One-dimensional solute transport in open channel flow from a stochastic systematic perspective

Abstract

Solute transport by river and stream flows in natural environment has significant implication on water quality and the transport process is full of uncertainties. In this study, a stochastic one-dimensional solute transport model under uncertain open-channel flow conditions is developed. The proposed solute transport model is developed by upscaling the stochastic partial differential equations through their one-to-one correspondence to the nonlocal Lagrangian–Eulerian Fokker–Planck equations. The resulting Fokker–Planck equation is a linear and deterministic differential equation, and this equation can provide a comprehensive probabilistic description of the spatiotemporal evolutionary probability distribution of the underlying solute transport process by one single numerical realization, rather than requiring thousands of simulations in the Monte Carlo simulation. Consequently, the ensemble behavior of the solute transport process can also be obtained based on the probability distribution. To illustrate the capabilities of the proposed stochastic solute transport model, various steady and unsteady uncertain flow conditions are applied. The Monte Carlo simulation with stochastic Saint–Venant flow and solute transport model is used to provide the stochastic flow field for the solute transport process, and further to validate the numerical solute transport results provided by the derived Fokker–Planck equations. The comparison of the numerical results by the Monte Carlo simulation and the Fokker–Planck equation approach indicated that the proposed model can adequately characterize the ensemble behavior of the solute transport process under uncertain flow conditions via the evolutionary probability distribution in space and time of the transport process.

Appendices

Appendix 1: Convergence of the Monte Carlo simulations

See Figs. 9 and 10.

Fig. 9

Change in flow velocity variance in space and time between different numbers (n₁, n₂) of Monte Carlo simulations by $\frac{{MC_{n1} - MC_{n2} }}{{MC_{n1} }}$ in Problem 1 a n₁ = 1000 and n₂ = 3000; b n₁ = 3000 and n₂ = 5000; c n₁ = 5000 and n₂ = 7000; d n₁ = 7000 and n₂ = 8000; e n₁ = 8000 and n₂ = 9000; f n₁ = 9000 and n₂ = 10000

Fig. 10

Change in flow velocity variance in space and time between different numbers (n₁, n₂) of Monte Carlo simulations by $\frac{{{\text{MC}}_{{{\text{n}}1}} - {\text{MC}}_{{{\text{n}}2}} }}{{{\text{MC}}_{{{\text{n}}1}} }}$ in Problem 2 (the ratio above 2% is shown) a n₁ = 1000 and n₂ = 3000; b n₁ = 3000 and n₂ = 5000; c n₁ = 5000 and n₂ = 7000; d n₁ = 7000 and n₂ = 9000

Appendix 2: Validation of the numerical scheme

As described in Sect. 2.3.1, the proposed Fokker–Planck equation is a special type of advection–dispersion equations (ADEs). For some ADEs under specific initial and boundary conditions, their corresponding analytical/exact solutions can be obtained. We provide the comparison between analytical and numerical solutions of a general one-dimensional advection–diffusion equation:

$$ \frac{{\partial {\mathbf{c}}}}{{\partial {\mathbf{t}}}} + {\mathbf{U}}\frac{{\partial {\mathbf{c}}}}{{\partial {\mathbf{x}}}} = {\mathbf{D}}\frac{{\partial^{2} {\mathbf{c}}}}{{\partial {\mathbf{x}}^{2} }} $$

For one-dimensional advection–diffusion equation in the infinite half-space with initial condition ${\text{c}}\left( {{\text{x}},{\text{t}} = 0} \right) = {\text{c}}_{0}$ and the boundary condition ${\text{c}}\left( {{\text{x}} = 0,{\text{t}} > 0} \right) = {\text{c}}_{\text{in}} ,{\text{c}}\left( {{\text{x}} = \infty ,{\text{t}} > 0} \right) = 0$, the analytical solution can be obtained as:

$$ {\text{c}}\left( {{\text{x}},{\text{t}}} \right) = {\text{c}}_{0} + \frac{{{\text{c}}_{{\rm in}} - {\text{c}}_{0} }}{2}\left( {{\text{erfc}}\left( {\frac{{{\text{x}} - {\text{Ut}}}}{{2\sqrt {\text{Dt}} }}} \right) + \exp \left( {\frac{{\text{Ux}}}{{\text{D}}}} \right){\text{erfc}}\left( {\frac{{{\text{x}} + {\text{Ut}}}}{{2\sqrt {\text{Dt}} }}} \right)} \right) $$

The corresponding solute transport parameters are given as: the channel length L = 10,000 m and flow velocity is 0.5 m/s. Initial solute concentration is 1 mg/L and the upstream inflow solute concentration is 2 mg/L. Dispersion coefficient is 50 m²/s and the total simulation time is 10000 s.

Figure 11 indicates the numerical scheme used in this study for one-dimensional transport problems can achieve satisfactory results when compared with the corresponding analytical solutions.

Fig. 11

Numerical results of the solute concentration through time at x = 1000 m, 3000 m, 5000 m by ULTIMATE QUICKEST (UQ) and the corresponding analytical solutions