Use of the Multi-Node Well (MNW) 

Package When Simulating Solute 

Transport with the MODFLOW 

Ground-Water Transport Process

Techniques and Methods 6-A15

U.S. Department of the Interior 

U.S. Geological Survey 
Cover: Conceptual cross-sectional view of a 

nonpumping multi-node well showing complex 

intraborehole flow (see figure 5 and related 

discussions for more details). 
Use of the Multi-Node Well (MNW) Package When 

Simulating Solute Transport with the MODFLOW 

Ground-Water Transport Process 

By L. F. Konikow and G. Z. Hornberger 

Techniques and Methods 6-A15

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

Gale A. Norton, Secretary 

U.S. Geological Survey 

P. Patrick Leahy, Acting Director 

U.S. Geological Survey, Reston, Virginia 

2006 

For product and ordering information: 

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Although this report is in the public domain, permission must be secured from the individual 

copyright owners to reproduce any copyrighted material contained within this report. 
CONTENTS 

ABSTRACT...................................................................................................................................1 

INTRODUCTION ..........................................................................................................................1 

WELL CONCENTRATIONS AND SOLUTE BUDGETS ...............................................................4 

HIGH-RATE PUMPING WELL ........................................................................................................4 

NONPUMPING WELL....................................................................................................................6 

LOW-RATE PUMPING WELL .......................................................................................................10 

FLOWING WELL ........................................................................................................................11 

SOLUTE BUDGET CALCULATIONS...............................................................................................11 

INPUT DATA CONSIDERATIONS .............................................................................................13 

MULTI-NODE WELL OBSERVATION PACKAGE (MNWO) ......................................................14 

OTHER OUTPUT CONSIDERATIONS ......................................................................................16 

TEST PROBLEM ........................................................................................................................17 

PROBLEM DESCRIPTION............................................................................................................17 

MODEL-SIMULATION RESULTS...................................................................................................20 

SELECTED INPUT DATA AND PRINTED RESULTS FOR TEST SIMULATION......................24 

SELECTED INPUT FILES.............................................................................................................24 

SELECTED OUTPUT FILES .........................................................................................................26 

SUMMARY AND CONCLUSIONS..............................................................................................32 

ACKNOWLEDGMENTS .............................................................................................................33 

REFERENCES CITED................................................................................................................33 

Figures 

 1. Conceptual cross-sectional view of a nonpumping vertical well that is connected to 

multiple layers in the model grid for a case where ground-water head increases 

with depth and intraborehole flow is consequently upwards................................................. 2 

 2. Conceptual cross-sectional view of a multi-node withdrawal well in which water 

enters the well at all well nodes............................................................................................ 3

 3. Conceptual cross-sectional view of a multi-node withdrawal well that has complex 

intraborehole flow ................................................................................................................. 3 

 4. Conceptual cross-sectional view of a multi-node injection well that has complex 

intraborehole flow ................................................................................................................. 4 

 5. Conceptual cross-sectional view of a nonpumping multi-node well that has complex 

intraborehole flow ................................................................................................................. 7 

CONTENTS iii 
 6. Schematic cross-sectional view through a MODFLOW grid showing a vertical 

nonpumping multi-node well open to two discrete intervals in layers 2-4 and 6-7................ 9 

 7. Schematic cross-sectional view through a MODFLOW grid showing a mostly 

horizontal nonpumping multi-node well open to columns 2-5 in layer 2 and 

column 1 in layer 1................................................................................................................ 9 

 8. Conceptual diagram showing geometry and boundaries for three-dimensional 

test problem with a nonpumping multi-node well.................................................................17 

 9. Map view of MODFLOW finite-difference grid showing location of transport subgrid 

and nonpumping multi-node well in the test problem ..........................................................19 

10. Calculated head distribution in vertical section near well on plane of symmetry ..................20 

11. Adjusted flow rates between the aquifer and the borehole with depth calculated 

with the Multi-Node Well Package in MODFLOW for the test problem ...............................21 

12. Calculated spreading of contaminant in presence of multi-node well showing relative 

concentrations in vertical section along row 30 of grid (representing the plane of 

symmetry): (A) initial conditions, (B) plume at 3 months, and (C) plume after 1 year .........22 

13. Three-dimensional perspective of model transport domain for the test problem 

showing simulated contaminant plume emanating from lower part of well 

after 1 year ..........................................................................................................................23 

14. Calculated depth-averaged concentration in nonpumping multi-node well in the 

test problem.........................................................................................................................23 

15. Concentration variations within the borehole of the nonpumping multi-node well in 

the test problem after 1 year................................................................................................23

Table 

 1. Selected physical and numerical parameters used in MODFLOW-GWT simulation 

of ground-water flow and transport in a three-dimensional, steady-state flow 

system containing a multi-node well....................................................................................18 

iv USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
Conversion Factors 

Inch/Pound to SI 

Multiply By To obtain 

Length 

foot (ft) 0.3048 meter (m) 

Area 

square foot (ft2) 0.09290 square meter (m2) 

Volume 

cubic foot (ft3) 0.02832 cubic meter (m3) 

Flow rate 

foot per day (ft/d) 0.3048 meter per day (m/d) 

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s) 

cubic foot per day (ft3

/d) 0.02832 cubic meter per day (m3/d) 

Hydraulic conductivity 

foot per day (ft/d) 0.3048 meter per day (m/d) 

v
PREFACE 

 This report describes modifications to MODFLOW-GWT, a three-dimensional 

solute-transport model developed by the U.S. Geological Survey (USGS). The 

modifications improve the capability of MODFLOW-GWT to accurately simulate 

transport in cases that involve fluxes to or from multi-node wells simulated with the 

MNW package. 

 The code and documentation for this revised MODFLOW-GWT model, including 

files for the test problem, are available for downloading over the Internet from a USGS 

software repository. The repository is accessible from a Web site for USGS groundwater models at http://water.usgs.gov/software/ground_water.html . When this code 

is revised or updated in the future, new versions or releases will be made available for 

downloading from this site. 

 Although extensive testing of the modifications of the code indicates that this 

model will yield reliable calculations for a variety of field problems, the user is cautioned 

that the accuracy of the model can be affected appreciably for certain combinations of 

parameter values. Users are encouraged to report any errors in this report or in the code 

to the contact listed on the appropriate software distribution Web page. 

vi USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT 
Use of the Multi-Node Well (MNW) Package 

When Simulating Solute Transport With the 

MODFLOW Ground-Water Transport 

Process 

By Leonard F. Konikow and George Z. Hornberger 

ABSTRACT 

 This report describes modifications to a U.S. Geological Survey (USGS) three-dimensional 

solute-transport model (MODFLOW-GWT), which is incorporated into the USGS MODFLOW

ground-water model as the Ground-Water Transport (GWT) Process. The modifications were 

made to create compatibility between the Multi-Node Well (MNW) Package for MODFLOW

and the MODFLOW-GWT model. This compatibility improves the capability of MODFLOWGWT to represent accurately solute transport in simulations that include multi-node wells 

because long-screen wells or long open boreholes that extend through multiple model layers can 

provide fast pathways for solutes to move from one location to another in a ground-water flow 

system. For nonpumping multi-node wells (used to simulate open boreholes or observation 

wells, for example), a simple routing and local mixing model was developed to calculate nodal 

concentrations within the borehole. A depth-averaged concentration is calculated for such 

nonpumping wells. For pumping multi-node wells (either withdrawal or injection) in which the 

flow between the well and the ground-water system is in the same direction at all nodes, the 

average concentration in the well is calculated as a flux-based mean assuming complete and 

instantaneous mixing in the wellbore of all inflows. For pumping multi-node wells (either 

withdrawal or injection) in which the flow between the well and the ground-water system is not 

unidirectional, the concentration distribution within the well is calculated using the same routing 

and local mixing model used for a nonpumping multi-node well, with the added assumption that 

the flux pumped in or out of the well is added or removed above the first well node. 

INTRODUCTION 

 The Multi-Node Well (MNW) 

Package (Halford and Hanson, 2002) allows 

MODFLOW-2000 (Harbaugh and others, 

2000) to simulate wells that extend beyond a 

single model node. That is, individual wells 

are not restricted to a single point in the 

finite-difference grid, but can now be more 

realistically associated with a discrete 

length. This association allows the 

simulated well to penetrate more than one 

model layer or to represent a horizontal well. 

 The MNW Package allows the user 

to specify a group of nodes that are 

associated with a single well. Specified 

fluxes can be defined for each node, which 

are then added to obtain a net rate for the 

well. The net flux can be negative 

(representing a withdrawal well), positive 

INTRODUCTION 1 
(representing an injection well), or zero 

(representing a nonpumping well or a longscreened observation well). The net flux 

represents the addition or removal of water 

from the ground-water system, and 

corresponds with the flow at the wellhead. 

 Regardless of the net flux, when a 

well is linked to multiple nodes of the finitedifference grid, then the flow between the 

model domain and the well can vary greatly 

in magnitude (and perhaps in direction) 

among the various nodes linked to the single 

well. If head losses inside a well are 

negligible compared to head losses between 

the well and the adjacent aquifer, it can then 

reasonably be assumed that the hydraulic 

head within the well will equilibrate to a 

single representative value. Because the 

heads in the aquifer at various model nodes 

encompassing a multi-node well will vary 

depending on local and regional aquifer 

properties and boundary conditions, a well 

can have nonuniform borehole flow (or 

intraborehole flow) that is greater than the 

net withdrawal or injection specified for the 

well. This intraborehole flow, in turn, can 

have a major effect on the water-quality 

distribution within the aquifer, as noted by 

Halford and Hanson (2002). 

 For the case of a nonpumping well, 

differing aquifer heads at the multiple well 

nodes will produce intraborehole flow, with 

ground water entering the well at one or 

more nodes and exiting from the well in 

other nodes (fig. 1). The inflows to the well 

will equal the outflows because the net flux 

is zero for a nonpumping well. (Note that in 

the convention used in figure 1, the 

subscripts “SNK” and “SRC” are defined and 

used in relation to the aquifer, so that flow 

into the well, for example, would correspond 

with outflow from the aquifer—a fluid sink 

with respect to the aquifer.) Solute will be 

transferred by advection through the well 

from aquifer cells having higher heads to 

aquifer cells having lower heads at rates 

proportional to the fluid fluxes. Where 

water enters the well, it will mix with water 

in the borehole and the mixture will flow 

towards well discharge nodes. If water 

enters at different concentrations at multiple 

nodes, the concentrations will change from 

node to node in the borehole at a given time. 

Solute transfer to the ground-water system at 

discharging nodes will change 

concentrations downgradient from a multinode well. 

 For multi-node pumping wells, the 

MNW Package inherently assumes that the 

pump intake is located above (closer to the 

land surface) the first node of the multi-node 

well (R.T. Hanson, U.S. Geological Survey, 

written commun., 2005). If the withdrawal 

or injection rate is sufficiently high so that 

Figure 1. Conceptual cross-sectional view of a nonpumping (QNET=0) vertical well that is connected 

to multiple layers in the model grid for a case where ground-water head increases with depth and 

intraborehole flow is consequently upwards. 

