Capillary Fluid Research Articles

Understanding gas transport mechanisms in shale gas reservoir: Pore network modelling approach

This report summarizes the recent findings on gas transport mechanisms in shale gas reservoir by pore network modelling. Multi-scale pore network model was developed to accurately characterize the shale pore structure. The pore network single component gas transport model was established considering the gas slippage and real gas property. The gas transport mechanisms in shale pore systems were elaborated on this basis. A multicomponent hydrocarbon pore network transport model was further proposed considering the influences of capillary pressure and fluid occurrence on fugacity balance. The hydrocarbon composition and pore structure influences on condensate gas transport were analyzed. These results provide valuable insights on gas transport mechanisms in shale gas reservoir. Cited as: Song, W., Yao, J., Zhang, K., Yang, Y., Sun, H. Understanding gas transport mechanisms in shale gas reservoir: Pore network modelling approach. Advances in Geo-Energy Research, 2022, 6(4): 359-360. https://doi.org/10.46690/ager.2022.04.11

Rheology of particulate suspensions with non-Newtonian fluids in capillaries.

Particulate suspensions occur in situations from blood flow to slurries in drilling applications. Existing investigations of these suspensions generally concentrate on the impact of particle volume fraction for suspensions in Newtonian fluids under free-flow conditions. Recently, particulate-polymer composites have been used in additive manufacturing (AM). Here, the polymer becomes a shear-thinning non-Newtonian fluid during extrusion, creating a particulate suspension. Motivated by the challenges in AM of particulate composites, this study investigates the rheology of suspensions of micrometre-sized particles in shear-thinning silicone while extruded through AM-scaled nozzles (millimetre-scale diameters). The suspensions were observed to follow a power-law behaviour and their rheology was investigated through the measured flow consistency () and behaviour () indices. The impact of the particle volume fraction () and the ratio () of the capillary inside diameter to the particle diameter on both indices were measured. was found to be only impacted by the suspension fluid type and . was found to be constant at large , but decreased and then increased to infinity with decreasing. Based on its behaviour, was categorized into two conditions and analysed separately with semi-empirical models. The impact of particle size distribution was also investigated.

Seepage–Fractal Model of Embankment Soil and Its Application

Over time and across space, the hydraulic conductivity, fractal dimension, and porosity of embankment soil have strong randomness, which makes analyzing seepage fields difficult, affecting embankment risk analysis and early disaster warning. This strong randomness limits the application of fractal theory in embankment engineering and sometimes keeps it in the laboratory stage. Based on the capillary model of porous soil, an analytical formula of the fractal relationship between hydraulic conductivity and fractal dimension is derived herein. It is proposed that the influencing factors of hydraulic conductivity of embankment soil mainly include the capillary aperture, fractal dimension, and fluid viscosity coefficient. Based on random field theory and combined with the embankment parameters of Shijiu Lake, hydraulic conductivity is discretized, and then the soil fractal dimension is approximately solved to reveal the internal relationship between hydraulic gradient, fractal dimension, and hydraulic conductivity. The results show that an increased fractal dimension will reduce the connectivity of soil pores in a single direction, increase the hydraulic gradient, and reduce the hydraulic conductivity. A decreased fractal dimension will lead to consistency of seepage channels in the soil, increased hydraulic conductivity, and decreased hydraulic gradient.

Comparative study on pore structure characteristics of marine and transitional facies shales: A case study of the Upper Permian Longtan Formation and Dalong Formation in the Lower Yangtze area, south China

