Transient Simulation of Groundwater Levels within a 

Sandbar of the Colorado River, Marble Canyon, Arizona, 

2004

By Thomas A. Sabol and Abraham E. Springer

Open-File Report 2013–1277

U.S. Department of the Interior

U.S. Geological Survey
ii

U.S. Department of the Interior

Sally Jewell, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2013

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Suggested citation:

Sabol, T.A., and Springer, A.E., 2013, Transient simulation of groundwater levels within a sandbar of the Colorado 

River, Marble Canyon, Arizona, 2004: U.S. Geological Survey Open-File Report 2013-1277, 22 p., 

http://dx.doi.org/10.3133/ofr20131277. 

ISSN 2331-1258 (online)

Any use of trade, product, or firm names is for descriptive purposes only and does not imply 

endorsement by the U.S. Government.

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.

Cover: Eddy sandbar at river mile 30.7R after recession of the 2004 high-flow experiment, Colorado River, Marble 

Canyon, Arizona. Visible are piezometers in the bar face regions of the downstream part of the sandbar.
iii

Contents

Abstract .......................................................................................................................................................................1

Introduction and Background ......................................................................................................................................1

Purpose and Scope.................................................................................................................................................4

Study Sites ..............................................................................................................................................................4

Data Collection............................................................................................................................................................5

Groundwater Model.....................................................................................................................................................7

Model Framework....................................................................................................................................................7

Hydraulic Conductivity .........................................................................................................................................9

Storage Coefficient ............................................................................................................................................11

Initial Head Values.............................................................................................................................................11

Boundary Conditions..........................................................................................................................................12

Model Calibration...................................................................................................................................................13

Calibration Criteria and Model Error...................................................................................................................13

Model Results........................................................................................................................................................14

Conclusion ................................................................................................................................................................18

References Cited ......................................................................................................................................................19

Figures

Figure 1. Map of the Colorado River in Glen, Marble and Grand Canyons, Arizona..............................................2

Figure 2. Continuous discharge record for the river mile 30.7 sandbar study site from November 20, 2004

 through January 20, 2005. .....................................................................................................................5

Figure 3. Plan view image of the river mile 30.7R, Colorado River in Marble Canyon, Arizona.............................6

Figure 4. Matched oblique photographic views of the river mile 30.7R sandbar study site, Colorado River

 in Marble Canyon, Arizona..................................................................................................................... 6

Figure 5. Idealized cross-section showing the eddy sandbar at river mile 30.7R, Colorado River in

 Marble Canyon, Arizona.........................................................................................................................9

Figure 6. Hydraulic conductivity (K) of model layers 1 and 2 within the three-dimensional groundwater model

 framework developed for this study, river mile 30.7R, Colorado River in Marble Canyon, Arizona......10

Figure 7. Plan view image of the river mile 30.7R eddy sandbar study site.........................................................12

Figure 8. Measured elevation of groundwater level in all piezometers at river mile 30.7R eddy sandbar site

 relative to river stage............................................................................................................................14

Figure 9. Graph plot showing measured and modeled groundwater level in piezometers...................................16

Figure 10. Graph plot showing measured and modeled groundwater level in piezometer 10................................17

Tables

Table 1. Locations and selected construction data for piezometers installed in the eddy sandbar at river

 mile 30.7R, Colorado River in Marble Canyon, Arizona. ........................................................................7

Table 2. Root mean squared error (RMSE) and mean absolute error (MAE) residual differences between the

 measured groundwater level in piezometers and the simulated model results at target locations .......15

Table 3. Root mean squared error (RMSE) and mean absolute error (MAE) for the measured groundwater

 level in piezometers and the simulated model results at piezometer 10...............................................18
iv

Conversion Factors

Inch/Pound to SI 

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

mile (mi) 1.609 kilometer (km)

Flow rate

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

SI to Inch/Pound 

Multiply By To obtain

Length

centimeter (cm) 0.3937 inch (in.)

meter (m) 3.281 foot (ft) 

kilometer (km) 0.6214 mile (mi)

Area

square meter (m2) 0.0002471 acre 

Flow rate

cubic meter per second (m3/s) 35.31 cubic foot per second (ft3/s)

Hydraulic conductivity

meter per day (m/d) 3.281 foot per day (ft/d)

Vertical coordinate information is referenced in meters to height above the GRS80 ellipse

defined by NAD83(1982).

Horizontal coordinate information is referenced to North American Datum of 1983, NAD83(1982).

Elevation, as used in this report, regers to distance above the vertical datum.
v

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Transient Simulation of Groundwater Levels within a 

Sandbar of the Colorado River, Marble Canyon, Arizona, 

2004

By Thomas A. Sabol1 and Abraham E. Springer2

Abstract

Seepage erosion and mass failure of emergent sandy deposits along the Colorado River in Grand 

Canyon National Park, Arizona, are a function of the elevation of groundwater in the sandbar, 

fluctuations in river stage, the exfiltration of water from the bar face, and the slope of the bar face. In 

this study, a generalized three-dimensional numerical model was developed to predict the time-varying 

groundwater level, within the bar face region of a freshly deposited eddy sandbar, as a function of river 

stage. Model verification from two transient simulations demonstrates the ability of the model to predict 

groundwater levels within the onshore portion of the sandbar face across a range of conditions. Use of 

this generalized model is applicable across a range of typical eddy sandbar deposits in diverse settings. 

The ability to predict the groundwater level at the onshore end of the sandbar face is essential for both 

physical and numerical modeling efforts focusing on the erosion and mass failure of eddy sandbars 

downstream of Glen Canyon Dam along the Colorado River. 

