SMIG -- OTIS/OTEQ Feature Article: March, 1996

Surface-water quality and flow Modeling Interest Group

Simulation Models for Conservative and Nonconservative Solute Transport in Streams

by Robert L. Runkel

USGS, Water Resources Division
Mail Stop 415, Denver Federal Center
Lakewood, CO 80225
Internet: runkel@usgs.gov
Phone: 303/236-4882 ext. 285
Fax: 303/236-4912

Editor's note:
This article was originally published as part of the proceedings of the 1995 meeting of the International Association of Hydrological Sciences. The full reference is as follows:

Runkel, R.L., 1995, Simulation models for conservative and nonconservative solute transport in streams, in Osterkamp, W.R., editor, Effects of scale on interpretation and management of sediment and water quality, proceedings of the 1995 meeting of the International Association of Hydrological Sciences, Boulder, Colorado, July 3-14, 1995: IAHS publication no. 226, p 153-159

Abstract

Solute transport in streams is governed by a suite of hydrologic and chemical processes. Interactions between hydrologic processes and chemical reactions may be quantified through a combination of field-scale experimentation and simulation modeling. Two mathematical models that are used to simulate conservative and nonconservative solute transport in streams are presented here. A model for conservative solutes that considers One-Dimensional Transport with Inflow and Storage (OTIS) may be used in conjunction with tracer-dilution methods to quantify hydrologic transport processes (advection, dispersion, lateral inflow, and transient storage). For nonconservative solutes, a model known as OTEQ may be used to quantify chemical processes within the context of hydrologic transport. OTEQ combines the transport mechanisms in OTIS with a chemical equilibrium submodel that considers complexation, precipitation/dissolution and sorption. OTEQ has been used to quantify the processes affecting trace metals in two streams in the Rocky Mountains of Colorado, USA.

Introduction

Many investigators are currently studying streams and the effect of basin loading on stream water quality. Of particular interest is the study of streams that are adversely affected by acid mine drainage and acid deposition. Because of acidic conditions, these streams often have elevated concentrations of trace metals, such as aluminum, copper, zinc, and iron. The fate and transport of solutes within these acidic environments is of importance as high concentrations of trace metals may be toxic to fish and other aquatic organisms.

Studies of acidic streams share two common goals: 1) to quantify the physical or hydrologic processes controlling solute concentrations, and 2) to quantify the dominant chemical and biological reactions. Both goals are generally addressed through a combination of field-scale experimentation and simulation modeling. Tracer-dilution methods, for example, are frequently used to quantify hydrologic transport processes. During a tracer-dilution experiment, a conservative solute (i.e., tracer) is injected at the upstream end of a stream reach. Tracer concentrations measured at several downstream locations are used to determine the volumetric flow rate and the additional inflow entering the stream via surface runoff and groundwater. The hydraulic properties of the stream are then determined by applying a conservative solute transport model (e.g. Broshears et al., 1993).

The importance of various chemical and biological reactions is assessed by conducting pH modification experiments (McKnight & Bencala, 1989; Kimball et al., 1994). Within a pH modification experiment, instream pH levels are temporarily altered to allow for the study of pH-dependent processes (e.g., precipitation, sorption). Data obtained during pH modification experiments may be used to identify the processes controlling solute concentrations. A key step in process identification is interpretation of the complex interactions between hydrologic processes and chemical reactions. Mathematical models for nonconservative or reactive solutes may be used to describe competing processes, such as sorption and precipitation.

This paper presents two models that have been developed to quantify process dynamics for conservative and nonconservative solutes. A model incorporating One-Dimensional Transport with Inflow and Storage (OTIS) is presented for the case of conservative solute transport. The OTIS model may be used to determine the hydrologic and hydraulic characteristics of the stream based on data obtained during tracer-dilution experiments. For nonconservative solute transport, a reactive solute transport model known as OTEQ is presented. OTEQ may be used to quantify the chemical processes responsible for changes in solute concentrations observed during pH modification experiments.

