fully documented
 in a separate report of the  same title
 (see  Project Report ordering informa-
 tion at back).

 Introduction
  The full report presents an analytical
solution for the upconing of an abrupt
salt-water/fresh-water interface below a
 pumping well. Dispersion phenomena
arising from the displacement of a mov-
 ing interface or a finite transition  zone
between the invading and displaced flu-
ids can be superimposed on the analyti-
cal solution for the position of an abrupt
interface. An interactive FORTRAN com-
puter code has been developed that en-
ables the user to modify  parameters
and to  control the computational se-
quence. This interactive approach en-
ables the  user to gain insight into the
effects of  geometry and physical prop-
erties on the rate and extent of upcon-
ing and salinization of a well.

Mathematical  Development
  McWhorter (1972) presented the
equations that describe the flow in satu-
rated aquifers which are underlain by a
zone of saline water and pointed out the
difficulties in  obtaining solutions to
these problems. The complexity of the
flow phenomenon has led many investi-
gators to idealize the system as a fresh-
water zone separated from  an underly-
ing salt-water zone by a sharp interface.
In  other words, the two  fluids are as-
sumed to be immiscible. Schmorak and
Mercado (1969) followed this approach
and accounted for the mixing of the two
fluids by superimposing the effects of
dispersion on the transient solution for
the position of an abrupt interface.
  For the  case of upconing beneath  a
pumping well partially penetrating a rel-
atively thick confined aquifer as shown
in  Figure 1,  Schmorak and Mercado
(1969) presented Bear and Dagan's so-
lution for the position of the interface as
a function of time and radial distance
from the pumping well as
 x(r.t) =
       2i;(Ap/p)Kxd
          1
1
                             (1)
       (1 + R2)1'2  [(1 + T)2 + R2]1'2

where R and T are dimensionless dis-

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                                                    Validity
                                                    of Linear
                                                    Approxi-
                                                    mation
                                                                  = Xc, + Zo
Figure 1.    Upconing of an abrupt interface below a pumping well
tance and time parameters defined by

                    , 1/2
and
              (AP/p)Kz
                                  (2)
                                  (3)
Other notations are defined as follows
(also refer to Figure 1):
     d = distance from  the bottom of
         the well to the  initial interface
         elevation (L)
  KX,KZ = horizontal and vertical perme-
         abilities, respectively (L/t)
     Q = well pumping rate (L3/t)
      r = radial distance from well axis
         (L)
      t = time elapsed  since start of
         pumping (t)
     X = rise of the interface above its
         initial position (L)
   Ap/p = dimensionless density differ-
         ence between the two fluids,
         
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elevation and  the dispersion concept
should be limited to the zone below the
critical depth. The complex mixing and
flow phenomena above the  critical
depth, near the well screen, and within
the well pipe are approximated empiri-
cally using the approach followed by
Schmorak and Mercado (1969).
  The  average dimensionless concen-
tration of the transition zone above the
critical rise, e(X > Xcr), is approximated
as one-half the concentration at the crit-
ical depth, or
       e(X > Xcr) = 0.5 e(Xcr)
(7)
The concentration in the pumped water,
ew,  is determined from dilution of the
average transition-zone concentration
above the critical depth with displaced
fluid, or
                    Xcr)
(8)
where cp is an interception coefficient, or
the fraction of transition zone fluid in
the total volume pumped.

Computer Program
  Two types of upconing problems are
considered. The first involves the  de-
scription of the  expected interface ele-
vation and the salinity of the pumped
water as a function of time for a given
pumping rate. The second problem ad-
dresses the maximum rate at which a
well can be pumped without exceeding
a specified salinity in the pumped water.
Both types of problems are included in
an interactive computer code. Data are
required under two  modes  of opera-
tion—"Basic Input Data" and "Edit".
  Basic input data are required to initi-
ate a new problem using the  UPCONE
program. The data entries include  the
problem title, the physical properties of
the aquifer and  the two fluids, and  the
geometry of the system. The user is
prompted for the required data through
a series of input commands.
  Once the basic input data have been
entered, the problem as currently  de-
fined is listed and the program enters
the "Edit" mode. The edit commands
can be used to redefine the problem,
execute elevation or pumping rate cal-
culations, or terminate the program.
  The program  has been written in an
unextended version of  FORTRAN 77
and has  been installed on microcom-
puters running under CP/M-80  and
MS-DOS as well as variety of minicom-
auters  and mainframe machines.  The
major modifications in code to  imple-
ment the program on a given system is
the assignment  of  logical devices.
Guidelines for these types of modifica-
tions are clearly identified in the source
code.

References
Bear J. and D. K. Todd. 1960. "The Tran-
  sition  Zone Between Fresh and  Salt
  Waters in  Coastal Aquifers." Contri-
  bution No. 29, Water Resources Cen-
  ter, University of California, Berkeley,
  CA.
McWhorter, D. B. 1972. "Steady and Un-
  steady Flow of Fresh Water in Saline
  Aquifers."  Water Management Tech-
  nical Report No. 20, Engineering Re-
  search Center, Colorado State Univer-
  sity, Fort Collins, CO.
Schmorak, S. and A. Mercado. 1969.
  "Upconing of Fresh Water-Sea Water
  Interface Below  Pumping Wells, Field
  Study,"  Water Resources Research,
  Vol. 5, No. 6, pp. 1290-1311.
        Jan Wagner and Douglas C. Kent are with Oklahoma State University, Stillwater,
          OK 74078.
        Carl G. Enfield is the EPA Project Officer (see below).
        The complete report, entitled "Upconing of a Salt/Fresh-Water Interface
          Below a Pumping Well," (Order No. PB 85-215341 /A S; Cost: $11.50, subject
          to change) will be available only from:
                National Technical Information Service
                5285 Port Royal Road
                Springfield, VA 22161
                Telephone: 703-487-4650
        The EPA Project Officer can be contacted at:
                Robert S. Kerr Environmental Research Laboratory
                U.S. Environmental Protection Agency
                P.O. Box 1198
                Ada, OK 74820

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Environmental Protection
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