The Permian shale reservoir in the Lower Yangtze area of South China is regarded as a target with a high potential for shale gas exploration. To investigate the pore structure characteristics and evaluate reservoir quality, 32 shale samples from the Upper Permian Longtan Formation and Dalong Formation in the Lower Yangtze area were studied using a suite of techniques including total organic carbon (TOC) content, X-ray diffraction (XRD), field emission scanning electron microscopy (FE-SEM), nitrogen (N2) and carbon dioxide (CO2) gas physisorption, mercury injection capillary pressure (MICP), petrographic observations, and fluid inclusion analysis. Petrographic observations reveal that the Longtan Formation is rich in vitrinite and inertinite, while the Dalong Formation contains a large amount of solid bitumen. Primary organic pores (POPs) are the original cellular pores that develop in inertinite. Vitrinite particles did not show visible pores. Solid bitumen infills fractures and occupies the interparticle (interp) space between the mineral grains in the Dalong Formation. A small number of speckled secondary organic pores (SOPs) were observed in the solid bitumen with diameters less than 10 nm. Both the TOC and quartz contents exhibit negative relationships with the pore volumes (PVs) for the Longtan and Dalong Formations, owing to the poor development of pores in the organic matter (OM) and occlusion of most mineral pores by migrated bitumen. There is a positive relationship between clay minerals and PVs owing to the sheet-like meso-pores and macro-pores in the clay minerals. The evolution of organic porosity is related to the differences in maceral composition and thermal maturity. Thermal cracking of residual oil and bitumen and the expulsion of gaseous hydrocarbons contributed to the formation of SOPs in marine shales. In contrast, vitrinite and inertinite in gas-prone shales do not display noticeable morphological changes across the entire thermal evolution process, owing to their low hydrocarbon generation potential.

Droplet motion on superhydrophobic/superhydrophilic wedge-shaped patterned surfaces with different micro-morphologies

Superhydrophobic/superhydrophilic wedge-shaped patterned surface (SSWPS) has promising application prospects in water transportation, fog harvesting, etc., and has therefore been widely prepared and studied. However, influences of micro-morphologies of the SSWPS on water transportation and fog harvesting, which may be of great significance, have rarely been reported. Here, we firstly investigated the influences of the micro-morphologies on the water transportation and fog harvesting processes. SSWPSs with different microgrooves were prepared by laser processing, and the influence mechanisms of the micro-morphologies on droplet motion behaviors were then investigated. The different micro-morphologies contributed to dissimilar capillary forces and fluid resistances during the transportation process, thus resulting in the different transport speeds and fog harvesting efficiencies. For regular microgrooves with 0° direction and larger depths, the capillary forces were relatively larger while the fluid resistances were smaller, which led to the faster water transport speeds of the SSWPS. Fog harvesting experiments on different surfaces demonstrated that the SSWPS with faster water transport speed exhibited much higher fog harvesting efficiency, and could achieve long-time continuous harvesting, which was expected to facilitate application of the SSWPS.

Management of urolithiasis in male Kankrej bovines

The study included 30 male Kankrej bovines showing symptoms of urinary obstruction. All the animals were clinically examined for the colour of conjunctival mucous membrane, capillary refill time, fluid thrill and distension of abdomen, and evaluated ultrasonographically. Ultrasonographic examination (transabdominal and trans-rectal) provided quick and definitive information about the status of urinary bladder. Surgical treatment was done in 17 bovines; post-scrotal urethrotomy in 9 bullocks, and tube cystostomy in 8 calves. Surgical intervention, urethrotomy or tube cystotomy, along with oral feeding of ammonium chloride yielded good results.

Control of capillary driven fluid flows for safe operation of spacecraft fluid supply systems using artificial porous media

In weightlessness, capillary forces become the main driving forces in the flow of liquids. Fuel supply of engines being started in weightlessness, water supply to life support systems in habitable modules, heat exchange using capillary pumps – all those processes rely heavily on capillary forces. Thus, safety and reliability of Space mission strongly depends on our skills in controlling capillary driven fluid flows. Adequate mathematical models for predicting capillary driven imbibition in artificial porous specimens are needed. The paper discusses the features of the flow of liquids through a porous medium under the action of capillary forces under microgravity conditions. An important feature of such flows is the hysteresis effect: in the case of multiple imbibition of a porous medium, the liquid flows in different ways through the dry medium and through the medium moistened in the process of previous imbibitions. This effect is observed during experiments on repeated imbibition of an artificial porous medium during parabolic flights of an Airbus A300-ZeroG aircraft organized by the European Space Agency. With each subsequent parabola, the liquid rises to a greater height. Modeling such processes requires special mathematical models described in this work. Comparison of the results of numerical modeling and experimental data provides a unique opportunity to choose empirical constants on an experimental basis thus developing a robust tool for predicting fluid behavior in hydraulic loops of Space vehicles.

The draining of capillary liquids from containers with interior corners aboard the ISS

In this work, we analyze liquid drains from containers in effective zero-g conditions aboard the International Space Station (ISS). The efficient draining of capillary fluids from conduits, containers, and media is critical in particular to high-value liquid samples such as minuscule biofluidics processing on earth and enormous cryogenic fuels management aboard spacecraft. The amount and rate of liquid drained can be of key concern. In the absence of strong gravitational effects, system geometry, and liquid wetting dominate capillary fluidic behavior. During the years 2010–2015, NASA conducted a series of handheld experiments aboard the ISS to observe “large” length scale capillary fluidic phenomena in a variety of irregular containers with interior corners. In this work, we focus on particular single exit port draining flows from such containers and digitize hours of archived NASA video records to quantify transient interface profiles and volumetric flow rates. These data are immediately useful for theoretical and numerical model benchmarks. We demonstrate this by making comparisons to lubrication models for slender flows in simplified geometries which show variable agreement with the data, in part validating certain geometry-dependent dynamical interface curvature boundary conditions while invalidating others. We further compare the data for the draining of complex vane networks and identify the limits of the current theory. All analyzed data is made available to the public as MATLAB files, as detailed within.

In situ local imaging and analysis of impregnation during liquid moulding of composite materials using synchrotron radiation computed laminography

Analysis of the impregnation process during injection experiments in fibre-reinforced polymer composites has been carried out via a three-dimensional in-situ imaging method, namely Synchrotron Radiation Computed Laminography (SR-CL). A resin transfer moulding device was designed to perform injection experiments at different flow rates while, at the same time, the evolution of the flow front was inspected by SR-CL. Injections with flow rates of 50 μL/min and 300 μL/min in plain woven E-glass fabrics were carried out, resulting in small capillary number fluid propagation with capillary forces dominant over viscous ones. Intratow and intertow voids were accurately segmented by using a machine learning algorithm that enabled the direct measurement of their time and spatial distribution from the raw XCT volumes. The saturation evolution, void content, shape and spatial distribution quantification are reported. Intratow void formation and transport mechanisms were accurately described, in particular those related with large void splitting including the subsequent flow dragging. Advantages and limitations of the use of SR-CL for the analysis of the impregnation process of dry preforms in liquid moulding are discussed in detail, and in particular, those aspects related with the time acquisition limitations.

Composition of Classification Models for Recognizing the Flow Velocity of Liquids in Capillaries

Purpose of research. Development of a technique for estimating the average flow rate of physiological fluids in capillaries from images obtained using laser speckle-contrast imaging. The technique includes obtaining experimental data in the form of an image of the fluid flow in a thin tube, their preliminary processing, including filtering and compressing data, as well as training and testing approximate models using modern machine learning methods.Methods. The experimental study of the fluid flow in the tube is based on the application of the laser speckle-contrast imaging method. The spatial speckle-contrast values are calculated from the obtained images. The obtained data are subjected to preliminary processing, including the data filtering out and extending to a steady flow mode, as well as compressing the obtained images using the principal component method, which allows reducing the dimension of the feature space. The problem of predicting the average velocity from the image of the fluid flow is solved as a classification problem based on the composition of decision trees constructed through the bagging procedure, as well as in the form of a random forest.Results. A technique for predicting the average velocity of liquid flow in a capillary from images obtained using the laser speckle-contrast imaging method has been developed. The accuracy of predicting the average velocity (or flow rate) based on the training sample was about 91%, on the validation and test samples - at least 81.5%.Conclusion. Based on the developed technique, it is planned to determine the kinematic characteristics of the parameters of physiological fluids flow, which will improve the inertial method of measuring the viscosity of the tested liquids developed earlier by the authors, getting rid of a number of assumptions about the velocity profile.

On the dynamics of a meniscus inside capillaries during imbibition and drainage processes: A generalized model, effect of inertia, and a numerical algorithm

