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Methods Note/

Representing Pump-Capacity Relations

in Groundwater Simulation Models

by L.F. Konikow

Abstract

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.

Introduction

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

pipelines.”

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;

(703) 648-5878; fax (703) 648-5274; lkonikow@usgs.gov

Received May 2009, accepted July 2009.

Journalcompilation©2009NationalGroundWaterAssociation.

No claim to original US government works.

doi: 10.1111/j.1745-6584.2009.00619.x

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 [2007]), 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

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Yield, in cubic feet per dayYield, in cubic feet per day

Total dynamic head, in feet Total dynamic head, in feet

End point

Intermediate point

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.

Numerical Implementation

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

of pumping.

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

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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

will resume.

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

be unconfined.

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

Pump-Capacity Relations

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

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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

and Reactivation

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

wells.

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.

Conclusions

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

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that the new smaller lift again falls within the operating

range of the pump.

Acknowledgments

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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