Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media

We present a spatial renormalization group algorithm to handle immiscibletwo-phase flow in heterogeneous porous media. We call this algorithmFRACTAM-R, where FRACTAM is an acronym for Fast Renormalization Algorithmfor Correlated Transport in Anisotropic Media, and the R stands for relativepermeability. Originally, FRACTAM was an approximate iterative process thatreplaces the L × L lattice of grid blocks, representing the reservoir,by a (L/2) × (L/2) one. In fact, FRACTAM replaces the original L× L lattice by a hierarchical (fractal) lattice, in such a way thatfinding the solution of the two-phase flow equations becomes trivial. Thistriviality translates in practice into computer efficiency. For N=L ×L grid blocks we find that the computer time necessary to calculatefractional flow F(t) and pressure P(t) as a function of time scales as N^1.7 for FRACTAM-R. This should be contrasted with thecomputational time of a conventional grid simulator N^2.3. The solution we find in this way is an accurateapproximation to the direct solution of the original problem.     

Regularity of strong solutions of one-dimensional SDE’s with discontinuous and unbounded drift

In this paper we develop a method for constructing strong solutions of one-dimensional Stochastic Differential Equations where the drift may be discontinuous and unbounded. The driving noise is the Brownian Motion and we show that the solution is Sobolev-differentiable in the initial condition and Malliavin differentiable. This method is not based on a pathwise uniqueness argument. We will apply these results to the stochastic transport equation. More specifically, we obtain a continuously differentiable solution of the stochastic transport equation when the driving function is a step function.     

Geometry of random sequential adsorption

By sequentially adding line segments to a line or disks to a surface at random positions without overlaps, we obtain configurations of the one- and two-dimensional random sequential adsorption (RSA) problem. We have simulated the one- and two-dimensional problem with periodic boundary condition. The one-dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation. In two dimensions the geometrical properties of the RSA configuration are discussed and in addition known results of the RSA process are reproduced. Various statistical distributions of the Voronoi-Dirichlet (VD) network corresponding to the RSA disk configuration are analyzed. In order to characterize pores in the RSA configuration, we introduce circular holes. There is a direct correspondence between vertices of the VD network and these holes, and also between direct/indirect geometrical neighbors and these holes. The hole size distribution is found to be a parabola. We also find general relations that connect the asymptotic behavior of the surface coverage, the correlation function, and the hole size distribution.     

A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.     

Comparison of nine different real-time PCR chemistries for qualitative and quantitative applications in GMO detection

Several techniques have been developed for detection and quantification of genetically modified organisms, but quantitative real-time PCR is by far the most popular approach. Among the most commonly used real-time PCR chemistries are TaqMan probes and SYBR green, but many other detection chemistries have also been developed. Because their performance has never been compared systematically, here we present an extensive evaluation of some promising chemistries: sequence-unspecific DNA labeling dyes (SYBR green), primer-based technologies (AmpliFluor, Plexor, Lux primers), and techniques involving double-labeled probes, comprising hybridization (molecular beacon) and hydrolysis (TaqMan, CPT, LNA, and MGB) probes, based on recently published experimental data. For each of the detection chemistries assays were included targeting selected loci. Real-time PCR chemistries were subsequently compared for their efficiency in PCR amplification and limits of detection and quantification. The overall applicability of the chemistries was evaluated, adding practicability and cost issues to the performance characteristics. None of the chemistries seemed to be significantly better than any other, but certain features favor LNA and MGB technology as good alternatives to TaqMan in quantification assays. SYBR green and molecular beacon assays can perform equally well but may need more optimization prior to use.     

A fast algorithm for estimating large-scale permeabilities of correlated anisotropic media

The problem of estimating large-scale permeabilities of reservoirs based on knowledge of the small-scale permeabilities is addressed. We present an accurate and fast algorithm to calculate the global permeabilities of two- or three-dimensional correlated and anisotropic block samples, thus providing a fast algorithm for obtaining grid block permeabilities for reservoir simulators from small scale data. The algorithm is tested on both two- and three-dimensional tube networks generated from real images and fractal forgeries modeling porous media. In almost all cases, the algorithm estimates the correct global permeability (calculated using exact but slow algorithms) of the network to better than 5%. The new algorithm is comparable in speed to conventional averaging techniques, such as the geometric mean, but the obtained estimates are always much better.List of Symbols K permeability of network (global permeability - K _e estimated permeability - K K×(L_x×L_y)/L_z - K permeability perpendicular to layering - K permeability parallel with layering - L _x, L_y, L_z Network size inx, y andz-directions - L Size of quadratic (cubic) network - Q global flux through network - U Q/(L_x×L_y), Darcy velocity (flux per unit area) - V volume of network - P pressure drop across network - a,b parameters in Equation (4) - P _i pressure at sitei - q _ij flux between nodesi andj - parameter of Pareto distribution - porosity - K(i) permeability at site (block)i - K _ij permeability of bond between nodesi andj - K _min minimumk(i) for sample - K _max maximumK(i) for sample - fluid viscosity     

Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation

In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions. We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz.