Diffusion Equation For Flow Research Articles

A Blood Supply Pathophysiological Microcirculatory Mechanism for Long COVID.

The term "Long COVID" is commonly used to describe persisting symptoms after acute COVID-19. Until now, proposed mechanisms for the explanation of Long COVID have not related quantitative measurements to basic laws. In this work, a common framework for the Long COVID pathophysiological mechanism is presented, based on the blood supply deprivation and the flow diffusion equation. Case-control studies with statistically significant differences between cases (post-COVID patients) and controls, from multiple tissues and geographical areas, were gathered and tabulated. Microvascular loss (ML) was quantified by vessel density reduction (VDR), foveal avascular zone enlargement (FAZE), capillary density reduction (CDR), and percentage of perfused vessel reduction (PPVR). Both ML and hemodynamic decrease (HD) were incorporated in the tissue blood supply reduction (SR) estimation. ML data were found from 763 post-COVID patients with an average VDR, FAZE, CDR, and PPVR of 16%, 31%, 14%, and 21%, respectively. The average HD from 72 post-COVID patients was 37%. The estimated SR for multiple tissues with data from 634 post-COVID patients reached a sizeable 47%. This large SR creates conditions of lower mass diffusion rates, hypoxia, and undernutrition, which at a multi-tissue level, for a long time, can explain the wide variety of the Long COVID symptoms. Disruption of peripheral tissue blood supply by the contribution of both ML and HD is proposed here to be the principal cause of the mechanism leading to Long COVID symptoms.

A normative blood velocity model in the exchange microvessels for discriminating health from disease: Healthy controls versus COVID-19 cases.

A usual practice in medicine is to search for "biomarkers" which are measurable quantities of a normal or abnormal biological process. Biomarkers can be biochemical or physical quantities of the body and although commonly used statistically in clinical settings, it is not usual for them to be connected to basic physiological models or equations. In this work, a normative blood velocity model framework for the exchange microvessels was introduced, combining the velocity-diffusion (V-J) equation and statistics, in order to define the normative range (NR) and normative area (NA) diagrams for discriminating normal (normemic) from abnormal (hyperemic or underemic) states, taking into account the microvessel diameter D. This is different from the usual statistical processing since there is a basis on the well-known physiological principle of the flow diffusion equation. The discriminative power of the average axial velocity model was successfully tested using a group of healthy individuals (Control Group) and a group of post COVID-19 patients (COVID-19 Group).

The velocity-diffusion equation in the exchange microvessels.

In human and animal microvascular networks, the exchange microvessels are the capillaries and postcapillary venules where material transport between the circulating blood and tissue takes place. For small-size molecules, this material transport is done by the physical mechanism of diffusion through the endothelium wall and the diffusion rate J in relation to blood volume flow Q is described by the flow-diffusion (Q-J) equation. However, the volume flow is not easy to be measured in vivo. The objective of this work was to transform the classical flow-diffusion equation into a new form with axial velocity V as an independent variable instead of volume flow Q. The new form was called the velocity-diffusion (V-J) equation and has the advantage that V can be measured directly in vivo by optical imaging techniques. The V-J equation could have important applications in the calculation of the mass diffusion rate of various substances in vivo.

RTA-Assisted Numerical History-Matching Workflow

Summary Due to strong nonlinearities in the governing diffusivity equation for flow in porous media, numerically assisted rate-transient analysis (RTA) techniques have been suggested for the analysis of multiphase production data from multifractured horizontal wells (MFHWs). However, these methods are based on some limiting assumptions that cannot be generalized for three-phase flow or when relative permeability is unknown. In this study, a new RTA-assisted history-matching technique is proposed to simultaneously match production data and diagnostic plots during the calibration process. In the proposed method, the objective function is modified to include the derivative of the integral of rate-normalized pressure for the primary phases. As such, in the history-matching process using compositional numerical simulation, the flow regimes are also matched, which can increase the reliability of the calibrated numerical model. This approach is applied to a challenging data set of production data from an MFHW completed in a Canadian shale reservoir hosting a near-critical gas condensate fluid. The results demonstrate that when the modified objective function is used, the history-matching scheme will reject models that cannot reproduce the flow regimes even if the production data are visually matched. Another benefit of this modified history-matching workflow is that, unlike other numerically assisted RTA techniques, it is not limited to any specific conceptual model or reservoir geometry. Further, interactions between parameters are accounted for during the calibration process. Including the derivative terms in the objective function can ensure a better history-matched model with improved forecast quality. However, comparing the convergence rates of the history-matching with the standard and modified objective functions indicates that adding the derivative terms comes with an additional computational cost requiring more iterations and a slower convergence rate. In this study, a modified objective function is introduced for the first time to enhance the numerical history-matching process to ensure the resulting calibrated model can also reproduce the observed transient flow regimes. This approach is easy to implement and is not limited to a specific model geometry or any input-output relationship.

