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.     

Displacement and Dissolution Characteristics of CO$$_{}$$/Brine System in Unconsolidated Porous Media

This study investigated the dynamic displacement and dissolution of \(\hbox {CO}_{2}\) in porous media at 313 K and 6/8 MPa. Gaseous (\(\hbox {gCO}_{2}\)) at 6 MPa and supercritical \(\hbox {CO}_{2 }(\hbox {scCO}_{2}) \) at 8 MPa were injected downward into a glass bead pack at different flow rates, following upwards brine injection. The processes occurring during \(\hbox {CO}_{2}\) drainage and brine imbibition were visualized using magnetic resonance imaging. The drainage flow fronts were strongly influenced by the flow rates, resulting in different gas distributions. However, brine imbibition proceeded as a vertical compacted front due to the strong effect of gravity. Additionally, the effects of flow rate on distribution and saturation were analyzed. Then, the front movement of \(\hbox {CO}_{2}\) dissolution was visualized along different paths after imbibition. The determined \(\hbox {CO}_{2}\) concentrations implied that little \(\hbox {scCO}_{2}\) dissolved in brine after imbibition. The dissolution rate was from \(10^{-8}\) to \(10^{-9}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) and from \(10^{-6}\) to \(10^{-8}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) for \(\hbox {gCO}_{2}\) at 6 MPa and \(\hbox {scCO}_{2 }\) at 8 MPa, respectively. The total time for the \(\hbox {scCO}_{2}\) dissolution was short, indicating fast mass transfer between the \(\hbox {CO}_{2}\) and brine. Injection of \(\hbox {CO}_{2}\) under supercritical conditions resulted in a quick establishment of a steady state with high storage safety.     

A new type of functional chemical sensitizer $$\hbox {MgH}_{2}$$ for improving pressure desensitization resistance of emulsion explosives

In millisecond-delay blasting and deep water blasting projects, traditional emulsion explosives sensitized by the chemical sensitizer \(\hbox {NaNO}_{2}\) often encounter incomplete explosion or misfire problems because of the “pressure desensitization” phenomenon, which seriously affects blasting safety and construction progress. A \(\hbox {MgH}_{2}\)-sensitized emulsion explosive was invented to solve these problems. Experimental results show that \(\hbox {MgH}_{2}\) can effectively reduce the problem of pressure desensitization. In this paper, the factors which influence the pressure desensitization of two types of emulsion explosives are studied, and resistance to this phenomenon of \(\hbox {MgH}_{2}\)-sensitized emulsion explosives is discussed.     

Comparative Study of Five Outcrop Chalks Flooded at Reservoir Conditions: Chemo-mechanical Behaviour and Profiles of Compositional Alteration

This study presents experimental results from a flooding test series performed at reservoir conditions for five high-porosity Cretaceous onshore chalks from Denmark, Belgium and the USA, analogous to North Sea reservoir chalk. The chalks are studied in regard to their chemo-mechanical behaviour when performing tri-axial compaction tests while injecting brines (0.219 mol/L \(\hbox {MgCl}_{2}\) or 0.657 mol/L NaCl) at reservoir conditions for 2–3 months (T = 130 \(^\circ \hbox {C}\); 1 PV/d). Each chalk type was examined in terms of its mineralogical and chemical composition before and after the mechanical flooding tests, using an extensive set of analysis methods, to evaluate the chalk- and brine-dependent chemical alterations. All \(\hbox {MgCl}_{2}\)-flooded cores showed precipitation of Mg-bearing minerals (mainly magnesite). The distribution of newly formed Mg-bearing minerals appears to be chalk-dependent with varying peaks of enrichment. The chalk samples from Aalborg originally contained abundant opal-CT, which was dissolved with both NaCl and \(\hbox {MgCl}_{2}\) and partly re-precipitated as Si-Mg-bearing minerals. The Aalborg core injected with \(\hbox {MgCl}_{2}\) indicated strongly increased specific surface area (from 4.9 \(\hbox {m}^{2}\hbox {/g}\) to within 7–9 \(\hbox {m}^{2}\hbox {/g}\)). Mineral precipitation effects were negligible in chalk samples flooded with NaCl compared to \(\hbox {MgCl}_{2}\). Silicates were the main mineralogical impurity in the studied chalk samples (0.3–6 wt%). The cores with higher \(\hbox {SiO}_{2}\) content showed less deformation when injecting NaCl brine, but more compaction when injecting \(\hbox {MgCl}_{2}\)-brine. The observations were successfully interpreted by mathematical geochemical modelling which suggests that the re-precipitation of Si-bearing minerals leads to enhanced calcite dissolution and mass loss (as seen experimentally) explaining the high compaction seen in \(\hbox {MgCl}_{2}\)-flooded Aalborg chalk. Our work demonstrates that the original mineralogy, together with the newly formed minerals, can control the chemo-mechanical interactions during flooding and should be taken into account when predicting reservoir behaviour from laboratory studies. This study improves the understanding of complex flow reaction mechanisms also relevant for field-scale dynamics seen during brine injection.     

