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.     

Experimental Study of Nonlinear Flow in Micropores Under Low Pressure Gradient

As throat radius decrease to micro-nanoscale, seepage in unconventional reservoirs such as ultra-low permeability and tight reservoirs differs from conventional ones. Flow experiment in micropores is a promising approach to study characteristics of microflow. In this paper, a visual experimental device was established. Water flow through micropores with radius of 1.38–10.03 \(\upmu \hbox {m}\) was investigated, under 0.033–16 MPa/m. The results showed that in microscale, water flow did not agree with Poiseuille equation. Flow rate was lower than theoretical value and showed nonlinear characteristics. In the near wall area, due to the attraction of solid wall, a stagnant fluid layer was formed. It occupied flow space and thus lowered flow rate. Its thickness declined with pressure gradient increasing, which led to nonlinear flow characteristics. When the pressure gradient was very high, the thickness stopped declining and kept constant. Afterward, the flow transited to linear. In pores with smaller radius, the steady stagnant layer was thinner, but took a larger proportion of the flow space. For tubes of \(r = 1.38, 4.81, 10.03\,\upmu \hbox {m}\), the thickness of steady stagnant layer was 0.11, 0.23, 0.27 \(\upmu \hbox {m}\), respectively.     

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.     

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.     

Ethylene–air detonation in water spray

Detonation experiments are conducted in a 52 \(\hbox {mm}\) square channel with an ethylene–air gaseous mixture with dispersed liquid water droplets. The tests were conducted with a fuel–air equivalence ratio ranging from 0.9 to 1.1 at atmospheric pressure. An ultrasonic atomizer generates a polydisperse liquid water spray with droplet diameters of 8.5–12 \(\upmu \hbox {m}\), yielding an effective density of 100–120 \(\hbox {g}/\hbox {m}^{3}\). Pressure signals from seven transducers and cellular structure are recorded for each test. The detonation structure in the two-phase mixture exhibits a gaseous-like behaviour. The pressure profile in the expansion fan is not affected by the addition of water. A small detonation velocity deficit of up to 5 % was measured. However, the investigation highlights a dramatic increase in the cell size (\(\lambda \)) associated with the increase in the liquid water mass fraction in the two-phase mixture. The detonation structure evolves from a multi-cell to a half-cell mode. The analysis of the decay of the post-shock pressure fluctuations reveals that the ratio of the hydrodynamic thickness over the cell size (\(x_{{\mathrm {HT}}}/{\lambda }\)) remains quite constant, between 5 and 7. A slight decrease of this ratio is observed as the liquid water mass fraction is increased, or the ethylene–air mixture is made leaner.     

Study of droplet flow in a T-shape microchannel with bottom wall fluctuation

Droplet generation in a T-shape microchannel, with a main channel width of 50 μm, side channel width of 25 μm, and height of 50 μm, is simulated to study the effects of the forced fluctuation of the bottom wall. The periodic fluctuations of the bottom wall are applied on the near junction part of the main channel in the T-shape microchannel. Effects of bottom wall's shape,fluctuation periods, and amplitudes on the droplet generation are covered in the research of this protocol. In the simulation,the average size is affected a little by the fluctuations, but significantly by the fixed shape of the deformed bottom wall, while the droplet size range is expanded by the fluctuations under most of the conditions. Droplet sizes are distributed in a periodic pattern with small amplitude along the relative time when the fluctuation is forced on the bottom wall near the T-junction,while the droplet emerging frequency is not varied by the fluctuation. The droplet velocity is varied by the bottom wall motion,especially under the shorter period and the larger amplitude. When the fluctuation period is similar to the droplet emerging period, the droplet size is as stable as the non-fluctuation case after a development stage at the beginning of flow, while the droplet velocity is varied by the moving wall with the scope up to 80% of the average velocity under the conditions of this investigation.     

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.     

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.     

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.

     

Sediment–Water Interface Flow with the Multiparticle Collision Dynamics

This paper presents an experimental study of particle transport in porous medium using a self-developed sand layer transportation–deposition testing system, aiming at delineating the detachment characteristics of deposited particles in porous medium. Two experimental modes, increase flow velocity and change flow direction, were adopted in this study. The tests were conducted using quartz powder as the particles and quartz sand as the porous media to study the response of detachment characteristics to changes in particle diameter (\(d_{s}\), with median diameter 18 and 41 \(\upmu \)m) and grain diameter (\(d_{p}\), with median diameter 0.36 and 1.25 mm). Breakthrough curves after the second peak were well described by a double exponential model with parameters of weight coefficient and detachment coefficient. This study shows that both modes can change the detach rate of deposited particles observably, and detach rate is affected by the value of flow velocity greatly.     

The role of angular momentum in the laminar motion of viscous fluids

In laminar flow, viscous fluids must exert appropriate elastic shear stresses normal to the flow direction. This is a direct consequence of the balance of angular momentum. There is a limit, however, to the maximum elastic shear stress that a fluid can exert. This is the ultimate shear stress, \(\tau _\mathrm{y}\), of the fluid. If this limit is exceeded, laminar flow becomes dynamically incompatible. The ultimate shear stress of a fluid can be determined from experiments on plane Couette flow. For water at \(20\,^{\circ }\hbox {C}\), the data available in the literature indicate a value of \(\tau _\mathrm{y}\) of about \(14.4\times 10^{-3}\, \hbox {Pa}\). This study applies this value to determine the Reynolds numbers at which flowing water reaches its ultimate shear stress in the case of Taylor–Couette flow and circular pipe flow. The Reynolds numbers thus obtained turn out to be reasonably close to those corresponding to the onset of turbulence in the considered flows. This suggests a connection between the limit to laminar flow, on the one hand, and the occurrence of turbulence, on the other.     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Imbibition into Highly Porous Layers of Aggregated Particles

