U.S. Department of the Interior

U.S. Geological Survey

Introduction

Solute transport in streams and rivers is governed by a suite

of hydrologic and geochemical processes. Knowledge of these

processes is needed when assessing the fate of contaminants that

are released into surface waters. The study of solute fate and

transport often is aided by solute transport models that mathematically describe the underlying processes. This fact sheet

describes a model that considers One-Dimensional Transport

with Inflow and Storage (OTIS). As shown by several example

applications, OTIS may be used in conjunction with field-scale

data to quantify hydrologic processes (advection, dispersion, and

transient storage) and certain chemical reactions (sorption and

first-order decay).

Model Description

OTIS may be used to characterize the fate and transport of

water-borne solutes in streams and rivers. Given a description of

watershed loading, OTIS determines the solute concentrations

that result from hydrologic transport and chemical transformation.

The primary assumption used within the model is that solute concentration varies only in the longitudinal direction; solute

concentration does not vary with width or depth. Given this

assumption, equations are developed for a one-dimensional system that consists of a series of stream segments. Conservation of

mass for each stream segment yields a set of differential equations that are solved using numerical methods. Solution of the

differential equations yields a general equation of the form:

 =

where

x is the distance,

t is time, and

C(x,t) is the solute concentration at distance x and time t.

This equation describes the spatial and temporal variation in

solute concentration as a function of several hydrologic and

geochemical parameters. As discussed by Runkel (1998),

parameter estimates may be obtained by nonlinear regression

using a modified version of OTIS known as OTIS-P.

Hydrologic Transport. Several hydrologic processes that govern

the downstream transport of solutes are considered in the model.

These processes include advection, dispersion, lateral inflow,

and transient storage. Advection, the downstream transport of

solute mass at a mean velocity, and dispersion, the spreading of

solute mass due to shear stress and molecular diffusion, are

considered in most mechanistic models of stream-water quality

and solute transport. Consideration of these important mechanisms leads to the familiar advection-dispersion equation.

Cxt ( ) , + Hydrologic

Transport

Chemical

Transformation

Within the OTIS model, additional terms are added to the

advection-dispersion equation to account for the effects of

transient storage and lateral inflow.

Transient storage has been noted in many streams where

solutes may be temporarily detained in small eddies and stagnant

pools of water that are stationary relative to the faster moving

water near the center of the channel (fig. 1A). In addition, significant portions of flow may move through the coarse gravel of the

streambed and porous areas within the streambank (fig. 1B). The

traveltime for solutes carried through these porous areas may be

substantially longer than that for solutes traveling within the

water column. These pools of water and porous areas of the

streambed are the two areas contributing to transient storage,

as shown in figure 1.

Lateral inflow is any water that is added to the stream due

to ground-water inflow, overland flow, interflow, or small springs.

These flows act to dilute (or concentrate) solutes in the stream

channel if they carry solute concentrations that are lower (or

higher) than the stream-solute concentration.

Two conceptual areas are defined within the model: the

main channel and the storage zone (fig. 2). The main channel is

defined as the portion of the stream in which advection and dispersion are the dominant hydrologic transport mechanisms. The

A

B

Transient Storage Mechanisms

Figure 1. Transient-storage mechanisms. Transient storage occurs

(A) when solutes enter small pockets of slow-moving water, and

(B) when solutes leave the main channel and enter the porous media

that make up the bed and banks of the channel. Arrows denote solute

movement between the main channel and the transient-storage zone.

Fact Sheet FS–138–99

January 2000

Using OTIS to Model Solute Transport

in Streams and Rivers
2

storage zone is defined as the portion of the stream that contributes to transient storage; that is, stagnant pools of water and

porous areas of the streambed. Water in the storage zone is considered immobile relative to water in the stream channel. The

exchange of solute mass between the main channel and the storage zone is modeled as a first-order mass transfer process.

water supplies from contamination. Given data from tracerinjection studies, stream transport models may be used to estimate the timing, magnitude, and duration of a pollutant cloud

that enters a stream due to an accidental spill.

In addition to providing management information, hydrologic parameters derived from tracer-injection data provide

insight into the physical characteristics of streams. Values of

the transient-storage parameters (As, α), for example, indicate

the degree of mixing due to stagnant pools and flow through

porous areas of the streambed. Further, model-derived estimates

of traveltime indicate the relevant time scales over which chemical reactions can potentially affect solute concentrations.

