: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. In porous media these eddies exist Acknowledgments only to a slight extent, if at all, because of the This review of the Theory of Dispersion was initiated small size of the pores; but in media consisting of while the author was associated with the Phoenix large grains, such as gravel, eddies may occur to Research Office of the U.S. Geological Survey. The some degree. In the more common types of media, author gratefully acknowledges the detailed discussions however, the existence of low gradients tends to and suggestions of R. H. Brown, H. H. Cooper, H. E. limit or preclude turbulence. Skibitzke, C. V. Theis, and others in the review of the 4. Mixing due to structural controls the apparent manuscript. mixing mechanism due to large-scale structural Historical Development variations in the granular material. These varia Perhaps the first to recognize the occurrence of dis tions control the direction of movement of a given persion phenomena was Slichter (1905) in his investiga fluid particle; hence, if an average concentration tion of the use of salt as a tracer in ground-water flow. is taken along a given plane parallel or transverse He noted that at a point away from the source the salt to the direction of flow, a large-scale mixing is concentration increased gradually rather than abruptly, observed. Whether this type of mixing can be as would be predicted by Darcy's expression. That is, a treated in a manner analogous to that of mechan gradation of salt concentration in the fluid occurred as ical dispersion is questionable, and, in any event, the front moved past a given point of measurement. such a treatment would likely require a complete Slichter explained this by noting that in flow through description of changes in the properties of the capillary tubes the velocity of the fluid varies across porous medium. The analysis would probably the cross section of each tube and that, because the soil require the use of the Lagrangian concept in complex is composed of a great number of these tubes, which each fluid particle must be traced along its the sum of the deviation in each tube is likely to cause path of flow. the mixing that he observed. Following the same line 5. Adsorption a process which differs from the others of reasoning, lateral dispersion dispersion perpen in that the amount of mass transported depends dicular to the direction of the average flow was on the physicochemical interaction of the trans attributed to the repeated branching of the individual ported substance and the solid of the medium. flow tubes within the solid matrix. This explanation is The existence of an unbalanced force field causes most commonly used today to describe the mechanical migration of a liquid-borne contaminant from the component of the dispersion phenomenon. liquid to the solid surface. In some instances the Earlier papers dealing with two-fluid systems have contaminant is permanently fixed on the surface, been concerned with the economical^ important whereas in others it moves continuously from problem of salt-water invasion in coastal aquifers. liquid to solid, or vice versa. Hence, the measure The first analytical descriptions and investigations ment of the liquid phase concentration shows a were based on the supposition that the fluids could be dimunition of the concentration. Also, the con treated as though they were immiscible. However, taminant does not travel at the same rate as the in the middle and later 1930's, a zone of diffusion fluid. which could not be explained by immiscible-fluid At present, there are basically two related methods of theory, was noted. This diffusion zone prompted analysis that are useful in depicting dispersion. The recognition of the dispersion process as a possible cause first is a microscopic study of a fluid particle which for its occurrence. This phenomenon was treated employs the concepts of statistical analysis. The theoretically by Kitagawa in 1934 when he made second is a microscopic analysis of dispersion or trans theoretical and experimental studies of dispersion in a port of mass which is expressed in terms of a heuristic porous system. The physical model he chose was that law analogous to Fick's first law of diffusion. The of dispersion of a salt tracer emitted from a line source microscopic analysis, which predicts the position of the within a horizontal flow field. The experimental model, fluid particle at any given time, is completely dependent on the mathematical model chosen to represent the like most laboratory models, was composed of sand. medium. The most satisfactory model to date is the The theoretical analysis was based on the assumption representation based on the assumption of pore chan that, at some distance from the source, the concentra nels located with complete randomness and multiply tion distribution would conform with the normal connected at some end points. On the other hand, the probability curve. THEORY OF DISPERSION IN A GRANULAR MEDIUM 13
