Flow Dynamics of \hbox {CO}_{2}/brine at the Interface Between the Storage Formation and Sealing Units in a Multi-layered Model

Pressure distribution and \(\hbox {CO}_{2}\) plume migration are two major interests in \(\hbox {CO}_{2}\) geologic storage as they determine the injectivity and storage capacity. In this study, we adopted a three-layer model comprising a storage formation and the over- and underlying seals and determined three distinct flow regions based on the vertical flux exchange of \(\hbox {CO}_{2}\) and native brine. Regions 1 and 2 showed \(\hbox {CO}_{2}\) flowing from the storage formation to adjacent seals with counter-flowing brine. The characteristics of these fluxes in Region 1 were governed by permeability change due to salt precipitation whereas buoyancy force controlled the flux pattern in Region 2. Region 3 showed brine flowing from storage formation toward the over- and underlying seals, which enabled the displaced brine to escape from the storage formation and make room for \(\hbox {CO}_{2}\) to store as well as reduce the pressure build-up. In the multi-layered model, the counter-flowing brine in flow Region 1 resulted in localized salt precipitation at the upper and lower boundary of storage formation. We assessed the bottom-hole pressure and \(\hbox {CO}_{2}\) mass in caprock with respect to reservoir size. While the formation thickness influenced the bottom-hole pressure in the early stage of injection, the horizontal extension of the reservoir was more influential to pressure build-up during the injection period, and to the stabilized pressure during the post-injection period. The \(\hbox {CO}_{2}\) mass in caprock gently increased during the injection period as well as during the post-injection period and reached about 4–5 % of injected \(\hbox {CO}_{2}\) . The percentage of escaped brine from the storage formation ranged from 80–100 % of the \(\hbox {CO}_{2}\) mass stored in the storage formation depending on the reservoir scale.     

The Effects of Brine Species on the Formation of Residual Water in a \hbox {CO}_{2}–Brine System

Geological storage of \(\hbox {CO}_{2}\) in deep saline aquifers is achieved by injecting \(\hbox {CO}_{2}\) into the aquifers and displacing the brine. Although most of the brine is displaced, some residual groundwater remains in the rock pores. We conducted experiments to investigate factors that influence how much of this residual water remains after \(\hbox {CO}_{2}\) is injected. A rock sample was saturated with brines of two different salts. Supercritical \(\hbox {CO}_{2}\) was injected into the samples at aquifer temperature and pressure, and the displaced water and water–gas mixtures were collected and measured. The results show that deionized water drains more completely than either of the two brines, and NaCl brine drains more completely than \(\hbox {CaCl}_{2}\) brine. The ranking of the irreducible water saturation at the end of the experiment is deionized \(\hbox {water}<\hbox {NaCl brine } <\hbox {CaCl}_{2}\) brine. The process of drainage can be divided into three stages according to the drainage flow rates; the Pushing Drainage, Portable Drainage, and Dissolved Drainage stages. This paper proposed a capillary model which is used to interpret the mechanisms that characterize these three stages.     

Semianalytical Solution for \text{ CO}_{2} Plume Shape and Pressure Evolution During \text{ CO}_{2} Injection in Deep Saline Formations