In this section, we illustrate the use of OTIS to quantify

hydrologic processes using tracer-injection data. Our first example uses data from Uvas Creek, a small pool-and-riffle stream in

northern California. Bencala and Walters (1983) describe a continuous, constant-rate injection of chloride into Uvas Creek.

Concentrations were monitored at several downstream locations,

and streamflow was estimated by tracer dilution.

Application of OTIS to the Uvas Creek chloride injection

requires estimates of stream cross-sectional area (A), transient

storage (As, α), and dispersion (D). Estimates of stream crosssectional area are related to traveltime as they control the

timing of the chloride profile, whereas estimates of the transientstorage parameters represent instream mixing as reflected

by the shape of the chloride profile. During a series of simulations, Bencala and Walters (1983) varied A, As, α, and D to

obtain a match between observed and simulated concentrations.

Final simulation results at two sampling locations are shown

in figure 3.

Conceptual Model:

Main Channel

Storage Zone

Downstream transport

Lateral

outflow

Lateral

inflow

Chemical

reaction

Chemical

reaction

(advection and dispersion)

Exchange due to

transient storage

Main Channel and Storage Zone

Consideration of the hydrologic processes discussed above

introduces several model parameters. The main channel crosssectional area (A) is used with estimates of streamflow to determine the average advective velocity of transported solutes.

Spreading of solute mass due to shear stress is described by the

dispersion coefficient (D). Solute mixing due to transient storage

is governed by the storage-zone cross-sectional area (As) and the

storage-zone exchange coefficient (α). Lateral inflows are characterized by the lateral inflow rate and the solute concentration

associated with the lateral inflow (CL).

Chemical Transformations. The hydrologic processes described

above affect the transport of all water-borne solutes. Additional

processes that act to add or remove solute mass are considered

for reactive solutes that undergo chemical transformation. OTIS

includes two types of chemical transformations: first-order

decay and kinetic sorption. Under first-order decay, removal of

solute mass is proportional to solute concentration. First-order

decay coefficients may be specified for the main channel (λ) and

(or) the storage zone (λs). Consideration of sorption within OTIS

is based on a distribution coefficient approach that allows for

sorption onto streambed sediments.

Model Applications

Mixing and Traveltime. Data from tracer-injection studies are

frequently used to quantify mixing and traveltime in streams and

rivers. These hydrologic characteristics are of importance to

water-resource managers who are responsible for protecting

Figure 2. Conceptual model includes the main channel and the

storage zone.

CHLORIDE CONCENTRATION, IN MILLIGRAMS PER LITER

TIME, IN HOURS

5 10 15 20 25 2

4

6

8

10

12

Uvas Creek

x = 105 meters

x = 433 meters

Estimates of A and As developed from the tracer data are

consistent with the pool-and-riffle characterization of Uvas

Creek reported by Bencala and Walters. The pools act to produce transient storage by temporarily detaining some of the

chloride. A dimensionless measure of this storage effect is

obtained by calculating the ratio of storage-zone area to

Figure 3. Simulated (solid lines) and observed (symbols)

chloride concentrations in Uvas Creek, California.
3

main-channel cross-sectional area (As/A). Values of As/A for various reaches of Uvas Creek range from 1.0 to 3.0. These values

indicate that the pool areas are large relative to the main channel.

Analysis of Uvas Creek data relied on a trial-and-error

approach wherein parameter estimates were manually adjusted

to produce an acceptable match between simulated and observed

tracer concentrations. In the following example, parameter estimates are obtained by nonlinear regression using OTIS-P.

Laenen and Risley (1997) describe several studies in Oregon

streams where rhodamine WT was used to determine traveltime.

In July 1992, a slug of rhodamine was added to the Clackamas

River at river mile 13.3 (RM 13.3). Water samples were collected at river miles 11.0, 9.5, and 8.0. Because rhodamine

was introduced as a slug (as opposed to a continuous injection),

data at the first sampling location (RM 11.0) are used to define

the upstream boundary condition (fig. 4). Given the boundary

condition, the traveltimes (as given by A) and mixing characteristics (as given by As, α, and D) for the two reaches ending at

river miles 9.5 and 8.0 may be determined.

parameter estimates using OTIS-P. As expected, application of

OTIS-P results in an improved simulation (solid lines, fig. 4) and

a revised set of parameter estimates (A = 30.4, 48.1 m2; As = 7.0,

14.4 m2; D = 2.0, 0.4 m2/s; α = 2.0 × 10−4

, 1.7 × 10−4/s).

