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Theory of Dispersion in a Granular Medium

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Theory of Dispersion in a Granular Medium :70 Theory of Dispersion in a Granular Medium GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-1 OD «! a i £ O I?i c MAY 2 6 1970 WRD Theory of Dispersion in a Granular Medium By AKIO OGATA GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-1 A review of the theoretical aspects of dispersion offluid flowing through a porous material UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970 UNITED STATES DEPARTMENT OF THE INTERIOR WALTER J. HICKEL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 45 cents (paper cover) CONTENTS Page Page Symbols... .-.-.._._...__....___.__.._.___ IV Field equation for two-fluid systems... .----.-..-._ 19 Abstract. --.___-_____..___-___--.__---_______.____. II Adsorption_ _ _____________________________________ 10 Introduction- __-___-___-_____--____--_______.______ 1 Mathematical treatment of the dispersion equation. _ _ _ _ 12 Acknowledgments..__....__._______.._._______ 2 Nature of the dispersion coefficient.___________________ 23 Historical development-_________________________ 2 Evaluation of dispersion coefficient by analytical Derivation of the basic field equations_________---___ 4 methods._-________.-_____---______----._-_._ 25 Isotropic dispersion. ..^......................... 5 Anisotropic dispersion.__________________________ 6 Evaluation of dispersion coefficient by experimental Some useful transformations.____________________ 7 methods. _-.__ .__.__________.-_.__..__ 27 Dispersion equation in cylindrical and spherical Summary....____..__,-._.._.__.--__. ._..___. 32 coordinates. ___.__-_-_.___..._-____.__-_-_. 8 References cited..-.._.__._,-____________ 33 ILLUSTRATIONS Page FIGURE 1. Diagram illustrating mass balance in a cubic element in space.-_-_-___-_--___--._-__-_- 15 2. Schematic diagram of coastal aquifer system.______________________ ..._._.__ . 9 3. Schematic diagram of transport in porous medium... __.__ __ 11 4. Definition sketch of one-dimensional system..___._____-____-_-___-_-.__.____-__..-,__ 12 5-8. Graphs showing 5. Concentration distribution; plane source at x=0 maintained at constant concentra­ tion. __ _._---__ ___.._. _._.____...__.____________________________ 14 6. Concentration distribution; region x<0 initially at constant concentration._______ 15 7. Steady-state concentration distribution; circular surface source at constant strength. 16 8. Steady-state concentration distribution; circular surface source maintained at con­ stant concentration._-__-__._-__.___-.__________-_.___-____.-___-_--_-.__ 17 9. Definition sketch of radial flow._______________________.._ _._ 17 10. Graph showing concentration distribution for radial-flow system._______________________ 19 11. Definition sketch of system depicting no dispersion in x direction.__-___---___--______-- 20 12. Approximate concentration profile due to lateral dispersion; radially symmetrical model._ 20 13. Approximate concentration profile due to lateral dispersion; two-dimensional model-....- 21 14. Graph showing comparison of examples 3, 4, and 6______.-_._.____.___.._______..___ 22 15. Concentration profile showing dispersion in an adsorbing medium with 5 S/dJ= bC... _______ 24 16. Concentration profile showing dispersion in an adsorbing medium with dS/dt=b(C mS), £_->2C/dz8 = 0, anddS ^....................................................... 25 df d* 17. Sketch showing random path chosen by a fluid particle moving through the canal system. _ 26 18. Sketch showing typical setup for column experiment._____.__..____._-__.___.-___--__- 28 19. Plot of the Schmidt number versus the Reynolds number._____________________________ 30 20. Plot of the Peclet number versus the Reynolds number.-_._....--._.__.. _ __ 31 21. Graph showing example of comparision between theoretical and experimental results.___._ 32 m SYMBOLS Q total rate of flow Inverse Laplace transform u, v, w velocities in the x, y, and z 1 fCC+tco =db directions, respectively (the 4mjc- Darcy velocity divided by porosity) erf (a;) error function defined by erf (z) C Concentration in mass of solute per unit volume of fluid C0 reference concentration C' ratio C/C0 Di dispersion coefficient in the i=x, erfc(x) complimentary error function y, z directions defined by 1 erf(z) F{ mass flux of dissolved compo­ J,(x) Bessel function of the first kind nent per unit area in the ith of order , direction Y,(x) Bessel function of the second x, y, z space dimensions of Cartesian kind of order system /»(*) modified Bessel function of the t time first kind of order , A cross-sectional area K,(x) modified Bessel function of the r radial space coordinate second kind of order , K hydraulic conductivity k intrinsic permeability J(x, y) 1-e-v f%-«/o[2foO M]ett, or p pressure Jo g gravitational constant 5 flux per unit area 1 -XH f " e-*-Vo[2 (tf) M]/i / porosity [2(zt)"]dt d diameter of the particle making up the solid matrix length