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.     

Theoretical analysis for the mechanical behavior caused by an electromagnetic cycle in ITER $$\hbox {Nb}_{3}\hbox {Sn}$$ cable-in-conduit conductors

The central solenoid (CS) is one of the key components of the International Thermonuclear Experimental Reactor (ITER) tokamak and which is often considered as the heart of this fusion reactor. This solenoid will be built by using \(\hbox {Nb}_{3}\hbox {Sn}\) cable-in-conduit conductors (CICC), capable of generating a 13 T magnetic field. In order to assess the performance of the \(\hbox {Nb}_{3}\hbox {Sn}\) CICC in nearly the ITER condition, many short samples have been evaluated at the SULTAN test facility (the background magnetic field is of 10.85 T with the uniform length of 400 mm at 1% homogeneity) in Centre de Recherches en Physique des Plasma (CRPP). It is found that the samples with pseudo-long twist pitch (including baseline specimens) show a significant degradation in the current-sharing temperature (Tcs), while the qualification tests of all short twist pitch (STP) samples, which show no degradation versus electromagnetic cycling, even exhibits an increase of Tcs. This behavior was perfectly reproduced in the coil experiments at the central solenoid model coil (CSMC) facility last year. In this paper, the complex structure of the \(\hbox {Nb}_{3}\hbox {Sn}\) CICC would be simplified into a wire rope consisting of six petals and a cooling spiral. An analytical formula for the Tcs behavior as a function of the axial strain of the cable is presented. Based on this, the effects of twist pitch, axial and transverse stiffness, thermal mismatch, cycling number, magnetic distribution, etc., on the axial strain are discussed systematically. The calculated Tcs behavior with cycle number show consistency with the previous experimental results qualitatively and quantitatively. Lastly, we focus on the relationship between Tcs and axial strain of the cable, and we conclude that the Tcs behavior caused by electromagnetic cycles is determined by the cable axial strain. Once the cable is in a compression situation, this compression strain and its accumulation would lead to the Tcs degradation. The experimental observation of the Tcs enhancement in the CS STP samples should be considered as a contribution of the shorter length of the high field zone in SULTAN and CSMC devices, as well as the tight cable structure.     

On the Validation of a Compositional Model for the Simulation of $$\text {CO}_2$$ Injection into Saline Aquifers

In this work we apply a recently proposed Bayesian Markov chain Monte Carlo framework (Akbarabadi et al. in Comput Geosci 19(6):1231–1250, 2015) to quantify uncertainty in the three-dimensional permeability field of a rock core. This process establishes the credibility of a compositional two-phase flow model to describe the displacement of brine by \(\text {CO}_2\) and \(\text {CO}_2\) storage in saline aquifers. We investigate the predictive capabilities of the compositional model in the context of an unsteady-state \(\text {CO}_2\)-brine drainage experiment at the laboratory scale, performed at field-scale aquifer conditions. We employ forward models consisting of a system of discretized partial differential equations along with relative permeability curves obtained by a curve fitting of experimental measurements. We consider a forward model to be validated when: (1) numerical simulations reveal that the Bayesian framework has accurately characterized the core’s permeability and (2) Monte Carlo predictions show excellent agreement between measured and simulated data. A large set of numerical studies with an accurate compositional simulator shows that forward models have been successfully validated. For such models, our numerical results show that we are able to capture all the dominant features and general trends of the \(\text {CO}_2\) saturation fields observed in the core. Our study is consistent with the design and findings of real experiments. Fluid properties, relative permeability data, measured porosity field, physical dimensions, and thermodynamic conditions are the same as those reported in Akbarabadi and Piri (Adv Water Resour 52:190–206, 2013). However, the measured saturation data are from flow experiments different from those reported in Akbarabadi and Piri (2013), and will be presented here.     