2 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
Figure 2. Conceptual cross-sectional view of a multi-node withdrawal well (QNET<0) in which 

water enters the well at all well nodes.

the flux between the aquifer and the well is 

in the same direction at every node of the 

well (as in figure 2), then intraborehole flow 

will have no effect on the solute 

concentration in the wellhead discharge for a 

withdrawal well or on the solute 

concentration in the recharge from an 

injection well. For such a unidirectional 

borehole-flow case with a multi-node 

injection well, there should be no 

concentration change within the borehole 

and the borehole concentration is always 

equal to the concentration in the fluid being 

injected (which must be defined by the 

user). For such a unidirectional-flow case 

with a multi-node withdrawal well (fig. 2), 

any potential concentration difference within 

the borehole would have no effect on the 

transport calculations for the ground-water 

system or on the concentration distribution 

in the ground-water system because flow at 

all nodes is from the aquifer into the well. 

It is feasible, however, that if the 

pumping rate is sufficiently small, that a 

pumping well can behave similarly to a 

nonpumping well in that substantial head 

differences in the ground-water system 

along the length of the borehole can result in 

a withdrawal well that has one or more 

nodes in which water enters the aquifer (fig. 

3) or an injection well that has one or more 

nodes in which water flows into the well 

from areas of relatively high head in the 

aquifer (fig. 4). As in the case of a 

nonpumping well, this nonuniform wellbore 

flow can also affect the water-quality 

patterns in the ground-water system. 

Figure 3. Conceptual cross-sectional view of a multi-node withdrawal well that has complex 

intraborehole flow. 

INTRODUCTION 3 
 The MNW Package of Halford and 

Hanson (2002) includes a simple approach 

to estimating water-quality effects with the 

MODFLOW model. This approach requires 

the user to associate a water-quality 

(concentration) value with each node. If 

flow is into a well at that node, it enters at 

the specified value of concentration (which 

remains constant over time during a stress 

period). If inflow occurs from multiple 

nodes, a flux-based averaging method is 

used to calculate the concentration in the 

well. This approach, however, ignores 

nonuniform initial spatial distributions of 

solute concentrations in the aquifer and the 

effects of related transient transport and 

dispersion processes. Hence, this simple 

approach has limited applicability. 

 If a model user is interested in 

calculating more accurately the effects of a 

multi-node well on water-quality patterns in 

a ground-water system or temporal changes 

in the average concentration in the borehole, 

then they should simulate solute transport in 

the aquifer simultaneously with flow 

simulation. This coupled simulation can be 

accomplished by activating the GroundWater Transport Process of MODFLOW

(the MODFLOW-GWT model) (Konikow 

and others, 1996; Kipp and others, 1998; 

Heberton and others, 2000; Konikow and 

Hornberger, 2003). 

Figure 4. Conceptual cross-sectional view of a multi-node injection well (QNET>0) that has 

complex intraborehole flow. 

 The purpose of this report is to 

document the inclusion of the MNW 

Package in the MODFLOW-GWT model. 

The report describes how solute 

concentrations in a multi-node well are 

calculated. This implementation does not 

require the creation of an additional input 

data set for the GWT model, but rather uses 

input data already required for the MNW 

Package. This report describes these 

linkages and the use of the input data for 

MNW Package as documented by Halford 

and Hanson (2002). The report also 

describes a new optional output package 

(MNWO Package) for concentration values 

in multi-node wells calculated when the 

MNW Package is used with MODFLOWGWT. 

WELL CONCENTRATIONS AND 

SOLUTE BUDGETS 

High-Rate Pumping Well 

 If the pumping rate in a multi-node 

injection well is sufficiently high so that 

flow leaves the borehole along its entire 

4 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
open or screened length, then it can be 

assumed that water within the borehole will 

become completely mixed at time scales 

smaller than the length of a typical time 

increment used to solve the equations of 

flow and transport in the ground-water 

system. Because the external fluid is the 

only source of solute to the borehole, there 

should be no changes or differences in 

solute concentration within the borehole and 

the specified external source concentration 

( ) would be the concentration in the 

fluid recharging the aquifer at all nodes of 

the well. 

CINJ ′

If the pumping rate in a withdrawal 

well is sufficiently high so that flow enters 

the borehole along its entire open or 

screened length, and the pump intake is 

located above the uppermost well node that 

interacts with the ground-water system (fig. 

2), then the solute concentration in the well 

discharge will be a function of the fluxweighted mean concentration of the inflows 

at all nodes of the well. This is consistent 

with an assumption of complete mixing 

within the borehole, allowing for differences 

in solute concentration entering the borehole 

because of variations in solute concentration 

in the ground-water system from node to 

node. By assuming (1) complete and 

instantaneous local mixing in the well, (2) 

that no reactions affect solute concentration, 

and (3) that wellbore storage is negligible, 

the concentration in water discharging from 

a high-rate multi-node withdrawal well (fig. 

2) can be computed using a simple mixing 

formula that computes a flux-weighted 

average concentration (similar to eq. 1 of 

Halford and Hanson, 2002) on the basis of 

fluid inflows as 

 

[ ]

∑[ ]

∑

=

= = n

i

snk i

n

i

i

t

aqsnk

t

well

Q

CQ

C

1

1 , (1) 

where Qsnk is the volumetric flow rate (in 

units of cubic length over time) at a node (or 

finite-difference cell) from the aquifer into 

the well (negative in sign according to 

MODFLOW convention because it 

represents a withdrawal from the aquifer), 

Caq is the concentration in the aquifer cell at 

the start of the time increment, n is the 

number of nodes in a well, i is an index for 

the number of individual withdrawal nodes 

connected to a multi-node well, the subscript 

snk is a directional indicator of flow relative 

to the aquifer, and the superscript t

represents the present time increment. The 

concentration in any flow out of the well 

(Cwell) during time increment t by 

withdrawal to the land surface is equal to the 

mixed concentration in the well during that 

time increment. 

 In multi-node wells simulated with 

the MNW Package, withdrawal from a well 

can be limited by drawdown constraints 

specified by the user. For example, in a 

withdrawal well this would be important if 

drawdown in the well causes the water level 

to fall below the level of the pump intake. 

In such cases, the withdrawal rate specified 

by the user will be overridden by a reduced 

value calculated on the basis of the constraints. Thus, the net withdrawal from the 

well (Qnet), defined as Qnet = Qsrc -│Qsnk│

over all nodes in a multi-node well (where 

Qsrc is the volumetric flow rate at a node 

from the well into the aquifer) may be less 

than the withdrawal specified by the user in 

the input data file (which is defined in the 

MNW input instructions as “the desired

volumetric pumping [withdrawal] or 

recharge [injection] rate,” or Qdes). Qnet is 

calculated as the sum of all fluxes between 

aquifer nodes and the multi-node well. 

Similarly, the user-specified injection rates 

will be reduced if injection head exceeds a 

specified limit. The MNW Package 

partitions Qnet among the various nodes 

WELL CONCENTRATIONS AND SOLUTE BUDGETS 5 
connected to a multi-node pumping well on 

the basis of relative heads and hydraulic 

conductances. 

Nonpumping Well 

 Flow will occur within the borehole 

of a nonpumping multi-node well (Qnet = 0) 

if aquifer heads differ between nodes 

because long well screens are equivalent to a 

high hydraulic conductivity inclusion. 

Internal flow in observation wells, 

abandoned open boreholes, or unpumped 

horizontal wells could be simulated with this 

approach (fig. 1). This intraborehole flow 

can also transport dissolved chemical 

constituents and affect the solute 

concentrations in the ground-water system. 

In a nonpumping borehole with 

intraborehole flow, it is observed that waterquality differences can be present within the 

borehole (Church and Granato, 1996, and 

Paillet and others, 2002). Such waterquality differences may occur in a 

nonpumping well because the intraborehole 

flow likely represents a slow laminar flow 

regime, and, therefore, it cannot be assumed 

that complete mixing occurs within the fluid 

volume contained in the borehole (as 

assumed for a high-rate pumping well). 

Rather, it may be more reasonable to assume 

that fluid entering the borehole at one 

location only mixes with the borehole fluid 

at that location, and the blended water then 

slowly displaces the fluid in the direction of 

the intraborehole flow. For example, in 

figure 1, the concentration in the well in 

layer 6 would be equal to the concentration 

in the aquifer in layer 6. The fluid and 

solute in the well in layer 6 then flows 

upwards to layer 5, where it mixes with 

more ground water entering the well at a 

possibly different concentration. Similar 

flow and mixing occurs in layer 4. 

However, because there is only outflow 

from the well above layer 4, the 

concentration in the well in layers 1, 2, and 

3 will equal the concentration leaving layer 

4 by upward flow. Therefore, for cases of 

nonpumping multi-node wells, solute 

concentration in water in the well is allowed 

to vary from node to node, and the nodal 

concentrations within a nonpumping multinode well will be calculated using a simple 

routing and local mixing model (similar to 

that used in the Streamflow-Routing (SFR1) 

Package of Prudic and others, 2004). 

 Head variations along the length of a 

borehole can be complex and may be more 

complicated in a multi-aquifer system, 

where intraborehole flow can change 

direction—perhaps multiple times (fig. 5). 

In this example, the node in layer 8 is 

considered a “strong source” with respect to 

the borehole (it is a “sink” for the aquifer) 

because the direction of intraborehole flow 

is away from this node towards both the 

upper and lower adjacent nodes. The 

concentration at this well node can be set 

equal to the concentration in the adjacent 

model layer 8 node because that is the only 

source of inflow to this node of the multinode well. The node in layer 3 is considered 

a “strong sink” for the borehole because the 

direction of intraborehole flow is into this 

node from both adjacent nodes. The 

concentration at this node cannot be 

calculated until after the concentrations in 

both adjacent nodes are defined by the 

routing algorithm. 

 The routing algorithm is always 

implemented after the flow equation is 

solved. Because complex intraborehole 

flow includes a fluid source term for the 

ground-water system, the concentration 

associated with that fluid source must be 

defined before the solute-transport equation 

for the ground-water system can be solved 

for that time increment. Therefore, the 

routing algorithm is implemented before the 

transport equation is solved, and it uses 

aquifer concentrations from the end of the 

previous time increment (or from the 

6 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
specified initial concentrations for the first 

time increment). This solution is analogous 

to the method used for computation of solute 

concentration in a lake with the LAK3 

Package (see eq. 17, Merritt and Konikow, 

2000). 

The first step in the routing 

algorithm is to compute the flow directions 

within the borehole on the basis of known 

fluxes between the aquifer and the well at all 

nodes of the multi-node well. All nodes in a 

nonpumping multi-node well are scanned 

and fluxes between nodes are tallied. Next, 

end nodes (the first and last nodes of the 

multi-node well) are checked to see if the 

direction of flow is into the well from the 

aquifer; if it is, the concentration at the well 

is set equal to the concentration in the 

aquifer at that node. Otherwise, if there is 

flow from the well into the aquifer, the 

concentration at the well will be set equal to 

the concentration in the adjacent well node, 

which by necessity is the only possible 

source of water to the end node in the well, 

after that concentration is defined. After the 

end nodes are checked, the concentration at 

well nodes representing “strong sources” to 

the well (sinks to the aquifer) are defined. 