Peridynamic model of deformation and failure for rock material under the coupling effect of multi-physical fields

This paper presents a thermo-hydro-mechanical coupling peridynamic (PD) model to simulate the deformation and failure of rock materials and the diffusion of the flow and temperature subjected to the coupling effect of multi-physical fields. By using the peridynamic least squares minimization (PD LSM) method, the thermal conduction and the flow diffusion equation in the coupling of thermo-mechanical, hydro-mechanical, and thermo-hydro-mechanical are refined in the PD form. The equation of motion for thermo-hydro-mechanical coupling is built in the framework of the bond-based PD model. The verification of the coupling PD model is demonstrated by comparison against the analytical solutions and previous numerical results in the problems of steady thermal conduction, thermal deformation, transient flow, hydraulic fracturing, thermal consolidation, and hydraulic fracturing considering the heat effect. The comparison results show that the proposed thermo-hydro-mechanical coupling PD model can capture the deformation and failure of rock materials under the coupling effect of multi-physical fields.

The nonlocal thermo‐hydro‐mechanical model for cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics

AbstractA novel nonlocal thermo‐hydro‐mechanical coupling model is proposed to investigate cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics (GPD). The nonlocal thermal conduction equation and the nonlocal flow diffusion equation are derived based on nonlocal vector calculus and the microscopic constitutive equation. Moreover, the coupling influences are considered based on the nonlocal equation and the coupling principle. The governing equation of the coupling system is established in the framework of GPD. The correctness of the proposed model is verified through comparing with the corresponding experimental data, and the numerical results are in good agreement with experimental results. Finally, the proposed model is applied to analyze the deformation and cracking behaviors of rock mass around tunnels, and the stress redistribution is revealed under thermo‐hydro‐mechanical coupling condition.

Research on the Treatment of Regional Air Pollution

According to the change of air quality in the Beijing-Tianjin-Hebei region, this paper analyzes the pollution characteristics of the air environment in this region, constructs the air quality evaluation model, and finds that the main pollution sources affecting the air quality in Beijing-Tianjin-Hebei region are coal combustion and automobile exhaust. By using the flow-diffusion equation, the air pollution concentration gradients of the Second Ring Road, the Fourth Ring Road, and the Sixth Ring Road in Beijing are given. Finally, this paper also puts forward specific countermeasures for the prevention and control of air pollution.

Finite point method of nonlinear convection diffusion equation

Aiming at the nonlinear convection diffusion equation with the numerical oscillations, a numerical stability algorithm is constructed. The basic principle of the finite point algorithm is given and the computational scheme of the nonlinear convection diffusion equation is deduced. Then, the numerical simulation of the one-dimensional and two-dimensional nonlinear convection-dominated diffusion equation is carried out. The relationship between the calculation result and the support domain size, step size and time is discussed. The results show that the algorithm has the characteristics of simplicity, stability and efficiency. Compared with the traditional finite element method and finite difference method, the new algorithm can attain a higher calculation accuracy. Simultaneously, it proves that the method given in this paper is effective to solve the nonlinear flow diffusion equation and can eliminate the numerical oscillations.

Recurrence of glioblastoma is associated with elevated microvascular transit time heterogeneity and increased hypoxia

Dynamic susceptibility contrast (DSC) perfusion MRI provide information about differences in macro- and microvasculature when executed with gradient-echo (GE; sensitive to macrovasculature) and spin-echo (SE; sensitive to microvasculature) contrast. This study investigated whether there are differences between macro- and microvascular transit time heterogeneity (MVTH and µVTH) and tissue oxygen tension (PO2mit) in newly-diagnosed and recurrent glioblastoma. Fifty-seven patients with glioblastoma (25 newly-diagnosed/32 recurrent) were examined with GE- and SE-DSC perfusion sequences, and a quantitative blood-oxygen-level-dependent (qBOLD) approach. Maps of MVTH, µVTH and coefficient of variation (MCOV and µCOV) were calculated from GE- and SE-DSC data, respectively, using an extended flow-diffusion equation. PO2mit maps were calculated from qBOLD data. Newly-diagnosed and recurrent glioblastoma showed significantly lower (P ≤ 0.001) µCOV values compared to both normal brain and macrovasculature (MCOV) of the lesions. Recurrent glioblastoma had significantly higher µVTH (P = 0.014) and µCOV (P = 0.039) as well as significantly lower PO2mit values (P = 0.008) compared to newly-diagnosed glioblastoma. The macrovasculature, however, showed no significant differences. Our findings provide evidence of microvascular adaption in the disorganized tumor vasculature for retaining the metabolic demands in stress response of therapeutically-uncontrolled glioblastomas. Thus, µVTH and PO2mit mapping gives insight into the tumor microenvironment (vascular and hypoxic niches) responsible for therapy resistance.

