Exit Boundaries Research Articles

Subcutaneous adipose tissue thickness determination using ultrasound signals processing: A phantom study

According to the top priority of health and fitness, the ultrasound (US) technique for measuring subcutaneous adipose tissue (SAT), as an indicator for body composition analysis, has received the most attention. So, tissue segmentation was performed to determine the boundaries between skin, SAT, and muscle in phantoms with different layers, and thicknesses that were fabricated to simulate tissue layers. The pre-processing of the signal was carried out using wavelet transform (WT) and Hilbert transform for denoising and detection. After the processing step, using WT for signal decomposition, a neural network was trained based on the scan line signals to detect the SAT layer. The coordinates of the convex probe were converted to linear to reduce the time and increase the accuracy for creating the target matrix. The algorithm was designed for automatic measurements of deep (normal), superficial (random), and noisy SAT thickness. Statistical evaluation was done to assess the system's skill in tissue classification and measurement of SAT thickness. Results revealed that most of the features related to coarse levels of detail coefficients extracted from wavelet decomposition levels can be used to build a classifier that can be applied successfully to differentiate between SAT and non-SAT tissue regions with a mean classification accuracy of 94.3% for 20 mm and 92.4% for 4 mm SAT in random mode with the error of estimate 0.05% and 0.07% respectively. Also, using 3 median filters and increasing their lengths from 3 to 7 improved the accuracy results to find SAT entry and exit boundaries.

Numerical investigations on the inlet and outlet behavior of foam flow process in porous media using stochastic bubble population balance model

Numerical investigations were carried out with special attentions on the inlet and outlet behaviors of foam flow process in a homogeneous porous media with confined entrance and exit boundaries. The Stochastic Bubble Population (SBP) balance model, which usually involves two parameters of the bubble generation rate Kg and the maximum bubble density n∞ for bubble number prediction, was employed to study the dynamic foam generation and development process in porous media. With introduction the Kfp parameter to take into account the foam phase relative permeability reduction effect, the control equations of the two-phase flow process in a two-dimensional computational domain were solved with the IMPES algorithm and the inlet and outlet behavior of the foam displacement process were scrutinized based on detailed analysis of the bubble density, water phase pressure and water phase saturation distributions in porous medium. Numerical results showed a clear entrance effect of higher water saturation, lower pressure gradient and developing bubble density in the inlet region of the porous media in the transient foam displacement process, whereas the confined exit results in elevating water saturation and decreasing pressure gradient towards the outlet of the core. Parametric studies on Kg,n∞, Kfp and inlet & outlet sizes D indicate higher bubble generation rate could remarkably suppress the entrance region in the foam seepage process, while the higher maximum bubble number did not lead to obvious change of the foam flow behavior in the inlet and outlet region, although it could produce higher absolute values of bubble density and pressure drop along the core. With lower Kfp values, the strong foam flooding behavior could be properly predicted with much higher pressure drop and lower water saturation. Two dimensional numerical results at various inlet and outlet sizes clearly show the confined geometry effect on foam flow characteristics with smaller sizes corresponding to longer inlet and outlet regions. Computational domain and numerical results were validated through comparisons with corresponding experimental setup and CT measurement results. It is observed that, the numerical simulation could reproduce not only qualitatively but also quantitatively the experimentally obtained water saturation distribution and pressure drop result. It is concluded that the mechanistic stochastic bubble population model could be reasonably employed on prediction of the inlet and outlet behavior in the transient foam flow process in the homogeneous porous media.

Exit boundaries of multidimensional SDEs

We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.