Introduction and Background

Emergent sandy deposits along the Colorado River in Grand Canyon National Park, Arizona, are 

a key fixture of the ecosystem downstream of Glen Canyon Dam (fig. 1; Rubin and others, 1998, 2002; 

Topping and others, 2006). Closure of the dam in 1963 reduced the supply of sand-sized sediment in the 

Colorado River at the upstream boundary of Grand Canyon National Park, located at the mouth of the 

Paria River, by about 94 percent from a mean annual sediment load of about 60 million metric tons to 

about 3 million metric tons (Topping and others, 2000a, 2006, 2010; Draut and others, 2005); the 

remaining 6 percent of the post-dam sand supply is contributed almost exclusively by the Paria River 

(Topping and others, 2006, 2010). In addition to the reduction in the downstream supply of sand-sized 

sediment, operation of the dam has altered the natural seasonal hydrograph such that sediment no longer 

accumulates seasonally on the main-channel bed of the river during low-flow periods (pre-dam, July–

March) and sediment is no longer redeposited seasonally in lateral recirculation eddy sandbars during 

high-flood-flow periods (pre-dam, April–June) (Howard and Dolan, 1981; Rubin and others, 2002; 

Topping and others, 2003, 2010; Hazel and others, 2006a). Although the post-dam seasonal hydrograph 

has been smoothed, operation of Glen Canyon Dam to provide peak on-demand hydroelectric power 

generation has caused large diurnal fluctuations in discharge that have increased the daily range in 

discharge 99.9 percent of the time relative to the pre-dam period of record (Topping and others, 2003). 

The combination of a reduced sediment supply, an increase in the frequency of floods with peak 

 

1 U.S. Geological Survey

2 Northern Arizona University 
2

discharges between 523.6 and 820.7 m3/s, and an increase in the daily range of discharge has facilitated 

the degradation of sediment storage along the Colorado River corridor through Marble and Grand 

Canyons (see, for example, Dolan and others, 1974; Laursen and others, 1976; Clark and others, 1991; 

Beus and others, 1992; Topping and others, 2000a, 2000b, 2003, 2010; Rubin and others, 2002; Hazel 

and others, 2006a). 

Figure 1. Map of the Colorado River in Glen, Marble and Grand Canyons, Arizona, showing the location of the 

Grand Canyon Monitoring and Research Center (GCMRC) river mile (RM) 30.7R study site. Horizontal 

coordinate information is referenced to North American Datum of 1983, NAD83(1982).

Sediment storage in the pre-dam Colorado River was driven seasonally by seasonal fluctuations 

in the main-stem discharge, with storage of sediment occurring in both the main-channel bed and the 

lateral recirculation eddy deposits. In the contemporary post-dam river, the predominant proportion of 

sediment in the river is stored mainly in eddies (see, for example, Schmidt, 1990; Rubin and others, 

2002; Hazel and others, 2006a; Topping and others, 2010). For example, in Marble Canyon, in the 99-

km reach beginning with km 0 at Lees Ferry, 51–94 percent of the sediment is stored within lateral 

recirculation eddies, despite the fact that eddies constitute only about 17 percent of the flow field surface 

area (Hazel and others, 2006a). These results indicate the importance of lateral recirculation eddy 

deposits as a resource for long-term storage of sediment along the Colorado River downstream of Glen 

Canyon Dam as the degradation of fine-sediment deposits progresses throughout the downstream 

ecosystem. Because of the infrequent inundation of the high-elevation eddy deposits (that is, parts of the 

deposit emergent above the 227 m3

/s river stage), these portions of the bar have a longer response time 

to typical diurnal dam operations. As a result, high-elevation eddy deposits potentially are the last 
3

bastion of storage in a system that is in a state of active scour during times when the bed of the channel 

is not enriched in fine sediment (Schmidt and others, 2004; Hazel and others, 2006a). 

Emergent bars create terrestrial habitats for riparian vegetation and low-velocity aquatic habitats 

for fish and other benthos, act as a source of fine sediment for aeolian processes that may be integral in 

the preservation of archeological resources, and are used by boaters and other visitors for recreation 

(Rubin and others, 2002; Draut and others, 2005; Kaplinski and others, 2005; Topping and others; 2006; 

Behn and others, 2010; Ralston, 2010). Given the current decline in sediment storage of the post-dam 

Colorado River in both Marble and Grand Canyons, and the importance of eddy sandbars within the 

downstream ecosystem, efforts have been made to understand both the long-term stability of the eddy 

deposits upon emergence and the efficacy of high-flow experiments to rebuild and maintain the eddy 

deposits (Hazel and others, 2010; Schmidt and Grams, 2011a; Alvarez and Schmeeckle, 2012). 

From 1966 to 1990, the Glen Canyon Dam hydropower plant was operated to deliver water to 

downstream users, as well as to optimally meet daily demand for electrical power by supplying 

electricity at peak times of the day (typically, midday in summer and midday and early evening in 

winter). During this time period, historical releases from the dam of as much as 892 m3/s (power plant 

capacity), accompanied by unrestricted ramp rates, resulted in large fluctuations in river stage 

downstream of the dam (U.S. Department of the Interior, 1995; Alvarez and Schmeeckle, 2012). 