Analytical Treatment of Pressure-Transient Solutions for Gas Wells With Wellbore Storage and Skin Effects by the Green's Functions Method

Summary Even when written in terms of a pseudopressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, nonlinear because the viscosity-compressibility product is pseudopressure-dependent. However, several techniques and analysis procedures neglect such nonlinearity. A new methodology for constructing solutions for gas reservoirs through the Green's functions (GF) technique was recently proposed in the literature. Such methodology handles the viscosity-compressibility product variation rigorously, and it was applied to solve several gas-well test problems successfully. However, wellbore storage and skin effects were not considered yet by this new approach. In this work, the GF technique is applied to obtain a new solution for an infinite, homogeneous, isotropic gas reservoir being produced through a single vertical well represented by a line-source with wellbore storage and skin. The solution, however, does not consider non-Darcy flow effects. Even though the wellbore storage introduces a new nonlinearity to an already nonlinear problem, this work presents two accurate approximate solutions compared with the results from a commercial numerical well-testing simulator. This work also shows that the wellbore pseudopressure dimensionless solution is a function of the correlating groups CDexp(2S) and tD/CD, exactly similar to the way that wellbore dimensionless liquid solutions are. Liquid and gas dimensionless solutions under these correlating groups are not equal, though.

Finite-Wellbore-Radius Solution for Gas Wells by Green's Functions

Summary Even when written in terms of a pseudopressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, nonlinear because the viscosity/compressibility product is pseudopressure-dependent. However, several techniques and analysis procedures neglect such nonlinearity. A new methodology for constructing solutions for gas reservoirs through the Green's-function (GF) technique was recently proposed in the literature. Such methodology handles the viscosity/compressibility product variation rigorously, and it was successfully applied to solve several gas-well-test problems. In those problems, the wellbore is always represented by a line source. This work extends the theory a little further by considering a finite-wellbore-radius (FWR) boundary condition for a single vertical well producing at constant rate from an isotropic homogeneous and infinite gas reservoir. The proposed solution does not consider non-Darcy-flow effects, wellbore storage, and skin. Results from our FWR solution are compared with a commercial finite-difference reservoir simulator that shows a very close agreement. We also compared the FWR solution to the correspondent line-source solution to study the difference between the two solutions. As expected, the pseudopressure solutions by use of line-source and FWR boundary conditions do not match at early times, but they do agree at long times, which is exactly how FWR and line-source well solutions for slightly compressible fluids behave. It seems that, even for gas-well problems, the wellbore can be satisfactorily represented by a line source without significant loss of generality. The line-source assumption greatly simplifies the mathematics and the computational effort. This aspect is especially attractive for complex nonlinear gas-well problems that remain to be solved by the GF approach.

The effects of capillary transit time heterogeneity (CTH) on brain oxygenation.

We recently extended the classic flow-diffusion equation, which relates blood flow to tissue oxygenation, to take capillary transit time heterogeneity (CTH) into account. Realizing that cerebral oxygen availability depends on both cerebral blood flow (CBF) and capillary flow patterns, we have speculated that CTH may be actively regulated and that changes in the capillary morphology and function, as well as in blood rheology, may be involved in the pathogenesis of conditions such as dementia and ischemia-reperfusion injury. The first extended flow-diffusion equation involved simplifying assumptions which may not hold in tissue. Here, we explicitly incorporate the effects of oxygen metabolism on tissue oxygen tension and extraction efficacy, and assess the extent to which the type of capillary transit time distribution affects the overall effects of CTH on flow-metabolism coupling reported earlier. After incorporating tissue oxygen metabolism, our model predicts changes in oxygen consumption and tissue oxygen tension during functional activation in accordance with literature reports. We find that, for large CTH values, a blood flow increase fails to cause significant improvements in oxygen delivery, and can even decrease it; a condition of malignant CTH. These results are found to be largely insensitive to the choice of the transit time distribution.

The Relationship between Tumor Blood Flow, Angiogenesis, Tumor Hypoxia, and Aerobic Glycolysis

Antiangiogenic therapies are being pursued as a means of starving tumors of their energy supply. Although numerous studies show that such therapies render tumors hypoxic, just as many studies have, surprisingly, shown improved tumor oxygenation. These contradicting findings challenge both the original rationale for antiangiogenic therapy and our understanding of the physiology of tissue oxygenation. The flow-diffusion equation, which describes the relation between blood flow and the extraction of freely diffusible molecules in tissue, was recently extended to take the heterogeneity of capillary transit times (CTH) into account. CTH is likely to be high in the chaotic microvasculature of a tumor, increasing the effective shunting of blood through its capillary bed. We review the properties of the extended flow-diffusion equation in tumor tissue. Elevated CTH reduces the extraction of oxygen, glucose, and cytotoxic molecules. The extent to which their net extraction is improved by antiangiogenic therapy, in turn, depends on the extent to which CTH is normalized by the treatment. The extraction of oxygen and glucose are affected to different extents by elevated CTH, and the degree of aerobic glycolysis-known as the Warburg effect-is thus predicted to represent an adaptation to the CTH of the local microvasculature.