The injection of supercritical carbon dioxide ( $\text{ CO}_{2})$ in deep saline aquifers leads to the formation of a $\text{ CO}_{2}$ rich phase plume that tends to float over the resident brine. As pressure builds up, $\text{ CO}_{2}$ density will increase because of its high compressibility. Current analytical solutions do not account for $\text{ CO}_{2}$ compressibility and consider a volumetric injection rate that is uniformly distributed along the whole thickness of the aquifer, which is unrealistic. Furthermore, the slope of the $\text{ CO}_{2}$ pressure with respect to the logarithm of distance obtained from these solutions differs from that of numerical solutions. We develop a semianalytical solution for the $\text{ CO}_{2}$ plume geometry and fluid pressure evolution, accounting for $\text{ CO}_{2}$ compressibility and buoyancy effects in the injection well, so $\text{ CO}_{2}$ is not uniformly injected along the aquifer thickness. We formulate the problem in terms of a $\text{ CO}_{2}$ potential that facilitates solution in horizontal layers, with which we discretize the aquifer. Capillary pressure is considered at the interface between the $\text{ CO}_{2}$ rich phase and the aqueous phase. When a prescribed $\text{ CO}_{2}$ mass flow rate is injected, $\text{ CO}_{2}$ advances initially through the top portion of the aquifer. As $\text{ CO}_{2}$ is being injected, the $\text{ CO}_{2}$ plume advances not only laterally, but also vertically downwards. However, the $\text{ CO}_{2}$ plume does not necessarily occupy the whole thickness of the aquifer. We found that even in the cases in which the $\text{ CO}_{2}$ plume reaches the bottom of the aquifer, most of the injected $\text{ CO}_{2}$ enters the aquifer through the layers at the top. Both $\text{ CO}_{2}$ plume position and fluid pressure compare well with numerical simulations. This solution permits quick evaluations of the $\text{ CO}_{2}$ plume position and fluid pressure distribution when injecting supercritical $\text{ CO}_{2}$ in a deep saline aquifer.     

Elucidating the Role of Interfacial Tension for Hydrological Properties of Two-Phase Flow in Natural Sandstone by an Improved Lattice Boltzmann Method

We investigated the interfacial tension (IFT) effect on fluid flow characteristics inside micro-scale, porous media by a highly efficient multi-phase lattice Boltzmann method using a graphics processing unit. IFT is one of the most important parameters for carbon capture and storage and enhanced oil recovery. Rock pores of Berea sandstone were reconstructed from micro-CT scanned images, and multi-phase flows were simulated for the digital rock model at extremely high resolution (3.2  \(\upmu \) m). Under different IFT conditions, numerical analyses were carried out first to investigate the variation in relative permeability, and then to clarify evolution of the saturation distribution of injected fluid. We confirmed that the relative permeability decreases with increasing IFT due to growing capillary trapping intensity. It was also observed that with certain pressure gradient \(\Delta P\) two crucial IFT values, \(\sigma _{1}\) and \(\sigma _{2}\) , exist, creating three zones in which the displacement process has totally different characteristics. When \(\sigma _{1}< \sigma < \sigma _{2}\) , the capillary fingering patterns are observed, while for \(\sigma < \sigma _{1}\) viscous fingering is dominant and most of the passable pore spaces were invaded. When \(\sigma > \sigma _{2}\) the invading fluid failed to break through. The pore-throat-size distribution estimated from these crucial IFT values ( \(\sigma _{1 }\) and \(\sigma _{2})\) agrees with that derived from mercury porosimetry measurements of Berea sandstone. This study demonstrates that the proposed numerical method is an efficient tool for investigating hydrological properties from pore structures.     

Towards the Development of an Evolution Equation for Flame Turbulence Interaction in Premixed Turbulent Combustion

Flame turbulence interaction is one of the leading order terms in the scalar dissipation \(\left (\widetilde {\varepsilon }_{c}\right )\) transport equation [35] and is thus an important phenomenon in premixed turbulent combustion. Swaminathan and Grout [36] and Chakraborty and Swaminathan [15, 16] have shown that the effect of strain rate on the transport of \(\widetilde {\varepsilon }_{c}\) is dominated by the interaction between the fluctuating scalar gradients and the fluctuating strain rate, denoted here by \(\overline {\rho }\widetilde {\Delta }_{c}= \overline {\rho {\alpha }\nabla c^{\prime \prime }S_{ij}^{\prime \prime }\nabla c^{\prime \prime }}\) ; this represents the flame turbulence interaction. In order to obtain an accurate representation of this phenomenon, a new evolution equation for \(\widetilde {\Delta }_{c}\) has been proposed. This equation gives a detailed insight into flame turbulence interaction and provides an alternative approach to model the important physics represented by \(\widetilde {\Delta }_{c}\) . The \(\widetilde {\Delta }_{c}\) evolution equation is derived in detail and an order of magnitude analysis is carried out to determine the leading order terms in the \(\widetilde {\Delta }_{c}\) evolution equation. The leading order terms are then studied using a Direct Numerical Simulation (DNS) of premixed turbulent flames in the corrugated flamelet regime. It is found that the behaviour of \(\widetilde {\Delta }_{c}\) is determined by the competition between the source terms (pressure gradient and the reaction rate), diffusion/dissipation processes, turbulent strain rate and the dilatation rate. Closures for the leading order terms in \(\widetilde {\Delta }_{c}\) evolution equation have been proposed and compared with the DNS data.     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\) , and let \(\mathcal {F}\) be a family of subsets of \(S\) . For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\) . The point \(x\) is forward \(\mathcal {F}\) -Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \) . The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\) -nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \) -prolongational limit set \(J\left( x,\mathcal {F}\right) \) . The paper contains applications to control systems.     

Investigation of Self-Similar Interface Evolution in Carbon Dioxide Sequestration in Saline Aquifers

In this work, we investigate numerically the injection of supercritical carbon dioxide into a deep saline reservoir from a single well. We analyze systematically the sharp-interface evolution in different flow regimes. The flow regimes can be parameterized by two dimensionless numbers, the gravity number, \(\Gamma \) and the mobility ratio, \(\lambda \) . Numerical simulations are performed using the volume of fluid method, and the results are compared with the solutions of the self-similarity equation established in previous works, which describes the evolution of the sharp interface. We show that these theoretical solutions are in very good agreement with the results from the numerical simulations presented over the different flow regimes, thereby showing that the theoretical and simulation models predict consistently the spreading and migration of the created \(\hbox {CO}_{2}\) plume under complex flow behavior in porous media. Furthermore, we compare the numerical results with known analytic approximations in order to assess their applicability and accuracy over the investigated parametric space. The present study indicates that the self-similar solutions parameterized by the dimensionless numbers \(\lambda , \Gamma \) are significant for examining effectively injection scenarios, as these numbers control the shape of the interface and migration of the \(\hbox {CO}_{2}\) plume. This finding is essential in assessing the storage capacity of saline aquifers.     

Gas Sorption and the Consequent Volumetric and Permeability Change of Coal II: Numerical Modeling

The permeability of coalbed methane reservoirs may evolve during the recovery of methane and injection of gas, due to the change of effective stress and gas adsorption and desorption. Experimental and numerical studies were conducted to investigate the sorption-induced permeability change of coal. This paper presents the numerical modeling part of the work. It was found that adsorption of pure gases on coal was well represented by parametric adsorption isotherm models in the literature. Based on the experimental data of this study, adsorption of pure \(\hbox {N}_2\) was modeled using the Langmuir equation, and adsorption of pure \(\hbox {CO}_2\) was well represented by the N-Layer BET equation. For the modeling of CO \(_2\) & N \(_2\) binary mixture adsorption, the ideal adsorbed solution (IAS) model and the real adsorbed solution (RAS) model were used. The IAS model estimated the total amount of mixture adsorption and the composition of the adsorbed phase based on the pure adsorption isotherms. The estimated total adsorption and adsorbed-phase composition were very different from the experimental results, indicating nonideality of the CO \(_2\) –N \(_2\) –Coal-adsorption system. The measured sorption-induced strain was linearly proportional to the total amount of adsorption despite the species of the adsorbed gas. Permeability reduction followed a linear correlation with the volumetric strain with the adsorption of pure \(\hbox {N}_2\) and the tested CO \(_2\) & N \(_2\) binary mixtures, and an exponential correlation with the adsorption of pure \(\hbox {CO}_2\) .     