Before Kitagawa's treatment became widely ac to simplify the flow regime so that an analysis was cepted, refinements of the displacement theory for a possible. The assumption that the medium can be coastal aquifer came about largely through the efforts described limited the analysis to an isotropic medium. of Wentworth (1948). He postulated a "rinsing theory" Most of the articles published assumed that the based on the hypothesis that the pores of the medium field equation, similar to Fick's second law of diffusion act as mixing cells and that the ocean tides cause the in a moving medium, was a good first-order approxima alternate filling of the pore by salt and fresh water, tion. Although the assumption that the transport thereby creating a partial mixing of the fresh and salt equation consists of two modes the Darcy component water. Wentworth derived an approximate expression and the dispersion component avoided the require for numerical computation. ment that the individual flow paths bo known, it re Accelerated field investigations in the latter part of quired that the average direction and magnitude of the the 1950's and in the early 1960's indicated that in flow be defined and that the dispersion coefficient be stability of the salt water-fresh water front posed a determined experimentally. Because the flow through a threat to the water-supply systems in many coastal columnar model is readily defined, this type of analysis regions. The results of these studies were recorded in is especially useful in predicting the behavior of reactor various publications; for example, Cooper (1959) and columns or ion-exchange columns employed in the Kohout (1960, 1961) described the conditions in chemical industry. Dankwerts (1953) gave a compre Florida with supplementary information on the Hawai hensive discussion of the analysis of a columnar model ian Islands to provide evidence of a dispersion-induced and described the types of flow found in experiments circulation of sea water into the zone of diffusion and on packed columns. In most industrial applications, the back to the sea. flow rate in the exchange process is much higher than The results of these investigations gave impetus to that of ground water in nature; therefore, the analysis the development of analytical methods designed to may not be useful in the field of hydrology. The differ predict the dispersive effects of ocean tides. Results of ential equation which describes the field is valid, but other analytical studies have been published by Car the nature and magnitude of the dispersion coefficient rier (1958), Jong (1959), Henry (1960), and Bear and are altered. Todd (1960). The limitations of these investigations An analysis which employs a "perfect mixture lie in the incomplete knowledge of the flow system in model" described, for example, by Aris and Amundson heterogeneous coastal aquifers and in the necessary (1957), may also be used for the higher fluid velocities. oversimplification of such a system. Hence, the theo This mathematical model assumed that each pore cell retical results were correlated only with those from a within the porous medium constituted a mixing laboratory model designed to conform with the mathe chamber, within which the two fluids were perfectly matical model. mixed. The analysis differed from that of the low-flow It was argued that, because of the discrepancy be model in that the laboratory experimental data in tween the medium of the mathematical model and the dicate a concentration breakthrough curve which was real medium, the mathematical model cannot be ex more nearly approximated by the Poisson distribution. pected to depict what actually took place. Because of In the middle and later 1950's there were many this several investigators have preferred a statistical studies of dispersion in sand-filled columns, which were model to develop expressions for the dispersion co related to hydrology. Day (1956) investigated a salt efficient. Two statistical models were used: the com water replacement system and based his analysis on pletely disordered flow of Scheidegger (1954) and the the diffusion equation and Scheidegger's disordered randomly oriented flow channels of Jong (1958) and model to explain data obtained in the laboratory. Saffman (1959, 1960). Experiments on laboratory Rifai and others (1956) published what was perhaps models, however, indicated a variance in the completely one of the most comprehensive laboratory and theo disordered model which could not be reconciled with retical studies to that time. The theoretical part of the laboratory data. An important limitation of their study employed both the capillary-tube and the Scheidegger's model was that it did not include a disordered-medium models. As was expected, the two directionally variable coefficient of dispersion. The models gave identical results because of the applica assumption of random fluid paths did, however, provide bility of the central limit theorem (Beran, 1957) in the for the directionally variable coefficient, because of the disordered-medium model. In essence, the application lateral components of the flow velocity. The difficulties of the central limit theorem indicated that the diffusion that precluded a model description of the aquifer were equation was valid. Both laboratory and theoretical apparent, and a variety of postulates were introduced results indicated that the dispersion coefficient in the 14 FLUID MOVEMENT IN EARTH MATERIALS
direction of flow was nearly proportional to the first To date, both theoretical and experimental investiga power of the average velocity, which confirmed tions have been confined primarily to unidirectional flow. Scheidegger's prediction. Although the experimental data indicated that disper The results of studies by Rifai and others have been sion was directionally dependent, they were not suffi largely confirmed by many investigators; however, cient to enable a statement as to how the ratio of the there are some indications that the dispersion co magnitudes of the two components of the coefficient can efficient is proportional to a power of the velocity which be predicted in advance of an experiment. To depict is slightly greater than one. It will be shown in a subse transverse dispersion, Ogata (1961) developed a simpli quent section that, if the flow similitude is specified by fied mathematical model for steady flow, principally the Reynolds number and the mass transport by the to analyze experimental results given in a later report Schmidt number, this power is very near