The Role of the Cerebral Capillaries in Acute Ischemic Stroke: The Extended Penumbra Model

The pathophysiology of cerebral ischemia is traditionally understood in relation to reductions in cerebral blood flow (CBF). However, a recent reanalysis of the flow-diffusion equation shows that increased capillary transit time heterogeneity (CTTH) can reduce the oxygen extraction efficacy in brain tissue for a given CBF. Changes in capillary morphology are typical of conditions predisposing to stroke and of experimental ischemia. Changes in capillary flow patterns have been observed by direct microscopy in animal models of ischemia and by indirect methods in humans stroke, but their metabolic significance remain unclear. We modeled the effects of progressive increases in CTTH on the way in which brain tissue can secure sufficient oxygen to meet its metabolic needs. Our analysis predicts that as CTTH increases, CBF responses to functional activation and to vasodilators must be suppressed to maintain sufficient tissue oxygenation. Reductions in CBF, increases in CTTH, and combinations thereof can seemingly trigger a critical lack of oxygen in brain tissue, and the restoration of capillary perfusion patterns therefore appears to be crucial for the restoration of the tissue oxygenation after ischemic episodes. In this review, we discuss the possible implications of these findings for the prevention, diagnosis, and treatment of acute stroke.

Analytical solutions to the one‐dimensional nonlinear diffusion equation for flow through porous media

The one‐dimensional nonlinear diffusion equation is solved approximately by an extension of the Neumann method for a step input to a semi‐infinite medium. The method of solution requires that the region under consideration be divided into an arbitrary number of zones, each zone having known constant diffusivities. The boundaries between zones move at rates that are initially unknown. Two problems are considered: (1) horizontal flow in an aquifer in which the transmissivity and storage coefficients are functions of hydraulic head and (2) horizontal absorption in an unsaturated soil for which the diffusivity is a function of moisture content. Computational results compare well with finite difference and other numerical solutions to the same problems. The technique has application to other nonlinear problems of the same type.

The Copper Oxide Rectifier

It is shown that the conductivity in the ohmic part of the cuprous oxide layer can be explained with the usual band picture of semiconductors only by assuming the presence of some donor-type impurities in addition to the usual acceptor type. The energy difference between the acceptors and the filled band is 0.3 electron volt, and the total number of impurity atoms is about ${10}^{14}$ to ${10}^{16}$ per ${\mathrm{cm}}^{3}$, the number of donors being less than but of the same order as the number of acceptors. Applying the Schottky theory of the space charge exhaustion layer, one finds from the dependence of capacity of the rectifier on bias voltage that the density of ion charge in the rectifying layer is of the same order of magnitude as the difference between the donors and acceptors found from the conductivity, thus furnishing a check for the theory. The field at the copper-cuprous oxide interface calculated from the space charge is about 2\ifmmode\times\else\texttimes\fi{}${10}^{4}$ volts/cm; the height of the potential at the surface as compared with the oxide interior is about 0.5 volt; and the thickness of the space charge layer about 5.0\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ cm. The diffusion equation for flow of current through this space charge region can be integrated to give the current in terms of the field at the interface and the applied potential across the space charge layer. Two currents are involved, one from the semiconductor to the metal (${I}_{s}$) and one from the metal to the semiconductor (${I}_{m}$) which is similar to a thermionic current into the semiconductor. The net current is, of course, $I={I}_{m}\ensuremath{-}{I}_{s}$. One can get this emission current (${I}_{m}$) by dividing the true current by the factor $1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\frac{e{V}_{a}}{\mathrm{kT}})$, where ${V}_{a}$ is the applied potential. This current depends on the absolute temperature and on the field at the copper-cuprous oxide interface. At high fields the logarithm of the current is proportional to the square root of the field, and at low fields the current decreases more rapidly indicating a patchy surface having small areas of low potential maximum from which all the comes when the field is large. This effective potential maximum measured from the Fermi level in the copper is about 0.5 ev, and the fraction of the total area effective ranges from ${10}^{\ensuremath{-}2}$ to ${10}^{\ensuremath{-}5}$ depending on how the rectifier was made. This last factor---the fraction of the area having this low potential maximum---is by far the most important variable, resulting in low reverse currents when the fraction is small and large reverse currents when the fraction is large.