Approximation of excessive backlog probabilities of two tandem queues

AbstractLetXbe the constrained random walk on ℤ+2having increments (1,0), (-1,1), and (0,-1) with respective probabilities λ, µ1, and µ2representing the lengths of two tandem queues. We assume thatXis stable and µ1≠µ2. Let τnbe the first time when the sum of the components ofXequalsn. LetYbe the constrained random walk on ℤ×ℤ+having increments (-1,0), (1,1), and (0,-1) with probabilities λ, µ1, and µ2. Let τ be the first time that the components ofYare equal to each other. We prove thatPn-xn(1),xn(2)(τ<∞) approximatespn(xn) with relative errorexponentially decayinginnforxn=⌊nx⌋,x∈ℝ+2, 0<x(1)+x(2)<1, x(1)>0. An affine transformation moving the origin to the point (n,0) and lettingn→∞ connect theXandYprocesses. We use a linear combination of basis functions constructed from single and conjugate points on a characteristic surface associated withXto derive a simple expression for ℙy(τ<∞) in terms of the utilization rates of the nodes. The proof that the relative error decays exponentially innuses a sequence of subsolutions of a related Hamilton‒Jacobi‒Bellman equation on a manifold consisting of three copies of ℝ+2glued to each other along the constraining boundaries. We indicate how the ideas of the paper can be generalized to more general processes and other exit boundaries.

The absolute of finitely generated groups: I. Commutative (semi)groups

We give a complete description of the absolute of commutative finitely generated groups and semigroups. The absolute (previously called the exit boundary) is a further elaboration of the notion of the boundary of a random walk on a group (the Poisson–Furstenberg boundary); namely, the absolute of a (semi)group is the set of all ergodic probability measures on the compactum of all infinite trajectories of a simple random walk which has the same so-called cotransition probability as the simple random walk. Related notions have been discussed in the probability literature: Martin boundary, entrance and exit boundaries (Dynkin), central measures on path spaces of graphs [see Vershik (J Math Sci 209(6):860–873, 2015)]. The main result of this paper, which is a far-reaching generalization of de Finetti’s theorem, is as follows: the absolute of a commutative semigroup coincides with the set of central measures corresponding to Markov chains with independent identically distributed increments. Topologically, the absolute is (in the main case) a closed disk of finite dimension.

Entrainment across a sheared density interface in a cavity flow

The entrainment of fluid across a sheared density interface has been examined experimentally in a purging cavity flow. In this flow, a long straight cavity with sloped entry and exit boundaries is located in the base of a straight open channel. Dense cavity fluid is entrained from the cavity into the turbulent overflow. The cavity geometry has been designed to ensure there is no separation of the overflow in the cavity region, with the goal of avoiding cavity-specific entrainment mechanisms as have been encountered in most previous experiments using similar arrangements. Results are obtained over a bulk Richardson number range $Ri_{b}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}D/\unicode[STIX]{x1D70C}_{0}U_{b}^{2}=1$ to 19, where $D$ and $U_{b}$ are the depth of the mixed layer and bulk velocity in the cavity, respectively. The experiments cover the Reynolds number range $Re=U_{b}D/\unicode[STIX]{x1D708}=7100$ to 15 100 and interface length to mixed layer depth ratios of 2.4 to 16. Particle image velocimetry and laser induced fluorescence measurements indicate the flow regime over this entire range is one dominated by the Holmboe wave instability. The non-dimensional entrainment rate, $E=u_{e}/U_{b}$, is shown to scale with the bulk Richardson number. We find that the entrainment scaling $E=CRi_{b}^{-1.38}$ applies over the entire experimental range, with no apparent dependence on interface length. The exponent in the scaling is similar to previous non-cavity-based sheared interface flows, however, the constant $C$ is up to an order of magnitude smaller. Close agreement is, however, obtained by instead correlating entrainment with the local gradient Richardson number centred on the interface, rather than bulk quantities. We obtain $E=0.0021Ri_{g}^{-0.63}$ for data over $10<Ri_{g}<50$, where $Ri_{g}=\langle g\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x2202}z\rangle /\langle (\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z)^{2}\rangle$. The density interface is much thinner and therefore more stable in the present flow configuration compared with other published results for the same bulk Richardson number. We suggest that our configuration ensures a sharp mixing layer profile at the upstream end of the cavity even at relatively low bulk Richardson numbers of $Ri_{b}=1$ and that the reduced mixing in the Holmboe wave regime allows the interface to retain its sharp character over the cavity length, resulting in weak sensitivity to cavity length.

