Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

Global Attractors: Topology and Finite-Dimensional Dynamics

Many dissipative evolution equations possess a global attractor with finite Hausdorff dimension d. In this paper it is shown that there is an embedding X of into , with N=[2d+2], such that X is the global attractor of some finite-dimensional system on with trivial dynamics on X. This allows the construction of a discrete dynamical system on which reproduces the dynamics of the time T map on and has an attractor within an arbitrarily small neighborhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.     

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

Der selbsterregte unsymmetrische Kreisel

Zusammenfassung Wir können alle bisherigen Erkenntnisse über die Selbsterregung des unsymmetrischen Kreisels dahin zusammenfassen, daß — abgesehen von den trivialen Fällen der Drehungen um jede der drei Hauptachsen und abgesehen von den permanenten Drehungen um beliebige Achsen — zu jeder achsenfesten zeitunabhängigen Selbsterregung achsenbewegliche Drehungen gehören, wie wir solche in Ziff. 2 bis 7 untersucht haben; ferner daß es, wieder abgesehen von jenen trivialen Fällen, nur den einzigen Typ der achsenfesten zeitabhängigen Selbsterregung von Ziff. 9 gibt, welche eine achsenfeste zeitabhängige Drehung zu beeinflussen, wenn auch weder zu erzeugen noch zu vernichten vermag; endlich daß der Momentvektor der Selbsterregung, welcher eine beliebige achsenfeste Drehung erzeugen oder vernichten oder in beliebiger Weise beeinflussen soll, im allgemeinen (also abgesehen von den trivialen Fällen und von dem Fall in Ziff. 9) zeitabhängig in einer körperfesten Ebene wandern muß. Die achsenbeweglichen Drehungen, welche von achsenfester zeitunabhängiger Selbsterregung erzeugt oder unterhalten werden, sind je nach der Lage (und Größe) des Momentes der Selbsterregung periodisch hinsichtlich der Bewegung des Drehvektors im Kreisel oder asymptotisch mit unbeschränkt wachsendem Drehvektor oder asymptotisch mit einem Drehvektor , der sich einem permanenten Drehvektor (von endlichem Betrag) annähert (oder auch sich von ihm entfernt).     

A new statistical approach to study and to predict wall turbulence

A new analysis method is developed to study the double- and triple-correlations of velocity fluctuations inside a stationary three-dimensional turbulent boundary layer (3D-TBL). Experimental eigenvalues and eigenvectors of measured Reynolds stress-tensors are obtained by diagonalization; a set of semi-empirical relationships is derived and these are interpreted (qualitatively) in terms of statistics of gas dynamics. Sample-averaged double- and triple-correlations are Monte Carlp (MC-) simulated, simultaneously, with 3 independent perturbed centered-Gaussians (trial probability density functions) along experimental eigenvectors. Comparisons with corresponding time-averaged measurements show excellent agreement for the double-correlations and qualitative agreement for the triple-correlations. Also, a statistical model for the double-correlations is presented: it can predict the -profiles inside the S-shaped wind tunnel at EPFL, given .     

Homologies of Intersections of Pseudomanifolds with Local Homotopic Conditions on Links of Strata

In terms of local homotopic properties of the links of strata of an n-dimensional PL-pseudomanifold X, we obtain a sufficient condition for the natural homomorphisms of the jth intersection homology groups with perversity multiindices and to be isomorphisms for all j i, where i < n – 3.     

Transient Free Convection from a Horizontal Surface in a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q ₁ on the surface. Then, at the heat flux on the surface is suddenly changed to q ₂ and maintained at this value for . Firstly, solutions which are valid for small and large are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state ( ) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times .     

On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.     

Flow in Random Porous Media

Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor (x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector (x) in terms of the gradient of the mean hydraulic head (x), an integrodifferential equation for (x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined.     

