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.     

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     

Stability of rarefaction waves in viscous media

We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, Burgers rarefaction wave, for initial perturbations w ₀ with small mass and localized as w ₀(x)= The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error.This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass (log (t)). These diffusion waves have amplitude (t ^-1/2logt) in linear degenerate transversal fields and (t ^-1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.     

The Experimental Approach to Effective Stress Law of Coal Mass by Effect of Methane

Effective stress law of all kinds of coal samples, including steam coal, fat coal, corking coal, thin coal and anthracite, under pore pressure of gas, is experimentally studied using a newly developed test machine. These samples are taken from Coal Mines in Wuda, Hebi, Yanzhou, Yangquan, Qingshui, and Gujiao in China. The experiment results show that, under pore pressure of gas, the tested coal samples comply with Biots effective stress law, where the Biots coefficient is not a constant, and is bilinear function of volumetric stress () and pore pressure (p), that is, We define four areas according to the numerical feature of , that is, functionless area of pore pressure, normal function area, fracturing function area, and quasi-soil function area. The effective stress law of coal mass introduced by this paper is a constitutive equation in the study of coupled solid and fluid. This has significance in the drainage and outburst of methane in coal seam.     

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 .     

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.     

Laminar source flow between parallel porous disks

An analysis is presented for laminar source flow between infinite parallel porous disks. The solution is in the form of a perturbation from the creeping flow solution. Expressions for the velocity, pressure, and shear stress are obtained and compared with the results based on the assumption of creeping flow.Nomenclature a half distance between disks - radial coordinate - r dimensionless radial coordinate, /a - axial coordinate - z dimensionless axial coordinate, /a - radial coordinate of a point in the flow - R dimensionless radial coordinate of a point in the flow, /a - velocity component in radial direction - u =a/, dimensionless velocity component in radial direction - velocity component in axial direction - v = a/}, dimensionless velocity component in axial direction - static pressure - p = (a ²/ ²), dimensionless static pressure - =p(r, z)–p(R, z), dimensionless pressure drop - V magnitude of suction or injection velocity - Q volumetric flow rate of the source - Re source Reynolds number, Q/4a - reduced Reynolds number, Re/r ² - critical Reynolds number - R _w wall Reynolds number, Va/ - viscosity - density - =/, kinematic viscosity - shear stress at upper disk - ₀ = (a ²/ ²), dimensionless shear stress at upper disk - shear stress ratio, ₀/( ₀)_inertialess - u = , dimensionless average radial velocity - u/u, ratio of radial velocity to average radial velocity - dimensionless stream function     

Momentum balance in highly distorted turbulent pipe flows

An in depth study into the development and decay of distorted turbulent pipe flows in incompressible flow has yielded a vast quantity of experimental data covering a wide range of initial conditions. Sufficient detail on the development of both mean flow and turbulence structure in these flows has been obtained to allow an implied radial static pressure distribution to be calculated. The static pressure distributions determined compare well both qualitatively and quantitatively with earlier experimental work. A strong correlation between static pressure coefficient Cp and axial turbulence intensity is demonstrated.List of symbols C _p static pressure coefficient = (pw-p)/1/2 - D pipe diameter - K turbulent kinetic energy - (r, , z) cylindrical polar co-ordinates. / 0 - R, y pipe radius, distance measured from the pipe wall - U, V axial and radial time mean velocity components - mean value of u - u, u/ , / - u, , w fluctuating velocity components - axial, radial turbulence intensity - turbulent shear stress - u friction velocity, (u ² = ₀/p) - ₀ wall shear stress - ^* boundary layer thickness A version of this paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

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.     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)     

Zum Mischen rheologisch inhomogener Stoffsysteme auf Einschneckenmaschinen

Zusammenfassung Das Mischen von Stoffen mit unterschiedlichen rheologischen Eigenschaften in Schneckenmaschinen ist in der Kunststoffauf- und -verarbeitung eine Standardaufgabe. Trotzdem gibt es hierfür kein zufriedenstellendes mathematisch-physikalisches Modell. Daher werden zunächst einfache Mischmodelle diskutiert. Auf der Basis dieser Modelle wird dann unter Berücksichtigung der Besonderheiten des Plastifizierextruderprozesses eine Mischgütebeziehung mathematisch formuliert. Die experimentelle Überprüfung erfolgt mit Hilfe der Grauwertanalyse extrudierter Zweistoffsysteme, bei denen ein Stoff mit Ruß eingefärbt war. Da der Mischprozeß hochgradig stochastisch ist, streuen die Meßergebnisse. Unter Berücksichtigung dieses Tatbestandes ist der theoretische Ansatz zufriedenstellend.

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Mixing of polymer resins with different rheological properties is a usual demand in plastics processing using screw extruders. A mathematical model describing this processing problem sufficiently is not known, however. Therefore, simple mixing models will be discussed. Based on these, a concept for the calculation of mixing homogeneity will be presented, including the particular requirement of the plasticating screw process. An experimental investigation utilizes the grey-value analysis of extruded two-component materials, which in one phase is carbon-black filled. Considering the fact that the mixing process is highly random, the theoretical model leads to a good level of aggreement with the scattering measurement data.