Lubricated point heavily loaded contacts of functionally graded materials. Part 1. Dry contacts

Over the last couple decades coatings attract more and more attention in practical applications. The present study addresses a question which is not well studied: how coated surfaces behave in lubricated contacts? In other words, this is a study of the effectiveness of functionally graded materials in heavily loaded point elastohydrodynamically lubricated contacts with straight lubricant entrainment. As a part of the study, some criteria of coating effectiveness are introduced and discussed. More specifically, the behavior of main parameters such as the lubrication film thickness and the frictional force in point heavily loaded lubricated contacts of functionally graded elastic materials are considered. The problem is studied based on the method of matched asymptotic expansions which allows us to split the problem into two separate problems: a dry contact problems for functionally graded elastic materials and an elastohydrodynamically lubricated problem for functionally graded materials. The elastohydrodynamically lubricated problem uses as input data not only the operational and physical parameters of the materials and lubricant but also the asymptotic behavior of the dry contact problem solution near the contact boundaries. Therefore, a sequence of two problems must be solved: the dry contact problems for functionally graded elastic materials and the elastohydrodynamically lubricated problem for functionally graded materials. Similar methods have been used for the analysis of an elastohydrodynamically lubricated problem for heavily loaded line contacts of functionally graded materials. The dry contact problem will be analyzed in Part 1 of the paper based on a semi-analytical bilateral method which produces correct asymptotic solutions for thin and thick coatings. The analytical expressions for contact pressure are obtained and analyzed for various combinations of coating thicknesses and elastic properties. The elastohydrodynamically lubricated problem will be considered in Part 2 of the paper based on the method of matched asymptotic expansions. In the analysis of the elastohydrodynamically lubricated problem, as in the case of homogeneous contact materials, it is shown that the whole contact region can be subdivided into three subregions: the central one which is adjacent to the other two regions occupied by the ends of the zones. The central region can be subdivided into the Hertzian region and then adjacent to it inlet and exit zones which, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. In the Hertzian region the elastohydrodynamically lubricated problem solution is very close to the solution of the corresponding dry (i.e. non-lubricated) contact problem for functionally graded elastic materials which have been analyzed. In the central region in the inlet and exit zones of a heavily loaded point elastohydrodynamically lubricated contact, the elastohydrodynamically lubricated problem for functionally graded elastic materials using certain scaling transforms can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials. Therefore, many of the well known properties of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials are also valid for heavily loaded point elastohydrodynamically lubricated contacts for functionally graded elastic materials. These asymptotically valid equations can be analyzed and numerically solved based on stable methods using a specific regularization approach, which were developed for lubricated line contacts. Also, this asymptotic analysis leads to an easy analytical derivation of formulas for the lubrication film thickness which take into account the inhomogeneity of the elastic materials. As a result of this analysis, some criteria for lubrication film thickness increase and friction force reduction are proposed. These criteria depend on lubricant properties as well as the properties of functionally graded elastic materials involved in lubricated contacts. Such a sequential solution of the elastohydrodynamically lubricated problem for functionally graded materials makes it perfectly clear what the dependence is of elastohydrodynamically lubricated contact parameters on the solid material (including the coating) and lubricant properties.

Lubricated point heavily loaded contacts of functionally graded materials. Part 2. Lubricated contacts