Nutrient Uptake. The examples above illustrate the use of OTIS

to quantify hydrologic processes. In this example, we show how

OTIS may be used to identify chemical reactions. McKnight and

Duff (1995) describe an experimental addition of chloride, phosphate, and nitrate into Green Creek, a glacial meltwater stream in

Antarctica. The objective of the study was to quantify stream

hydrology and to study nutrient uptake by algal mats that cover the

bed of Green Creek. As in the previous examples, a conservative

tracer (chloride) is used to define the hydrologic system. Hydrologic parameters based on the analysis of chloride transport then

are used to evaluate the conservative transport of nitrate and phosphate. A conservative simulation of nitrate is shown as a dashed

line in figure 5. As shown in the figure, observed nitrate concentrations are significantly lower than simulated concentrations, based

on conservative transport. This discrepancy indicates that nitrate

concentrations are attenuated by geochemical processes.

Under the assumption that nutrient uptake is a first-order process, the first-order decay coefficients (λ, λs) in OTIS are used to

quantify the loss of nitrate. Given the hydrologic characterization,

OTIS-P is used to estimate uptake in the main channel (λ) and

in the storage zone (λs). Reach-specific estimates of λ range

from 4 × 10−5

 to 4 × 10−4; estimates of λs range from 3 × 10−6 to

2 × 10−3. Simulation results based on these parameters are shown

as a solid line in figure 5. Additional mass-balance calculations

based on λ and λs indicate that 84 to 93 percent of the nitrate loss

occurs in the main channel where algal uptake is the likely mechanism of nitrate loss. The remaining loss occurs in the storage zone

where possible mechanisms include microbial uptake and nitrate

reduction. In contrast, similar analyses indicate that phosphate loss

occurs only in the main channel (λs = 0). OTIS thus provides a

framework for determining the magnitude and location of processes controlling nutrient concentrations in Green Creek.

Use of OTIS-P requires a set of initial estimates for the

parameters of interest. Here we set A and D for both reaches

equal to 50 square meters (m2) and 10 m2/s, respectively. The

exchange coefficient, α, is initially set to 0.0 such that transient

storage is not considered. Preliminary OTIS results based on

these initial parameter estimates are shown as dashed lines in

figure 4. These preliminary profiles indicate that our initial estimates of stream cross-sectional area are too large; the simulated

traveltime is overestimated at both RM 9.5 and RM 8.0. In addition, the symmetric simulation profile at RM 8.0 is in contrast to

the asymmetry of the observed data. This observation indicates

that transient storage may be an important mixing mechanism.

Both of these discrepancies may be addressed by developing

10 12 14 16 18

0

5

10

15

20

25 RM 11.0 (boundary condition)

RM 9.5

RM 8.0

RHODAMINE WT CONCENTRATION, IN MILLIGRAMS PER LITER

TIME, IN HOURS

Clackamas River

21 23 25 27 29

0

20

40

 NITRATE CONCENTRATION, IN MICROMOLES PER LITER

TIME, IN HOURS

Green Creek Figure 4. Upstream boundary condition (RM 11.0), observed

concentrations (symbols), simulated concentrations based on

initial parameter estimates (dashed lines), and simulated

concentrations based on final parameter estimates (solid lines)

for the Clackamas River, Oregon.

Figure 5. Observed (symbols) and simulated nitrate concentrations in Green Creek, Antarctica, 226 meters from the injection

point. Simulation results are shown for conservative transport

(λ = λs = 0; dashed line) and first-order loss (λ, λs > 0; solid line).
Printed on recycled paper 4

Trace Metal Removal. In addition to the time-variable simulations shown in the previous examples, OTIS may be used to

determine the steady-state solute concentrations that result from

a constant loading scenario. In this application, we consider the

spatial concentration profile of dissolved iron in a small stream.