Solute Transport Models

Conservative Transport - OTIS

The OTIS solute transport model was developed to simulate the transport of solutes in streams and rivers in which one-dimensional transport may be assumed. Although the model was developed primarily for conservative solutes, nonconservative behavior may be simulated through the specification of a first-order decay or production rate. Several physical mechanisms govern the downstream transport of solutes and are considered in the model. These mechanisms 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. 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 zones of water that are stationary relative to the faster moving water near the center of the channel. In addition, significant portions of the flow may move through the coarse gravel of the streambed and the porous areas within the streambank. The travel time for solutes carried through these porous areas may be significantly longer than that for solutes traveling within the water column. Lateral inflow is any water that is added to the stream due to groundwater inflow, overland flow, interflow, or small springs. These flows act to dilute (or concentrate) solutes in the stream channel if they carry solute concentrations which are lower (or higher) than the stream solute concentration.

The OTIS model is formed by writing mass balance equations for two conceptual areas: the stream channel and the storage zone. The stream channel is defined as that portion of the stream in which advection and dispersion are the dominant transport mechanisms. The storage zone is defined as the portion of the stream that contributes to transient storage, i.e. stagnant pockets 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 stream channel and the storage zone is modeled as a first-order mass transfer process. Conservation of mass for the stream channel and storage zone yields (Bencala & Walters, 1983; Runkel & Broshears, 1991):

(1)

(2)

where A is the stream channel cross-sectional area [L²], A_S is the storage zone cross-sectional area [L²], C is the in-stream solute concentration [M L^-3], C_L is the solute concentration in lateral inflow [M L^-3], C_S is the storage zone solute concentration [M L^-3], D is the dispersion coefficient [L² T^-1], Q is the volumetric flowrate [L³ T^-1], q_LIN is the lateral inflow rate [L³ T^-1 L^-1], t is the time [T], x is the distance [L], a is the storage zone exchange coefficient [T^-1], l is the in-stream first-order decay coefficient [T^-1], and l_S is the storage zone first-order decay coefficient [T^-1]. Equations (1) and (2) describe the spatial and temporal variation in solute concentration within the stream channel and the storage zone.

Several model features provide a flexible and efficient framework from which to consider solute transport problems. Noteworthy features include:

Simulation Flexibility. The user may request the output of simulation results at any number of arbitrary locations along the stream. Simulations for multiple conservative solutes may be performed during a single program execution. Nonconservative solutes subject to first-order transformations may also be considered. The governing equations may be solved for both dynamic (time-varying) and steady-state conditions.
Consideration of Unsteady Flow Regimes. Many solute transport models assume steady flow regimes where flow rates and cross-sectional areas are constant in time. OTIS allows for the consideration of unsteady flow regimes where time-varying flows and areas are specified in an external flow file. This allows the transport model to be linked with flow routing models (Runkel & Restrepo, 1993).
User Interface. OTIS is available as a stand-alone program and as a module within the Modular Modeling System (MMS, Leavesley et al., 1992). MMS is a modeling framework that provides a graphical user interface (GUI). The GUI includes a spreadsheet for data entry and other options for viewing simulation results.
Efficiency. The governing differential equations are solved using the Crank-Nicolson method and the decoupling procedure described by Runkel & Chapra (1993). These methods provide an efficient solution that results in fast execution times for most solute transport problems.

OTIS is ideally suited for the analysis of tracer data. Broshears et al. (1993) present an application of OTIS to a tracer-dilution experiment conducted on a small mountain stream. During the experiment, a solution of lithium chloride was injected into the stream for 52 hours. Water samples were collected at several downstream locations. Model parameters were estimated using tracer data and a series of simulation runs as described in Broshears et al. (1993). Simulation results and tracer data for two sampling locations are shown in Fig. 1.

Nonconservative Transport - OTEQ

The OTIS solute transport model provides a useful tool for the analysis of conservative solutes and nonconservative solutes that are subject to simple first-order transformation. Many cases arise, however, in which solutes are affected by chemical and biological processes that are not considered in such a simple model. Trace metals, for example, may be affected by precipitation/dissolution, sorption and oxidation-reduction reactions. The OTEQ model described in this section was developed to consider these complex reactions within the context of hydrologic transport.