Scientific findings toward the end of this time period concluded that large stage fluctuations resulting 

from the unrestricted operation of Glen Canyon Dam were eroding sandbars and deteriorating the 

downstream ecosystem (U.S. Department of the Interior, 1988). As a partial response to these findings, 

the Grand Canyon Protection Act (GCPA) was passed in 1992; the “Operation of Glen Canyon Dam—

Final Environmental Impact Statement” was prepared; and the Record of Decision for alternative damoperation strategies for meeting the GCPA requirements was signed by the Secretary of the Interior in 

1996 (U.S. Department of the Interior, 1996; Schmit and Schmidt; 2011; Alvarez and Schmeeckle, 

2012). As a result of the 1996 Record of Decision, which imposes restrictions on fluctuating discharges 

from Glen Canyon Dam, a modified low fluctuating flow (MLFF) operation criteria was implemented 

that constrained minimum (141.5 m3

/s by night, 226.5 m3/s by day) and maximum (707.5 m3/s) releases, 

maximum daily fluctuations (226.5 m3/s within 24 hours), and discharge ramp rates [113.2 (m3/s)/h 

rising limb, 42.5 (m3/s)/h falling limb] (U.S. Department of the Interior, 2008; Alvarez and Schmeeckle, 

2012). Under the current MLFF operating regime of Glen Canyon Dam, typical diurnal fluctuations in 

discharge of the Colorado River result in daily stage changes of about 1.5 m in Marble Canyon (U.S. 

Department of the Interior, 1996; Hazel and others, 2006b; Alvarez and Schmeeckle, 2012). 

The erosion of sandbars is thought to occur as a dynamic interplay between seepage erosion, 

mass failure, and tractive forces within the water column (such as turbulent flow and waves) acting on 

the toe of the bar and the submerged seepage face (Beus and others, 1992; Budhu, 1992; Cluer, 1992, 

1995; Bauer and Schmidt, 1993; Budhu and Gobin, 1994, 1995; Alvarez and Schmeeckle, 2012). On the 

rising limb of the diurnal stage fluctuation, and during times of steady high flow, water infiltrates the 

sandbar and recharges the unsaturated and variably saturated pores of the sandbar aquifer skeleton. 

Conversely, during the downramp cycle, rapid decreases in stage lead to an elevated water table in 

emergent bars that, in turn, results in elevated pore water pressures and the exfiltration of water 

accompanying the drainage of water stored in the aquifer skeleton of the sandbar. Seepage erosion 

occurs when the exfiltration of water causes the entrainment and transport of unconsolidated sediment 

along the sandbar face; in some instances, seepage erosion initiates the process of mass failure (Beus 

and others, 1992; Cluer, 1992; Bauer and Schmidt, 1993; Budhu and Gobin, 1994, 1995; Fox and others, 

2006; Alvarez and Schmeeckle, 2012).

Since the 1990s, considerable scientific research has focused on the restoration and preservation 

of eddy sandbar deposits throughout the Colorado River ecosystem in Marble and Grand Canyons 

(Schmidt and Grams, 2011a, 2011b; Alvarez and Schmeeckle, 2012). High-flow experiments (1996, 

2004, 2008), with discharges about 50 percent greater than peak hydropower plant operating range, were 
4

conducted to increase the area and volume of eroded eddy sandbar resources (Schmidt and Grams, 

2011a, 2011b); the response of the high-elevation eddy sandbars to these experiments was positive in 

both area and volume (Hazel and others, 2010). However, readjustment of the newly aggraded eddy 

sandbars to typical diurnal dam operation led to system-wide erosion as the equilibrium beach face slope 

evolved; equilibrium slope values for the subaerial part of a typical Grand Canyon eddy sandbar beach 

face range from 11 to 14 degrees (Schmidt and Graf, 1990; Budhu and Gobin, 1994; Hazel and others, 

2010; Alvarez and Schmeeckle, 2012). 

The evolution of subaerial beach face sediments, from a freshly deposited steep sandbar face to a 

shallower equilibrium slope within the zone of fluctuating river stage, is governed by an interplay 

between seepage erosion, mass failure, and turbulent sediment transport within the water column (Beus 

and others, 1992; Bauer and Schmidt, 1993; Budhu and Gobin, 1994, 1995; Alvarez and Schmeeckle, 

2012). Seepage erosion and mass failure are primarily a function of the elevation of groundwater in the 

sandbar, the stage fluctuations along the sandbar face, the exfiltration of water from the sandbar face, 

and the slope of the sandbar face (Budhu and Gobin, 1994, 1995; Alvarez and Schmeeckle, 2012). 

Downstream of Glen Canyon Dam, the duration of Colorado River peak discharge typically is 

insufficient for groundwater levels in sandbars to equilibrate with peak river stage, resulting in an 

observed lag between groundwater fluctuations in sandbars and the hydrograph of the river (Cluer, 1992, 

1995; Budhu and Gobin, 1995; Carpenter and others, 1995a, 1995b; Alvarez and Schmeeckle, 2012). 

Understanding the time-varying relation between groundwater levels in the sandbar, the regulated 

hydrograph of the river, and dam operations is critical for any model, numerical or physical, where the 

onshore, groundwater-level boundary condition is required. 

Purpose and Scope

The purpose of this report is to develop a generalized numerical model for the prediction of timevarying groundwater levels, within the bar face region of a freshly deposited eddy sandbar, as a function 

of river stage. This report describes the development and calibration of a numerical model using an 

existing finite-element software package. The ability to predict the groundwater level at the onshore end 

of a sandbar face is essential for physical and numerical modeling efforts focusing on the erosion and 

mass failure of eddy sandbars downstream of Glen Canyon Dam along the Colorado River within Grand 

Canyon National Park (fig. 1). The model framework developed in this report was based on data 

collected from a single eddy sandbar (river mile [RM] 30.7R); however, use of this generalized model 

may be applicable across a range of typical eddy sandbar deposits at multiple study sites.