The paper presents a study of the effectiveness of functionally graded materials in heavily loaded point elastohydrodynamically lubricated contacts with straight lubricant entrainment. As part of the study, some criteria of coating effectiveness are introduced and discussed. The approach used for achieving this goal is based on the recently developed asymptotic solution method for steady isothermal elastohydrodynamically lubricated problems of heavily loaded point contacts, as well as on a semi-analytical method for the solutions of contact problems for elastic functionally graded materials without the presence of lubrication in the contacts (dry contacts). For line elastohydrodynamically lubricated contacts such a problem has already been studied. The solution process can be subdivided into two stages: the solution of the contact problems for dry contacts and a solution of the problem for lubricated contacts. The solution of dry contact problems for functionally graded materials is described in detail in Part 1 of the paper. As in the case of homogeneous contact materials in the analysis of the elastohydrodynamically lubricated problem for functionally graded elastic materials, it is shown that the whole contact region can be subdivided into three subregions: the central one which is adjacent to the other two regions occupied by the ends of the zones. The central region can be subdivided into the Hertzian region and adjacent to it the inlet and exit zones which, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. In the Hertzian region the elastohydrodynamically lubricated problem solution is very close to the solution of the corresponding dry (i.e. non-lubricated) contact problem for functionally graded elastic materials which have been analyzed in Part 1 of the paper. In the central region in the inlet and exit zones of the heavily loaded point elastohydrodynamically lubricated contact, the elastohydrodynamically lubricated problem for functionally graded elastic materials using certain similarity transforms can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of the heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials. Therefore, many of the well known properties of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials are also valid for heavily loaded point elastohydrodynamically lubricated contacts for functionally graded elastic materials. These asymptotically valid equations can be analyzed and numerically solved based on stable methods using a specific regularization approach which was developed for lubricated line contacts. Also, this asymptotic analysis leads to the easy analytical derivation of equations for the lubrication film thickness which take into account the inhomogeneity of the elastic materials. As a result of this analysis, criteria for an increase in lubrication film and friction force reduction are proposed. These criteria depend on lubricant properties, as well as the properties of functionally graded elastic materials involved in lubricated contacts.

Numerical Simulation of Turbulent Blood Flow in the System of Coronary Arteries with Stenosis

In this paper, we propose a mathematical model of turbulence flow of fluid through a deformable channel to study the pulsatile blood flow in the coronary system with arterial stenosis. Blood is assumed to be an incompressible non- Newtonian fluid and its motion is considered as turbulent and modelled by the mass and momentum conservations with turbulent mixing energy and specific dissipation rate. The mechanical deformation of the arterial wall is modelled by a hyperelastic differential equation. The pulsatile behaviour during each heartbeat is assigned on the entrance and exit boundaries. Numerical simulation based on the Finite Element method for the solution of arterial wall deformation, and the Arbitrary Lagrangian Eulerian Finite Volume method for the turbulence fluid-flow solution is used to investigate the effect of stenosis severity at the proximal part of the left anterior descending artery on the blood velocity, the pressure distribution and the wall shear stresses along the flow direction.

Speculative Futures Trading under Mean Reversion

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, or exponential Ornstein–Uhlenbeck model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor’s timing option to enter or exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short.

Speculative Futures Trading Under Mean Reversion

This paper studies the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR), or exponential Ornstein-Uhlenbeck (XOU) model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing option to enter or exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short.

Equipped Graded Graphs, Projective Limits of Simplices, and Their Boundaries

In this paper, we develop a theory of equipped graded graphs (or Bratteli diagrams) and an alternative theory of projective limits of finite-dimensional simplices. An equipment is an additional structure on the graph, namely, a system of “cotransition” probabilities on the set of its paths. The main problem is to describe all probability measures on the path space of a graph with given cotransition probabilities; it goes back to the problem, posed by E. B. Dynkin in the 1960s, of describing exit and entrance boundaries for Markov chains. The most important example is the problem of describing all central measures, to which one can reduce the problems of describing states on AF-algebras or characters on locally finite groups. We suggest an unification of the whole theory, an interpretation of the notions of Martin, Choquet, and Dynkin boundaries in terms of equipped graded graphs and in terms of the theory of projective limits of simplices. In the last section, we study the new notion of “standardness” of projective limits of simplices and of equipped Bratteli diagrams, as well as the notion of “lacunarization.”