Expansivity and Cone-fields in Metric Spaces

Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function \(f:X\rightarrow X\) is uniformly expansive on a set \(\varLambda \subset X\) if there exist \(\varepsilon >0\) and \(\alpha \in (0,1)\) such that for any two orbits \(\hbox {x}:\{-N,\ldots ,N\} \rightarrow \varLambda \) , \(\hbox {v}:\{-N,\ldots ,N\} \rightarrow X\) of \(f\) we have $$\begin{aligned} \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n) \le \varepsilon \implies d(\hbox {x}_0,\hbox {v}_0) \le \alpha \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n). \end{aligned}$$ It occurs that a function is uniformly expansive iff there exists a generalized cone-field on \(X\) such that \(f\) is cone-hyperbolic.     

Experimental Characterization of Porosity Structure and Transport Property Changes in Limestone Undergoing Different Dissolution Regimes

Limestone dissolution by $\hbox {CO}_2$ -rich brine induces critical changes of the pore network geometrical parameters such as the pore size distribution, the connectivity, and the tortuosity which govern the macroscopic transport properties (permeability and dispersivity) that are required to parameterize the models, simulating the injection and the fate of $\hbox {CO}_2$ . A set of four reactive core-flood experiments reproducing underground conditions ( $T = 100\,^{\circ }\hbox {C}$ and $P = 12$ MPa) has been conducted for different $\hbox {CO}_2$ partial pressures $(0.034 < P_{\mathrm{CO}_2}< 3.4\; \hbox {MPa})$ in order to study the different dissolution regimes. X-ray microtomographic images have been used to characterize the changes in the structural properties from pore scale to Darcy scale, while time-resolved pressure loss and chemical fluxes enabled the determination of the sample-scale change in porosity and permeability. The results show the growth of localized dissolution features associated with high permeability increase for the highest $P_{\mathrm{CO}_2}$ , whereas dissolution tends to be more homogeneously distributed for lower values of $P_{\mathrm{CO}_2}$ . For the latter, the higher the $P_{\mathrm{CO}_2}$ , the more the dissolution patterns display ramified structures and permeability increase. For the lowest value of $P_{\mathrm{CO}_2}$ , the preferential dissolution of the calcite cement associated with the low dissolution kinetics triggers the transport that may locally accumulate and form a microporous material that alters permeability and produces an anti-correlated porosity–permeability relationship. The combined analysis of the pore network geometry and the macroscopic measurements shows that $P_{\mathrm{CO}_2}$ regulates the tortuosity change during dissolution. Conversely, the increase of the exponent value of the observed power law permeability–porosity trend while $P_{\mathrm{CO}_2}$ increases, which appears to be strongly linked to the increase of the effective hydraulic diameter, depends on the initial rock structure.     

\mathcal {H}_{\infty } filtering for sample data systems with stochastic sampling and Markovian jumping parameters

This paper deals with the problem of \(\mathcal {H}_{\infty }\) filtering for sample data systems that possess random jumping parameters described by a finite-state Markov process with stochastic sampling. Multiple stochastic sampling periods are considered in which each sampling period is assumed to be time varying that switches between two different values in a random way with given probability. The aim of this paper is to design a filter such that the filtering error system is stochastically stable with a prescribed \(\mathcal {H}_{\infty }\) disturbance attenuation level. Sufficient conditions for the existence of \(\mathcal {H}_{\infty }\) filters are expressed in terms of linear matrix inequalities (LMIs), which can be solved by using Matlab LMI toolbox. Numerical examples are given to illustrate the effectiveness of the proposed result including a realistic Transmission Control Protocol network model.     

