Representing Pump-Capacity Relations
in Groundwater Simulation Models
by L.F. Konikow
The yield (or discharge) of constant-speed pumps varies with the total dynamic head (or lift) against which
the pump is discharging. The variation in yield over the operating range of the pump may be substantial. In
groundwater simulations that are used for management evaluations or other purposes, where predictive accuracy
depends on the reliability of future discharge estimates, model reliability may be enhanced by including the effects
of head-capacity (or pump-capacity) relations on the discharge from the well. A relatively simple algorithm has
been incorporated into the widely used MODFLOW groundwater flow model that allows a model user to specify
head-capacity curves. The algorithm causes the model to automatically adjust the pumping rate each time step
to account for the effect of drawdown in the cell and changing lift, and will shut the pump off if lift exceeds a
critical value. The algorithm is available as part of a new multinode well package (MNW2) for MODFLOW.
The capacity of a pump installed in a well to deliver
water depends on several factors, including the size of
the pump and the power of the motor, as well as the lift,
or vertical distance over which the water must be raised.
Boonstra and Soppe (2007) relate pump efficiency and
pump performance to the total dynamic head, which they
state “is made up of (1) the water-level depth inside the
pumped well ...; (2) the above ground lift; and (3) head
losses due to friction and turbulence in the discharge
There are a number of reasons why well yields
and pump performance might decrease over time. Some
involve damage or deterioration to the pump or well
screens. Others simply are related to changing heads over
time. Driscoll (1986, p. 583) gives an example for a deep-
well turbine pump where “the total head would be as low
as 60 feet (18.3 m) during a season of high water level or
U.S. Geological Survey, 431 National Center, Reston, VA 20192;
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Received May 2009, accepted July 2009.
No claim to original US government works.
minimum withdrawal of water; but during another season,
the total head might be 100 feet (30.5 m) because the
water level in the aquifer has decreased or interference
from adjacent wells has increased. Under these conditions,
the rate of pumping would range from nearly 1,340 gpm
[gallons per minute] (7,300 m3/day) down to about 620
gpm (3,380 m3/day).”
Conceptually, after pumping starts, the water level
in the well will decline over time and the lift (and total
dynamic head) required to discharge at a fixed point and
elevation above the land surface will increase. As the total
dynamic head increases, more work is required to lift and
discharge a unit volume of water, so the discharge from
a standard constant-speed pump will tend to decrease.
The methods described in this note are not applicable to
a variable-speed pump designed to maintain a constant
discharge under conditions of changing lift.
Most pump manufacturers provide performance
curves for their products that typically include a head-
capacity (or pump-capacity) curve relating the total
dynamic head to the discharge rate (Boonstra and Soppe
2007). A hypothetical example set of performance curves
having representative shapes is shown in Figure 1A. Near
the design capacity or maximum flow rate of the pumps
(termed “runout” by Pritchard ), the curves are
steeper and there is a relatively small change in discharge
106Vol. 48, No. 1–GROUND WATER–January-February 2010 (pages 106–110) NGWA.org
Yield, in cubic feet per dayYield, in cubic feet per day
Total dynamic head, in feet Total dynamic head, in feet
Linear step-wise approximation
Figure 1. Plots showing (A) hypothetical but representative performance (pump-capacity) curves for three models (and sizes)
of vertical turbine pump, with the top curve representing the largest pump and (B) points defined for approximation of top
curve using linear interpolation.
for a unit change in total dynamic head. As the lift
increases, however, the curves tend to flatten out and there
may be a relatively large change in discharge for a unit
change in total dynamic head—until a point is reached
where the pump can no longer provide water and the dis-
charge goes to zero. This point is called the “shut-off
head” (Pritchard 2007; Driscoll 1986, p. 585).
In applying a groundwater flow model, there may
be cases where it is deemed valuable to incorporate
the relation between pump capacity and lift because as
drawdown increases with time, the well yield will be
reduced. Where historical data on discharge from wells
are based on metering or other estimates of the total
volume produced over a given time period, incorporating
these relations may provide little or no added value
for model calibration because there are no observations
of temporal changes in production as a function of
drawdown. However, if the groundwater flow model is
used to make predictions of future behavior of the flow
system, evaluation of management scenarios, or for small-
scale studies near a pumping center, the use of these head-
capacity curves may add more realism and defensibility
to predictions of future conditions.
This article describes the incorporation of head-
capacity curves for constant-speed pumps into the MOD-
FLOW groundwater flow simulation model (Harbaugh
et al. 2000; Harbaugh 2005). This new feature is part of
a new multinode well (MNW2) package (Konikow et al.
2009), but is also applicable to single-node wells.
The new MNW2 package provides the user an option
to specify a performance curve (head-capacity curve) for
each well. Use of this option will generally lead to a grad-
ual automatic adjustment of the well discharge rate over a
large range in water levels (although the rate of adjustment
depends on the slope of the performance curve).