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b Schneckenkanalbreite - B Bandbreite der Grauwerte - c Konstante - mittlere Konzentration, bezogen auf die Grauwertbandbreite - h Höhe, Gangtiefe, Schneckenkanalhöhe - h ₀ Gangtiefe der Einzugszone - h ₁ Gangtiefe der Ausstoßzone - L Länge - gemittelte Schmelzebettlänge - n Exponent des Potenzfließgesetzes - s Standardabweichung der Grauwerte bezogen auf die Grauwertbandbreite - S Standardabweichung der Grauwerte - t Verweilzeit - t ₁ kürzeste Verweilzeit - mittlere Verweilzeit - ₀ Umfangsgeschwindigkeit - mittlere Geschwindigkeit - V Volumenstrom - w Dicke eines Kontrollelements - w Ausstreichdicke eines Kontrollelements - x Koordinate - Mittelwert der Grauwerte - y Koordinate - Scherdeformationswinkel - Scherdeformation - mittlere Scherdeformation - Schergeschwindigkeit - Viskosität - ₁ dimensionslose kürzeste Verweilzeit - dimensionsloser Volumenstrom - _LSM laminarer Schermischgrad - _{LSM, the} theoretischer laminarer Schermischgrad - _{LSM, exp} experimenteller laminarer Schermischgrad - ² Varianz der Verweilzeit im Schmelzebett - Schubspannung - Gangsteigungswinkel der Schnecke - ø Volumenanteil - dimensionslose Kennzahl     

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.     

Modeling of contaminant transport resulting from dissolution of nonaqueous phase liquid pools in saturated porous media

A mathematical model for transient contaminant transport resulting from the dissolution of a single component nonaqueous phase liquid (NAPL) pool in two-dimensional, saturated, homogeneous porous media was developed. An analytical solution was derived for a semi-infinite medium under local equilibrium conditions accounting for solvent decay. The solution was obtained by taking Laplace transforms to the equations with respect to time and Fourier transforms with respect to the longitudinal spatial coordinate. The analytical solution is given in terms of a single integral which is easily determined by numerical integration techniques. The model is applicable to both denser and lighter than water NAPL pools. The model successfully simulated responses of a 1,1,2-trichloroethane (TCA) pool at the bottom of a two-dimensional porous medium under controlled laboratory conditions.Notation a,a ₁ defined in (45a) and (45b), respectively - b defined in (45c) - b vector of true model parameters (n×1) - vector of estimated model parameters (n×1) - c liquid phase solute concentration (solute mass/liquid volume), M/L³ - c _s aqueous saturation concentration (solubility), M/L³ - C dimensionless liquid phase solute concentration, equal toc/c _s - molecular diffusion coefficient, L²/t - ^e effective molecular diffusion coefficient, equal to / ^*, L²/t - D _x longitudinal hydrodynamic dispersion coefficient, L²/t - D _z hydrodynamic dispersion coefficient in the vertical direction, L²/t - e random vector with zero mean (m×1) - erf[x] error function, equal to (2/ ^1/2) - f vector of fitting errors or residuals (m×1) - Fourier operator - _-1 Fourier inverse operator - g vector of model simulated data (m×1) - k mass transfer coefficient, L/t - average mass transfer coefficient, L/t - K _d partition or distribution coefficient (liquid volume/solids mass), L³/M - pool length, L - _o distance between the pool and the origin of the specified Cartesian coordinate system, L - Laplace operator - _-1 Laplace inverse operator - m number of observations - M Laplace/Fourier function defined in (38) - n number of model parameters - N Laplace/Fourier function defined in (39) - p defined in (46) - Pe _x Péclet number, equal toU _x /D _x - Pe _z Péclet number, equal toU _x /D _z - q defined in (47) - R retardation factor - s Laplace transform variable - S objective function - Sh local Sherwood number, equal tok/ _e - Sh _o overall Sherwood number, equal to l/ _e - t time,t - T dimensionless time, equal toU _x t/ - u dummy integration variable - u vector of independent variables - U _x average interstitial velocity, L/t - x spatial coordinate in the longitudinal direction, L - X dimensionless longitudinal length, equal to (x–)/ - y vector of observed data (m×1) - z spatial coordinate in the vertical direction, L - Z dimensionless vertical length, equal toz/ - Fourier transform variable - defined in (37) - defined in (50) - porosity (liquid volume/aquifer volume), L³/L³ - defined in (52a) and (52b), respectively - decay coefficient, t^–1 - dimensionless decay coefficient, equal to /U _x - bulk density of the solid matrix (solids mass/aquifer volume), M/L³ - dummy integration variable - ^* tortuosity     

Superstable Manifolds of Semilinear Parabolic Problems

We investigate the dynamics of the semiflow φ induced on H₀¹(Ω) by the Cauchy problem of the semilinear parabolic equation

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.     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

A Jordan Curve Spanned by a Complete Minimal Surface

In this paper we construct complete (conformal) minimal immersions which admit continuous extensions to the closed disk, . Moreover, is a homeomorphism and is a (non-rectifiable) Jordan curve with Hausdorff dimension 1. It turns out that the set of Jordan curves constructed by the above procedure is dense in the space of Jordan curves of with the Hausdorff metric.