The co-occurrence of gravity-driven drainage and forced convective drying in a macroporous medium is investigated in this study. The drainage and drying processes of fully saturated porous asphalt (PA) specimens placed in a custom-made mini wind tunnel are documented with neutron radiography (NR). Six PA specimens of dimensions \(180\times 10\times 30\,\hbox {mm}^{3}\) with a maximum aggregate size of 8 or 11 mm are used in the experiments. In the first few minutes of each experiment, there is significant moisture loss in all the specimens due to gravity-driven drainage. Most of the residual water retention is observed at the bottom region of the specimens due to the strong impact of gravity-driven drainage in the upper regions. The specimens are subjected to many hours of airflow at their top surface; however, forced convection from turbulent airflow near the upper part of the specimens is found to have a minor influence on moisture loss when there are no water clusters in the upper regions of the specimens. This points to the strong resistance to evaporation in PA as a result of the large vapor diffusion lengths. By combining neutron radiography and microcomputer tomography (X-ray \(\upmu \)-CT) images, saturated and unsaturated flows in the pores are distinguished. Fluid flow path during air entry and water redistribution is further analyzed by reconstructing the real three-dimensional pore geometry of the specimens from X-ray \(\upmu \)-CT scans.     

Research on viscosity of metal at high pressure

A new experimental technique, the flyer-impact method, is proposed in this article to investigate the viscosity coefficient of shocked metals. In this technique, a shock wave with a sinusoidal perturbation on the front is induced by the sinusoidal profile of the impact surface of the sample by use of a two-stage light-gas gun, and the oscillatory damping process of the perturbation amplitude is monitored by electric pins. The damping processes of aluminum at 78 and 101 GPa and iron at 159 and 103 GPa are obtained by this technique, which supplement the existing data by measuring the viscosity coefficient via a dynamic high-pressure method. Applying the formula of Miller and Ahrens to fit the experimental data, the shear viscosity coefficients of aluminum at 78 and 101 GPa are \(1350\,\pm \,500\) and \(1200\,\pm \,500~\hbox {Pa}\,\hbox {s}\), respectively, and those of iron at 159 and 103 GPa are \(1150\,\pm \,1000\) and \(4800\,\pm \,1000~\hbox {Pa}\,\hbox {s}\), respectively. The values measured by the flyer-impact method, approximately \(10^{3}~\hbox {Pa}\, \hbox {s}\), are consistent with those measured by Sakharov’s method, while still greatly differing from those measured by static high-pressure methods. In dynamic high-pressure experiments, the shear viscosity is related to dislocation motion in the solid material, while that in static high-pressure experiments is related to the diffusion motion of atoms or molecules in liquids. Therefore, there are different physical meanings of shear viscosity in dynamic and static high-pressure experiments, and there is no comparability among these results.     

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.     

Application of an Inverse Simulator of Single-Phase Flow Analysis for Leakage Pathway Estimation in a Multiphase System for Geologic $$\hbox {CO}_{2}$$ Storage

When \(\hbox {CO}_{2}\) is injected in a brine reservoir, brine or \(\hbox {CO}_{2}\) can be discharged into a permeable formation saturated with brine above the storage reservoir along a leakage pathway, if present. In most situations, the overlying formation can act as a single-phase aquifer with only brine leakage before \(\hbox {CO}_{2}\) leaks. This study examines the applicability of a developed inverse code for single-phase fluids to detect leakage pathway locations in view of the overlying formation using pressure anomalies induced by leaks. Before applying inverse analysis, forward modeling is performed using the TOUGH2 model to determine brine and \(\hbox {CO}_{2}\) leakage in a homogeneous conceptual model, and the simulated pressure profiles at monitoring wells are used as measurements in the inverse model. In the inverse code, an important consideration is that the vertical hydraulic conductivity and cross-sectional area of a leakage pathway that are inherent to a leakage term in the mass balance equation are integrated as a single parameter to estimate the leakage pathway locations. This method eliminates the impact of the uncertainty of the leakage pathway size on the accuracy of leakage pathway estimation. The inverse model examines the effect of the number of monitoring wells, monitoring periods and \(\hbox {CO}_{2}\) leakage into the overlying formation on the accuracy of leakage pathway estimation according to eleven application examples. The comparison between the results of the single-phase inverse code and iTOUGH2 code illustrates that the single-phase inverse model can be applicable to the leakage pathway estimation in a multiphase flow system.     

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.     

SIF-based fracture criterion for interface cracks

The complex stress intensity factor K governing the stress field of an interface crack tip may be split into two parts, i.e.,■ and s~(-iε), so that K = ■ s~(-iε), s is a characteristic length and ε is the oscillatory index. ■ has the same dimension as the classical stress intensity factor and characterizes the interface crack tip field. That means a criterion for interface cracks may be formulated directly with■, as Irwin(ASME J. Appl. Mech. 24:361–364, 1957) did in 1957 for the classical fracture mechanics. Then, for an interface crack,it is demonstrated that the quasi Mode I and Mode II tip fields can be defined and distinguished from the coupled mode tip fields. Built upon SIF-based fracture criteria for quasi Mode I and Mode II, the stress intensity factor(SIF)-based fracture criterion for mixed mode interface cracks is proposed and validated against existing experimental results.     

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.