The stream is St. Kevin Gulch, a headwater stream in the Rocky

Mountains of Colorado. St. Kevin Gulch receives acidic, metalrich waters from a series of springs that emanate from the toe of

a large mine dump. Instream metal concentrations increase and

pH levels decrease in the vicinity of the dump. During a synoptic

study, water samples were collected at numerous instream locations and the metal-rich springs. Figure 6 depicts the observed

profile of dissolved iron concentration. Two features of the

spatial profile are of interest. First, there is a large increase in

dissolved iron at 400 m; second, there is an abrupt decrease in

dissolved iron concentration downstream from 500 m. The

purpose of this application is to quantify the processes causing

the changes in dissolved iron concentration.

A simple approach to quantifying the chemical reactions

is to determine the first-order rate at which iron is lost from the

system. This is accomplished by estimating a first-order decay

coefficient for the main channel (λ). A first-order decay coefficient of 1 × 10−4/s results in the simulated profile shown in

figure 6 (solid line). Although the decay-rate approach yields

an excellent fit to the data, it does not explicitly identify the

mechanisms causing the iron loss. Inspection of the spatial pH

profile indicates an increase in pH downstream from 500 m

that could result in the precipitation of hydrous iron oxides.

This observation is confirmed by additional calculations using

a chemical equilibrium model.

— Robert L. Runkel

References

Bencala, K.E., and Walters, R.A., 1983, Simulation of solute transport

in a mountain pool-and-riffle stream—A transient storage model:

Water Resources Research, v. 19, no. 3, p. 718–724.

Broshears, R.E., Bencala, K.E., Kimball, B.A., and McKnight, D.M.,

1993, Tracer-dilution experiments and solute-transport simulations for a mountain stream, Saint Kevin Gulch, Colorado:

U.S. Geological Survey Water-Resources Investigations

Report 92–4081, 18 p.

Laenen, Antonius, and Risley, J.C., 1997, Precipitation-runoff and

streamflow-routing model for the Willamette River Basin,

Oregon: U.S. Geological Survey Water-Resources Investigations Report 95–4284, 252 p.

McKnight, D.M., and Duff, J., 1995, Hyporheic zone interactions controlling nutrient uptake by algal mats in an Antarctic stream:

San Francisco, 1995 Fall Meeting of the American Geophysical

Union, December 11–15, 1995, Eos Transactions, AGU Fall

Meeting Supplement, v. 76, no. 46, p. 226.

Runkel, R.L., 1998, One-dimensional transport with inflow and

storage (OTIS)—A solute transport model for streams and

rivers: U.S. Geological Survey Water-Resources Investigations

Report 98–4018, 73 p.

0 500 1,000 1,500 2,000 0

50

100

150

200

250

 IRON CONCENTRATION, IN MICROMOLES PER LITER

DISTANCE, IN METERS

St. Kevin Gulch

Figure 6. Observed (symbols) and simulated iron concentrations

in St. Kevin Gulch, Colorado. Simulation results are shown for

conservative transport (λ = λs = 0; dashed line) and first-order loss

(λ = 1 × 10−4; solid line).

The first step in the analysis is to define the hydrology of the

system. Broshears and others (1993) describe the addition of a

conservative tracer that coincided with the synoptic study. Data

from the tracer addition are used to determine flow rates and the

hydrologic parameters. Observed data from the springs are used to

set the lateral inflow concentrations, CL. OTIS then is used to

develop a spatial profile under the assumption that dissolved iron

is conservative (nonreactive). Results from this simulation are

shown in figure 6 (dashed line). The close correspondence

between observed and simulated iron concentrations in the upper

portions of St. Kevin Gulch (0–500 m; fig. 6) indicates that the initial increase in iron is due to strictly hydrologic factors; that is,

iron loading from the metal-rich springs. Simulated iron concentrations decrease downstream from 500 m due to the addition of

water from a relatively dilute tributary but do not match the

decrease in the observed data. This discrepancy indicates that

chemical reactions affect dissolved iron concentrations downstream from 500 m.

Sources of Additional Information

The OTIS solute transport model is available on the

World Wide Web (http://webserver.cr.usgs.gov/otis). Funding for OTIS development is provided by the USGS Toxic

Substances Hydrology Program (http://toxics.usgs.gov/

toxics). For additional information, contact:

Rob Runkel

U.S. Geological Survey

Box 25046, MS 415

Denver Federal Center

Denver, CO 80225–0046

runkel@usgs.gov