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.     

Finite plasticity in $$\varvec{P}^\top \! \varvec{P}$$. Part I: constitutive model

We address a finite-plasticity model based on the symmetric tensor \(\varvec{P}^\top \! \varvec{P}\) instead of the classical plastic strain \(\varvec{P}\). Such a structure arises by assuming that the material behavior is invariant with respect to frame transformations of the intermediate configuration. The resulting variational model is lower dimensional, symmetric and based solely on the reference configuration. We discuss the existence of energetic solutions at the material-point level as well as the convergence of time discretizations. The linearization of the model for small deformations is ascertained via a rigorous evolution-\(\Gamma \)-convergence argument. The constitutive model is combined with the equilibrium system in Part II where we prove the existence of quasistatic evolutions and ascertain the linearization limit (Grandi and Stefanelli in 2016).     

Design of a steering control law for an autonomous underwater vehicle using nonlinear $$\mathscr {H}_{\infty }$$ state feedback technique

This paper proposes a new robust nonlinear \(\mathscr {H}_{\infty }\) state feedback (NHSF) controller for an autonomous underwater vehicle (AUV) in steering plane. A three-degree-of-freedom nonlinear model of an AUV has considered for developing a steering control law. In this, the energy dissipative theory is used which leads to form a Hamilton–Jacobi–Isaacs (HJI) inequality. The nonlinear \(\mathscr {H}_{\infty }\) control algorithm has been developed by solving HJI equation such that the AUV tracks the desired yaw angle accurately. Furthermore, a path following control has been implemented using the NHSF control algorithm for various paths in steering plane. Simulation studies have been carried out using MATLAB/Simulink environment to verify the efficacies of the proposed control algorithm for AUV. From the results obtained, it is concluded that the proposed robust control algorithm exhibits a good tracking performance ensuring internal stability and significant disturbance attenuation.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Observer-based quantized sliding mode $${\varvec{\mathcal {H}}}_{\varvec{\infty }}$$ control of Markov jump systems

An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty }\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty }\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.     

Foam Generation Hysteresis in Porous Media: Experiments and New Insights

Foam application in subsurface processes including environmental remediation, geological carbon-sequestration, and gas-injection enhanced oil recovery (EOR) has the potential to enhance contamination remediation, secure \(\hbox {CO}_{2}\) storage, and improve oil recovery, respectively. Nanoparticles are a promising alternative to surfactants in creating foam in harsh environments. We conducted \(\hbox {CO}_{2}\)-in-brine foam generation experiments in Boise sandstones with surface-treated silica nanoparticle in high-salinity conditions. All the experiments were conducted at the fixed \(\hbox {CO}_{2}\) volume fraction and fixed flow rate which changed in steps. The steady-state foam apparent viscosity was measured as a function of injection velocity. The foam flowing through the cores showed higher apparent viscosity as the flow rate increased from low to medium and high velocities. At very high velocities, once foam bubbles were finely textured, the foam apparent viscosity was governed by foam rheology rather than foam creation. A noticeable hysteresis occurred when the flow velocity was initially increased and then decreased, implying multiple (coarse and strong) foam states at the same superficial velocity. A normalized generation function was combined with CMG-STARS foam model to cover full spectrum of foam behavior in the experiments. The new model successfully captures foam generation and hysteresis trends in presented experiments in this study and data from the literature. The results indicate once foam is generated in porous media, it is possible to maintain strong foam at low injection rates. This makes foam more feasible in field applications where foam generation is limited by high injection rates that may only exist near the injection well.     

