Abstract
In this work, we develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. The model takes into consideration several major physical, chemical, and biological mechanisms. The resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. We present both a partial analytic approach towards solving the nonlinear system and finding the main type of dynamics of microbes, and a full-scale numerical simulation. Following the auxiliary equation method for nonlinear reaction-diffusion equations, we obtain a closed form traveling wave solution for the Richards equation. Using the propagating front solution for the pressure head, we reduce the transport equation to an ODE along the moving frame and obtain an analytic solution for the history of bacteria concentration for a specific test case. To solve the system numerically, we employ upwind finite volume method for the transport equations and stabilized explicit Runge–Kutta–Legendre super-time-stepping scheme for the Richards equation. Finally, some numerical simulation results of an infiltration experiment are presented, providing a validation and backup to the analytic partial solutions for the transport and growth of bacteria in the soil, stressing the occurrence of front moving solitons in the nonlinear dynamics.
Highlights
We develop a mathematical model for transport and growth of microbes by natural water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations with the Richards equation. e model takes into consideration several major physical, chemical, and biological mechanisms. e resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically
Results and Discussion. e proposed model was applied to simulate the transport and growth of microbes with the transport of growth-limiting substrate through an unsaturated finite column. e growth of microorganisms is often controlled by a single substrate and the specific growth rate is assumed to be a function of the concentration of that rate limiting substrate
In our mathematical model, we used the hypothetical values for microbes and growth-limiting substrate at the top of the soil column, and the total microbial count, moisture content count, and substrate content count down the soil column were calculated. e model shows that the initial bacterial count, i.e., 1.1 × 103 cfu/ml, increases down the soil column in presence of substrate, i.e., 10 mg/100 ml, with the moisture of the porous medium
Summary
We develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. e model takes into consideration several major physical, chemical, and biological mechanisms. e resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. Results of different experiments conducted on soils and other subsurface materials have shown that many environmental factors affect the transport and outcome of microbes in porous media, for example, the chemical composition of interstitial solution, water flow velocity, and physical properties of the solid matrix [6, 8]. Physical processes such as growth, decay, adsorption, straining, advection, motility, diffusion, and dispersion are known to be involved with the transport and outcome of microbes [9]. If the flux of fluid is given by Darcy’s law for flow through a porous medium, the transport equation becomes the Richards equation
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