Investigation of the Geometrical Dispersion Regime in a Single Fracture Using Positron Emission Projection Imaging

The transport properties of a natural fracture crossing a limestone block of 36 cm × 26 cm × 60 cm is studied using positron emission projection imaging. This non-invasive technique allows to measure the spatial distribution of the activity of a radioactive solution (here irradiated-copper-EDTA solution) within the fracture. The fracture aperture is measured from the spatial distribution of the activity as the fracture is completely filled with the tracer. The experiment consists in injecting the tracer at a constant flow rate in the plane of the fracture filled with an identical non-radioactive solution. Every 10 min, a two-dimensional grey scale image of the concentration field is recorded. The heterogeneity of the tracer distribution increases with time in relation with the spatial heterogeneity of the aperture field, and favours only slightly the region of larger aperture. The correlation length of the aperture distribution is larger than the correlation length of the concentration distribution of the tracer within the sample. Consequently, the concentration distribution cannot be modelled using a classical advection–dispersion equation; the mixing process has not reached a stationary Fickian dispersion regime in the finite size domain of the experiment. Nevertheless, the transversally averaged concentration profiles evaluated along the flow direction x rescale adequately with an advective variable , where is the mean velocity and t the time. This result is explained in the context of the geometrical dispersion regime where the mixing dispersion zone grows proportionally with time. Different approaches are proposed to characterise this anomalous dispersion regime.     

On the Denseness of the Set of Unsolvable Cauchy Problems in the Set of All Cauchy Problems in the Case of an Infinite-Dimensional Banach Space

We prove the following statement: Theorem 1. Let E and be an arbitrary infinite-dimensional Banach space and a continuous mapping, respectively. Then, for every and > 0, there exists a continuous mapping such that

Grundzüge einer Stabilitätstheorie für elastische Systeme unter nichtkonservativer Belastung

Zusammenfassung Um den Einfluß von nichtkonservativen Kräften zu erfassen, muß die Stabilitätstheorie nichtkonservativer elastischer Systeme auf dem Prinzip der virtuellen Verrückung aufgebaut werden, das von genügender Allgemeinheit ist. Als mathematische Methode steht nach Klärung von Konvergenzfragen das Galerkinsche Verfahren zur Verfügung, das dem ursprünglichen Problem der Stabilität elastischer Systeme das Problem der Stabilität der Bewegung eines mechanischen Systems von unendlich vielen Freiheitsgraden als gleichwertig zuordnet und die Algebraisierung des Stabilitätsproblems erlaubt. Damit wird die Ljapunowsche Theorie anwendbar. Als Wichtigstes ist die Berechtigung der Linearisierung zu untersuchen, wozu der Einfluß der Dämpfung festzustellen ist. Es ergibt sich, daß bei Berücksichtigung von Dämpfung die Stabilität stets asymptotisch ist, so daß die Linesarisierung gerechtfertigt wird. Darüber hinaus zeigt es sich, daß die Struktur der Matrizen und von entscheidender Bedeutung für die Zuständigkeit eines der in Frage kommenden Stabilitätskriterien ist: Ist symmetrisch und eine beliebige positive Diagonalmatrix bzw. schiefsymmetrisch, oder symmetrisierbar bzw. unsymmetrisch und noch gleichzeitig (die Dämpfungskoeffizienten alle gleich), so wirkt in diesen Fällen die Dämpfung günstig, und die Stabilitätskriterien des ungedämpften Vorganges, das statische bzw. das kinetische, bleiben daher hinreichend. Man darf dann auch ohne Dämpfung rechnen. Ist dagegen unsymmetrisch und eine positive Diagonalmatrix mit verschiedenen Elementen, so darf in solch einem Fall nicht ohne Dämpfung gerechnet werden, weil gegenüber dem ungedämpften Vorgang ein Sprung in den Stabilitätsbedingungen auftritt, durch welchen das Ergebnis der Rechnung ohne Dämpfung in Frage gestellt wird.     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

Relatively Compact Orbits and Compact Attractors for a Class of Nonlinear Evolution Equations

In the present paper we consider the nonlinear evolution equation u+AuG(u), where A:D(A)XX is m-accretive with (I+A)^–1 compact for some >0, and is continuous, and we prove that the orbit is relatively compact if and only if u is uniformly continuous, and both u and G^u are bounded on . In the same spirit, we derive conditions for orbits of bounded sets to have compact attractors. Some consequences and an example from age-structured population dynamics illustrate the effectiveness of the abstract result.     