OTEQ is an equilibrium-based transport model formed by combining the physical transport mechanisms in OTIS with a chemical equilibrium submodel. The chemical submodel uses the numerical framework MINTEQ (Westall et al., 1976; Allison et al., 1991), a chemical equilibrium model distributed by the U.S. Environmental Protection Agency. Given total analytical concentrations of chemical components, MINTEQ computes the distribution of chemical species that exist within a batch reactor at equilibrium. Specific reactions considered include complexation, precipitation/dissolution and sorption. Use of the chemical equilibrium submodel is based on the "Local Equilibrium Assumption," wherein chemical reactions are considered sufficiently fast relative to hydrologic processes (Rubin, 1983). When the assumption of equilibrium is inappropriate, kinetic controls may be placed on specific reactions, such as sorption and dissolution from the streambed.

A conceptual diagram illustrating the coupling of hydrologic transport and equilibrium chemistry is given as Fig. 2. Within the model, the stream is represented as a series of segments or control volumes. Hydrologic processes transport solute mass from one segment to the next. This downstream movement is described by the differential equations governing transport [i.e., Equations (1) and (2)]. Within each segment, chemical equilibrium is assumed, and solute mass is distributed between dissolved, sorbed and precipitated forms. This chemical partitioning is described by the chemical equilibrium submodel.

A complete description of the governing equations and solution techniques underlying the OTEQ model is presented by Runkel et al. (1996a). In short, the total solute concentration, T, equals the sum of five distinct phases. The first three phases are the dissolved, precipitated and sorbed concentrations present in the water column (C, P_w and S_w). These phases are mobile and are subject to downstream transport. The final two phases are the precipitated and sorbed concentrations present on the streambed (P_b and S_b). Dissolved mass in the water column forms precipitates when the solution becomes oversaturated with respect to the defined solid phases. Any precipitated mass resides in the water column until the precipitate settles to the streambed or redissolution occurs. Precipitates dissolve when the solution becomes undersaturated. Dissolved species sorb (desorb) to (from) solid phases present in the water column or on the streambed.

Model equations for the stream channel are developed by considering conservation of mass for the total solute concentration. This yields:

(3)

(4)

(5)

where T_w is the water-borne solute concentration [ML^-3], T_wL is the water-borne solute concentration in the lateral inflow [ML^-3], T_wS is the water-borne solute concentration in the storage zone [ML^-3], f_b is a source/sink term describing dissolution from the streambed [ML^-3T^-1], g_b is a source/sink term describing sorption/desorption from the streambed [ML^-3T^-1], v_p1 is the settling velocity for water-column precipitates [LT^-1], v_s1 is the settling velocity for water column sorbates [LT^-1], d₁ is the effective settling depth [L], and the model parameters are as defined for the OTIS model.

Equation (3) states that any change in the total solute concentration is due to the physical transport of the water-borne solute concentration. The water-borne solute concentration is equal to the total solute concentration minus the immobile phases (P_b and S_b):

(6)

Equations (4) and (5) state that any change in the immobile precipitate and sorbed concentrations is due to settling and the effects of chemical reaction (dissolution and sorption/desorption). Source/sink terms describing these chemical reactions are functions of the equilibrium submodel. Model equations for the storage zone are presented by Runkel et al. (1996a).

OTEQ has been used to quantify the dominant processes affecting trace metals in two acidic streams in the Rocky Mountains of Colorado (Runkel et al., 1996b; Broshears et al., 1994). Modeled processes include precipitation/dissolution, degassing, and kinetically controlled sorption/desorption.

Software Availability

Source code and executable binaries for the OTIS solute transport model are available over the Internet computer network via anonymous ftp. The Internet address for the OTIS home page is http://co.water.usgs.gov/otis/. Executable binaries are provided for several hardware platforms (SUN, Data General, DEC and IBM workstations, MS-DOS-based personal computers). Source code is provided for users with other hardware requirements. Persons without Internet access may obtain the code by contacting the author.