Conjugate heat transfer during oscillatory laminar flow in porous media

Hydrodynamics and conjugate heat transfer in porous media subject to unidirectional-steady state as well as steady-periodic flow conditions were numerically studied. Two-dimensional flows in porous media composed of periodically configured arrays of square cylinders were simulated using a computational fluid dynamics tool, with sinusoidal time variation of inlet and exit boundaries and conjugate heat transfer between the solid and fluid domains. The simulated domain contained seven unit cells, with each unit cell representing a square cylinder. Simulations showed that the end effects did not persist more than two unit cells. Simulations were conducted for flow oscillation frequencies of 0–60Hz, and low and high velocity amplitudes (representing tidal fluid penetration amplitudes of 8 and 2 times the hydraulic diameter, respectively, at inlet) for a 75% porous domain. Using the simulation results, pore-scale volume-average Darcy permeability, Forchheimer coefficient, and Nusselt number associated with the standard volume-average porous media momentum and thermal energy equations were calculated. These parameters were calculated in instantaneous as well as cycle-averaged forms. The Forchheimer coefficient and Nusselt number were significantly different in unidirectional-steady and oscillatory flow conditions. These parameters strongly depend on the flow oscillation frequency and amplitude. Furthermore, significant phase lag occurs among velocity, pressure, temperature and heat transfer processes. The results confirm that unidirectional-steady flow models and correlations are not suitable for applications involving flow oscillations.

Asymptotic analysis of lubricated heavily loaded point contacts

ABSTRACTThe paper is devoted to the development of an asymptotic approach to solution of the steady isothermal problem of elastohydrodynamic lubrication (EHL) for heavily loaded point contacts. It is shown that the whole contact region can be subdivided into three subregions: the central one that is adjacent to the other two regions occupied by the ends of the horseshoe‐shaped pressure/gap distribution zone. The central region, in turn, can be subdivided into the Hertzian region and its adjacent inlet and exit zones that, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. Moreover, in the central region, in the inlet and exit zones of heavily loaded point EHL contact, the EHL problem can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of heavily loaded line EHL contacts. The latter means that many of the well‐known properties of heavily loaded line EHL contacts are also valid for heavily loaded point EHL contacts. These asymptotically valid equations can be analysed and numerically solved based on the stable methods using a specific regularisation approach that were developed for lubricated line contacts. Cases of pre‐critical and over‐critical lubrication regimes are considered. The by‐product of this asymptotic analysis is an easy analytical derivation of formulas for the lubrication film thickness for pre‐critical and over‐critical lubrication regimes. The method is validated by the results of some experimental and numerical studies published by a number of researches. Copyright © 2013 John Wiley & Sons, Ltd.

Translation by Ribosomes with mRNA Degradation: Exclusion Processes on Aging Tracks

We investigate the role of degradation of mRNA on protein synthesis using the totally asymmetric simple exclusion process (TASEP) as the underlying model for ribosome dynamics. mRNA degradation has a strong effect on the lifetime distribution of the mRNA, which in turn affects polysome statistics such as the number of ribosomes present on an mRNA strand of a given size. An average over mRNA of all ages is equivalent to an average over possible configurations of the corresponding TASEP—both before steady state and in steady state. To evaluate the relevant quantities for the translation problem, we first study the approach towards steady state of the TASEP, starting with an empty lattice representing an unloaded mRNA. When approaching the high density phase, the system shows two distinct phases with the entry and exit boundaries taking control of the density at their respective ends in the second phase. The approach towards the maximal current phase exhibits the surprising property that the ribosome entry flux can exceed the maximum possible steady state value. In all phases, the averaging over the mRNA age distribution shows a decrease in the average ribosome density profile as a function of distance from the entry boundary. For entry/exit parameters corresponding to the high density phase of TASEP, the average ribosome density profile also has a maximum near the exit end.

Boundaries of local no-return sets for control systems

For control systems described by ordinary differential equations and given subsets A of the state space, the points on the boundary of A through which entrance and exit is possible are studied. We assume that solutions that leave A and remain close to it cannot return. Continuity properties of the entrance and exit boundaries when the control range is changed are described.

Effect of nonuniform boundary conditions on steady flow in saturated homogeneous cylindrical soil columns

Laboratory column experiments involving steady flow in homogeneous soil are often analyzed assuming that the flow is spatially uniform in any plane transverse to the longitudinal axis aligned with the column centerline. Axisymmetric steady flow in such a column was analyzed to determine the impact of radially nonuniform boundary conditions at the column entrance and exit planes. A general solution to the governing Laplace equation was derived taking into account arbitrary functional forms of the imposed head and flux boundary conditions. Specific solutions were deduced for smoothly varying and abrupt disturbances at the boundaries. The solutions were used to derive expressions for the length scale over which the induced flow nonuniformities are dissipated within the column. For soil columns with an aspect ratio (column radius/length) less than about 1 3 , the maximum dissipation length scale is in all cases less than 3 2 R , where R is the column radius. For practical purposes it is sufficient to take R as the dissipation length scale. Consequently, no matter what the radial variation in the boundary condition, flow will be uniform within the column if at each end a baffle zone with length equal to R is incorporated into the soil column design. The results can be applied to homogeneous anisotropic soil via a simple scaling. Published experimental results showing nonuniform flow near the entrance and exit boundaries were found to be consistent with the theoretical results.