Influence of \text{ CF }_{3}\text{ H } and \text{ CCl }_{4} additives on acetylene detonation

The influence of $\text{ CF }_{3}\text{ H }$ and $\text{ CCl }_{4}$ admixtures (known as detonation suppressors for combustible mixtures) on the development of acetylene detonation was experimentally investigated in a shock tube. The time-resolved images of detonation wave development and propagation were registered using a high-speed streak camera. Shock wave velocity and pressure profiles were measured by five calibrated piezoelectric gauges and the formation of condensed particles was detected by laser light extinction. The induction time of detonation development was determined as the moment of a pressure rise at the end plate of the shock tube. It was shown that $\text{ CF }_{3}\text{ H }$ additive had no influence on the induction time. For $\text{ CCl }_{4}$ , a significant promoting effect was observed. A simplified kinetic model was suggested and characteristic rates of diacetylene $\text{ C }_{4}\text{ H }_{2}$ formation were estimated as the limiting stage of acetylene polymerisation. An analysis of the obtained data indicated that the promoting species is atomic chlorine formed by $\text{ CCl }_{4}$ pyrolysis, which interacts with acetylene and produces $\text{ C }_{2}\text{ H }$ radical, initiating a chain mechanism of acetylene decomposition. The results of kinetic modelling agree well with the experimental data.     

Predominant Gas Transport in Microporous Hydrotalcite–Silica Membrane

The prepared microporous hydrotalcite (HT)–silica membrane was found to exhibit the molecular sieving characteristic of pristine silica material and high $\mathrm{CO}_{2}$ adsorption capacity of HT. The combined properties made enhanced $\mathrm{CO}_{2}$ permeability and separability from $\mathrm{CH}_{4}$ possible. The gas transport in the membrane was predominantly surface adsorption. The porous membrane overcame the Knudsen limitation and yielded the highest separation selectivity of 120 at 40 % $\mathrm{CO}_{2}$ feed concentration, $30\,^{\circ }\mathrm{C}$ operating temperature, and 100 kPa pressure difference.     

Residual Trapping Beneath Impermeable Barriers During Buoyant Migration of \mathrm{CO}_2

The presence of impermeable barriers in a reservoir can significantly impede the buoyant migration of $\mathrm{CO}_2$ injected deep into a heterogeneous geological formation. An important consequence of the presence of these impermeable barriers in terms of the long-term storage of $\mathrm{CO}_2$ is the residual trapping that takes place beneath the barriers, which acts to both increase the storage potential of the reservoir and improve the storage security of the $\mathrm{CO}_2$ . Analytical results for the total amount of $\mathrm{CO}_2$ trapped in a reservoir with an uncorrelated random distribution of impermeable barriers are obtained for both two and three-dimensional cases. In two dimensions, it is shown that the total amount of $\mathrm{CO}_2$ contained in this fashion scales as $n^{5/4}$ , where $n$ is the number of barriers in the vertical direction. In three dimensions, the trapped amount scales as $n^c$ , where $5/4 \le c \le 2$ depending on the aspect ratio of the barriers. The analytical two-dimensional results are compared with results of detailed numerical simulations, and good agreement is observed.     

Visual Investigation of Improvement in Extra-Heavy Oil Recovery by Borate-Assisted $$\hbox {CO}_{2}$$-Foam Injection