Study Sites

The study site is a reattachment-type sandbar (Schmidt and Graf, 1990) in an eddy of 

recirculating flow at RM 30.7R along the Colorado River in Marble Canyon within Grand Canyon 

National Park (GRCA) (fig. 1). By longstanding convention, locations along the Colorado River in 

GRCA are referenced to river miles. Marble Canyon extends from RM 0 to the mouth of the Little 

Colorado River near RM 62; Grand Canyon extends from the mouth of the Little Colorado River to the 

Grand Wash Cliffs near RM 277. Data used in this study were collected at the RM 30.7R eddy sandbar 

during two different discharge regimes regulated by Glen Canyon Dam, following a period of partial 

scour, reworking, and deposition of sediment during the Glen Canyon Dam high-flow experiment of 

November, 2004 (fig. 2; Topping and others, 2006; Melis and others, 2007, 2011; Melis, 2011; Melis 

and others, 2012).
5

Figure 2. Continuous discharge record for the river mile 30.7 sandbar study site from November 20, 2004 through 

January 20, 2005. Shown is the discharge record during (1) the 2004 high-flow experiment (HFE), (2) the return 

to modified low fluctuating flow (MLFF) operation criteria, and (3) the beginning of nonnative fish suppression 

flows (NFSF). Throughout the 39-day period after the downramp of the 2004 HFE, the discharge of water 

moving past the river mile 30.7R study site was regulated by the MLFF operational criteria of Glen Canyon 

Dam. Beginning on January ,3 2005, through the end of the discharge record shown, the discharge of water 

moving past the river mile 30.7R study site was part of the NFSF experimental test release from Glen Canyon 

Dam. The period of NFSF discharge shown illustrates the typical diurnal weekday and week end operational 

flow releases from Glen Canyon Dam characteristic of the NFSF experimental test at the river mile 30.7R study 

site; maximum daily range in discharge during this time period (up to 424.5 cubic meters per second within a 24 

hour period) exceeded the MLFF maximum daily fluctuation range of 226.5 cubic meters per second within 24 

hours.

Data Collection

In November 2004, during the recession of a high-flow experiment release from Glen Canyon 

Dam, the emergent part of a freshly deposited high-elevation eddy sandbar at RM 30.7R was 

instrumented with transducers to measure changes in pressure head across the subaerial extent of the 

sandbar relative to fluctuations in river stage at the bar face (fig. 3, cover photo). A total of 16 15-PSI 

MiniTroll TM (InSitu Corp, Fort Collins, Colorado) vented pressure transducers were individually 

deployed in 16 hand-driven, 1-1⁄4-inch threaded black steel pipe piezometers. Each piezometer was fitted 

with Johnson model no. 813 (Johnson Screens, Saint Paul, Minnesota), 304 stainless steel, 0.006 slot 

wire wrap sand points; the screened interval of the sand point was 31.8 cm. Piezometers were developed 

to remove fines clogging the screened interval during late November 2004, and again on December 31, 

2004, using a peristaltic pump. 

Following recession of the high-flow experiment, the topography of the newly deposited highelevation eddy sandbar and the top of each piezometer casing were surveyed with vertical heights 

referencing the ellipsoid on December 5, 2004; the survey again was conducted on March 8, 2005 (J. 

Hazel, Northern Arizona University, written commun., 2006). The level reference value for the 

transducers was set using the measured distance from the top of casing ellipsoid height to the water 

surface in the piezometer (table 1). On December 31, 2004 and January 1, 2005, slug tests were 

performed to measure hydraulic conductivity of the sediments recently deposited during the November 

2004 high-flow experiment in the lateral recirculation eddy at RM 30.7R. Recovery data from 

displacement of the slug were recorded and analyzed using the Bouwer and Rice (1976) method. 
6

Figure 3. Plan view image of the river mile 30.7R, Colorado River in Marble Canyon, Arizona, eddy sandbar study 

site showing the approximate horizontal extent of pre-dam sediments within the structure of the sandbar at 

depth. Tan polygon depicts the extent of the sandbar presumed to be pre-dam sediment at depth. The pre-dam 

sediment within the structure of the bar is typically covered by a mantle of newer sediment ranging in thickness 

from 0.5 to several meters; mantle of younger sediment overlying the tan pre-dam deposit polygon is not shown 

in this figure. Green polygon depicts the subaerial extent of deposits, above the 141.5 cubic meters per second 

river stage elevation, from the 2004 high-flow experiment; the full thickness of the sandbar within the area 

delineated by the green polygon is thought to consist almost entirely of deposits form the 2004 high-flow 

experiment. Dashed line is the inferred location of the erosional contact between pre-dam sediments and the 

2004 high-flow experiment deposits. Location of piezometers containing pressure transducers within the active 

model shown in red, number indicates the designation (table 1). Sedimentological data used for this figure is 

based off of trench and pit data collected in March, 2005 by David M. Rubin (D. Rubin, U.S. Geological Survey, 

written commun., 2005).
7

Table 1. Locations and selected construction data for piezometers installed in the eddy sandbar at river mile 

30.7R, Colorado River in Marble Canyon, Arizona. 