of elementary canal y0 thickness of the aquifer density (also r/d) 8 concentration per unit volume specific weight of solid material viscosity b, m constants related to linear kinematic viscosity _ adsorption rates stream function n number of steps in a random transformed coordinates as walk analysis specified exp exponential T ui\x X D/ux Per udlDr, radial Peclet number Laplace transform L{f(t) } Pe, ua/Dr, axial Peclet number FLUID MOVEMENT IN EARTH MATERIALS THEORY OF DISPERSION IN GRANULAR MEDIUM By AKIO OQATA ABSTRACT 2. The laboratory problem for confirmation of (1) and This report is a presentation of the theoretical aspects of the its extension to include parameters that cannot investigations and a few confirmatory laboratory studies of the be included in the mathematical model. dispersion process. The basic assumptions associated with the 3. The field that is, the natural aquifer problem development of the dispersion equation, based on the applica­ bility of a heuristic expression similar to Pick's law, are described. which introduces larger variations in parameters. In general, these assumptions limit its application to a homo­ Much progress has been made in segment (1) in recent geneous medium in which the dispersion process is anisotropic. years, and because of availability of electronic com­ The present level of knowledge is made apparent through the puters, complex mathematical equations can now be presentation of several solutions for the differential equation readily analysed. The problem (segment 3) is not that approximates the dispersion process. Adsorption and its effect on the fate of a contaminant is given only limited discus­ clearly understood because of large variations in the sion because of the lack of formal studies, on a macroscopic parameters controlling the mixing mechanisms. Seg­ scale, applicable to the hydrologic regimen. The equations ment (2), on the other hand, is effectively used as a approximating dispersion in a two-fluid system (salt-fresh method to confirm mathematical analysis, but as yet, water) are also presented. is limited in its use as a physical model of the real system. Bachmat (1967) pointed out that both the INTRODUCTION geometric and time-scale ratio for the model and The rapidly growing need to dispose of radioactive prototype must be 1:1. waste products, and the increased rate of pollution of the This paper is limited to the theory of dispersion or ground-water resource by other chemical compounds, category (1) as described above. Because existing make it increasingly apparent that a much more de­ literature on the subject is extensive, a thorough review tailed description of flow in porous media must be is beyond the scope of this paper. sought. Because of the dangers inherent in the presence The mechanisms associated with the transport of a of contaminants in the water supply, their fate and dissolved substance in porous media are listed below. mode of travel downstream from their sources must be Some of these mechanisms are of little significance in predicted. The fate of a contaminant depends on both natural ground-water bodies, but they are important the macroscopic and microscopic behavior of the fluid in exchange systems that are used in the chemical under the existing flow conditions and the physico- industry. The mechanisms are as follows: chemical conditions within the environment of the 1. Molecular diffusion the transport of mass in its granular material. The discussion to follow presents a ionic or molecular state due to differences in simplified treatment of dispersion phenomenon, and a concentration of a given species in space. The mathematical expression, in the form of a differential gross transport obeys Fick's first and second laws equation, is developed. As the medium becomes more of diffusion (Bird and others, 1960, p. 502). complex, the flow description becomes indeterminate 2. Mechanical dispersion the mixing mechanism that (Skibitzke and Robinson, 1963), since the description is present because of the variations in the micro­ of the transport mechanism, owing to convection a.nd scopic velocity within each flow channel and from dispersion, requires first a correlation between proper­ one channel to another. Microscopically, there is ties of the porous medium and the flow conditions. no mixing; however, if the average concentration The present-day study of dispersion may be separated of a given volume of fluid is considered, an ap­ into three categories (C. V. Theis, written commun., parent dilution or spreading is present. The gross Oct. 1967): transport, as in eddy diffusion, is given by an 1. The mathematical problem. expression similar to Fick's law. li 12 FLUID MOVEMENT IN EARTH MATERIALS 3. Eddy diffusion the mixing process that is due to heuristic approach requires that a partial differential the random fluctuation of fluid mass or the occur­ equation similar to all transport systems be solved. rence of eddies in the condition described as turbulent flow.
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