An Approximate Solution for Fluid Infiltration near a Circular Opening in Unsaturated Media

A regular perturbation technique is employed to approximate the solution for fluid infiltration from a circular opening into an unsaturated medium. Introducing two empirical constitutive relations and relating the permeability k and water content with pore fluid pressure p, a nonlinear diffusion equation in terms of pore pressure is established. After rearranging the nonlinear diffusion equation, a parameter perturbation on is performed and an approximate solution with an error of is obtained, which correlate to a condition in which = . This approximate solution is verified by a finite difference solution and compared also with a linear solution in which the diffusivity is constant. It is shown that the perturbation solution with terms up to and including first-order can give a reasonably accurate solution for the parameter range for p ₀ selected in this paper. The solution procedure provided in this paper also avoids the numerical problem normally encountered for a small time solution. The solution may also be used to overcome difficulties arising in solution procedure by the similarity transformation (Boltzmann), commonly conducted on diffusion equation, which cannot be applied for a finite wellbore problem.     

Interval stochastic matrices: A combinatorial lemma and the computation of invariant measures of dynamical systems

The concept of an interval stochastic matrix is introduced. We prove a combinatorial theorem which describes the network flow associated with an interval matrix. The semi-invariant vectors of are characterized in terms of eigenvectors with unit eigenvalue of stochastic matrices . These results are then applied to the approximation and machine computation of invariant measures of dynamical systems.Funded under Australian Research Council Grant A 8913 2609.     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

A study of stationary crack-tip deformation fields in thin sheets by computer vision

The in-plane deformation fields near a stationary crack tip for thin, single edge-notched (SEN) specimens, made from Plexiglas, 3003 aluminum alloy and 304 stainless steel, have been successfully obtained by using computer vision. Results from the study indicate that (a) in-plane deformations ranging from elastic to fully plastic can be obtained accurately by the method, (b) for U, and , the size of the HRR dominant zone is much smaller than forV and , respectively. Since these results are in agreement with recent analytical work, suggesting that higher order terms will be needed to accurately predict trends in the data, it is clear that the region where the first term in the asymptotic solution is dominant is dependent on the component of the deformation field being studied, (c) the HRR solution can be used to quantity only in regions where theplastic strains strongly dominate the elastic strain components (i.e., when ); forV, the HRR zone appears to extend somewhat beyond this region, (d) the displacement componentU does not have the HRR singularity anywhere within the measurement region for either 3003 aluminum or 304 SS. However, the displacement componentV agrees with the HRR slope up to the plastic-zone boundary in 3003 aluminum ( ) and over most of the region where measurements were obtained ( ) in 304 SS and (e) the effects of end conditions must be included in any finite-element model of typical SEN specimen geometries to accurately calculate theJ integral and the crack-tip fields.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Non-persistence of Homoclinic Connections for Perturbed Integrable Reversible Systems

The dynamics of an analytic reversible vector field (X,) is studied in with one real parameter close to 0; X=0 is a fixed point. The differential D_x (0,0) generates an oscillatory dynamics with a frequency of order 1—due to two simple, opposite eigenvalues lying on the imaginary axis—and it also generates a slow dynamics which changes from a hyperbolic type—eigenvalues are —to an elliptic type—eigenvalues are —as passes trough 0. The existence of reversible homoclinic connections to periodic orbits is known for such vector fields. In this paper we study a particular subclass of such vector fields, obtained by small reversible perturbations of the normal form. We give an explicit condition on the perturbation, generically satisfied, which prevents the existence of a homoclinic connections to 0 for the perturbed system. The normal form system of any order admits a reversible homoclinic connection to 0, which then does not survive under perturbation of higher order. It will be seen that normal form essentially decouples the hyperbolic and elliptic part of the linearization to any chosen algebraic order. However, this decoupling does not persist arbitrary reversible perturbation, which finally causes the appearance of small amplitude oscillations.