Modeling of Solute Transfer in a River with Transient Storage Zones Using a Network of Equivalent Electrical Circuits

Document Type : Research Article


1 Ph.D. Student of Water Structures, Department of Water Structures Engineering, Tarbiat Modares University, Tehran, Iran.

2 Professor in Sediment and River Engineering, Water Structures Engineering Department, Tarbiat Modares University

3 Professor in Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran.

4 Professor in Department of Applied Physics, Polytechnic University of Cartagena, Cartagena, Spain.


Human activities everyday release a huge amount of domestic, industrial and agricultural waste into water bodies and continuously change the ecosystem conditions in the world. Considering the harmful effects of these pollutants entering water resources, study about pollution transfer in streams and predicting the pollutant concentration at downstream points seem to be important. For this purpose, the well-known classical advection-dispersion equation (ADE) was presented as the first attempt for describing mass transfer and energy transfer in physical systems. This equation is useful for channels with relatively prismatic and uniform cross-sections.
Experimental studies carried out in rivers show that ADE is no longer applicable for natural streams, especially mountain pool and riffle streams because of their irregular cross-sections. Afterward, some more accurate models, referred dead zone models or transient storage models, were suggested by several researchers for predicting solute concentration in natural rivers and calibrated using tracer approach. Such models cause more realistic concentration-time distributions which have lower picks and longer residence time. Solving such models, for which in most cases the analytical solution doesn’t exist, needs numerical methods –methods which usually deal with complexity and is time-consuming.
In this study, we have applied Network Simulation Method (NSM) –a powerful and efficient computational method for simulating systems governed by differential equations based on the electric circuit concepts and the analogy between the governing differential equations of hydrodynamic and electrical phenomena– which according to the previous studies simulates desirably the transport of mass in natural streams, to solve two transient storage models. The method consists of two phases of designing and simulating. In designing phase, the system of differential equations corresponding to the prototype must be discretized spatially over the studied domain and then, for each term of the discretized equations the equivalent electrical devices are chosen. These electrical elements are connected based on the algebraic sign of the terms to satisfy Kirchhoff’s current low. Regarding the mathematical models, in most studies, electric potential and electric current are equivalent to the value of the unknown variable and its flux, respectively. The last step of designing the electro-analogical model is the implementation of initial conditions and boundary conditions of unknown variables using appropriate dependent and/or independent, voltage and/or current sources. Simulating this equivalent electrical network is performed through an appropriate electrical-computational circuit code, such as PSpice code. PSpice, which is a powerful circuit analysis software, uses the Newton-Raphson iterative algorithm to solve this set of nonlinear equations and performing the transient analysis.
In this paper, NSM is firstly verified by simulating a transient storage transport model developed by Bencala and Walters (1983) for unsteady conservative solute transfer in pool and riffle streams. This model includes two equations for solute concentration in the main channel and in the storage zone and involves one storage zone. The analytical solution for this model has been presented in Laplace domain by Kazezyılmaz-Alhan (2008) considering a hypothetical channel with a constant cross-sectional area, flow velocity, and dispersion coefficient and for two types of upstream boundary conditions including a continuous injection and a pulsating solute injection. The results of this verification were desirable.
Then, the accuracy and efficiency of NSM were compared with Finite Volume Method (FVM) –a widely used numerical method in computational fluid dynamics- through simulating an unsteady reactive solute transport using a nested two-storage zone transport model developed by Kerr et al. (2013). This model consists of three equations and involves two storage zones including the surface and hyporheic storage zones interacting together. The results of simulating a hypothetical solute transport problem with this nested model indicate a good match between these two methods with near-zero error indices. The computational time needed for NSM and FVM were 117 seconds and 505 seconds, respectively. So, NSM is much faster. Furthermore, the implementation of boundary conditions in NSM is direct, easier and more flexible.
Therefore, NSM is proposed as a precise and efficient alternative for numerical methods in solving one-dimensional coupled differential equations of unsteady transport, simultaneously and providing benchmarks without complex mathematical calculations. Because of its analogical based concept, it can be used as a predicting and monitoring tool for transport phenomena instead of using troublesome physical hydraulic models to perform the water quality studies with less time, low expense and higher accuracy. Hence, in critical conditions, including a sudden spill of a high-hazardous contaminant in a specified point of the river or increasing the concentration of a chemical element to its maximum level, the monitoring and controlling measures at different parts of the river can be carried out with an acceptable accuracy and speed to improve the water quality.


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  • Receive Date: 10 February 2019
  • Revise Date: 09 June 2019
  • Accept Date: 29 July 2019
  • First Publish Date: 22 November 2019