In imbibition or drainage processes, a fluid displaces another immiscible one. If the displacing fluid is wetting, this is an imbibition process and is drainage if otherwise. While imbibition can proceed without the action of external force (e.g., pressure), drainage cannot unless sufficient external force is applied. One of the most important phenomena in this regard is related to the estimation of the location of the meniscus inside the tube and its velocity with time. This has been the topic of extensive research works for which analytical expressions exist for some special cases including the case in which the displaced fluid is air. Recently, a generalization to this approach has been developed, which accounts for the more general scenario in which the displaced fluid assumes considerable viscosity and density contrasts compared with the displacing one. However, in this recently developed model, and even in most of the previously studied special cases, an inherent assumption was made to ignore inertial effects. While this assumption is reasonable given the relatively slow advancement of the meniscus in capillaries, it results in the velocity to jump at the start of the imbibition process to a relatively higher value before declining as the meniscus advances. In fact, in actual imbibition experiments, velocity develops from zero to a maximum value in a short period of time before it declines as the meniscus continues to advance. In this work, a generalized model is developed, which accounts for the inertia of the fluids inside the tube. A nonlinear ordinary differential equation is developed, which accounts for the acceleration of the fluid and the contrasts of viscosity and density of the two fluids in capillaries. A numerical algorithm is also developed where the differential equation is linearized to facilitate the numerical solution. Verifications of the numerical algorithm are conducted to build confidence in the computational approach.

Mechanisms Governing Anaphylaxis: Inflammatory Cells, Mediators, Endothelial Gap Junctions and Beyond.

Anaphylaxis is a severe, acute, life-threatening multisystem allergic reaction resulting from the release of a plethora of mediators from mast cells culminating in serious respiratory, cardiovascular and mucocutaneous manifestations that can be fatal. Medications, foods, latex, exercise, hormones (progesterone), and clonal mast cell disorders may be responsible. More recently, novel syndromes such as delayed reactions to red meat and hereditary alpha tryptasemia have been described. Anaphylaxis manifests as sudden onset urticaria, pruritus, flushing, erythema, angioedema (lips, tongue, airways, periphery), myocardial dysfunction (hypovolemia, distributive or mixed shock and arrhythmias), rhinitis, wheezing and stridor. Vomiting, diarrhea, scrotal edema, uterine cramps, vaginal bleeding, urinary incontinence, dizziness, seizures, confusion, and syncope may occur. The traditional (or classical) pathway is mediated via T cells, Th2 cytokines (such as IL-4 and 5), B cell production of IgE and subsequent crosslinking of the high affinity IgE receptor (FcεRI) on mast cells and basophils by IgE-antigen complexes, culminating in mast cell and basophil degranulation. Degranulation results in the release of preformed mediators (histamine, heparin, tryptase, chymase, carboxypeptidase, cathepsin G and tumor necrosis factor alpha (TNF-α), and of de novo synthesized ones such as lipid mediators (cysteinyl leukotrienes), platelet activating factor (PAF), cytokines and growth factors such as vascular endothelial growth factor (VEGF). Of these, histamine, tryptase, cathepsin G, TNF-α, LTC4, PAF and VEGF can increase vascular permeability. Recent data suggest that mast cell-derived histamine and PAF can activate nitric oxide production from endothelium and set into motion a signaling cascade that leads to dilatation of blood vessels and dysfunction of the endothelial barrier. The latter, characterized by the opening of adherens junctions, leads to increased capillary permeability and fluid extravasation. These changes contribute to airway edema, hypovolemia, and distributive shock, with potentially fatal consequences. In this review, besides mechanisms (endotypes) underlying IgE-mediated anaphylaxis, we also provide a brief overview of IgG-, complement-, contact system-, cytokine- and mast cell-mediated reactions that can result in phenotypes resembling IgE-mediated anaphylaxis. Such classifications can lead the way to precision medicine approaches to the management of this complex disease.

Effects of controlled ovarian stimulation on vascular barrier and endothelial glycocalyx: a pilot study

PurposeControlled ovarian stimulation significantly amplifies the number of maturing and ovulated follicles as well as ovarian steroid production. The ovarian hyperstimulation syndrome (OHSS) increases capillary permeability and fluid extravasation. Vascular integrity intensely is regulated by an endothelial glycocalyx (EGX) and we have shown that ovulatory cycles are associated with shedding of EGX components. This study investigates if controlled ovarian stimulation impacts on the integrity of the endothelial glycocalyx as this might explain key pathomechanisms of the OHSS.MethodsSerum levels of endothelial glycocalyx components of infertility patients (n=18) undergoing controlled ovarian stimulation were compared to a control group of healthy women with regular ovulatory cycles (n=17).ResultsPatients during luteal phases of controlled ovarian stimulation cycles as compared to normal ovulatory cycles showed significantly increased Syndecan-1 serum concentrations (12.6 ng/ml 6.1125th–19.1375th to 13.9 ng/ml 9.625th–28.975th; p=0.026), indicating shedding and degradation of the EGX.ConclusionA shedding of EGX components during ovarian stimulation has not yet been described. Our study suggests that ovarian stimulation may affect the integrity of the endothelial surface layer and increasing vascular permeability. This could explain key features of the OHSS and provide new ways of prevention of this serious condition of assisted reproduction.