Editorial to the Special Issue on Reconstruction of Porous Media and Materials and Its Applications

The single-well chemical tracer test (SWCTT) has emerged in the past decades as a method for measuring oil saturation prior to and/or after EOR operations, to measure the recovery performance in-situ. To use this technology, the partition coefficients of the selected tracers are essential for estimating the level of residual oil at the targeted single well. Commonly, injection of short chain alcohols and ethyl acetate, a reactive tracer, is carried out for the tracer slug, mainly based on site-specific reservoir conditions, to accurately determine the level of oil saturation in-situ. However, injection of ethyl formate has been less common due to its fast hydrolysis rate under elevated temperature, which increases the challenges in data interpretation. Therefore, a systematic study for using ethyl formate under mid-range temperature \((<60\,^{\circ }\hbox {C})\), as commonly found in mature oil field in the USA, shows the potential to be applied for SWCTT. As part of the design effort for a series of EOR field tests to manage the project risk, we particularly assessed the relationships between the partition coefficients of reactive tracers and subsurface conditions such as salinity, temperatures, type of electrolytes, and the equivalent alkane carbon number (EACN) of the crude oil experiments was performed under various reservoir conditions as a function of actual site characteristics at the targeted high saline formations. In brief, our data clearly show that the (oil/water) partition coefficient of ethyl formate increases steadily with increasing NaCl concentrations, ranging from 10,000 (0.17 M) to 250,000 mg/L (4.28 M). A similar upward trend was observed for increasing temperature between 25 and \(52\,^{\circ }\hbox {C}\); however, the partition coefficient decreases inversely with increasing the crude oil EACN over the range from 8 to 12, which are common for domestic oil samples. It was also showed that brine with high NaCl concentration yielded higher partition coefficients. In contrast, brine with high \(\hbox {CaCl}_{2}\) and \(\hbox {BaCl}_{2}\) concentration yielded lower values. And \(\hbox {MgCl}_{2}\) performed somewhat unusual trend in our tests. These results further indicate that the partition coefficient of the reactive tracer, ethyl formate, is sensitive to change in salinity, temperatures, type of electrolytes and EACN, as observed for other chemical tracers. In addition, based on the hydrolysis rate of ethyl formate under various reservoir conditions, the appropriate window of shut-in time can be pre-determined before initiating the field test. We believe that the ability of better understanding the partition coefficients and predicting the shut-in time beforehand could drastically reduce the risks of SWCTT operations.     

A novel approach for generating multi-direction multi-double-scroll attractors

Attention is focused in this work on quasiperiodic motion of nonlinear systems whose spectrum contains uniformly spaced sideband frequencies with a distance \(\omega _{d}\) apart, around a frequency \(\omega \) with \(\omega \gg \omega _{d}\) and its integer multiples, which are referred to as carrier frequencies. The ratio of the two frequencies \(\omega \) and \(\omega _{d}\) is an irrational number. A new method based on the traditional incremental harmonic balance (IHB) method with multiple timescales, referred to as Lau method, where two timescales, \(\tau _{1}=\omega t\) (a fast timescale) and \(\tau _{2}=\omega _{d}t\) (a slow timescale), are introduced, is presented to analyze quasiperiodic motion of nonlinear systems. An amplitude increment algorithm is adapted to deal with cases where the two frequencies \(\omega \) and \(\omega _{d}\) are    unknown a priori, in order to automatically trace frequency response of quasiperiodic motion of nonlinear systems and accurately calculate all frequency components and their corresponding amplitudes. Results of application of the present IHB method to quasiperiodic free vibration of a hinged–clamped beam with internal resonance between two transverse modes are shown and compared with previously published results with Lau method and those from numerical integration. While differences are noted between results predicted by the present IHB method and Lau method, excellent agreement is achieved between results from the present IHB method and numerical integration even in cases of strongly nonlinear vibration. The present IHB method is also used to analyze quasiperiodic free vibration of high-dimensional models of the hinged–clamped beam.     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Detonation propagation in hydrogen–air mixtures with transverse concentration gradients

The influence of transverse concentration gradients on detonation propagation in \(\hbox {H}_2\)–air mixtures is investigated experimentally in a wide parameter range. Detonation fronts are characterized by means of high-speed shadowgraphy, OH* imaging, pressure measurements, and soot foils. Steep concentration gradients at low average \(\hbox {H}_2\) concentrations lead to single-headed detonations. A maximum velocity deficit compared to the Chapman–Jouguet velocity of 9 % is observed. Significant amounts of mixture seem to be consumed by turbulent deflagration behind the leading detonation. Wall pressure measurements show high local pressure peaks due to strong transverse waves caused by the concentration gradients. Higher average \(\hbox {H}_2\) concentrations or weaker gradients allow for multi-headed detonation propagation.     