From the concentrations initially defined at 

end nodes and at strong sources, the solute 

fluxes can be routed to adjacent nodes and 

the concentration in those adjacent nodes 

defined on the basis of a simple mixing 

formula as 

Figure 5. Conceptual cross-sectional view of a nonpumping multi-node well that has complex 

intraborehole flow. 

 

[ ]

[ ] i snk i

i

t

aqsnk

t

ii t

i

QQ

CQCQ

C

−

−

=

−

− −−

1

1

11

, (2) 

where Ci is the concentration at well node i

(corresponding to layer i in the example 

vertical well shown in figure 5, but this oneto-one correspondence between layer 

number and well node number is not a 

requirement), Qi-1 is the intraborehole flow 

between well node i-1 and node i (positive 

for flow towards higher values of i), 

[QsnkCaq]i is the solute mass inflow from 

node i in the aquifer, and the superscript t-1 

represents the previous time increment. The 

bracketed terms on the right side of eq. 2 are 

subtracted because the sign convention for 

the flow term is that Qsnk is negative for 

inflow to the well (discharge from the 

ground-water system). This node-to-node 

routing process is repeated until no 

additional nodes can have their 

WELL CONCENTRATIONS AND SOLUTE BUDGETS 7 
concentrations defined this way. Then 

nodes that are strong sinks for the wellbore 

(flow into the node from both directions) are 

checked, and if not already defined by the 

routing from strong sources, their nodal 

concentrations are defined next. 

 Although not illustrated, it would 

also be possible that the head in the aquifer 

adjacent to several nodes of a multi-node 

well would be exactly identical. For such a 

condition, we would expect that there would 

be no intraborehole flow over that segment 

of the multi-node well. However, at the 

location of such well nodes, there could still 

be a lateral flow in that model layer because 

of a regional hydraulic gradient within the 

aquifer. For any node of a multi-node well 

for which there is no flow to or from both 

adjacent well nodes, it is assumed that there 

is no intraborehole mixing with fluid in 

adjacent nodes of the well and that the 

concentration at such a node in the well is 

equal to the concentration in the aquifer 

adjacent to this node. Thus, as a final step in 

the routing process to calculate the solute 

concentration distribution in a nonpumping 

multi-node well, a check is made for nodes 

in which there is no intraborehole flow 

entering or leaving that segment of the 

multi-node well; if such a condition is 

found, the concentration in the well at that 

node is then set equal to the concentration in 

the aquifer at that location. 

 After the concentration has been 

defined at all nodes of a multi-node well, the 

average concentration in the well can be 

defined. However, because there is not 

complete mixing in a nonpumping well, the 

average concentration must be defined on a 

basis other than flux-weighted mixing. For 

a nonpumping multi-node well, the average 

concentration can be defined as a 

concentration that would be measured in a 

volume of water collected using a sampling 

device that continuously obtains and 

integrates discrete samples at every location 

as it is lowered (or raised) through the entire 

length of the borehole at a constant rate. 

This would represent a depth-averaged 

concentration in a vertical well (or lengthaveraged concentration in a nonvertical 

well). This average concentration is not 

used in any calculations, but is reported for 

information purposes in the output files for 

the Multi-Node Well Observation Package 

described below. 

 The method used to calculate the 

total length of the well and the length of the 

segment of the well associated with each 

node (and each nodal concentration) is 

illustrated by examples in figures 6 and 7. 

The method can account for a well that has 

multiple discrete sections of well screen. 

The following working assumptions are 

made: 

• A well is assumed to fully penetrate 

each cell in which a well node is 

located. 

• Where a multi-node well is open to 

the aquifer, the open interval of the 

well is assumed to pass through the 

block-centered node in the finitedifference cell. 

• The length between two successive 

nodes is defined as the distance 

between those two nodes. Half the 

length of an internodal section is 

assigned to each node. Thus, if two 

nodes are not adjacent to each other 

(such as well nodes 3 and 4 in fig. 6), 

it is possible that the length of well 

associated with a node can be longer 

than the maximum cell dimension. 

Proper calculation of lengths, 

therefore, requires that nodes for 

each multi-node well be listed in 

sequential order in the input data file. 

• For the first node, the length of the 

water column in the well is assumed 

to extend from the location of the 

block-centered node vertically 

8 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
WELL CONCENTRATIONS AND SOLUTE BUDGETS 9 

Figure 6. Schematic cross-sectional view through a MODFLOW grid showing a vertical nonpumping 

multi-node well open to two discrete intervals in layers 2-4 and 6-7. The borehole is assumed to 

extend to the top of the model grid to compute total borehole length. 

Figure 7. Schematic cross-sectional view through a MODFLOW grid showing a mostly horizontal 

nonpumping multi-node well open to columns 2-5 in layer 2 and column 1 in layer 1. The borehole 

is assumed to extend vertically from the first node to the top of the model to compute total borehole 

length. 

upward to either the top of the 

uppermost active grid layer or to the 

elevation of the head in the first node 

of the well, whichever is lower (see 

node 1 in figures 6 and 7). Because 

of this assumption, it is important 

that when defining the sequence of 

nodes in a nonpumping multi-node 

well, that the user assure that the first 

node in the sequence represents the 

well node located closest to the 

wellhead. In figure 6, the length of 

well associated with node 1 would be 

equal to the sum of the thicknesses 

of the top two layers. 

• The segment length associated with 

the final end node is assumed to 
equal half the internodal length to the 

previous well node, whether adjacent 

or not, plus the distance from the end 

node to the terminating cell wall at 

the intersection with the extended 

line of the final internodal segment. 

This is calculated as the distance 

from the intersection of this line with 

one of the cell faces to the node 

center. For the relatively simple case 

illustrated in Figure 6, the length of 

the final (5th) segment would be 

equal to the thickness of layer 7, and 

for the example in Figure 7, it would 

equal the width of column 5. 

After all the lengths are calculated, they are 

used as weights in computing the average 

concentration in the well, as in the following 

equation 

( )

∑( )

∑

=

= = n

i

i

n

i

ii

L

CL

C

1

1 , (3) 

where C is the length-averaged 

concentration in the well and Li is the length 

of the ith segment of the well. 

Low-Rate Pumping Well 

 If the pumping rate is viewed as 

transitional between zero and some 

arbitrarily high rate, the intraborehole flow 

pattern can then be viewed as representing a 

continuum between a complex multidirectional intraborehole flow pattern 

possible with a nonpumping multi-node well 

(fig. 5) and a simple unidirectional flow 

pattern for a high-rate pumping well (fig. 2). 

Increasing the pumping rate causes the flow 

pattern to shift towards a simple 

unidirectional mode. However, at 

sufficiently low pumping rates, the flow 

pattern may still be as complex as for a 

nonpumping case. If the withdrawal (or 

injection) rate specified for a pumping well 

is insufficient to induce unidirectional flow 

between all well nodes and the 

corresponding aquifer cells, the resulting 

complex intraborehole flow pattern can 

generate a nonuniform distribution of solute 

concentrations in the borehole, similar to 

that in a nonpumping multi-node well. For 

example, Izbicki and others (2005) collected 

depth-dependent water-quality samples in a 

well under pumping conditions and 

demonstrated large changes in chloride 

concentration with depth. Therefore, the 

assumption of complete mixing in a pumped 

borehole may not be valid, and water 

entering the ground-water system from a 

node in a withdrawal well (such as layer 3 in 

figure 3), for example, may not have the 

same solute concentration as water 

discharging through a pump at the wellhead. 

Similarly, water entering the ground-water 

system from one node in a multi-node 

injection well may not have the same solute 

concentration as water entering the groundwater system from another node (see figure 

4). 

The model will automatically detect 

if a pumping multi-node well has a complex 

multidirectional flow pattern between the 

borehole and the ground-water system. If it 

does, the model will calculate the 

concentration distribution in the wellbore 

using the same routing routine described 

above for nonpumping wells. This will 

assure that water leaving the well, whether 

to the ground-water system or to the land 

surface, will have a concentration consistent 

with a local (single node) mixing 

assumption rather than a less rigorous 

assumption of total instantaneous mixing 

within the entire borehole. To calculate the 

solute distribution within the borehole using 

the routing routine, the additional flux 

between the well and the land surface (Qnet) 

is assigned to the upper bound of the first 

well node (consistent with the MNW 

10 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
Package assumption that a pump is always 

located above the screened or open interval 

of the well). In effect, the pump can be 

considered to represent node “zero” in the 

string of well nodes. 

 The concentration everywhere in the 

borehole of a multi-node injection well will 

not necessarily equal the concentration of 

the external source fluid ( ), which is 

specified by the user, if aquifer water enters 

an injection well at one or more well nodes 

even while external water is injected (fig. 4). 

Because the concentration at any node 

where water enters the well is known from 

the solution to the transport equation at the 

end of the previous time increment, the 

concentration distribution in a multi-node 

injection well (and in the outflow from the 

well, where present) can be computed 

directly from eq. 2 and the associated 

routing routine described above. The only 

addition to the routing routine in the case of 

a low-rate injection well is to treat the 

injection from the external source (at a 

known flow rate and concentration) as 

derived from an additional node (node zero) 

above the first well node, which can only 

enter the first node of the well and is routed 

down from there with flow and 

concentration adjusted as appropriate 

according to eq. 2. 

CINJ ′

Flowing Well 

 A flowing artesian well can be 

simulated with the MNW Package. An 

artificially high discharge rate, Qdes, and a 

minimum head equal to the wellhead 

elevation are specified. The desired 

discharge cannot be maintained and 

discharge is calculated with a specified-head 

condition instead (K.J. Halford, U.S. 

Geological Survey, written commun., 2005). 

Although there may be no pump in a 

flowing artesian well, the hydraulic 

interaction with the ground-water system is 

the same as if the well were pumped at the 

calculated discharge rate. Therefore, solute 

transport in a flowing artesian well is 

simulated the same as in a pumping well, 

and solute routing in the borehole will be 

applied if the flow between the well and the 

ground-water system is not in the same 

direction at all nodes. 

Solute Budget Calculations 

 The solute mass balance computed 

by MODFLOW-GWT includes an 

itemization of all budget items that represent 

mass entering or leaving the transport model 

domain. Solute mass entering or leaving the 

transport model domain through all multinode wells is itemized separately. Fluid and 

solute in the borehole of a multi-node well is 

considered to be external to the model 

domain for this calculation. Thus, the solute 

flux (or loading) into or out of the model 

domain through multi-node pumping wells 

is not included directly in the overall solute 

mass balance computed and printed by 

MODFLOW-GWT. The mass balance for 

the transport model is instead computed on 

the basis of summations of individual fluxes 

at all nodes of all multi-node wells. This 

mass-balance computation is consistent with 

the way in which fluid mass balances and 

budgets are computed in MODFLOW for 

multi-node wells with the MNW Package. 