Drainage Capillary Pressure Distribution and Fluid Displacement in a Heterogeneous Laminated Sandstone

AbstractWe applied three‐dimensional X‐ray microtomography to image a capillary drainage process (0–1,000 kPa) in a cm‐scale heterogeneous laminated sandstone containing three distinct regions with different pore sizes to study the capillary pressure. We used differential imaging to distinguish solid, macropore, and five levels of subresolution pore phases associated with each region. The brine saturation distribution was computed based on average CT values. The nonwetting phase displaced the wetting phase in order of pore size and connectivity. The drainage capillary pressure in the highly heterogeneous rock was dependent on the capillary pressures in the individual regions as well as distance to the boundary between regions. The complex capillary pressure distribution has important implications for accurate water saturation estimation, gas and/or oil migration and the capillary rise of water in heterogeneous aquifers.

High accuracy capillary network representation in digital rock reveals permeability scaling functions

Permeability is the key parameter for quantifying fluid flow in porous rocks. Knowledge of the spatial distribution of the connected pore space allows, in principle, to predict the permeability of a rock sample. However, limitations in feature resolution and approximations at microscopic scales have so far precluded systematic upscaling of permeability predictions. Here, we report fluid flow simulations in pore-scale network representations designed to overcome such limitations. We present a novel capillary network representation with an enhanced level of spatial detail at microscale. We find that the network-based flow simulations predict experimental permeabilities measured at lab scale in the same rock sample without the need for calibration or correction. By applying the method to a broader class of representative geological samples, with permeability values covering two orders of magnitude, we obtain scaling relationships that reveal how mesoscale permeability emerges from microscopic capillary diameter and fluid velocity distributions.

MOF-enzyme hybrid nanosystem decorated 3D hollow fiber membranes for in-situ blood separation and biosensing array

Modified metal–organic frameworks (MOFs) doping with enzymes exhibit high enzyme stability and catalytic performance, which is a research hotspot in the field of enzyme-based sensing. Although the MOF-enzyme constitutes a 3D structure in the nanoscale, the macroscopic assembly configuration still stays in 1D or 2D structures, limiting sensing applications towards complex biological targets. Herein, the MOF-enzyme hybrid nanosystem was assembled into 3D porous conductive supports via a controllable physical embedding method, displaying high enzymatic loading, stability and cascade catalytic performance. The modified MOFs combing with enzymes served as a sensing reaction system, and the conductive hollow fiber membranes (HFMs) served as a functional platform. The multifunctional device integrates pumpless hydrodynamic transport, interconnected conductive polymer, and blood separation modules, showing fast capillary fluid flow, trace sampling (3 μL), high selectivity and accuracy. The linear sensing range was in 2–24 mM glucose, 0.05–6 mM lactic acid, and 0.1–10 mM cholesterol, respectively, with sensitivities of 24.2, 150, 73.6 nA mM−1. Furthermore, this strategy of modular assembly of biosensing array can easily implement multiplex metabolites detection simultaneously.

A generalized analytical model for estimating the rate of imbibition/drainage of wetting/nonwetting fluids in capillaries

Displacing one fluid by another immiscible with it is a phenomenon of greatest importance and exists in various applications from large scale oil productions to small scale microfluidics. Modeling this phenomenon is crucial, particularly, with respect to estimating penetration rates. Although considerable amounts of research works have been conducted to understand the physics involved in this phenomenon, they remain adherent to particular applications. This includes, for example, imbibition scenarios in which a wetting fluid imbibes inside a capillary tube even without the need to initiate the penetration. On the other hand, drainage scenarios, in which a nonwetting fluid displaces a wetting one, requires such initiation by providing external boosting. There are several external factors that influence the rate at which the meniscus advances inside a capillary tube. The most common such forces include pressure force, capillary force, and gravity force. Several of the modeling approaches have considered the case in which the displaced fluids are gases (e.g., air), particularly in applications related to infiltration studies in porous media. This allowed researchers to ignore the frictional resistance of the displaced fluid and only considers the other denser fluid. However, there exists other applications in which this is not correct and it may not be appropriate to ignore the displaced fluid. In this work, a new generalized model is introduced that accounts for all the physics involved in this process and also does not ignore the displaced fluid. For validation, it is shown that the derived model reduces to all the special cases for which analytical solutions exist, (including that of a single-phase flows). Furthermore, comparisons with computational fluid dynamics simulation (CFD) of drainage/imbibition scenarios show very good match with the results of the derived model, which builds confidence in the modeling approach.

