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Temperature di erence in the soil and its environments is a common phenomenon. Soil permeability changes in parallel to temperature, mostly due to water viscosity variations in di erent temperatures. A more realistic estimation of the seepage value through and beneath hydraulic structures leads to their more e cient design. In this paper, the heat conduction equation is solved by the least-squares mesh-free method to calculate the temperature distribution in soil. Distribution of permeability coe cients can vary irregularly and may lead to some di culties in mesh-based methods. In these methods, soil permeability changes in each mesh, and ner mesh or some kind of interpolation is required to contribute to the solution procedure. Since there is no need to form elements or grids in a mesh-free method, it can handle this irregular variation simply. Herein, the seepage equation is solved by the same least squares mesh-free method. The method is integral-free, simple, and e cient in calculation due to its sparse and positive de nite matrices. The scheme is validated by solving a simpli ed version of the governing equations. Problems that are more complicated are dealt with to investigate the phenomenon numerically.

Fitting an explicit curve over some discrete data extracted from a rheometer is the usual way of writing a rheological model for generalized Newtonian fluids. These explicit models may not match totally with the extracted data and may ignore some features of the rheological behavior of the fluids. In this paper, a cubicspline curve fitting is used to fit a smooth curve from discrete rheological data. Spline interpolation avoids the problem of Runge's phenomenon, which occurs in interpolating using high degree polynomials. The formulation for applying presented rheological model is described in the context of least squares meshfree technique. One problem is solved to show validity of the scheme: a fluid with rather complex rheology model is considered and solved by both conventional explicit and proposed implicit models to show the advantages of the presented method.

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