The solute flux (both for a time increment 

and cumulative) into or out of the model 

domain through multi-node pumping wells 

is, however, calculated and made available 

for inspection in optional separate output 

files that record data for each well (see 

MNWO Package below). 

The solute mass removed from the 

transport domain in a multi-node well is 

based on the computed nodal values of Qsnk

times the appropriate aquifer concentration 

( ) summed over all nodes as * Caq

WELL CONCENTRATIONS AND SOLUTE BUDGETS 11 
 ∑ , (4) 

=

= Δ n

i

out iaqsnk tCQM

1

* ][

where Mout is the solute mass discharged 

from the aquifer into the well during a 

transport time increment, n is an index of the 

number of nodes in the multi-node well, and 

Δt is the length of the time increment. The 

time level for the appropriate aquifer 

concentration in eq. 4 depends on whether 

the concentration at the sink node of the 

multi-node well was computed using eq. 1 

(for a high-rate pumping well) or eq. 2 (for a 

nonpumping or low-rate pumping well). In 

the former case, , 

whereas in the latter case . 

The use of an averaged concentration for the 

former case is the same approach used for a 

fluid sink, such as a simple single-node well 

in the WEL Package of MODFLOW (see 

Konikow and others, 1996, p. 25). 

2/][][ * 1

i

t

aq

t

aqiaq CCC − +=

i

t CC aqiaq ][][ * −1 =

The solute mass added to the 

transport domain from a multi-node well is 

calculated as a summation over all nodes as 

 ∑ , (5) 

=

= Δ

n

i

i

t

in wellsrc tCQM

1

][

where Min is the solute mass entering the 

aquifer by flow out of the well during a 

transport time increment and [Cwell]i is the 

concentration in the well at the ith well node, 

where fluid is entering the aquifer from the 

well. The values of Qnet, Qsnk, and Qsrc are 

updated for each time step used in solving 

the flow equation, and values of Cwell, Caq, 

Mout, and Min are updated for each time 

increment used to solve the solute-transport 

equation (it is possible that more than one 

time increment would be used to solve the 

transport equation for each time step used to 

solve the flow equation). The value of 

[Cwell]i would equal for a high-rate 

injection well and would be computed by 

routing for a nonpumping or low-rate 

pumping well. 

CINJ ′

 Identifying Qnet is important for the 

solute-budget considerations because the 

sum of all nodal flows into or out of one 

multi-node well (Qsnk or Qsrc) does not 

necessarily represent the fluid flux out of or 

into the ground-water system, respectively. 

This result occurs because in the low-rate 

case, some of the flux between the well and 

the model domain can represent internal 

flow through the well between different 

layers or cells of the model rather than flow 

between the model domain and an external 

fluid source or sink. Therefore, although the 

nodal values of Qsnk and Qsrc are used in 

calculating the solute budgets for multi-node 

wells within the transport model domain, the 

solute added or removed from the groundwater system must be calculated on the basis 

of Qnet and the appropriate concentration. 

For a multi-node withdrawal well the 

appropriate concentration is Cwell, and for a 

multi-node injection well the appropriate 

concentration is CINJ ′ (fig. 4), a value 

specified by the user in the input data set. 

The model computes the solute mass 

removed from the transport domain in a 

multi-node withdrawal well on the basis of 

the computed value of Qnet as 

 [ ] , (6) 1

t

netout netΔ= CtQM well

where Mnetout is the solute mass removed 

from the ground-water system with the well 

discharge. Because the pump is always 

assumed to be located above the first node 

of the well, the concentration in the well 

discharge at the wellhead will always be 

equal to the concentration at the first well 

node, which will have the same value 

whether it’s computed by routing and local 

mixing or by assuming complete mixing 

within the borehole. These calculations are 

12 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
accumulated over time and written as 

supplemental information with the solute 

budget, but are not included directly in the 

calculation of residuals. 

 The solute mass added to the 

transport domain from a multi-node 

injection well is calculated from Qnet as 

 , (7) netin net CtQM INJ Δ= ′

where Mnetin is the solute mass injected into 

the aquifer from the well during a time 

increment. As is done for the net mass 

discharged from multi-node wells, the 

values of net mass injected are accumulated 

over time and written as supplemental 

information with the solute budget, but are 

not included directly in the calculation of 

residuals. 

 Accurate calculations of solute flux 

and changes in concentration related to 

intraborehole flow in multi-node wells 

require that the fluid fluxes be accurately 

computed by the MNW Package. The 

FORTRAN code developed and originally 

released with the documentation of the 

MNW Package is written using singleprecision variables. This may not yield 

sufficient numerical precision in some cases. 

For example, a test case for a nonpumping 

multi-node well in a regional flow field 

yielded a sum of inflows equal to 5.4079 

m3

/s, and a sum of outflows equal to 5.3962 

m3

/s, an imbalance (and error) of about 0.2 

percent. This imbalance in fluid flux would 

then cause an error in the solute-transport 

model. Therefore, the MNW code was 

modified by changing all variables used in 

calculating the flux into or out of multi-node 

wells to double-precision variables. After 

this modification, the same test case was run 

again, and both sums were then calculated to 

equal exactly 5.4005 m3

/s. Thus, the 

double-precision version of the MNW 

Package is incorporated into MODFLOW

(Version 1.15.01) and into MODFLOWGWT. 

INPUT DATA CONSIDERATIONS 

 The use of the MNW Package for 

flow does not require the use of a separate 

GWT Package or the use of an additional 

input data set for transport. Instead, GWT 

uses standard MNW Package input data. 

 Data for each node of a multi-node 

well are read in a list that associates a 

number of variables with each of the nodes, 

including the specified volumetric flow rate 

(Qdes) and a water-quality value (QWval) 

(see Halford and Hanson, 2002, p. 14). If a 

well is associated with multiple nodes, it is 

conceptually simplest if the user specifies a 

flow for only the first of the multiple nodes 

and zero flow for the other nodes linked 

with this individual well. This nonzero 

nodal value is then assumed to apply to the 

multi-node well as a whole. This input style 

is strongly recommended for clarity and 

understanding of model input. However, the 

MNW Package allows the user to list a 

nonzero value for Qdes for as many nodes of 

a multi-node well as desired—there are no 

formal restrictions. In such an instance, the 

desired volumetric flow rate for the well 

([Qdes]Well) is then recomputed as the 

summation of all the individual values for 

all nodes associated with the well as 

 [] [] ∑ , (9) 

=

=

n

i

Qdes Well Qdes i

1

where [Qdes]i

 are the specified volumetric 

flow rates for individual nodes composing a 

multi-well node. If [Qdes]Well < 0.0, the well 

is a withdrawal well. If [Qdes]Well > 0.0, the 

well is an injection well. If [Qdes]Well = 0.0, 

even though individual nodes might have 

nonzero values of Qdes associated with 

them, the well is considered to be a nonpumping well. 

 In the list of nodes defining the 

multi-node well, nodes may be listed in any 

INPUT DATA CONSIDERATIONS 13 
order—it doesn’t make any difference with 

respect to flow calculations. However, as 

noted above, when calculating solute 

concentrations in a nonpumping multi-node 

well, defining the order of nodes is critical, 

and the data list must be constructed and 

ordered so that the first node listed 

represents the node closest to the wellhead 

(node 1 in figures 5 and 6), the last node 

listed represents the node furthest from the 

wellhead (node 5 in figures 5 and 6), and all 

nodes be listed in sequential order from the 

top to the bottom of the well (corresponding 

to the order of first to last well nodes). 

 The numerical solution of the solutetransport equation requires that whenever 

and wherever fluid enters the model domain, 

that a source concentration be specified for 

that fluid. Therefore, for a multi-node 

injection well, the user must specify what 

the solute concentration is in the source 

fluid. If an injection rate for a well is 

associated with the first node of a multi-well 

injection well, which is the proper 

procedure, then the source concentration 

should be specified for that node. However, 

if flows and source concentrations are 

defined for multiple nodes, and if that multinode well is defined as an injection well by 

eq. 9, then the source concentration for the 

injected fluid ( ) will be assumed to be 

equal to the value of QWval associated with 

the first node in a particular multi-node well; 

all other values of QWval in the list will be 

ignored in a solute-transport simulation. 

CINJ ′

 The input instructions for the MNW 

Package require that a value for QWval be 

specified for every node of a multi-node 

well, regardless of the value of the 

volumetric flow rate (Qdes). When the 

GWT Process is active, however, the 

solution to the solute-transport equation will 

yield values of concentration to be 

associated with nodes in which the 

volumetric flow rate to a well is less than or 

equal to zero. Thus, all values of QWval for 

nodes where Qdes ≤ 0 are read by the code 

but ignored in its internal computations. 

The user must include values for this 

variable even when it will not be used by the 

model. Users should specify QWval = 0.0 

for these nodes. 

 The MODFLOW-GWT model allows 

the user to solve the transport equation over 

a subset of the primary grid used to solve the 

flow equation—this subdomain is called a 

transport subgrid. For specifying the 

locations of nodes in a multi-node well, 

GWT requires that a multi-node well be 

either entirely outside of the transport 

subgrid or entirely within the transport 

subgrid. If the model detects that a multinode well includes some nodes inside the 

transport subgrid and some nodes outside 

the transport subgrid, a warning message 

will be printed and execution of the code 

will stop. 

MULTI-NODE WELL 

OBSERVATION PACKAGE 

(MNWO) 

 The calculated concentration (and 

optional additional information) for each 

multi-node well located within the transport 

subgrid can be recorded at every time 

increment. The specific nature of the recorded 

and written calculated concentration in the 

well will depend on the flow. For withdrawal 

wells (Qnet < 0), the MNWO Package will 

record the calculated concentration in the well 

discharge. For injection wells (Qnet > 0), the 

MNWO Package will record the fluxweighted average concentration in the 

borehole; for a high-rate injection well, this 

would be equal to the user-specified source 

concentration (CINJ ′ ). For nonpumping wells 

and low-rate injection wells, the MNWO 

Package will record the length-weighted 

average concentration in the borehole. At 

14 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
each multi-node well designated for 

observation, the time and concentration in the 

well after each transport time increment will 

be written to a separate output file (or files) to 

facilitate graphical postprocessing of the 

calculated data. The input file for specifying 

multi-node wells for observation is read if the 

file type (Ftype) “MNWO” is included in the 

GWT name file. 

 Multi-node wells can be designated 

for recording observed concentrations (and 

other data) through use of the site 

identification label (or name). The 

“MNWsite” parameter is an optional label 

that can be specified in the input data set for 

the MNW Package (see Halford and 

Hanson, 2002, p. 15). The MNWO 

Package, however, will only work for multinode wells that have had a unique site label 

specified in the MNW input data. 

 Although it is expected that a multinode well will include more than one node 

in the grid, it is possible and allowable for a 

single-node well to be included in the list of 

multi-node wells read by the MNW 

Package. If a single-node injection well is 

specified for observation in this Package, it 

will simply record the user-specified sourcefluid concentration, which is constant during 

a stress period. It will not record the 

concentration in the aquifer; if those are 

desired, concentrations calculated at specific 

nodes in the grid can be retrieved using the 

standard Observation Well (OBS) Package 

available for the MODFLOW-GWT model. 