Well id Well location Top of casing Screen elevation Hydraulic conductivity Screened interval

Easting Northing Elevation Top Bottom Model Model layer

(m) (m) (m) (m) (m) (m/d)

3D 219535.890 611732.214 858.865 852.635 852.318 0.08 2

3S 219535.486 611732.649 858.410 853.723 853.405 13.8 1

4D 219531.553 611739.648 859.594 853.386 853.068 6.8 1

4S 219531.287 611740.128 858.643 853.983 853.666 6.8 1

5 219526.182 611748.163 859.626 854.884 854.566 6.8 1

7 219545.636 611768.546 860.320 855.627 855.309 6.8 1

8S 219531.718 611778.616 860.201 855.492 855.175 6.8 1

9D 219540.767 611787.724 859.059 854.402 854.085 6.8 1

9S 219541.666 611787.661 859.228 856.086 855.769 13.8 1

10 219553.932 611805.708 858.602 855.466 855.149 13.8 1

US1 219560.145 611816.863 857.186 854.047 853.730 n/a 1 

Groundwater Model

Two-dimensional numerical models have been used to predict changes in the phreatic surface in 

eddy sandbars, resulting from various transient dam discharge regimes along the Colorado River, with 

“good” results for an individual cross-section of the sandbar face (Budhu and Contractor, 1994; Budhu 

and Gobin, 1995). Potential spatial variations in crossbar topography and internal structural 

heterogeneity of the recirculating eddy sandbar deposits can be overlooked in two-dimensional analyses 

(Rubin and others, 1990; Schmidt, 1990; Springer and others, 1999; Alvarez and Schmeeckle, 2012). 

Therefore, in this study, we considered the response of the groundwater level in the sandbar across the 

entire three-dimensional bar face region. A finite-element model (FEFLOW 5.2, Institute for Water 

Resources Planning and Systems Research Inc. [WASY], Berlin, Germany) was used to develop a 

generalized three-dimensional model for predicting time-varying groundwater levels within the sandbar 

face as a function of river stage fluctuations. Model verification, using two transient simulations, against 

a range of groundwater levels observed during two different discharge regimes regulated by Glen 

Canyon Dam (fig. 2), demonstrates the ability of the model to predict the groundwater level in the 

sandbar face region for a wide range of soil parameters in fully saturated, variably saturated, and 

unsaturated conditions. 

Model Framework

Model parameters were defined for saturated media with transient flow in an unconfined aquifer. 

The uppermost surface of the model was developed directly from topographic survey data collected on 

December 5, 2004 (J. Hazel, Northern Arizona University, written commun., 2006). The topographic 

response of the RM 30.7R sandbar to the 2004 high-flow experiment (HFE) is shown in figure 4, and the 

subsequent discharge regimes regulated by Glen Canyon Dam are shown in figure 2. The model 

framework was discretized into two, three-dimensional slices representative of the general 

hydrogeological structure of the RM 30.7R eddy sandbar study area; discretization of the model 

framework was based on soil properties determined during field data collection (fig. 5, table 1). The 

uppermost surface was set as free and movable, which allows the surface of the top layer (layer 1) to 

follow the groundwater surface using the BASD (Best-Adaptation-to-Stratigraphic -Data) moving grid 

technique (WASY, Berlin, Germany). The top of layer 2 was defined as unspecified which allows it to 
8

be distributed according to the moving groundwater surface and the hydraulic conductivity of the 

sediment. The bottom of the model was fixed and unmovable. 

Figure 4. Matched oblique photographic views of the river mile 30.7R sandbar study site, Colorado River in 

Marble Canyon, Arizona, taken from left river bank illustrating the topographic condition of the study area (A) 

before the 2004 HFE, (B) 10-hours after the cessation of the 2004 HFE downramp, (C) 37-days post 2004 HFE 

downramp, (D) 55-days post 2004 HFE downramp, and (E) 57-days post 2004 HFE downramp. Direction of 

downstream flow is from right-to-left. Little change in the topographic condition of the RM 30.7R sandbar (B, C) 

is observed under modified low fluctuating flow (MLFF) operation criteria (fig. 2); however, during times of 

experimental nonnative fish suppression flows (NFSFs) (fig. 2), when maximum 24-hour range in discharge is 

increased above MLFF operational criteria, changes in the topographic condition of the sandbar become 

evident (D, E). During the NFSF experimental flows, a change in two dynamic variables occurred: (1) a nearly 

factor of 2X increase in the maximum 24-hour range of discharge at the study site (424.5 cubic meters per 

second per day versus 226.4 cubic meters per second per day), and (2) an increase in the subaqueous extent 

of the sandbar exposed to tractive forces as a result of the increase in the 24-hour discharge range. During this 

experiment, the hourly downramping rate for fluctuations was also increased by a factor of 2X, from 42.5 cubic 

meters per second per hour to 84.9 cubic meters per second per hour which also resulted in an increase in the 

time the river was held at peak flow each day through January, February and March, 2005.
9

Figure 5. Idealized cross-section showing the eddy sandbar at river mile 30.7R, Colorado River in Marble 

Canyon, Arizona, view upstream. Diagram shows both the location of piezometers and the inferred location of 

pre-dam and younger deposits within the framework of the sandbar. Values of hydraulic conductivity used to 

simulate zones of pre-dam and 2004 deposits during the model runs are shown. See Table 1 for the values of 

hydraulic conductivity used at each target piezometer during development of the model framework.