The significant reduction in heavy oil viscosity when mixed with \(\hbox {CO}_{2}\) is well documented. However, for \(\hbox {CO}_{2}\) injection to be an efficient method for improving heavy oil recovery, other mechanisms are required to improve the mobility ratio between the \(\hbox {CO}_{2}\) front and the resident heavy oil. In situ generation of \(\hbox {CO}_{2}\)-foam can improve \(\hbox {CO}_{2}\) injection performance by (a) increasing the effective viscosity of \(\hbox {CO}_{2}\) in the reservoir and (b) increasing the contact area between the heavy oil and injected \(\hbox {CO}_{2}\) and hence improving \(\hbox {CO}_{2}\) dissolution rate. However, in situ generation of stable \(\hbox {CO}_{2}\)-foam capable of travelling from the injection well to the production well is hard to achieve. We have previously published the results of a series of foam stability experiments using alkali and in the presence of heavy crude oil (Farzaneh and Sohrabi 2015). The results showed that stability of \(\hbox {CO}_{2}\)-foam decreased by addition of NaOH, while it increased by addition of \(\hbox {Na}_{2}\hbox {CO}_{3}\). However, the highest increase in \(\hbox {CO}_{2}\)-foam stability was achieved by adding borate to the surfactant solution. Borate is a mild alkaline with an excellent pH buffering ability. The previous study was performed in a foam column in the absence of a porous medium. In this paper, we present the results of a new series of experiments carried out in a high-pressure glass micromodel to visually investigate the performance of borate–surfactant \(\hbox {CO}_{2}\)-foam injection in an extra-heavy crude oil in a transparent porous medium. In the first part of the paper, the pore-scale interactions of \(\hbox {CO}_{2}\)-foam and extra-heavy oil and the mechanisms of oil displacement and hence oil recovery are presented through image analysis of micromodel images. The results show that very high oil recovery was achieved by co-injection of the borate–surfactant solution with \(\hbox {CO}_{2}\), due to in-situ formation of stable foam. Dissolution of \(\hbox {CO}_{2}\) in heavy oil resulted in significant reduction in its viscosity. \(\hbox {CO}_{2}\)-foam significantly increased the contact area between the oil and \(\hbox {CO}_{2}\) significantly and thus the efficiency of the process. The synergy effect between the borate and surfactant resulted in (1) alteration of the wettability of the porous medium towards water wet and (2) significant reduction of the oil–water IFT. As a result, a bank of oil-in-water (O/W) emulsion was formed in the porous medium and moved ahead of the \(\hbox {CO}_{2}\)-foam front. The in-situ generated O/W emulsion has a much lower viscosity than the original oil and plays a major role in the observed additional oil recovery in the range of performed experiments. Borate also made \(\hbox {CO}_{2}\)-foam more stable by changing the system to non-spreading oil and reducing coalescence of the foam bubbles. The results of these visual experiments suggest that borate can be a useful additive for improving heavy oil recovery in the range of the performed tests, by increasing \(\hbox {CO}_{2}\)-foam stability and producing O/W emulsions.     

Delay-dependent stability analysis and \mathcal {H}_{\infty } control for LPV systems with parameter-varying state delays

This paper develops the stability analysis and delay-dependent \(\mathcal {H}_{\infty }\) control synthesis for linear parameter-varying (LPV) systems with time-varying state delays. On the basis of the Finsler’s lemma, sufficient conditions on \(\mathcal {H}_{\infty }\) performance analysis are formulated in terms of parameterized linear matrix inequalities. The interesting annihilator matrix is constituted by time-varying parameters of LPV systems to reduce the conservatism. A numerical example is presented to confirm the efficiency of the proposed method.     

Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression

A recent field theory of elastic bodies undergoing non-smooth submacroscopic geometrical changes (disarrangements) provides a setting in which, for a given homogeneous macroscopic deformation \(F\) of the body, there are typically a number of different states \(G\) of smooth, submacroscopic deformation (disarrangement phases) available to the body. A tensorial consistency relation and the inequality \(\det G\le \det F\) that guarantees that \(F\) accommodates \(G\) determine the totality of disarrangement phases \(G\) corresponding to \(F\) , and it is natural to seek for a given \(F\) those disarrangement phases that minimize the Helmholtz free energy (stable disarrangement phases). We introduce these concepts in the particular context of continuous bodies comprised of many small elastic bodies (elastic aggregates) and in the context where disarrangements do not contribute to the Helmholtz free energy (purely dissipative disarrangements). In this setting, the Helmholtz free energy response \(G\longmapsto \varPsi (G)\) of the pieces of the aggregate determines the totality of disarrangement phases corresponding to \(F\) , which necessarily includes the phase \(G=F\) (compact phase) in which every piece of the aggregate undergoes the given macroscopic deformation \(F\) . When the response function \(\varPsi \) is isotropic and smooth, and when \(\varPsi \) possesses standard semiconvexity and growth properties, the body also admits phases of the form \(G=\zeta _{\min }R\) (loose phases) with \(R\) an arbitrary rotation, provided that \(\zeta _{\min }R \) satisfies the accommodation inequality \(\zeta _{\min }^{3}\le \det F\) . Loose phases, when available, achieve the global minimum \(\varPsi (\zeta _{\min }R)\) of the free energy and consequently are stable and stress-free. When \( \varPsi (G)\) has the specific form \(\varPsi _{\alpha \beta }(G)=(\alpha /2)(\det G)^{-2}+(\beta /2)tr(GG^{T})\) , with \(\alpha \) , \(\beta \) given elastic constants, we determine all of the disarrangement phases corresponding to \(F\) . These include not only the compact and loose phases, but also disarrangement phases \(G\) in which the stress \(D\varPsi (G)\) is uniaxial or planar. Our main result (“stability implies no-tension”) is the assertion that every stable disarrangement phase for \(\varPsi _{\alpha \beta }\) cannot support tensile tractions, and our treatment of elastic aggregates thus provides a natural setting for the emergence of no-tension materials whose response in compression is non-linear. Existing treatments of no-tension materials assume at the outset that the body cannot support tension and that the response in compression is linear.     

Estimability Analysis for Optimization of Hysteretic Soil Hydraulic Parameters Using Data of a Field Irrigation Experiment

To improve the quality of parameter optimization, estimability analysis has been proposed as the first step before inverse modeling. When using field data of irrigation experiments for the determination of soil hydraulic parameters, wetting and drying processes may complicate optimization. The objectives of this study were to compare estimability analysis and inverse optimization of the soil hydraulic parameters in the models with and without considering hysteresis of the soil water retention function. Soil water pressure head data of a field irrigation experiment were used. The one-dimensional vertical water movement in variably saturated soil was described with the Richards equation using the HYDRUS-1D code. Estimability of the unimodal van Genuchten–Mualem hydraulic model parameters as well as of the hysteretic parameter model of Parker and Lenhard was classified according to a sensitivity coefficient matrix. The matrix was obtained by sequentially calculating effects of initial parameter variations on changes in the simulated pressure head values. Optimization was carried out by means of the Levenberg-Marquardt method implemented in the HYDRUS-1D code. The parameters \(\alpha , K_{s}, \theta _{s}\) , and \(n\) in the nonhysteretic model were found sensitive and parameter \(\theta _{s}\) strongly correlated with parameter \(n\) . When assuming hysteresis, the estimability was decreased with soil depth for \(K_{s}\) and \(\alpha ^{d}\) , and increased for \(\theta _{s}\) and n. Among the shape parameters, \(\alpha ^{w}\) was the most estimable. The hysteretic model could approximate the pressure heads in the soil by considering parameters from wetting and drying periods separately as initial estimates. The inverse optimization could be carried out more efficiently with most estimable parameters. Despite the remaining weaknesses of the local optimization algorithm and the inflexibility of the unimodal van Genuchten model, the results suggested that estimability analysis could be considered as a guidance to better define the optimization scenarios and then improved the determination of soil hydraulic parameters.     

On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

We study the energy decay of the turbulent solutions to the Navier–Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate \({O(t^{-\alpha}), t \rightarrow \infty, \alpha \in [0, 5/2]}\) , are exactly characterized by their initial conditions belonging into the homogeneous Besov space \({\dot{B}^{-\alpha}_{2, \infty}}\) . Similarly, for a solution u and \({p \in [1, \infty]}\) the integral \({\int_{0}^{\infty} \|t^{\alpha/2} u(t)\|^p \frac{1}{t} dt}\) is finite if and only if the initial condition of u belongs to the homogeneous Besov space \({\dot{B}_{2, p}^{-\alpha}}\) . For the case \({\alpha \in (5/2, 9/2]}\) we present analogical results for some subclasses of turbulent solutions.