Similarly, if a single-node withdrawal well 

is specified for observation in this package, 

it will record the values of aquifer 

concentration at the node corresponding to 

the location of this well (in this case, an 

identical record would be obtained using the 

OBS Package [see Konikow and others, 

1996, p. 77]). 

FOR EACH SIMULATION, IF MNWO PACKAGE USED: 

1. Data: MNWOBS 

MNWOBS Number of multi-node wells for which concentrations in wells are to be saved. 

FOR EACH MULTI-NODE WELL TO BE MONITORED: 

2. Data: MNWsite UNIT MNWOflag 

MNWsite Name of multi-node well, as identified by using “SITE:” option in input data set 

for MNW Package (see Halford and Hanson, 2002, p. 15). 

UNIT Unit number for output file. 

MNWOflag A flag to indicate what information for this particular multi-node well is written to 

its output file. If MNWOflag = 0, time and concentration in the well are saved 

and written to a separate output file. If MNWOflag = 1, additional columns of 

information about mass flux will be printed. If the well was specified as a 

nonpumping well (Qnet = 0.0), intraborehole mass flux through this well during 

each time increment and cumulative intraborehole mass flux through this well 

will be written in columns 3 and 4. Otherwise, for pumping wells, the mass 

flux removed from (or injected into) the ground-water system for the time 

increment (as calculated using eq. 6 or eq. 7, as appropriate) and cumulatively 

will be written in columns 3 and 4, respectively. Additionally, the mass flux 

into the well from the ground-water system during the time increment (from eq. 

MULTI-NODE WELL OBSERVATION PACKAGE (MNWO) 15 
4), the cumulative mass flux into the well, the mass flux out of the well and 

into the ground-water system during the time increment (from eq. 5), and the 

cumulative mass flux out of the well will be written in columns 5 through 8. If 

MNWOflag = 2, time and concentration in the well and additional data on the 

calculated concentration at every well node are saved and written to a separate 

output file (but no data on mass flux is recorded). If MNWOflag = 3, all data 

(time, concentration in the well, mass flux information, and concentration at 

individual nodes) are saved and written to the output file. 

Notes: 

MNWsite is a text string limited to 11 alphanumeric characters. A unique unit number must be 

specified for each observation well in Record 2 and matched to a DATA file type and file name in the 

GWT name file. 

OTHER OUTPUT 

CONSIDERATIONS

 The MNW Package normally prints 

information about the multi-node wells to 

the main MODFLOW listing file. This 

information is printed by the program after 

the solution to the flow equation for a given 

time step is achieved; in the output file, the 

information is written prior to (and located 

above) the calculated budget information for 

that time step. When GWT is active, 

however, these output actions are normally 

taken prior to solving the transport equation 

for that same time step. Consequently, the 

standard approach in the MNW Package to 

write relevant output information would not 

have access to data on calculated 

concentrations for that time step. Therefore, 

for cases in which the MNW Package is 

used in a transport simulation, the printing 

of the MNW information has been delayed 

until after the transport equation has been 

solved. The only effect this delay has on the 

main MODFLOW output file (or listing file) 

is to place the MNW information below the 

flow budget information (instead of above 

it). Water-quality information included in 

this output will then include concentration 

values calculated by the transport model. 

 Similarly to that described above, 

water-quality information included in other 

optional MNW output files will be based on 

updated concentration values calculated by 

GWT. The “BYNODE” file will include the 

concentration value in the well at every node 

of each multi-node well (as opposed to the 

concentration in the aquifer at the node). In 

the case of pumping multi-node wells, where 

complete mixing is assumed, the 

concentration will be the same at all nodes in a 

particular well. In the case of nonpumping 

multi-node wells, however, the concentration 

can vary between nodes in a single well. The 

“QSUM” output file includes summary 

information about each multi-node well 

(rather than for each node of a multi-node 

well). For this file, the output has been 

modified so that the written water-quality 

values will represent the average 

concentration in the well. A flux-weighted 

average is reported for pumping wells, which 

is consistent with the complete mixing model. 

A depth-weighted average concentration is 

reported for nonpumping wells. The “WEL1” 

output file has not been modified to recognize 

concentration values calculated by GWT. 

16 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
TEST PROBLEM 

Problem Description 

 A test problem was developed to 

evaluate the model and to demonstrate its 

value. This problem is a slightly modified 

version of the one documented by Reilly and 

others (1989). Their numerical experiments 

demonstrated that appreciable wellbore flow 

can occur in observation wells screened 

through multiple layers, even in 

homogeneous aquifers having small vertical 

head differences (less than 0.01 ft between 

the top and bottom of the screen). 

 The hypothetical unconfined groundwater system represents regional flow that is 

predominantly lateral, but includes some 

vertical components because of diffuse areal 

recharge (at a rate of 0.004566 ft/d) and a 

constant-head boundary condition at the 

surface of the right side of the regional 

ground-water system that controls discharge 

(fig. 8). No-flow boundaries are on all other 

external boundaries. The system is 

substantially longer (10,000 ft) than it is 

thick (205 ft) or wide (200 ft); the width was 

selected to eliminate any important effect of 

the position of that lateral no-flow boundary 

on the solution in the area of the well. A 

nonpumping borehole with a 60-ft screen is 

located close to the left side of the system 

(about 252 ft from that boundary). Other 

properties of the system and the model are 

Figure 8. Conceptual diagram showing geometry and boundaries for three-dimensional test 

problem with a nonpumping multi-node well. 

TEST PROBLEM 17 
listed in table 1. Reilly and others (1989) 

simulated the regional system with a twodimensional cross-sectional model, arguing 

that the width of the cross section was 

irrelevant for their analysis, and applied a 

local (approximately a 100-ft by 100-ft area) 

three-dimensional flow model in the vicinity 

of the wellbore. Their local model was 

discretized vertically into 5-ft layers and 

used a variably spaced areal grid with a 

minimum spacing of about 0.33 ft by 0.33 ft 

around the borehole, and they represented 

the borehole using a relatively high vertical 

hydraulic conductivity (as compared to the 

aquifer), the value of which was based on 

equivalence of Darcy’s law to the equation 

for laminar pipe flow (Reilly and others, 

1989, p. 272). 

 The approach used here was to 

simulate the regional flow system with a 

three-dimensional model for a width 

sufficient to minimize any effects of that 

dimension on the flow field close to the 

borehole. Because a vertical plane of 

symmetry is present, and passes through the 

well, only one-half of the domain outlined 

by Reilly and others (1989) was simulated 

here (with model dimensions being 10,000 ft 

long by 100 ft wide by 205 ft deep). The 

solute-transport model was applied to a local 

area around the well using a transport 

subgrid. Within the transport subgrid, which 

included 20 rows, 40 columns, and 41 layers 

of cells, a uniform areal cell spacing of 2.5 ft 

was used. Outside of the uniformly spaced 

subgrid, the lateral grid spacing was 

increased geometrically to a maximum 

spacing of 50.25 ft in the row (x-) direction 

and 9.55 ft in the column (y-) direction (see 

fig. 9). The vertical discretization (∆z) was 

5 ft everywhere in the model domain. 

 The well was assumed to have a 60-

ft well screen and was open to layers 2 

through 13 (that is, connected to 12 

vertically aligned nodes of the grid) in the 

bounding row of cells. Reilly and others 

Table 1. Selected physical and numerical parameters used in 

MODFLOW-GWT simulation of ground-water flow and transport in a 

three-dimensional, steady-state flow system containing a multi-node 

well. [ft/d, feet per day; ft, feet; ft2

/d, feet squared per day] 

 Parameter Value 

Horizontal hydraulic conductivity 250 ft/d 

Vertical hydraulic conductivity 50 ft/d 

Well radius 0.133 ft 

Well “skin” 2.6 

Effective porosity 0.20 

Longitudinal dispersivity 0.0 ft 

Horizontal transverse dispersivity 0.0 ft 

Vertical transverse dispersivity 0.0 ft 

Diffusion coefficient 0.0 ft2

/d 

Recharge rate 0.004566 ft/d 

CELDIS (Courant number criteria) 1.0 

NPTPND (initial number of particles per node) 8 or more1

1

 The value of NPTPND was increased to 275 in a block of cells 

surrounding the well. 

18 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
(1989) reported that their well was 

represented by a cell having areal 

dimensions of 0.333 ft by 0.333 ft, which 

yields a cross-sectional area of 0.111 ft2. 

Because it is assumed that the well lies on 

the plane of symmetry in the grid, a well 

radius is assigned in the model that yields an 

equivalent cross-sectional area to one-half of 

the cross-sectional area of the well in the 

simulation of Reilly and others (0.0555 ft2

). 

For a well with a circular casing, this 

equivalent cross-sectional area would 

require a well radius of 0.133 ft. It also was 

assumed that there would be a linear wellloss coefficient represented by a 

nondimensional skin coefficient (see 

Halford and Hanson, 2002). The value of 

the “skin” was adjusted to achieve a vertical 

profile of flows between the aquifer and the 

well that closely matched that of Reilly and 

others (1989, fig. 2). 

 Reilly and others (1989) were 

concerned with “two situations that 

represent extreme possibilities for the 

misrepresentation of aquifer contamination 

from well samples.” They did not, however, 

actually simulate solute transport in their 

analysis of these two cases. For this report, 

only their first situation was evaluated, in 

which they postulate a contaminant plume in 

the shallow part of the aquifer intersects the 

upper part of the well screen in the 

monitoring wellbore. For this case, an 

initial mass of contaminant was placed in 

layer 1, immediately upgradient from the 

borehole, as an initial condition for the 

solution of the transport equation in the 

presence of a multi-node well. The solute 

was assigned an initial concentration of 100 

in the 32 cells of layer 1 located within rows 

26 to 30 and columns 33 to 40. The 

simulation was then run for a stress period 

length of 1.0 year, assuming steady-state 

flow prevailed. 

Figure 9. Map view of MODFLOW finite-difference grid showing location of transport subgrid 

and nonpumping multi-node well in the test problem. 

 Heads were calculated using the 

PCG2 solver in MODFLOW-2000. 

Concentrations were calculated using the 

MOCWT method in MODFLOW-GWT, 

which is a variant of the method of 

characteristics that uses volume-weighted 

particles to track advection. Selected 

numerical parameters are also listed in table 

1. In applying the MOCWT method, a 

background particle density of 8 particles 

per cell was used, but a higher density of 

particles (up to 275 particles per cell) was 

used in the vicinity of the well. 

TEST PROBLEM 19 
Model-Simulation Results 

 The flow model converged to a 

steady-state head distribution in 225 

iterations with a 0.00 percent discrepancy. 

The head in the well was calculated to be 

4.932 ft and the head distribution in the 

aquifer near the nonpumping multi-node 

well indicates that water should flow from 

the aquifer into the upper part of the 

borehole and discharge back into the aquifer 

through the lower part of the well (fig. 10). 