On sound propagation from a slanted side branch into an infinitely long rectangular duct

The transmission of sound from a slanted side branch into an infinitely long rectangular duct is studied numerically using the method of finite element with absorptive domain exit boundaries. The sound transmission coefficients associated with various acoustic modes are investigated in details. The results show that the plane wave assumption is only valid at very low frequency. It is also found that the intensities of the higher modes are stronger than that of the plane wave once they are excited. Besides, a critical side-branch slant angle is found over which a significant change of sound propagation mode takes place. This affects substantially the energy distribution between various acoustic modes inside the main duct. A simplified model is proposed to explain the phenomenon and the relationship of this critical angle with the width ratio between the side branch and the main duct is established.

Femoral seating position of the EndoButton in single incision anterior cruciate ligament reconstruction: an anatomical study

Arthroscopic EndoButton fixation of anterior cruciate ligament (ACL) grafts over the femoral cortex has become popular in recent years. However, elongation of fixation materials has lead to tunnel enlargement, graft tunnel motion and instability. Synovial fluid passages along femoral tunnels following ACL reconstruction may result in destruction of tissues. The purpose of the present study was to identify the seating position of the EndoButton in regard to the boundaries of the knee capsule in single-incision arthroscopic ACL reconstruction. A total of 20 cadaveric knees were dissected and arthroscopic drill guides were used to create tibial and femoral tunnels. The distances between the exit points and boundaries of the suprapatellar bursa at three different degrees of knee flexion were measured. The average distances from the exit points to the superior boundaries of the suprapatellar bursa were 6.89 +/- 5.40 mm (range, 16.87-1.21). However, in knee flexion of greater than 90 degrees , tunnel exits were placed below the superior boundaries of the suprapatellar bursa (-7.08 +/- 3.69 mm, range, -3.24 to -12.87). In order to place the EndoButton extraarticularly, a safe degree of knee flexion during femoral tunnel drilling was defined as 80 degrees .

Transport of conservative solutes in simulated fracture networks: 2. Ensemble solute transport and the correspondence to operator‐stable limit distributions

In networks where individual fracture lengths follow a fractal distribution, ensemble transport of conservative solute particles at the leading plume edge exhibit characteristics of operator‐stable densities. These densities have, as their governing equations of transport, either fractional‐order or integer‐order advection‐dispersion equations. Model selection depends on the identification of either multi‐Gaussian or operator‐stable transport regimes, which in turn depends on the power law exponent of the fracture length distribution. Low to moderately fractured networks with power law fracture length exponents less than or equal to 1.9 produce solute plumes that exhibit power law leading‐edge concentration profiles and super‐Fickian plume growth rates. For these network types, a multiscaling fractional advection‐dispersion equation (MFADE) provides a model of multidimensional solute transport where different rates of power law particle motion are defined along multiple directions. The MFADE model is parameterized by a scaling matrix to describe the super‐Fickian growth process, in which the eigenvectors correspond to primary fracture group orientations and the eigenvalues code fracture length and transmissivity. The approximation of particle clouds by a multi‐Gaussian (a subset of the operator stable) for densely fractured networks with finite variance fracture length distributions can be ascribed to the classical ADE where Fickian scaling rates pertain along orthogonal plume growth directions. Fracture networks show long‐term particle retention in low‐velocity fractures so that coupling of the equations of motion with retention models such as continuous time random walk or multirate mobile/immobile will increase accuracy near the source. Particle arrival times at exit boundaries for multi‐Gaussian plumes vary with spatial density. Generally, arrival times are faster in sparsely fractured domains where transport is governed by a few very long fractures.