Figure 2. Schematic cross section of an unconfined aquifer
showing a multinode well open to parts of three model layers,
and the relation of the lift (or total dynamic head) to the
reference elevation (hlift) and the water level in the well
(hWELL) at two different times. Q at time 1 (A) is greater
than Q at time 2 (B) with greater drawdown after a period
Pump-capacity adjustments are based on the elevation
of the outflow (discharge) location only. Therefore, the
user must specify a reference elevation (hlift) correspond-
ing to the elevation of the discharge point (Figure 2). The
model will then automatically compute the lift (or total
dynamic head) based on the difference between the ref-
erence elevation and the most recently calculated water
level in the well (hWELL). At a later time, while pump-
ing continues and drawdown increases, the lift increases
(Figure 2B). The pump discharge may therefore decrease
because more energy is required to lift water a greater dis-
tance. If one also wants to account for head loss because
of friction and turbulence in the pipes, the reference eleva-
tion can be artificially increased proportionately to obtain
the desired total effect.
Details about the MNW2 numerical scheme are
described by Bennett et al. (1982), Halford and Hanson
(2002), Neville and Tonkin (2004), and Konikow et al.
(2009). If the pump-capacity option is activated, at the
NGWA.orgL.F. Konikow GROUND WATER 48, no. 1: 106–110107
beginning of each iteration cycle MNW2 updates the net
discharge from the well on the basis of the most recent
value of the water level in the well and the user-defined
pump-capacity curve. Consequently, the discharge rate
from the well may vary from one time step to the next
during a particular stress period. In contrast, the standard
well package in MODFLOW imparts a discharge rate that
is constant during the entire length of a stress period.
The MNW2 user must enter a table of values repre-
senting discrete points on the head-capacity curve for the
pump. The two end points of the provided curve should
represent values of total dynamic head corresponding
with zero discharge and the maximum design discharge,
respectively. In addition, a minimum of one additional
intermediate point on the curve must be specified. MNW2
applies linear interpolation to estimate the yield for any
value of total dynamic head between defined points. The
use of four intermediate points leads to a very accurate
approximation over the entire range of heads (Figure 1B).
Because the pump-capacity curves may be nonlin-
ear and, where gently sloping, small changes in lift may
induce large changes in discharge, the overall numeri-
cal solution may become unstable, fail to converge, or
oscillate. To minimize such numerical problems, several
steps are taken in the code; these steps are described in
detail by Konikow et al. (2009). Even with these prepro-
grammed measures to facilitate convergence, numerical
problems may still occur. In such cases, the user may
have to change numerical solution tolerances, reduce the
time-step size, increase the allowable number of iterations,
adjust the pump-capacity relations, or turn off the option
to use pump-capacity relations.
The model assumes that for any total dynamic head
equal to or less than the minimum head end point (on
the right side of the curve in Figure 1B), the discharge
will equal the maximum-operating discharge. For any total
dynamic head equal to or greater than the maximum head
end point (on the left end of the curve), the model assumes
that the discharge equals zero. If the discharge is thereby
set to zero, and at a later time or subsequent iteration the
water level in the well rises sufficiently that the lift does
not exceed the maximum total dynamic head, pumping
Demonstration of Capabilities
Description of Hypothetical Groundwater System
(Modified Reilly Problem)
To test and illustrate the use of pump-capacity (head-
capacity) curves, a hypothetical groundwater system was
developed. The problem is based on one described by
Reilly et al. (1989) and slightly modified by Konikow
and Hornberger (2006a, 2006b). This same test problem is
further modified here to help evaluate the pump-capacity
calculations and to illustrate their potential value.
The hypothetical, unconfined groundwater system
represents regional flow that is predominantly horizontal
but includes some vertical components. The system is sub-
stantially longer (10,000 feet) than it is thick (205 feet)
or wide (200 feet). A borehole with a 60-feet screen is
located close to the no-flow boundary on the upgradient
side of the system. The well is assumed to have a skin
that is about 1.6 feet thick with a hydraulic conductivity
that is 5% of that of the aquifer. Other properties of the
system and the model are described in detail by Reilly
et al. (1989), Konikow and Hornberger (2006a, 2006b),
and Konikow et al. (2009).
Konikow and Hornberger (2006a, 2006b) simulated
this three-dimensional hypothetical regional flow system
using a variably spaced areal grid (over half the domain
space because of the presence of a plane of symmetry).
In the local area around the well, a relatively fine and
uniform areal cell spacing of 2.5 feet by 2.5 feet was
used. The vertical discretization was 5 feet everywhere
in the model domain, and the top layer was assumed to
To demonstrate the effects of using pump-capacity
curves, the Reilly problem was modified so that the long
borehole had a pump with a characteristic performance
curve that followed the upper curve in Figure 1A; this
curve was discretized using four intermediate points,
as shown in Figure 1B. The pumping rate was set at
−7800 ft3/d, which equals the maximum capacity of the
pump, so that the drawdown would be large enough to
illustrate clearly the effects of using pump-capacity curves
to limit discharge.
Test Problem 1: Effect of Using
In the first test, a 300-d transient stress period
followed the initial steady-state stress period. The 300
d were divided into 20 time steps using a time-step
multiplier of 1.2. The reference elevation for calculating
lift (hlift) was set equal to 10.0 feet (note that the elevation
of the top surface of the grid is 0.0 feet).