To date, the OTEQ solute transport model has been used exclusively as an in-house research tool. Further development, testing and documentation efforts are required prior to releasing the code as a public domain software package.

Acknowledgments

This work was completed as part of the U. S. Geological Survey Toxic Substances Hydrology Program. The author acknowledges the assistance of Kenneth Bencala, Robert Broshears, Briant Kimball, and Diane McKnight.

References

Allison, J.D., Brown, D.S. & Novo-Gradac, K.J. (1991) MINTEQA2/PRODEFA2, A geochemical assessment model for environmental systems: Version 3.0 User's Manual Rep. EPA/600/3-91/021, U. S. Environ. Prot. Agency, Washington, D. C.

Bencala, K.E. & Walters, R.A. (1983) Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model. Water Resour. Res., 19(3), 718-724.

Broshears, R.E., Bencala, K.E., Kimball, B.A. & McKnight, D.M. (1993) Tracer-dilution experiments and solute-transport simulations for a mountain stream, Saint Kevin Gulch, Colorado. USGS Wat. Res. Invest. Rpt. 92-4081.

Broshears, R.E., Runkel, R.L. & Kimball, B.A. (1994) Development and application of a reactive solute transport model for trace metals in mountain streams. In: Toxic Substances and the Hydrologic Sciences (ed. by A. Dutton)(Proc. AIH Annual Conf., Austin, TX, April 1994), 19-34. AIH Publ.

Kimball, B.A., Broshears, R.E., McKnight, D.M. & Bencala, K.E. (1994) Effects of instream pH modification on transport of sulfide-oxidation products. In: Environmental Geochemistry of Sulfide Oxidation (ed. by C.N. Alpers & D.W. Blowes) (ACS National Meeting, Washington, DC, August 1992) 224-243. ACS Symposium Series 550.

Leavesley, G.H., Restrepo, P.J., Stannard, L.J. & Dixon, M.J. (1992) The modular hydrologic modeling system - MHMS (paper presented AWRA Annual Conf., Reno, NV, November 1992).

McKnight, D.M. & Bencala, K.E. (1989) Reactive iron transport in an acidic mountain stream in Summit County, Colorado: A hydrologic perspective. Geochim. Cosmochim. Acta, 53, 2225-2234.

Rubin, J. (1983) Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions. Water Resour. Res., 19(5), 1231-1252.

Runkel, R.L. & Broshears, R.E. (1991) One dimensional transport with inflow and storage (OTIS): A solute transport model for small streams. Tech. Rep. 91-01, Center for Adv. Decision Support for Water and Environ. Syst., Univ. of Colo., Boulder.

Runkel, R.L. & Chapra, S.C. (1993) An efficient numerical solution of the transient storage equations for solute transport in small streams. Water Resour. Res., 29(1), 211-215.

Runkel, R.L. & Restrepo, P.J. (1993) Solute transport modeling under unsteady flow regimes: An application of the Modular Modeling System. In: Water management in the `90s: a time for innovation (ed. by K. Hon) (Proc. Wat. Res. Planning &Mgmt. Div., ASCE, Seattle, WA, May 1993).

Runkel, R.L., Bencala, K.E., Broshears, R.E. & Chapra, S.C. (1996a) Reactive solute transport in streams, 1. Development of an equilibrium-based model. Water Resour. Res., 32(2), 409-418.

Runkel, R.L., McKnight, D.M., Bencala, K.E. & Chapra, S.C. (1996b) Reactive solute transport in streams, 2. Simulation of a pH modification experiment. Water Resour. Res., 32(2), 419-430.

Westall, J.C., Zachary, J.L. & Morel, F.M.M. (1976) MINEQL: A computer program for the calculation of chemical equilibrium composition in aqueous systems. Tech. Note 18, Water Qual. Lab., Dep. of Civ. Eng., Mass. Inst. of Technol., Cambridge.

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