Sedimentological data used for this figure is based off of trench and pit data collected in March, 2005 by David 

M. Rubin (D. Rubin, U.S. Geological Survey, written commun., 2005).

Hydraulic Conductivity

Recovery data from the slug tests were analyzed, using the Bouwer and Rice (1976) method for 

unconfined aquifers with partially penetrating wells, to constrain the spatial distribution of hydraulic 

conductivity (K) in the model framework and to define the horizontal boundary between model layers 

(fig. 5, table 1). Throughout the calibration process, parameters of hydraulic conductivity were varied 

slightly to achieve model calibration. The hydraulic conductivity values at individual target locations in 

each layer were interpolated across the model layer to develop a best-fit approximation of the 

anisotropic and heterogeneous hydrogeologic framework that exists throughout the sandbar structure 

(fig. 6).
10

Figure 6. Hydraulic conductivity (K) of model layers 1 and 2 within the three-dimensional groundwater model 

framework developed for this study, river mile 30.7R, Colorado River in Marble Canyon, Arizona. Figure shows 

interpolation of hydraulic conductivity across model layer 1 between piezometers screened in pre-dam 

sediments (K~6.8 meters per day) and piezometers screened in post-dam sediment deposits from the 2004 

high-flow experiment (K~13.8 meters per day). Hydraulic conductivity of 0.08 meters per day used throughout 

model layer 2. Finite element mesh shown in white.

The uppermost layer of the model, layer 1, contained 9 piezometers screened in sediment higher 

than the 852.9-m bottom boundary elevation of the layer. Analyses of the slug test data indicate the 

presence of two spatially discrete zones of hydraulic conductivity within the layer 1 framework (figs. 5–

6, table 1). The interior area of the sandbar (about 800 m2

) is composed primarily of pre-dam lower 

hydraulic conductivity sediments, high in silt and clay content (30–40 percent silt and clay), and 

overlain by a thin mantle of sediment deposited during a 2,800 m3/s release from Glen Canyon Dam in 

June 1983 (figs. 3, 5). Piezometers 5, 7, and 8S, screened in this zone of pre-dam sediment, had average 

hydraulic conductivity values of 6.8 m/d (table 1). A comparison of the horizontal position and the 

screened interval elevation of piezometers 5, 7, and 8S with a detailed examination of internal structures 

exposed in a 60 m long sequence of trenches and pits (D. Rubin, U.S. Geological Survey, written 
11

commun., 2005) confirmed that piezometers 5, 7, and 8S were all screened within the pre-dam low 

hydraulic conductivity zone of sediments in layer 1 (figs. 3, 5). An average hydraulic conductivity value 

of 6.8 m/d was also used for piezometers 4S, 4D, and 9D (fig. 6, table 1). Although trench and pit data 

(D. Rubin, U.S. Geological Survey, written commun., 2005) could not unequivocally confirm the locus 

of the screened interval of the piezometers (4S, 4D, 8S) within the pre-dam deposits at RM 30.7R, the 

nearness of these piezometers to the pre-dam deposits at depth and an analysis of field slug tests data 

confirmed the approximate hydraulic conductivity values used at piezometers 4S, 4D, and 8S (figs. 3, 6). 

The exterior part of the model framework in layer 1 consists of a thick sequence of relatively 

high hydraulic conductivity fluvial sediments, deposited during the 1,160 m3

/s peak of the 2004 highflow experiment, which overlay a sharp erosional contact with the low hydraulic conductivity pre-dam 

sediments in the onshore interior area of the sandbar (figs. 3, 5, 6). The high hydraulic conductivity 

sediments in the exterior part of the sandbar are bounded by a time-varying specified-head boundary 

(river stage) along the streamside edge of the active model area. Piezometers screened in this zone of 

freshly deposited sediment on the outer part of the bar (3S, 9S, 10) had hydraulic conductivity values of 

13.8 m/d (figs. 3, 6, table 1). Only one piezometer, 3D, was screened within layer 2. The relatively low 

hydraulic conductivity value, 0.08 m/d, indicates the presence of silt- and clay-rich lenses at depth that 

shows the degree of variability that can occur within the already heterogeneous structure of the sandbar 

(table 1).

Storage Coefficient

Values of specific yield (Sy), storativity (S), and porosity (n) were defined for predictive transient 

modeling simulations. The storage coefficient (S) is the volume of water a permeable unit will absorb or 

expel per unit surface area per unit change in head. In most unconfined aquifers, storativity (S) is 

assumed to be nearly equal to the specific yield (Sy), that is, S=Sy (Fetter, 1994). The conceptual 

distribution of specific yield (Sy=n-Sr) varied at RM 30.7R because the eddy sandbar consists of a core 

of older pre-dam and 1983 deposits at depth mantled by new deposits from the 2004 high-flow 

experiment around the perimeter (figs. 3, 5, 6). In places, pre-dam and 1983 deposits contain 30–40 

percent silt and clay, which greatly increases the specific retention (Sr). Finer sand typically has a higher 

specific retention (Sr). As the percent of Sr relative to porosity (n) becomes greater, Sy becomes smaller. 

Using porosity values between 25 percent and 35 percent for model layer 1, the Sy of pre-dam deposits 

was estimated between 2 and 5 percent, while the Sy of newly deposited 2004 high-flow experiment 

sediment was greater than 10 percent. Through calibration, the sensitivity of the transient model to 

storage coefficient became apparent and a best-fit approximation for Sy of 2 percent was used across the 

model area in layer 1. The storage coefficient (S) for layer 2, considered confined by the model, was 

estimated at 0.0005 (Anderson and Wossener, 1992). 