Direct Numerical Simulation of Head-On Quenching of Statistically Planar Turbulent Premixed Methane-Air Flames Using a Detailed Chemical Mechanism

A three-dimensional compressible Direct Numerical Simulation (DNS) analysis has been carried out for head-on quenching of a statistically planar stoichiometric methane-air flame by an isothermal inert wall. A multi-step chemical mechanism for methane-air combustion is used for the purpose of detailed chemistry DNS. For head-on quenching of stoichiometric methane-air flames, the mass fractions of major reactant species such as methane and oxygen tend to vanish at the wall during flame quenching. The absence of \(\text {OH}\) at the wall gives rise to accumulation of carbon monoxide during flame quenching because \(\text {CO}\) cannot be oxidised anymore. Furthermore, it has been found that low-temperature reactions give rise to accumulation of \(\text {HO}_{2}\) and \(\mathrm {H}_{2}\mathrm {O}_{2}\) at the wall during flame quenching. Moreover, these low temperature reactions are responsible for non-zero heat release rate at the wall during flame-wall interaction. In order to perform an in-depth comparison between simple and detailed chemistry DNS results, a corresponding simulation has been carried out for the same turbulence parameters for a representative single-step Arrhenius type irreversible chemical mechanism. In the corresponding simple chemistry simulation, heat release rate vanishes once the flame reaches a threshold distance from the wall. The distributions of reaction progress variable c and non-dimensional temperature T are found to be identical to each other away from the wall for the simple chemistry simulation but this equality does not hold during head-on quenching. The inequality between c (defined based on \(\text {CH}_{4}\) mass fraction) and T holds both away from and close to the wall for the detailed chemistry simulation but it becomes particularly prominent in the near-wall region. The temporal evolutions of wall heat flux and wall Peclet number (i.e. normalised wall-normal distance of \(T = 0.9\) isosurface) for both simple and detailed chemistry laminar and turbulent cases have been found to be qualitatively similar. However, small differences have been observed in the numerical values of the maximum normalised wall heat flux magnitude \(\left ({\Phi }_{\max } \right )_{\mathrm {L}}\) and the minimum Peclet number \((Pe_{\min })_{\mathrm {L}}\) obtained from simple and detailed chemistry based laminar head-on quenching calculations. Detailed explanations have been provided for the observed differences in behaviours of \(\left ({\Phi }_{\max }\right )_{\mathrm {L}}\) and \((Pe_{\min })_{\mathrm {L}}\). The usual Flame Surface Density (FSD) and scalar dissipation rate (SDR) based reaction rate closures do not adequately predict the mean reaction rate of reaction progress variable in the near-wall region for both simple and detailed chemistry simulations. It has been found that recently proposed FSD and SDR based reaction rate closures based on a-priori DNS analysis of simple chemistry data perform satisfactorily also for the detailed chemistry case both away from and close to the wall without any adjustment to the model parameters.     

A nonlocal species concentration theory for diffusion and phase changes in electrode particles of lithium ion batteries

A nonlocal species concentration theory for diffusion and phase changes is introduced from a nonlocal free energy density. It can be applied, say, to electrode materials of lithium ion batteries. This theory incorporates two second-order partial differential equations involving second-order spatial derivatives of species concentration and an additional variable called nonlocal species concentration. Nonlocal species concentration theory can be interpreted as an extension of the Cahn–Hilliard theory. In principle, nonlocal effects beyond an infinitesimal neighborhood are taken into account. In this theory, the nonlocal free energy density is split into the penalty energy density and the variance energy density. The thickness of the interface between two phases in phase segregated states of a material is controlled by a normalized penalty energy coefficient and a characteristic interface length scale. We implemented the theory in COMSOL Multiphysics\(^{\circledR }\) for a spherically symmetric boundary value problem of lithium insertion into a \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\) cathode material particle of a lithium ion battery. The two above-mentioned material parameters controlling the interface are determined for \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\), and the interface evolution is studied. Comparison to the Cahn–Hilliard theory shows that nonlocal species concentration theory is superior when simulating problems where the dimensions of the microstructure such as phase boundaries are of the same order of magnitude as the problem size. This is typically the case in nanosized particles of phase-separating electrode materials. For example, the nonlocality of nonlocal species concentration theory turns out to make the interface of the local concentration field thinner than in Cahn–Hilliard theory.