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This communication deals with the properties of heat source/sink in a magneto-hydrodynamic flow of a non-Newtonian fluid immersed in a porous medium. Shrinking phenomenon along with the permeability of the wall is considered. Mathematical modelling is performed to convert the considered physical process into set of coupled nonlinear mathematical equations. Suitable transformations are invoked to convert the set of partial differential equations into nonlinear ordinary differential equations which are tackled numerically for the solution computations. It is noted that dual solutions for various physical parameters exist which are analyzed in detail., Competing Interests: The authors have declared that no competing interests exist.

In a suite of water-saturated sandstones, we have recently demonstrated that irreducible water saturation can be well estimated using relaxation time only, in the form of any of several "averages" giving more emphasis to short times than does the geometric mean time. The best estimate of permeability came from fits giving more emphasis to slightly longer times. In this paper we present estimates of irreducible water saturation and permeability using approximations (here called Robusta approximation) to the fractional power average relaxation time (Tp)1/p. The advantage of this approximation is that it does not involve previous computation of exponential components; therefore, it does not depend on the choice of the inversion method, and it permits a very fast computation. The Robusta approximation gives some of the best correlations using p = -0.55 for irreducible water saturation and p = +0.18 for permeability.

A semi-continuous relaxation model is constructed using sums of gamma functions and non-negative least squares for the inversion of Carr-Purcell-Meiboom-Gill (CPMG) echo data. No regularization is necessary for this approach, and yet the solution is stable even with noisy data. Test results derived from 60 echo trains are presented. Computational advantages of the method are presented.