A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

Establishment and application of logging saturation interpretation equation in vuggy reservoirs

Vuggy reservoirs are the most common, albeit important heterogeneous carbonate reservoirs in China. However, saturation calculations using logging data are not well developed, whereas Archie method is more common. In this study, electrical conduction in a vuggy reservoir is theoretically analyzed to establish a new saturation equation for vuggy reservoirs. We found that vugs have a greater effect on saturation than resistivity, which causes inflection in the rock-electricity curve. Using single-variable exPeriments, we evaluated the effects of rug size, vug number, and vug distribution on the rock-electricity relation. Based on the general saturation model, a saturation equation for vuggy reservoirs is derived, and the physical significance of the equation parameters is discussed based on the seepage-electricity similarity. The equation parameters depend on the pore structure, and vugs and matrix pore size distribution. Furthermore, a method for calculating the equation parameters is proposed, which uses nuclear magnetic resonance （NMR） data to calculate the capillary pressure curve. Field application of the proposed equation and parameter derivation method shows good match between calculated and experimental results, with an average absolute error of 5.8%.

Localization of normalized solutions for saturable nonlinear Schr?dinger equations

In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.