Effect of water occurrence on shale seepage ability

Due to the strong water absorption of clay minerals, the original water-bearing state of shale reservoirs changes after hydraulic fracturing, and the seepage capacity is affected. Based on nuclear magnetic resonance (NMR) technology and fluid state experiment of water-bearing shale, the distribution characteristics of water in clay minerals and shale were obtained, and the influence of water occurrence on shale seepage capacity was quantitatively analyzed. The results show that water is mainly adsorbed on the surface of clay pores in the state of bound fluid. In the early stage of shale water absorption, water mainly occurs in the form of bound water film on the pore surface of clay bound fluid with 0.02 μm < r < 0.46 μm. In the later stage of shale water absorption, water mainly occurs in the form of bound water film and capillary column in the pores with 0.02 μm < r < 5.79 μm. Water saturation is the main control factor affecting shale seepage capacity. When the water saturation of shale is about 60%, the shale permeability can be reduced by about 50% on average. At the same time, the effect of water occurrence on shale seepage capacity is more significant under low water saturation. Under low water content ( S w <25%), the existence of bound water film on the pore surface of clay bound fluid is the main reason for the decrease of shale seepage capacity. Under high water content (35%< S w <65%), the bound water film and the capillary water column in the pores of capillary bound fluid jointly lead to the decrease of shale seepage capacity, and the effect of bound water film on shale seepage capacity is more significant. This study clarifies the water distribution characteristics of shale with different water saturations, clarifies the influence of water occurrence on shale seepage capacity, and provides reference for shale gas development. • Water in shale is mainly bound by clay and partly by capillary during imbibition. • Water saturation is the main control factor affecting shale seepage capacity. • The effect of bound water film on shale seepage capacity is more significant. • The influence of water on shale seepage capacity was quantitatively analyzed.

Negative Pressure Pulmonary Edema

Negative pressure pulmonary edema (NPPE) develops when a patient makes a forced inspiration against a closed upper airway resulting in transudation of pulmonary capillary fluid into the interstitium. Young, healthy athletic males are more susceptible to this complication, and NPPE is most frequently encountered during oral and maxillofacial surgeries. Patients who have undergone septorhinoplasty surgery have a significant risk for NPPE, unlike other cosmetic surgeries. Herein, we describe two cases of postoperative NPPE following a septorhinoplasty. Early symptoms of NPPE are severe respiratory distress, tachypnea, agitation, and crushed strawberry-colored bloody sputum during the early postextubation period. An early diagnosis of NPPE is very important, and patient recovery is quick when an early diagnosis is made.

Novel solitary wave solution of the nonlinear fractal Schrödinger equation and its fractal variational principle

PurposeThe nonlinear Schrödinger equation plays a vital role in wave mechanics and nonlinear optics. The purpose of this paper is the fractal paradigm of the nonlinear Schrödinger equation for the calculation of novel solitary solutions through the variational principle.Design/methodology/approachAppropriate traveling wave transform is used to convert a partial differential equation into a dimensionless nonlinear ordinary differential equation that is handled by a semi-inverse variational technique.FindingsThis paper sets out the Schrödinger equation fractal model and its variational principle. The results of the solitary solutions have shown that the proposed approach is very accurate and effective and is almost suitable for use in such problems.Practical implicationsNonlinear Schrödinger equation is an important application of a variety of various situations in nonlinear science and physics, such as photonics, the theory of superfluidity, quantum gravity, quantum mechanics, plasma physics, neutron diffraction, nonlinear optics, fiber-optic communication, capillary fluids, Bose–Einstein condensation, magma transport and open quantum systems.Originality/valueThe variational principle of the Schrödinger equation without the Lagrange multiplier method in the sense of the fractal calculus is developed for the first time in the literature to the best of the author's understanding.