Phosphate transport in soil and ground water

There is some serious concern about phosphorous contamination of groundwater from application of inorganic and organic fertilizers. So, a study of phosphate transport in soil and groundwater is badly needed and we intend to carry out such a study. For phosphate transport the general form of the mathematical model can be written as,

coupled to

where C is the solute concentration in the solution phase, S is the solute associated with solid phase,  is the volumetric water content,  is the bulk density of solid, v is the average pore water velocity, D is the dispersion coefficient,  is the adsorption rate constant and  is the de-sorption rate constant. f(C) is the nonlinear isotherm which is normally taken as either Freundlich,  or Langmuir, , isotherm, where and N are positive constants with N < 1. In a recent article (Notodarmojo et al., 1991) such a phosphorus transport model was studied but, assuming time dependent sorption such as

where k, m and n are constants. It was shown that exact analytical solutions are possible for phosphate concentration only in a couple of special cases (n = m = 1 and ). However, an analysis of this model shows that it is very sensitive to influent phosphorus concentration. One of our objectives is to develop ideas in order to solve for phosphate concentration without making any simplifying apriori assumptions or neglecting terms. The study will investigate the possibility of constructing traveling wave solutions for solute concentration C. Such solutions are the limit profiles for a continuous feed problem when the inflow concentration stabilizes to a fixed value. The methodology will involve introducing the traveling wave coordinate

(where  is the wave speed) and analyzing the resulting coupled nonlinear differential equations in C and S. Infact, by employing appropriate mathematical operations, one will be able to eliminate the variable S from the coupled system to obtain a second order nonlinear differential equation just in the aqueous concentration C. The coefficient of the second derivative term of this equation will be D/v reflecting the coupled multiplicative effect of dispersion (D) and advection (v). Obviously, when advection is large compared to dispersion, i.e., v is large in relation to D, this coefficient will be small. However, at instances where the second derivative is very large, one can not neglect the second derivative term because those instances will correspond to problems. We propose to study such boundary layer problems for aqueous concentration using singular perturbation techniques with D/v as our small parameter . The singular perturbation technique involves obtaining an  and an and matching them in an appropriate fashion at the interface. We have experience in using such techniques for reaction-diffusion problems (Wollkind, Manoranjan and Zhang, 1994).

We also intend to use phase plane analysis to get better qualitative understanding of the problem. Already, we have carried out such a study successfully for the case when the Langmuir isotherm can be replaced by a logistic isotherm (Manoranjan and Gomez, 1995). Here, we were able to show that it is possible to have a homoclinic wave for the aqueous concentration under certain conditions and were able to obtain the exact solution of that homoclinic wave. In figures 1, 2 & 3 we present some of the typical phase plane diagrams related to this study, where c' is the first derivative of the aqueous concentration with respect to the traveling wave coordinate.

Figure 1: The phase orbit which is a closed loop through the origin corresponds to the homoclinic orbit (traveling pulse).

Figure 2: The closed loop is destroyed and a spiral behavior is obtained when the conditions are different from that of figure 1

Figure 3: This shows the phase trajectory corresponding to the heteroclinic wave (traveling front).

We are encouraged by these preliminary results and feel that similar phase plane techniques can be successfully applied to solute transport with either Langmuir or Freundlich isotherm.