WATER RESOURCES RESEARCH, VOL. 29, NO. 1, PAGES 211-215, JANUARY 1993 An Efficient Numerical Solution of the Transient Storage Equations for Solute Transport in Small Streams ROBERT L. RUNKEL AND STEVEN C. CHAPRA Center for Advanced DecisionSupport for Water and Environmental Systems, Universi.ty of Colorado, Boulder Severalinvestigators have proposed solute transport models that incorporate the effects of transient storage. Transient storageoccursin small streams when portions of the transported solutebecome isolatedin zonesof water that are immobile relativeto water in the main channel (e.g., pools,gravel beds). Transient storageis modeledby adding a storage term to the advection-dispersion equation describing conservation of mass for the main channel. In addition, a separate mass balance equation is written for the storage zone. Although numerous applications of the transient storage equations may be found in the literature, little attention hasbeenpaidto the numerical aspects of the approach. Of particular interest is the coupled nature of the equations describing mass conservation for the main channel and the storage zone. In the work described herein, an implicit finite difference technique is developed thatallows for a decoupling of thegoverning differential equations. Thisdecoupling method may beapplied to other sets of coupled equations such as those describing sediment-water interactions for toxic contaminants. For the case at hand, decoupling leads to a 50% reduction in simulation run time. Computational costsmay be further reduced through efficient application of the Thomas algorithm. These techniques may beeasily incorporated into existing codes and new applications in which simulation run time is of concern. INTRODUCTION Several investigators have proposed surface watersolute transport models that incorporate the effects of transient storage [Thackston and Krenkel, 1967; Thackston and Schnelle, 1970; Valentine and Wood, 1977; Nordin and Troutman, 1980; Jackman et al., 1984; BencaIa andWalters, 1983; BencaIa et al., 1990; Kim et al., 1990]. These transient storage (or "dead zone") models consider a physical mech- anism wherein solute massis exchanged between the main channeland an immobile storage zone. This quasi-two- dimensional process is represented by adding a storage term to the conventional advection-dispersion equation. Transient storage occurs in small streams when portions of thetransported solute become isolated from the main channel in small pools and in the gravel underbed. As a result of this storage phenomenon, the magnitude of atypical solute tracer pulse is attenuated and its position delayed. To account for attenuation, two distinct zones areconsidered. The first zone represents the main channel that is usually considered when modeling advective surface-water systems. Processes influencing solute concentrations in this zone include advection, dispersion, lateral inflow, transient stor- age, and chemical reaction. A second area,the storage zone, represents recirculating pools, underflow channels and other areas thatare immobile relative to flow in themain channel. Only the processes of storage andchemical reaction are considered in thestorage zone. Thezones are linked by an exchange termthatacts to transfer mass between thetwo regimes. The transient storage equations presented herein are gen- erally applicable to streams and riversin which one- dimensional solute transport may be assumed. As described by Fischer etal. [1979], one-dimensional analysis is valid for systems in which solute mass is uniformly distributed over Copyright 1993 by the American Geophysical Union. Papernumber 92WR02217. 0043-1397/93/92WR-02217502.00 the stream's cross-sectional area. Although this uniformity rarelyoccurs in nature, it is a reasonable assumption for streams of small to moderate width and depth. Although numerous applications of the transient storage equations may befound in theliterature, littleattention has been paid to the numerical aspects of the approach. Of particular interest is the coupled nature of the equations describing mass conservation forthestream channel and the storage zone. This paper details animplicit finite difference technique that decouples the governing equations. Addi- tional numerical issues are also discussed. GOVERNING DIFFERENTIAL EQUATIONS ANDFINITE DIFFERENCE APPROXIMATIONS Conservationof mass for the stream and storage zone segments isgiven by(1)and (2), respectively [Bencala and Walters, 1983; Runkel and Broshears, 1991]: OC QoC 1 o ( o.•_xC ) qLIN ..... + -- • AD + (CL - C) Ot A Ox A Ox • + a(Cs- C) - xC (1) dC s A ..... a •(C-Cs)-XsCs dt As (2) where A stream channel cross-sectional area [L 2]; As storage zone cross-sectional area [L2]' C in-stream solute concentration [M L-3]; C L solute concentration in lateral inflow [M L-3]. Cs storage zone solute concentration [M L -3 ]. D dispersion coefficient [L 2 T- •]; Q volumetric flow rate [L 3 T-• ]; q LINlateral inflow rate [L 3 T-• L- • ]; t time IT]; x distance ILl; a storage zone exchange coefficient [T-•]; 211