The results (solid lines in Figure 3) show that the
net discharge remained unchanged at the desired rate
until the fourth time step. During the first three time
Time, in days
Head in well, in feet
Well discharge, in ft3/day
Figure 3. Results for test problem 1 of applying the pump-
capacity relations to the modified Reilly problem in which
the desired discharge equals −7800 ft3/d for a 300-d tran-
sient stress period (solid lines). For reference, dashed lines
show responses without considering pump-capacity relation.
108 L.F. Konikow GROUND WATER 48, no. 1: 106–110NGWA.org
steps, the calculated lift was sufficiently small that the
maximum discharge of the pump was allowed. In time
step 4, the water level in the well fell below that for runout
at −3.65 feet (yielding a lift exceeding 13.65 feet), and
the discharge was reduced in accordance with the pump-
capacity curve. From the time step 4 on, the net discharge
from the well was reduced gradually each time step as the
head in the well declined (and the pumping lift increased)
continually throughout the simulation. By the end of the
simulation, the discharge had been reduced by about 15%
(from −7800 ft3/d to about −6620 ft3/d).
For comparison, Figure 3 also shows (as dashed
lines) the responses of the system if the pump-capacity
relation had not been considered (i.e., if well discharge
was constant). Without the effects of the pump-capacity
relation, the drawdown in the well would have been
increased by an additional 3.8 feet (about 23%). If this
simulation had been a predictive run for evaluating future
pumpage, substantial error could have been introduced if
the pump-capacity relations had not been considered.
Test Problem 2: Well Shut-Down
A second test was evaluated to assure that the pump-
capacity curves can shut a pump off if the lift increases
substantially, as well as allow it to be turned back on
if the head in the well subsequently rises sufficiently. In
this variation of the previous test, heads were simulated
for two 365-d transient stress periods (after an initial
steady-state stress period). During the first transient stress
period, three single-node pumping wells located near
the multinode well were set at discharge rates equal
to −4000 ft3/d each; and during the second transient
stress period, these three wells were shut off so that
heads would recover. Both transient stress periods were
simulated using 15 time steps and a time-step multiplier
of 1.2. Everything else was unchanged from the previous
test. The results (Figure 4) show that the simulated net
discharge from the multinode well was reduced relative
to the specified maximum discharge during every time
step, until it was reduced to zero during the 12th time
step when the head in the well dropped below the value
yielding a lift greater than the maximum lift for this pump.
During the first time step of the second transient stress
period, the heads recovered quickly and sufficiently such
that the pump in the multinode well was reactivated. The
computed net discharge continued to increase as the water
levels rose in response to shutting off the three nearby
These two tests indicate that the pump-capacity rela-
tions work as expected on a conceptual basis. Additional
tests (not described herein) indicated that under some cir-
cumstances, oscillatory behavior and/or nonconvergence
occurred, but these problems were eliminated or min-
imized by adjusting numerical parameters or time-step
size. To reiterate, the use of pump-capacity relations is
optional in the MNW2 package (Konikow et al. 2009),
and the user can deactivate it during any one or all stress
periods of a simulation.
For a constant-speed pump, the discharge from a
well depends on the total dynamic head (or lift) against
which the pump works. As the lift increases, the discharge
decreases. If the lift becomes too great, the pump will
shut off. Representing these effects in a groundwater flow
simulation model can result in more realistic predictions
of the future behavior of a groundwater system.
A method to simulate the relation between well yield
and lift has been developed and incorporated into the
MODFLOW groundwater simulation model as an optional
part of a new MNW2 package (Konikow et al. 2009).
The results of testing use of the pump-capacity curves
in a hypothetical groundwater system (modified Reilly
problem) show that the method can realistically reduce the
pumping rate as drawdown progresses and lift increases
over time. The testing also showed that the pump will shut
off when the lift exceeds the maximum capacity of the
pump and that the pump will restart at a later time if the
water level in the well subsequently rises sufficiently such
Well discharge, in ft3/day
Time, in days
Head in well, in feet
Figure 4. Results for test problem 2 of applying the pump-capacity relations to the modified Reilly problem in which the
desired discharge equals −7800 ft3/d for two 365-d transient stress periods with three nearby wells pumping at −4000 ft3/d
during the first transient stress period. When the head in the well drops below −23.75 feet, the lift is greater than the
maximum capacity of the pump, the pump is shut off, and the well discharge becomes zero.
NGWA.orgL.F. Konikow GROUND WATER 48, no. 1: 106–110 109
that the new smaller lift again falls within the operating
range of the pump.
The author wishes to thank G.Z. Hornberger for
his continuing assistance and collaboration in developing
the code for the new multinode well (MNW2) package,
which includes the pump-capacity relations. The author
also appreciates the helpful review comments on this
manuscript provided by T.E. Reilly and Tracy Nishikawa
of the U.S. Geological Survey, Adam Siade (UCLA), and
journal reviewers David Abbott and John Guswa.
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