A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

Comments on “Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization”

This paper explores a nonlinear Cournot oligopoly with n firms displaying general isoelastic demand. The marginal profits-based gradient rule and the expectation rule Local Monopolistic Approximation were employed in two Cournot oligopoly games. Nash equilibrium stability analysis is carried out on each of the two games to throw light on the effects of demand elasticity and other parameters on the dynamics of the game. Our results show that the influence of demand elasticity on stability depends on firms’ expectation rules.     

Comments on “Design of sampled data state estimator for Markovian jumping neural networks with leakage time-varying delays and discontinuous Lyapunov functional approach”

This paper first points out some fatal errors in the paper (Nonlinear Dyn 73:1367–1383, 2013). Then the main results of the above paper are corrected with some suitable modifications. The new feedback gain matrix is used to prove the estimation of the neuron states through numerical simulations with the same examples as discussed in the above paper.     

Comment on “Precise Control for vibration of rigid-flexible mechanical arm”

The results presented in a paper by Zichen et al. [1] has some flaws. The idea in [1] is to combine assumed modes method using an integration method with optimal control adaptive law for a spatial rigid-flexible mechanical arm as an underactuated mechanical systems (UMS). The nonlinear dynamic equation formulation from reference [6] is incorrectly used in [1]. Furthermore, the control law approach in [1] was to achieve a similar result for a UMS rigid manipulator as the one published by Spong [2] and a proper citation must have been credited.     

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.     

Comment on “A theorem for a shear stress-free plane boundary in Stokes flow”

The results proved in a recent paper by Akhtar and Sen [Mech. Res. Commun. 38 (2011) 529–531] are not new and are already known in the literature. In fact, the main expressions – presented in the form of a theorem – in the cited article follow as a special case of the more general results for a two-fluid Stokes flow with a planar interface proved earlier by Palaniappan [Acta Mechanica 139 (2000) 1–13]. We briefly demonstrate the reduction of particular cases of the solution given by Palaniappan which also reveals the duplication of another special solution elsewhere.     

Comments on “Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays”

In this note some points for paper [Huabin Chen, Chuanxi Zhu, Peng Hu, Yong Zhang, Delayed-state-feedback exponential stabilization for uncertain Markovian jump systems with mode-dependent time-varying state delays, Nonlinear Dyn. (2012), doi:10.1007/s11071-012-0324-3] are presented.     

A note on paper “Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media”

The paper is devoted to correcting structure of the representation for Rayleigh waves at non-semisimple degeneracy of the fundamental matrix. The incorrect representation was given in a paper “Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media” by Cerv and Plesek (2013) .     

Comments on “Abrupt-Interface Solution for Carbon dioxide Injection into Porous Media” by M. Dentz and D. Tartakovsky

Several numerical simulations were conducted to compare the results with analytical solutions given by “Abrupt-Interface Solution for Carbon Dioxide Injection into Porous Media” by M. Dentz and D. Tartakovsky, “Injection and Storage of CO₂ in Deep Saline Aquifers: Analytical Solution for CO₂ Plume Evolution During Injection” by J. M. Nordbotten et al.     

Comments on “Non-existence of Shilnikov chaos in continuous-time systems”

Comments on "Non-existence of Shilnikov chaos in continuous-time systems" are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.     

Discussion on“General Solutions of Axial Symmetrical Ring Shells”

There is one point to be mentioned about the decay constant of homogeneoussolution in the paper“General Solutions of Axial Symmetrical Ring Shells”(No.3.1980 of this Journal).There is an infinite number of rootsλ.The general form ofthese roots is     

Comments on “Confluent edge conditions for the electromagnetic wave at the edge of a wedge bounded by material sheets” by Mithat Idemen

It is pointed out some inaccuracies in Idemen’s method [Idemen (2000, 2011)] while having investigated electromagnetic field near the edge of a resistive half-plane. The limiting for the field expansion only to the first power of logarithmic function is not valid and rigorously leads to the trivial solution. It is needed to include the logarithmic functions to the field expansion in any cases when among the solutions of the characteristic equation there are roots that differ from one another by any integer.     

Minimum-drag cascade design for an “impeller-directing vanes-unchoked convergent nozzle” system

The variational problem of designing the slender airfoils of the plane cascade of an impeller-directing vanes-unchoked convergent nozzle system in a supersonic ideal (inviscid and non-heat-conducting) gas flow with a subsonic normal velocity component is solved in the linear approximation. The minimized functional is the axial component of the force acting on the system. The conditions to be met by the optimized airfoils include the specification of the projection of the force applied to the cascade onto the axis running parallel to its front, and the free stream condition. Flows past rather sparse cascades, whose discontinuous characteristics departing from the lower surface of an airfoil, do not impinge on the upper surface of the adjacent airfoil are considered.By analogy with an isolated cascade [1], the upper and lower generators of the optimum airfoils are constructed of straight-line segments. In the calculated example the quadrangular airfoils whose generator segments are inclined in accordance with the double-sided extremum conditions are shown to be optimum.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 162–165, May–June, 1996.