Development of fuzzy differential quadrature numerical method and its application for uncertainty quantification of solute transport model

Abstract

Uncertainty analysis of solute transport model is an essential component for decision-making problem in the regulatory framework for safe discharge of liquid effluent in environment. Performance assessment of any industry is based on the violation of discharge limit stipulated on yearly basis by any regulatory body. Very often, regulatory bodies provide deterministic (point) solution of the solute concentration as limit of discharge which is derived using the closed-form solution. However, uncertainty of the model parameters of the governing advection dispersion equation with specific initial and field boundary conditions restricts user to obtain deterministic or closed-form solution of the solute transport model. This necessitates the numerical solution of the solute-transport model to fulfill the task of uncertainty quantification. The present paper proposes an innovative numerical method of solving solute-transport model in the presence of imprecise uncertain input parameters such as dispersivity and velocity of flow. Dispersivity and velocity of flow, both being measurable field parameters, generally posses an epistemic behavior, that is, imprecise. Therefore, uncertainty of these parameters is taken into account as triangular fuzzy number for its simplicity. Solute transport equation with these fuzzy (uncertain) parameters is numerically solved using differential quadrature method and fuzzy vertex method. Therefore, the present version of uncertainty modeling is named as fuzzy differential quadrature scheme of uncertainty modeling. The paper discusses the details of the differential quadrature method and fuzzy vertex method. The stability of the newly developed methodology of uncertainty modeling is assessed by comparing the numerical solution with possible analytical solution.