This paper presents a comparative study of volume average predictions between low Reynolds number (LRN) turbulence models: Abe-Kondoh-Nagano (AKN), Lam-Bremhorst, Yang-Shih, standard k-, and k-!. A porous medium, which represents conditions in which the flow path changes rapidly, was defined as an infinite array of square cylinders. In addition, to explore the effect of particle size on the rapid expansion and contraction of the flow paths, the diameter ratio (DR) of the square cylinders was systematically varied from 0.2 to 0.8. This generalization revealed new insights into the flow. The Reynolds number (ReD) covered a turbulent range of 500 to 500 103, and the porosity was varied from 0.27 to 0.8. The correlations of the turbulent kinetic energy (k), its dissipation rate ("), and macroscopic pressure gradient as a function of , which are useful in macroscopic turbulence modeling, are presented. The results show that the AKN model yields better predictions of the volume-averaged flow parameters because it is better suited to reproduce recirculation zones. For all the diameter ratios, at high , the distances between walls are high, and the interstitial velocities are low. Consequently, wake flows are produced, and energy losses by friction are moderate. As the flow becomes increasingly bound, the wakes are suppressed and disrupted, and k and " increase owing to shear layer interactions and frictional forces. Distinctive low-velocity recirculation patterns appear inside pores depending on DR.
Assessment of rock formation permeability is a complicated and challenging problem that plays a key role in oil reservoir modeling, production forecast, and the optimal exploitation management. Generally, permeability evaluation is performed using porosity-permeability relationships obtained by integrated analysis of various petrophysical measurements taken from cores and wireline well logs. Dependence relationships between pairs of petrophysical variables, such as permeability and porosity, are usually nonlinear and complex, and therefore those statistical tools that rely on assumptions of linearity and/or normality and/or existence of moments are commonly not suitable in this case. But even expecting a single copula family to be able to model a complex bivariate dependency seems to be still too restrictive, at least for the petrophysical variables under consideration in this work. Therefore, we explore the use of the Bernstein copula, and we also look for an appropriate partition of the data into subsets for which the dependence strucure was simpler to model, and then a conditional gluing copula technique is applied to build the bivariate joint distribution for the whole data set.
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