The MNW Package calculates the flow 

between the aquifer and each node of the 

borehole and writes these data to a separate 

output file (the MNW BYNODE output 

file). These flows along the length of the 

borehole are plotted in figure 11, and the 

results show an excellent match to those 

published by Reilly and others (1989, fig. 2). 

Inflow to the well is greatest near the top of 

the well screen and outflow is greatest near 

the bottom of the well screen. The 

calculated total flow into the borehole was 

9.79 ft3

/d, which compares closely with 9.63 

ft3

/d reported by Reilly and others (1989). 

(Note that to adjust for using symmetry to 

simulate half the original domain size, the 

indicated flow rates were calculated by 

doubling flows calculated with the reduced 

domain size.) 

Although Reilly and others (1989) 

hypothesized that the intraborehole flow 

Figure 10. Calculated head distribution in vertical section near well on plane of symmetry (shown 

in Fig. 8). For clarity, only upgradient 5 percent of domain is shown (flow model domain extends 

to 10,000 feet in x-direction). 

20 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
would have a substantial effect on 

contaminant spreading, they did not provide 

any quantitative analysis to demonstrate that 

effect. The revised MODFLOW-GWT code 

documented in this report enables such a 

quantitative analysis to be performed. The 

results show that a contaminant plume 

initially located only in the upper 5 ft of the 

aquifer rapidly spread to depths of 40 to 60 

ft in the aquifer because of downward flow 

in the borehole open to the deeper parts of 

the aquifer (fig. 12). The initial slug of 

contaminant in layer one moves downward 

over time in response to the local velocity 

field and is reduced in size and mass in layer 

1 because of capture by the well and because 

there is no source adding new contaminants 

at the water table. The reduction in 

contaminant mass in the shallowest part of 

the aquifer is matched by the increase in 

mass in the deeper part of the aquifer—to 

depths where the contaminant would not 

have been expected if not for the pathway 

provided by the open borehole. In 1 year, 

high concentrations (greater than 1.0 percent 

of the source concentration) had spread 

about 21 ft downgradient from the borehole, 

12 ft upgradient from the borehole, and 15 ft 

laterally from the borehole at depths of 55 to 

60 ft in the aquifer (figs. 12 and 13). The 

concentration profiles shown in figures 12B 

and 12C include some apparent dispersion 

induced by the smoothing function inherent 

in the visualization software. 

 The MNWO Package was used to 

track the depth-averaged relative 

concentration in the multi-node well, as well 

as solute mass flux through the borehole. 

The changes in concentration with time are 

shown in figure 14, which was plotted 

directly from data in the MNWO output file. 

The oscillations in the breakthrough curve 

are related to the discrete nature of the 

particles used to track advective transport in 

the method of characteristics. In this case, 

particles with both low and high 

concentrations converged on the uppermost 

well nodes from all directions. Because a 

discrete and limited number of particles 

were used, the particles with low and high 

relative concentrations entered well cells at 

different ratios each time increment, 

resulting in oscillations over time in the 

mixed (or averaged) concentrations in the 

well nodes that represent fluid sinks to the 

aquifer. The oscillations are greatest when 

dispersion is zero (as in this example); 

oscillations can be reduced with higher 

dispersion and when using a greater density 

of particles. 

Figure 11. Adjusted flow rates between the 

aquifer and the borehole with depth 

calculated with the Multi-Node Well Package 

in MODFLOW for the test problem. 

 In the example described above, the 

MNWO output file records the depthaveraged concentration in the well because 

the well was a nonpumping one. For the 

case of a nonpumping multi-node well, the 

user may be interested in evaluating the 

model-calculated variation in concentration 

in the well. These data are contained in the 

“BYNODE” MNW output file 

(“wellbore.byn” in this example problem). 

These results for this example are shown in 

TEST PROBLEM 21 
figure 15, and the profile reflects that inflow 

has the highest relative concentration in the 

uppermost well node and that the 

concentration doesn’t change in the deeper 

part of the borehole where there is outflow 

from the well to the aquifer. 

Figure 12. Calculated spreading of contaminant in presence of multi-node well showing relative 

concentrations in vertical section along row 30 of grid (representing the plane of symmetry): (A) 

initial conditions (C0=100), (B) plume at 3 months, and (C) plume after 1 year. Note that only 

part of the grid in the column direction is shown; concentrations less than 1.0 are not shown. 

22 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
Figure 13. Three-dimensional perspective of model transport domain for the test problem 

showing simulated contaminant plume emanating from lower part of well after 1 year 

(concentrations < 1.0 not shown). 

Figure 14. Calculated depth-averaged 

concentration in nonpumping multi-node 

well in the test problem. 

Figure 15. Concentration variations within 

the borehole of the nonpumping multinode well in the test problem after 1 year. 

TEST PROBLEM 23 
SELECTED INPUT DATA AND 

PRINTED RESULTS FOR TEST 

SIMULATION 

 This simulation is designed to 

demonstrate the capability of the MultiNode Well (MNW) Package when used in 

conjunction with the Ground-Water 

Transport (GWT) Process of MODFLOW

(MODFLOW-GWT). As described above, 

the test case simulates a regional flow field 

with a single nonpumping multi-node well. 

Selected sections of several key input and 

output data files are shown below—

sometimes with annotations; gaps in the 

listings are indicated by an ellipsis. Not all 

files are shown. A complete set of these 

files is available for distribution at the Web 

sites described in the Preface. Contents of 

some files are enclosed in a border and 

explanations are noted outside of the border; 

for other files, explanations are sometimes 

included as comments following a 

semicolon on the line being explained. Font 

sizes in the following listings are sometimes 

reduced so that lines will fit within page 

margins. Information pertaining specifically 

to the MNW Package is highlighted in 

yellow. 

Selected Input Files 

 Following (enclosed in a border) are the contents of the MODFLOW name file for the test 

simulation; explanations are noted outside of border: 

File name: wellbore.nam

LIST 14 wellbore.lst ← Output file for MODFLOW-2000

DIS 15 wellbore.dis ← Input file for Discretization Package

BAS6 18 wellbore.bas ← Input file for Basic Package

OC 19 wellbore.oc ← Input file for Output Control option

LPF 20 wellbore.lpf ← Input file for Layer-Property Flow Package

DATA(BINARY) 2 calced.bhd OLD ← Initial head data

RCH 21 wellbore.rch ← Input file for Recharge Package

PCG 22 wellbore.pcg ← Input file for PCG Package

MNW1 23 wellbore.mnw ← Input file for Multi-Node Well (MNW) Package 

GWT 24 wellbore.gwt ← Name file for Ground-Water Transport Process

DATA(BINARY) 4 wellbore.bhd REPLACE ← Output file for calculated heads (binary)

DATA(BINARY) 33 wellbore.bud REPLACE ← Output file for cell-by-cell budget data

 ↑ ↑ ↑ ↑ 

 1 2 3 4 

1 Ftype (the type of file) 

2 Unit number 

3 File name (name chosen to reflect contents of file) 

4 Option to replace any existing file with this name 

24 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
 Following are the contents of the Multi-Node Well (MNW) Package input file for the test 

simulation; explanations are noted outside of border (see Halford and Hanson, 2002, for more 

detailed explanation): 

File name: wellbore.mnw 

12 -1 0 Reference SP = 1 ← 1

SKIN ← 2

FILE:wellbore.byn BYNODE:11 ← 3

FILE:wellbore.qsu QSUM:12 ← 4

12 ← 5

 2 30 41 0. 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 3 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 4 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 5 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 6 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 7 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 8 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

 9 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

10 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

11 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

12 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

13 30 41 0. MN 0.00E+000 1.333000000000E-001 2.60E+000 0.00E+000 0.00E+000 0 SITE:Well1 ← 6

1 MXMNW, IWL2CB, IWELPT, reference stress period. 

2 LOSSTYPE. 

3 Optional output: BYNODE file. 

4 Optional output: QSUM file. 

5 ITMP (number of active well nodes for this stress period). 

6 Well-node information. The first node is linked to the others by inclusion of the MN tag on each subsequent line; 

this creates a multi-node well. Data columns are: Layer, Row, Column, Qdes, (MN), QWval, Rw, Skin, Hlim, 

Href, Iqwgrp, SITE. 

Following (enclosed in a border) are the contents of the Ground-Water Transport (GWT) 

Process name file; explanations are noted outside of border:

File name: wellbore.gwt

CLST 25 wellbore.out ← Output file for Ground-Water Transport Process

MOCWT 26 wellbore.moc ← Input file for GWT using volume-weighted MOC transport solver

IPDA 88 wellbore.ipda ← Input file for spatially varying initial particles

BFLX 27 wellbore.bflx ← Input file for boundary flux package

CRCH 28 wellbore.crc ← Input file for concentration of recharge

CNCA 29 wellbore.cna ← Output file for calculated concentrations

VELA 30 wellbore.vla ← Output file for calculated velocities

MNWO 33 wellbore.mno ← Input file for MNW observation well records

DATA 47 wellbore.mn1 ← Output file for MNW observation well records

 ↑ ↑ ↑ 

 1 2 3 

1 Ftype (the type of file) 

2 Unit number 

3 File name (name chosen to reflect contents of file) 

SELECTED INPUT DATA AND PRINTED RESULTS FOR TEST SIMULATION 25 
Following (enclosed in a border) are the contents of the Multi-Node Well Observation 

(MNWO) Package input file for the test simulation; explanations are noted as comments at the 

end of each record: 

File name: wellbore.mno

 1 ; MNWOBS, MNW Observation well data 

 Well1 47 1 ; site, unit number, MNWOflag 

 Selected Output Files 

 Following are the contents of the main MODFLOW output file for the test simulation: 

File name: wellbore.lst

 MODFLOW-2000 

 U.S. GEOLOGICAL SURVEY MODULAR FINITE-DIFFERENCE GROUND-WATER FLOW MODEL 

 VERSION 1.15.01 04/05/2005 

 This model run combines GLOBAL and LIST output into this single file. 

 GLOBAL LISTING FILE: wellbore.lst 

 UNIT 14 

 OPENING wellbore.dis 

 FILE TYPE:DIS UNIT 15 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.bas 

 FILE TYPE:BAS6 UNIT 18 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.oc 

 FILE TYPE:OC UNIT 19 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.lpf 

 FILE TYPE:LPF UNIT 20 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING calced.bhd 

 FILE TYPE:DATA(BINARY) UNIT 2 STATUS:OLD 

 FORMAT:BINARY ACCESS:SEQUENTIAL 

 OPENING wellbore.rch 

 FILE TYPE:RCH UNIT 21 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

26 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
 OPENING wellbore.pcg 

 FILE TYPE:PCG UNIT 22 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.mnw 

 FILE TYPE:MNW1 UNIT 23 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.gwt 

 FILE TYPE:GWT UNIT 24 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.bhd 

 FILE TYPE:DATA(BINARY) UNIT 4 STATUS:REPLACE 

 FORMAT:BINARY ACCESS:SEQUENTIAL 

 OPENING wellbore.bud 

 FILE TYPE:DATA(BINARY) UNIT 3 STATUS:REPLACE 

 FORMAT:BINARY ACCESS:SEQUENTIAL 

 THE FREE FORMAT OPTION HAS BEEN SELECTED 

 DISCRETIZATION INPUT DATA READ FROM UNIT 15 

 # Data read from D:\Data Sets\MNW\wellbore\GWT_MNW\wellbore.dis 

 # File generated on 5/25/2005 10:11:10 AM 

 41 LAYERS 30 ROWS 272 COLUMNS 

 1 STRESS PERIOD(S) IN SIMULATION 

 MODEL TIME UNIT IS DAYS 

 MODEL LENGTH UNIT IS FEET 

 

 U.S. GEOLOGICAL SURVEY 

 Ground-Water Transport Process (GWT) 

 GWT (Version 2.4 BETA) 12/05/2005 

 GWT BASIC INPUT READ FROM UNIT 26 

 GWT OUTPUT ON FILE UNIT 25 

 THE GROUND-WATER TRANSPORT PROCESS IS ACTIVE 

 THE OBSERVATION PROCESS IS INACTIVE 

 THE SENSITIVITY PROCESS IS INACTIVE 

 THE PARAMETER-ESTIMATION PROCESS IS INACTIVE 

 MODE: FORWARD 

 ... 