Initial Head Values

During the model calibration process, initial hydraulic head values across the model area were 

interpolated from water levels recorded at individual piezometers for a common time step. Initial head is 

the desired starting boundary condition for the model simulations. The initial head condition is taken 

from a time step when the groundwater level in all piezometers throughout the sandbar is best 

equilibrated with a particular peak or trough stage in the river hydrograph. For subsequent predictive 

modeling simulations, initial head was estimated using previously observed relations between river stage 

and groundwater elevation data in piezometers during analogous regulated flow regimes.
12

Boundary Conditions

The outer boundary of the active model area in contact with the Colorado River was delineated 

by the subaerial extent of the eddy bar higher than the 141.5 m3/s stage elevation as surveyed on 

December 5, 2004 (Hazel and others, 2006b; J. Hazel, Northern Arizona University, written commun., 

2006). A specified head, first type Dirichlet, boundary condition was used to describe the fixed potential 

head along the perimeter of the model that comes in contact with river (fig. 7). The temporal changes in 

river stage elevation along the river boundary were simulated with a time-varying function during the 

model run. The relation between stage and time was interpolated from the river stage piezometer US1 

for the model calibration runs and from an ADVM at RM 30.3 for the predictive runs. Site-specific 

stage-discharge relations developed by Hazel and others (2006b) were used to facilitate this process. 

 

Figure 7. Plan view image of the river mile 30.7R eddy sandbar study site showing the location of piezometers 

used as targets for calibration and transient verification of the groundwater model developed for this study, river 

mile 30.7R, Colorado River in Marble Canyon, Arizona. Shown are the locations of the Specified Head 

Boundary and the No Flow Boundary around the perimeter of the Finite-Element Mesh within the active model 

area. Outer boundary of the active model area is delineated by the subaerial extent of the eddy sandbar above 

the 141.5 cubic meters per second river stage elevation.
13

Along the onshore model boundary between the contact of the sandbar with the bedrock and 

talus slope, a specified flux of zero was assigned to the boundary nodes (fig. 7). Recharge, another type 

of specified flux boundary condition, was not incorporated in the model. Despite its application during a 

period of above-average precipitation, the model was not sensitive to recharge; the specified head river 

stage boundary condition is the main control on the location of the phreatic surface in the model. 

Model Calibration

Calibration Criteria and Model Error

A calibrated three-dimensional numerical model of Colorado River eddy sandbars can produce 

model outputs that replicate, although non-uniquely, observed changes in the phreatic surface of the 

sandbar as a function of river stage. Evaluation of the degree of calibration, or goodness-of-fit, is based 

on the residual differences between observed and simulated model results at target locations. 

Calibration criteria were established so that the residual between the transient model simulation 

and the observed field condition should be no more than 15 percent of the variability in the total head 

change across the model domain, expressed as:

 , (1) 

where hm is measured head in meters, hs is the modeled head in meters, and hr is the range in 

measured head (hm) across the model area in meters. Mean absolute error (MAE) and 

the root mean squared error (RMSE) combine individual residuals and quantify the 

average error across the model domain as: 

 , (2)

 , (3) 

where n is the number of calibration values. Because MAE and RMSE provide an average 

error from specific target locations, they may be biased to these locations and may not 

necessarily represent the spatial distribution of error across the entire active model 

area.

The transient model was calibrated to water surface elevation data collected at 11 piezometers 

between January 3, 2005 and January 18, 2005 (fig. 8). Upon final download of the transducers, the 

mean absolute error of the residual between the measured depth to water in each piezometer and the 

value recorded by the transducer was 0.01 m for all targets. When all possible sources of systematic 

error associated with the measurement of the water surface elevation in each piezometer were accounted 

for, the error envelope for all target data used in the model calibration and verification was plus-orminus 0.35 m, or 4 percent of the maximum range in head change across the sandbar at any given time 

step. The primary uncertainty stems from the precision of the optical survey measurements along the 

Colorado River in Grand Canyon, which is reported at “0.05 m or better” (Hazel and others, 2006b).

[ ] rsm ≤− 15%/hhh

MAE ( )ihh/n s

n

i

= ∑ m −

=1

1

( )

50

1

2 1

. n

i

RMSE sm ihh/n 















 = ∑ −

=
14

Figure 8. Measured elevation of groundwater level in all piezometers at river mile 30.7R eddy sandbar site 

relative to river stage (US1), Colorado River in Marble Canyon, Arizona, during the time period of calibration 

and transient verification of the model, January 3–18, 2005. 

Model Results

The transient model for the entire eddy sandbar area at RM 30.7R was calibrated to water surface 

elevation data collected at 11 piezometers between January 3, 2005 and January 18, 2005 (fig. 8). The 

mean absolute error of the residual between the measured and modeled groundwater elevations at each 

target location, expressed as a percent of the total observed head change across the model area (1.66 m), 

was 14.6 percent among all 11 targets (table 2). The calibrated transient model is considered a good 

predictor of the phreatic surface across the entire three-dimensional domain of the RM 30.7R eddy 

sandbar model area.
15

Table 2. Root mean squared error (RMSE) and mean absolute error (MAE) residual differences between the 

measured groundwater level in piezometers and the simulated model results at target locations, Colorado River 

in Marble Canyon, Arizona, for transient model verification purposes, January 3–18, 2005 (see Table 1 for 

locations).

[RMSE and MAE are also presented as a percent of the total measured head change across the model domain at target 

locations].

