On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions

A governing equation of stable random walks is developed in one dimension. This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (a) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Levy's a-stable densities that resemble the Gaussian except that they spread proportional to time 1/a , have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Levy motion would grow faster than Fickian plume, at a rate of time 2/a , where 0 , a # 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Levy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space.

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The fractional‐order governing equation of Lévy Motion

This textbook provides a basic understanding of the formative processes of igneous and metamorphic rock through quantitative applications of simple physical and chemical principles. The book encourages a deeper comprehension of the subject by explaining the petrologic principles rather than simply presenting the student with petrologic facts and terminology. Assuming knowledge of only introductory college-level courses in physics, chemistry, and calculus, it lucidly outlines mathematical derivations fully and at an elementary level, and is ideal for intermediate and advanced courses in igneous and metamorphic petrology. The end-of-chapter quantitative problem sets facilitate student learning by working through simple applications. They also introduce several widely-used thermodynamic software programs for calculating igneous and metamorphic phase equilibria and image analysis software. With over 350 illustrations, this revised edition contains valuable new material on the structure of the Earth's mantle and core, the properties and behaviour of magmas, recent results from satellite imaging, and more.

Principles of igneous and metamorphic petrology

Building upon the award-winning second edition, this comprehensive textbook provides a fundamental understanding of the formative processes of igneous and metamorphic rocks. Encouraging a deeper comprehension of the subject by explaining the petrologic principles, and assuming knowledge of only introductory college-level courses in physics, chemistry, and calculus, it lucidly outlines mathematical derivations fully and at an elementary level, making this the ideal resource for intermediate and advanced courses in igneous and metamorphic petrology. With over 500 illustrations, many in color, this revised edition contains valuable new material and strengthened pedagogy, including boxed mathematical derivations allowing for a more accessible explanation of concepts, and more qualitative end-of-chapter questions to encourage discussion. With a new introductory chapter outlining the “bigger picture,” this fully updated resource will guide students to an even greater mastery of petrology.

Principles of Igneous and Metamorphic Petrology

Dispersion in Porous Media

Published papers indicate that most investigators use the coordinate transformation (x ut) in order to solve the equation tor dispersion of a moving fluid in porous media. Further, the boundary conditions O=0 at x=«> and 0=00 at x= «> for Z>0 are used, which results in a symmetrical concentration distribution. This paper presents a solution of the differential equation that avoids this transformation, thus giving rise to an asymmetrical concentration distribution. It is then shown that this solution approaches that given by symmetrical boundary conditions, provided the dispersion coefficient D is small and the region near the source is not considered. INTRODUCTION In recent years considerable interest and attention have been directed to dispersion phenomena in flow through porous media. Scheidegger (1954), deJong (1958), and Day (1956) have presented statistical means to establish the concentration distribution and the dispersion coefficient. A more direct method is presented here for solving the differential equation governing the process of dispersion. It is assumed that the porous medium is homogeneous and isotropic and that no mass transfer occurs between the solid and liquid phases. It is assumed also that the solute transport, across any fixed plane, due to microscopic velocity variations in the flow tubes, may be quantitatively expressed as the product of a dispersion coefficient and the concentration gradient. The flow in the medium is assumed to be unidirectional and the average velocity is taken to be constant throughout the length of the flow field. BASIC EQUATION AND SOLUTION Because mass is conserved, tl^e governing differential equation is determined to be d<7 (1) v where D=dispersion coefficient C= concentration of solute in the fluid u= average velocity of fluid or superficial velocity/ porosity of medium x= coordinate parallel to flow y,z coordinates normal to flow 2=time. In the event that mass transfer takes place between the liquid and solid phases, the differential equation becomes _ 5(7 d(7 &F where F is the concentration of the solute in the solid phase. The specific problem considered is that of a semiinfinite medium having a plane source at z=0. Hence equation 1 becomes Initially, saturated flow of fluid of concentration, (7=0, takes place in the medium. At t Q, the concentration of the plane source is instantaneously changed to (7=(70 . Thus, the appropriate boundary conditions are <7(co,£)=0; *> The problem then is to characterize the concentration as a function of x and t. To reduce equation 1 to a more familiar form, let (4) A-l 586211 61 2 A-2 FLUID MOVEMENT IN EARTH MATERIALS Substituting equation 4 into equation 1 gives The boundary conditions transform to = Co exp ( It is thus required that equation 5 be solved for a timedependent influx of fluid at 2=0. The solution of equation 5 may be obtained readily by use of Duhamel's theorem (Carslaw and Jaeger, 1947, p. 19): If C=F(x,y,z,t) is the solution of the diffusion equation for semi-infinite media in which the initial concentration is zero and its surface is maintained at concentration unity, then the solution of the problem in which the surface is maintained at temperature <j>(t) is

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Akio Ogata

Papers

A solution of the differential equation of longitudinal dispersion in porous media