MNW1 -- MULTI-NODE WELL PACKAGE, VERSION 1, 8/13/2002. 

 INPUT READ FROM UNIT 23 

MAXIMUM OF 12 WELLS 

CELL-BY-CELL FLOWS WILL BE PRINTED WHEN ICBCFL NOT 0 

 The heads at the beginning of SP: 1 will be the default reference elevations. 

Flow rates will not be estimated after the9999th iteration 

 A BYNODE data input file will be written to wellbore.byn on unit 11 

SELECTED INPUT DATA AND PRINTED RESULTS FOR TEST SIMULATION 27 
 A QSUM data input file will be written to wellbore.qsu on unit 12 

334560 ELEMENTS IN X ARRAY ARE USED FOR MNW1

 ... 

 VOLUMETRIC BUDGET FOR ENTIRE MODEL AT END OF TIME STEP 1 IN STRESS PERIOD 1 

 ------------------------------------------------------------------------------ 

 CUMULATIVE VOLUMES L**3 RATES FOR THIS TIME STEP L**3/T 

 ------------------ ------------------------ 

 IN: IN: 

 --- --- 

 STORAGE = 0.0000 STORAGE = 0.0000 

 CONSTANT HEAD = 0.0000 CONSTANT HEAD = 0.0000 

 RECHARGE = 1642883.8750 RECHARGE = 4501.0518 

 MNW = 1786.9209 MNW = 4.8957 

 TOTAL IN = 1644670.7500 TOTAL IN = 4505.9473 

 OUT: OUT: 

 ---- ---- 

 STORAGE = 0.0000 STORAGE = 0.0000 

 CONSTANT HEAD = 1642883.7500 CONSTANT HEAD = 4501.0513 

 RECHARGE = 0.0000 RECHARGE = 0.0000 

 MNW = 1786.9209 MNW = 4.8957 

 TOTAL OUT = 1644670.6250 TOTAL OUT = 4505.9468 

 IN - OUT = 0.1250 IN - OUT = 4.8828E-04 

 PERCENT DISCREPANCY = 0.00 PERCENT DISCREPANCY = 0.00 

 TIME SUMMARY AT END OF TIME STEP 1 IN STRESS PERIOD 1 

 SECONDS MINUTES HOURS DAYS YEARS 

 ----------------------------------------------------------- 

 TIME STEP LENGTH 3.15360E+07 5.25600E+05 8760.0 365.00 0.99932 

 STRESS PERIOD TIME 3.15360E+07 5.25600E+05 8760.0 365.00 0.99932 

 TOTAL TIME 3.15360E+07 5.25600E+05 8760.0 365.00 0.99932 

 MNW PERIOD = 1 STEP = 1 

 Entry LAY ROW COL Q H-Well H-Cell DD QW-Avg s-LINEAR s-NonLINEAR 

 1 2 30 41 -1.71315 4.93222 4.93308 4.93222 32.7160 0.853341E-03 0.00000 

 2 3 30 41 -1.24043 4.93222 4.93284 4.93222 27.7536 0.617890E-03 0.00000 

 3 4 30 41 -.906686 4.93222 4.93267 4.93222 21.2355 0.451660E-03 0.00000 

 4 5 30 41 -.613199 4.93222 4.93253 4.93222 18.3247 0.305481E-03 0.00000 

 5 6 30 41 -.340668 4.93222 4.93239 4.93222 17.0279 0.169739E-03 0.00000 

 6 7 30 41 -.815486E-01 4.93222 4.93226 4.93222 16.7443 0.406772E-04 0.00000 

 7 8 30 41 0.168571 4.93222 4.93214 4.93222 16.7443 -.839019E-04 0.00000 

 8 9 30 41 0.413198 4.93222 4.93202 4.93222 16.7443 -.205745E-03 0.00000 

 9 10 30 41 0.656765 4.93222 4.93190 4.93222 16.7443 -.327061E-03 0.00000 

 10 11 30 41 0.906529 4.93222 4.93177 4.93222 16.7443 -.451463E-03 0.00000 

 11 12 30 41 1.18058 4.93222 4.93163 4.93222 16.7443 -.587961E-03 0.00000 

 12 13 30 41 1.57003 4.93222 4.93144 4.93222 16.7443 -.781940E-03 0.00000 

 Multi-Node Rates & Average QW 

 Site Identifier ENTRY: Begin - End Q-Total H-Well DD QW-Avg 

Well1 1 12 -.374478E-10 4.93222 4.93222 20.3967 

28 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
Following are the contents of the main Ground-Water Transport Process output file for 

the test simulation: 

File name: wellbore.out

 

 

 U.S. GEOLOGICAL SURVEY 

 Ground-Water Transport Process (GWT) 

 GWT (Version 2.4 BETA) 12/05/2005 

 LISTING FILE: wellbore.out UNIT 25 

 OPENING wellbore.mocIPDA 

 FILE TYPE:MOCWT UNIT 26 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.ipda 

 FILE TYPE:IPDA UNIT 88 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.bflx 

 FILE TYPE:BFLX UNIT 27 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.crc 

 FILE TYPE:CRCH UNIT 28 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.cna 

 FILE TYPE:CNCA UNIT 29 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.vla 

 FILE TYPE:VELA UNIT 30 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.mno 

 FILE TYPE:MNWO UNIT 33 STATUS:OLD 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 OPENING wellbore.mn1 

 FILE TYPE:DATA UNIT 47 STATUS:UNKNOWN 

 FORMAT:FORMATTED ACCESS:SEQUENTIAL 

 GWT BASIC INPUT READ FROM UNIT 26 

 Test simulation for GWT/MNW 

 Wellbore project 

 MAPPING OF SOLUTE-TRANSPORT SUBGRID IN FLOW GRID: 

 FIRST LAYER FOR SOLUTE TRANSPORT = 1 LAST LAYER FOR SOLUTE TRANSPORT = 41 

 FIRST ROW FOR SOLUTE TRANSPORT = 11 LAST ROW FOR SOLUTE TRANSPORT = 30 

 FIRST COLUMN FOR SOLUTE TRANSPORT= 21 LAST COLUMN FOR SOLUTE TRANSPORT = 60 

 UNIFORM DELCOL AND DELROW IN SUBGRID FOR SOLUTE TRANSPORT 

SELECTED INPUT DATA AND PRINTED RESULTS FOR TEST SIMULATION 29 
 NO. OF LAYERS = 41 NO. OF ROWS = 20 NO. OF COLUMNS = 40 

 NO SOLUTE DECAY 

 NO MOLECULAR DIFFUSION 

 

 BOUNDARY FLUX PACKAGE ACTIVATED 

 NO ET IN SIMULATION: IEVTTP NOT READ 

 IRCHTP= 1 

 RECHARGE APPLIED AS BOUNDARY FLUX ON TOP FACE 

 ...

SITE ID FOR 1 MNW WELL DESIGNATED FOR OBSERVATION: 

 WELL # SITE ID UNIT MNWOflag 

 1 WELL1 47 1 

DATA FOR MNW WELLS DESIGNATED FOR OBSERVATION WILL BE WRITTEN ON UNIT NUMBERS LISTED ABOVE 

 MNWO OUTPUT: 

 MNWOflag = 0: TIME, CONCENTRATION AT WELL 

 MNWOflag = 1: TIME, CONCENTRATION AT WELL, PLUS EXTENDED MNW MASS FLUX INFO 

 MNWOflag = 2: TIME, CONCENTRATION AT WELL, PLUS CONCENTRATIONS AT WELL NODES 

 MNWOflag = 3: TIME, CONCENTRATION AT WELL, MASS FLUX INFO, CONC. AT WELL NODES 

 ...

 MNW CONCENTRATIONS FOR THIS STRESS PERIOD 

 (ONLY FOR MNW NODES WITHIN TRANSPORT SUBGRID) 

 EXTERNAL 

 WELL NO. LAYER ROW COLUMN VOL/T SOURCE CONC SITEID 

 ------------------------------------------------------------------- 

 1 2 41 30 -1.7131E+00 0.0000E+00 Well1 

 2 3 41 30 -1.2404E+00 Well1 

 3 4 41 30 -9.0669E-01 Well1 

 4 5 41 30 -6.1320E-01 Well1 

 5 6 41 30 -3.4067E-01 Well1 

 6 7 41 30 -8.1549E-02 Well1 

 7 8 41 30 1.6857E-01 Well1 

 8 9 41 30 4.1320E-01 Well1 

 9 10 41 30 6.5676E-01 Well1 

 10 11 41 30 9.0653E-01 Well1 

 11 12 41 30 1.1806E+00 Well1 

 12 13 41 30 1.5700E+00 Well1 

 ...

 SOLUTE BUDGET AND MASS BALANCE FOR TRANSPORT SUBGRID 

 VALUES CALCULATED AT END OF: 

 STRESS PERIOD 1 OUT OF 1 

 FLOW TIME STEP 1 OUT OF 1 

 TRANSPORT TIME INCREMENT 35 OUT OF 35 

 ELAPSED TIME = 3.6500E+02 

 

30 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
 External MNW mass flux (based on wellhead flow) (L**3)(M/VOL) 

 ------------------------------------------------------------- 

 Cumulative mass flux into aquifer by MNWs = 0.0000E+00 

 Cumulative mass flux out of aquifer by MNWs = 0.0000E+00 

 

 CHEMICAL MASS IN STORAGE: 

 INITIAL: MASS DISSOLVED = 3.9739E+04 MASS SORBED = 0.0000E+00 

 PRESENT: MASS DISSOLVED = 3.9744E+04 MASS SORBED = 0.0000E+00 

 ...