Well id RMSE MAE RMSE as % of total MAE as % of total

of residual of residual observed head change observed head change

(m) (m) across model area1 across model area1

3D 0.216 0.167 13.00% 10.01%

3S 0.164 0.117 9.88% 7.02%

4D 0.465 0.439 27.95% 26.39%

4S 0.305 0.269 18.35% 16.15%

5 0.343 0.313 20.60% 18.80%

7 0.297 0.228 17.86% 13.73%

8S 0.378 0.340 22.73% 20.47%

9D 0.399 0.298 23.97% 17.89%

9S 0.391 0.295 23.51% 17.72%

10 0.284 0.212 17.07% 12.75%

US1 0.000 0.000 0.02% 0.02%

Average 0.295 0.243 17.72% 14.60%

1

Total head change across model area during simulation: 1.663 m

The sandbar face is the critical region affected by the processes of seepage erosion and mass 

failure in newly deposited eddy sandbars along the Colorado River. A major control on the exfiltration 

of groundwater from the sandbar and seepage erosion along the bar face is the elevation and location of 

the groundwater level within the sandbar, onshore from the bar face (Budhu and Contractor, 1994; 

Alvarez and Schmeeckle, 2012). To better emphasize the usefulness of the model within the sandbar 

face region conceptual refinement of the model was conducted to focus solely on the targets within the 

bar face area of critical concern for issues of seepage erosion and mass failure (piezometers 3S, 3D, and 

10). In this usage, the boundary of sandbar face region is about10 meters onshore, horizontally, from the 

shoreline of the river at peak stage. The model provides an excellent simulation of the groundwater 

level within the bar face region that is continually affected by diurnal fluctuation in the hydrograph. The 

mean absolute error of the residual between the measured and modeled groundwater elevations at 3 

piezometers (3S, 3D, and 10) within the sandbar face region is about10 percent; total observed head 

change across the model area is 1.663 m (fig. 9, table 2). 
16

Figure 9. Graph plot showing measured and modeled groundwater level in piezometers (A) 3S, (B) 3D, and (C) 

10, within the bar face region of the eddy sandbar at river mile 30.7R, Colorado River in Marble Canyon, 

Arizona, during transient verification of the model, January 3–18, 2005. Measured river stage at US1 is also 

shown.
17

Hydrologic data collected in piezometers at sandbar RM 30.7R were used to provide out-ofsample verification of the model as a predictive tool within the bar face region of the eddy sandbar. A 

time-varying function was used to simulate the river stage data observed at US1, from December 7, 

2004 to December 20. 2004, at the specified head model boundary (fig. 10). The mean absolute error of 

the residual between the measured and modeled groundwater elevation during this simulation at 

piezometer 10, within the sandbar face region, is 8.67 percent (fig. 10, table 3). This out-of-sample 

goodness-of-fit confirms that the generalized three-dimensional model developed in this study 

accurately predicts time-varying groundwater levels, within the sandbar face of an eddy sandbar, as a 

function of highly variable fluctuations in river stage. 

Figure 10. Graph plot showing measured and modeled groundwater level in piezometer 10, within the bar face 

region of the eddy sandbar at river mile 30.7R, Colorado River in Marble Canyon, Arizona, during transient 

verification of the model, December 7–20, 2004. Measured river stage at US1 is also shown. 
18

Table 3. Root mean squared error (RMSE) and mean absolute error (MAE) for the measured groundwater level in 

piezometers and the simulated model results at piezometer 10, located within the river mile 30.7R sandbar 

face, during a transient model simulation, December 7–20, 2004.

Well ID RMSE MAE RMSE as % of total MAE as % of total

of Residual of Residual observed head change observed head change

(m) (m) across model area1 across model area1

10 0.088 0.075 10.06% 8.67%

US1 0.000 0.000 0.03% 0.03%

1

Total head change across model area during simulation: 0.87 m

As newly deposited sandbars evolve, temporal fluctuations in hydraulic conductivity, 

permeability, and effective stress can occur throughout the aquifer skeleton (Semmens, 1999; Springer 

and others, 1999). The two transient simulations (in-sample and out-of-sample) of the groundwater level 

in the bar face region yield mean absolute errors of less than 10 percent. During both simulations, the 

only model input parameter that was adjusted was the time-varying specified head boundary for river 

stage; all soil properties were kept constant. The two transient model runs demonstrate the ability of the 

model to simulate groundwater levels in the bar face region for observed data collected throughout a 

period where temporal fluctuations in soil properties were likely occurring, 14 and 40 days after the 

deposition of new 2004 high-flow sediments. Given (1) the likely temporal fluctuations in hydraulic 

conductivity and permeability as a function of effective stress during these transient model runs, and (2) 

that the model properly simulated (MAE < 10 percent) all observed bar face groundwater levels without 

any parameter adjustment, it reasonably may be concluded that the model developed in this study is not 

sensitive to small changes in soil parameters. Therefore, the model accurately simulates the groundwater 

level in the sandbar face under a range in hydrogeologic conditions at the sandbar throughout time. 

Conclusion

A generalized numerical model was developed to predict groundwater levels within the bar face 

region of eddy sandbars, along the Colorado River downstream of Glen Canyon Dam, as a function of 

changes in river stage owing to unsteady dam releases. Two transient simulations (in-sample and out-ofsample) of the groundwater level in the bar face region, as a function of time-varying stage, yield mean 

absolute errors of less than 10 percent. The low mean absolute errors between the measured and 

modeled groundwater elevations in bar face piezometers are an indication of the model’s ability to 

replicate observed changes in the groundwater level within the sandbar as a function of river stage. The 

model implemented in this study accurately simulates the groundwater level in the bar face region under 

a range in hydrogeologic conditions at the sandbar aquifer throughout time. 

The generalized model developed during this study is applicable across a range of typical eddy 

sandbar deposits in diverse settings. The general predictive capability provided by this study meets the 

basic needs of researchers to estimate time-varying groundwater levels at the onshore end of a sandbar 

face for physical and numerical modeling of beach face stability, particularly during times of sandbar 

adjustment in response to seasonal, monthly, or weekly changes in diurnal fluctuating flow regimes. 
19

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