 CHANGE IN MASS STORED = 4.6619E+00 

 CUMULATIVE SOLUTE MASS (L**3)(M/VOL) 

 ---------------------- 

 IN: 

 --- 

 CONSTANT HEAD = 0.0000E+00 

 SUBGRID BOUNDARY = 0.0000E+00 

 RECHARGE = 0.0000E+00 

 MULTI-NODE WELLS = 2.2566E+04 

 TOTAL IN = 2.2566E+04 

 OUT: 

 ---- 

 CONSTANT HEAD = 0.0000E+00 

 SUBGRID BOUNDARY = 0.0000E+00 

 RECHARGE = 0.0000E+00 

 MULTI-NODE WELLS = -2.2566E+04 

 TOTAL OUT:WEIGHTED PTS= -2.2566E+04 

 PARTICLE-BASED RESIDUAL = -4.6626E+00 

 PERCENT DISCREPANCY = 2.0662E-02 RELATIVE TO MASS FLUX OUT, BASED ON PTS. 

 Following are the contents of the MNWO output file for the test simulation. The 

concentration data (“WELL-CONC”) included in this output file are also plotted in figure 14. 

File name: wellbore.mn1 

 DATA FOR MNW WELL DESIGNATED FOR OBSERVATION 

 SITE Well1 

 TIME WELL-CONC MASS FLUX IN BOREHOLE MASS INTO BOREHOLE MASS OUT OF BOREHOLE 

 PRESENT t CUM. PRESENT t CUM. PRESENT t CUM. 

 1.0429E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 

 2.0857E+01 5.9829E-01 2.2592E+01 2.2592E+01 -2.2592E+01 -2.2592E+01 2.2592E+01 2.2592E+01 

 3.1286E+01 3.3502E+00 1.2651E+02 1.4910E+02 -1.2651E+02 -1.4910E+02 1.2651E+02 1.4910E+02 

 4.1714E+01 4.3551E+00 1.6445E+02 3.1356E+02 -1.6445E+02 -3.1356E+02 1.6445E+02 3.1356E+02 

 5.2143E+01 6.5589E+00 2.4767E+02 5.6123E+02 -2.4767E+02 -5.6123E+02 2.4767E+02 5.6123E+02 

 6.2571E+01 7.7873E+00 2.9406E+02 8.5529E+02 -2.9406E+02 -8.5529E+02 2.9406E+02 8.5529E+02 

... 

SELECTED INPUT DATA AND PRINTED RESULTS FOR TEST SIMULATION 31 
 3.1286E+02 2.1011E+01 8.4828E+02 1.8247E+04 -8.4828E+02 -1.8247E+04 8.4828E+02 1.8247E+04 

 3.2329E+02 2.0844E+01 8.4451E+02 1.9092E+04 -8.4451E+02 -1.9092E+04 8.4451E+02 1.9092E+04 

 3.3371E+02 2.1459E+01 8.9331E+02 1.9985E+04 -8.9331E+02 -1.9985E+04 8.9331E+02 1.9985E+04 

 3.4414E+02 2.1013E+01 8.7358E+02 2.0859E+04 -8.7358E+02 -2.0859E+04 8.7358E+02 2.0859E+04 

 3.5457E+02 2.0525E+01 8.5248E+02 2.1711E+04 -8.5248E+02 -2.1711E+04 8.5248E+02 2.1711E+04 

 3.6500E+02 2.0397E+01 8.5488E+02 2.2566E+04 -8.5488E+02 -2.2566E+04 8.5488E+02 2.2566E+04 

 Following are the contents of the BYNODE output file for the test simulation. The 

concentration data (QW-Avg) included in this output file represent nodal values in the multinode well and are also plotted in figure 15. 

File name: wellbore.byn 

SiteID Entry NODE Total_Time Q H-Well H-Cell QW-Avg 

... 

Well1 1 16089 365.00000 -1.7131450 4.9322224 4.9330757 32.716038 

Well1 2 24249 365.00000 -1.2404266 4.9322224 4.9328403 27.753590 

Well1 3 32409 365.00000 -.90668571 4.9322224 4.9326740 21.235537 

Well1 4 40569 365.00000 -.61319940 4.9322224 4.9325278 18.324675 

Well1 5 48729 365.00000 -.34066847 4.9322224 4.9323921 17.027941 

Well1 6 56889 365.00000 -.81548553E-01 4.9322224 4.9322630 16.744301 

Well1 7 65049 365.00000 0.16857103 4.9322224 4.9321385 16.744301 

Well1 8 73209 365.00000 0.41319797 4.9322224 4.9320166 16.744301 

Well1 9 81369 365.00000 0.65676482 4.9322224 4.9318953 16.744301 

Well1 10 89529 365.00000 0.90652868 4.9322224 4.9317709 16.744301 

Well1 11 97689 365.00000 1.1805792 4.9322224 4.9316344 16.744301 

Well1 12 105849 365.00000 1.5700321 4.9322224 4.9314404 16.744301 

EndOfFile__EndOfFile__EndOfFile_ 

 Following are the contents of the QSUM output file for the test simulation showing 

summary values for the entire multi-node well. 

File name: wellbore.qsu 

SiteID Entry Total_Time Qin Qout Qsum H-Well QW-Avg 

Well1 00001-00012 

Well1 00001-00012 365.00000 -.37447823E-10 0.0 -.37447823E-10 4.9322224 20.396748 

EndOfFile__EndOfFile__EndOfFile_ 

SUMMARY AND CONCLUSIONS 

 The Multi-Node Well (MNW) 

Package allows MODFLOW to simulate 

wells that extend beyond a single model 

node. Because multi-node wells can allow 

nonuniform flow to occur within the 

borehole, solute can be transported directly 

from one model node to another through the 

well. In some cases, this intraborehole 

transport can facilitate the movement of 

contaminants through a ground-water 

system and thereby needs to be recognized 

when calculating changes in concentration 

in the system. The MODFLOW-GWT

solute-transport model was modified to 

32 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT
provide compatibility with the MNW 

Package. If the flow between a multi-node 

well and the model domain is in the same 

direction at all well nodes, the solute 

concentration in the borehole is assumed to 

be uniform and consistent with an 

assumption of complete mixing within the 

borehole during a given time increment. If 

the flow between a multi-node well and the 

model domain is complex and into the well 

at some nodes while out of the well at other 

nodes, the solute concentration in the 

borehole is computed using a routing 

algorithm that only assumes local mixing at 

each node in the well during a given time 

increment. 

A test problem was designed to 

simulate ground-water flow and solute 

transport in a ground-water system that 

contains a contaminant plume near its top 

surface and a borehole extending through 

multiple model layers (represented as a 

multi-node well). In the test problem, the 

flow model converged to a steady-state head 

distribution with a 0.00 percent discrepancy. 

Model-simulated flows along the length of 

the borehole compare closely with 

previously reported results. MODFLOWGWT was used to analyze solute transport 

through the borehole and demonstrate that 

intraborehole flow has a substantial effect on 

simulated contaminant spreading. Changes 

in contaminant concentration with time 

indicate that the presence of the long 

borehole enabled a secondary contaminant 

plume to develop at a greater depth in the 

ground-water system than would have 

occurred if the well did not exist or had been 

effectively plugged or sealed. Coincident 

with the solute transfer to greater depths 

through the borehole is a reduction in solute 

mass in the shallowest part of the groundwater system. 

 This report documents the 

assumptions, methods, and input data 

requirements for using the MODFLOWGWT model to simulate solute transport in a 

ground-water system containing a multinode well. This enhancement improves the 

capability of MODFLOW-GWT to 

accurately simulate transport in cases that 

involve fluxes to or from multi-node wells. 

The test problem confirmed the numerical 

accuracy of the revised model. 

ACKNOWLEDGMENTS 

 The authors appreciate the helpful 

review comments provided by USGS 

colleagues Keith Halford, Brian Clark, 

Randy Hanson, Tom Reilly, and Angel 

Martin, Jr. Halford also helped the authors 

to better understand the source code for the 

MNW Package. 

REFERENCES CITED 

Church, P.E., and Granato, G.E., 1996, Bias 

in ground-water data caused by well-bore 

flow in long-screen wells: Ground Water, 

v. 34, no. 2, p. 262-273. 

Halford, K.J., and Hanson, R.T., 2002, 

MODFLOW-2000, User guide for the 

drawdown-limited, Multi-Node Well 

(MNW) Package for the U.S. Geological 

Survey’s modular three-dimensional 

ground-water flow model, Versions 

MODFLOW-96 and MODFLOW-2000: 

U.S. Geological Survey Open-File Report 

02-293, 33 p. 

Harbaugh, A.W., Banta, E.R., Hill, M.C., 

and McDonald, M.G., 2000, 

MODFLOW-2000, The U.S. Geological 

Survey modular ground-water model—

User guide to modularization concepts 

and the ground-water flow process: U.S. 

Geological Survey Open-File Report 00-

92, 121 p. 

REFERENCES CITED 33 
Heberton, C.I., Russell, T.F., Konikow, L.F., 

and Hornberger, G.Z., 2000, A threedimensional finite-volume EulerianLagrangian Localized Adjoint Method 

(ELLAM) for solute-transport modeling: 

U.S. Geological Survey Water-Resources 

Investigations Report 00-4087, 63 p. 

Izbicki, J.A., Christensen, A.H., Newhouse, 

M.W., Smith, G.A., and Hanson, R.T., 

2005, Temporal changes in the vertical 

distribution of flow and chloride in deep 

wells: Ground Water, v. 43, no. 4, p. 531-

544. 

Kipp, K.L., Jr., Konikow, L.F., and 

Hornberger, G.Z., 1998, An implicit 

dispersive transport algorithm for the 

U.S. Geological Survey MOC3D solutetransport model: U.S. Geological Survey 

Water-Resources Investigations Report 

98-4234, 54 p. 

Konikow, L.F., Goode, D.J., and 

Hornberger, G.Z., 1996, A threedimensional method of characteristics 

solute-transport model (MOC3D): U.S. 

Geological Survey Water-Resources 

Investigations Report 96-4267, 87 p. 

Konikow, L.F., and Hornberger, G.Z., 2003, 

Use of boundary fluxes when simulating 

solute transport with the MODFLOW 

ground-water transport process: U.S. 

Geological Survey Open-File Report 03-

303, 17 p. 

Merritt, M.L., and Konikow, L.F., 2000, 

Documentation of a computer program 

to simulate lake-aquifer interaction using 

the MODFLOW ground-water flow 

model and the MOC3D solute-transport 

model: U.S. Geological Survey WaterResources Investigations Report 00-

4167, 146 p. 

Paillet, F.L., Williams, J.H., Oki, D.S., and 

Knutson, K.D., 2002, Comparison of 

formation and fluid-column logs in a 

heterogeneous basalt aquifer: Ground 

Water, v. 40, no. 6, p. 577-585. 

Prudic, D.E., Konikow, L.F., and Banta, 

E.R., 2004, A new streamflow-routing 

(SFR1) package to simulate streamaquifer interaction with MODFLOW2000: U.S. Geological Survey OpenFile Report 2004-1042, 95 p. 

Reilly, T.E., Franke, O.L., and Bennett, 

G.D., 1989, Bias in groundwater 

samples caused by wellbore flow: 

Journal of Hydraulic Engineering, v. 

115, no. 2, p. 270-276. 

34 USE OF MULTI-NODE WELL PACKAGE WHEN SIMULATING SOLUTE TRANSPORT