one. Studies (Ogata, 1964a; also Skibitzke, 1964). Harleman and of friction losses in consolidated media (Muskat, 1946), Rumer (1962) used a similar mathematical model to indicated that the Reynolds number, unless modified correlate results of their experiments. Many results to some extent, did not adequately specify the simili published in chemical journals will not be mentioned tude conditions. Efforts to extend the Reynolds number because they involved flow rates higher than those of concept to flow in other than isotropic unconsolidated interest here. Articles have appeared also in petroleum material have not been successful. journals, but most of these articles are reviews of Experimental evidence indicated that the magnitude previous works generally with additional experimental of the dispersion coefficient was not equal in all direc data to supplement results of others so that empirical tions. In fact, the very nature of the flow system makes expressions can be developed. it highly unlikely that the magnitudes would be the same. Hence, Jong (1958) and Saffman (1959, 1960) DERIVATION OF THE BASIC FIELD EQUATIONS developed a model of random flow paths to compute The equation derived in this section is a statement of the magnitude of the dispersion in directions both the law of conservation of mass. For simplicity, the parallel and transverse to the flow. The mere existence development will be in terms of Cartesian (x, y, z) of individual grains in porous material gave rise to coordinates. The field equations can then be obtained microscopic velocity components transverse to the readily in terms of other coordinate systems through direction of the average flow, thereby creating the known transformation. lateral movement of the contaminant and the velocity The description of the mechanism of mass transport dependency of the dispersion coefficient. through a complex porous medium necessitates the This concept was extended by Bear (1961) in a paper adoption of a fictitious model such that average condi on the tensor form of the dispersion coefficient. He tions can be more easily expressed. The first general verified the tensor characteristic of the coefficient assumption is that Darcy's law applies. This is corollary experimentally by observing the dispersion of a tracer to the assumption that a piston-type flow occurs. In from a point source. Although there was little doubt other words, the model is such that all particles of that the coefficient was a tensor, Bear's experiment did water move at the same rate of speed through the not correct for the spread due to the volume of the porous medium. Because of this assumption, which tracer injected. On the basis of experience in other neglects microscopic variations in fluid velocity, a fields, such as heat flow and molecular diffusion, it can refinement must be employed to approximate the real be assumed that the generalized transport coefficient system. For the dispersion mechanism, this refinement may be characterized as a tensor. involves a transport component that is analogous to The magnitude of the dispersion coefficient was Fick's first law. The justification for the use of this dependent not only on the fluid velocity but also on the type of expression is given by Taylor (1953) in his characteristics of the porous medium. Many experi study of laminar flow through pipes. ments reported in the literature show this to be true. The dispersion coefficient that appears hi the disper Among these were experiments by Orlob and Radha- sion component is assumed to be approximated by a krishna (1958), who explored the effect of entrapped constant, whereas it generally is to some extent de air in a granular material, and by Biggar and Nielsen pendent on the concentration and the type of dis (1960), who examined sands and other types of na solved or suspended substance. The development of turally occurring soils to determine the effect of in- the differential equation, however, is based on the homogeneity and variations of particle sizes. Also, assumption that the concentration is sufficiently small Skibitzke and Robinson (1963) obtained qualitative that the dispersion coefficient is independent of con laboratory results that demonstrated the importance of centration. No chemical reaction is assumed to occur heterogeneity of the medium. between the solid and the liquid phases. This supposes THEORY OF DISPERSION IN A GRANULAR MEDIUM
that within the fluid system no loss or addition of Similarly, expressions in the other two directions are matter can take place. written dC ISOTROPIC DISPERSION Fv=vC-JD To establish the mathematical statement of the conservation of mass, consider a cubic element hi the three-dimensional Cartesian space, as represented in figure 1. The two modes of transport of a fluid within From figure 1, the total amount of solute entering a cubic element hi space is F* dzdy+Fv dzdx+F, dxdy. The total amount leaving the cubic element is
dy\ dzdx bx
where the partial terms indicate the spatial change of the fluid mass hi the specified direction. Accordingly, the difference in the amount entering and leaving the cubic element is
Because there is no loss in the mass of the liquid, the difference between the amount entering and leaving FIGURE 1. Mass balance in a cubic element in space. must be equal to the amount of mass accumulated within the element. The rate of mass change is repre a porous medium have been described as (a) convection sented mathematically by the expression and (b) dispersion. In mathematical terms these modes for direction x (say) :are expressed: Transport by convection='wC'eLl The equating of the difference of outflow and inflow to the amount of dissolved substance accumulated within Transport by dispersion=D,,. ^-dC dA> the cubic element leads to the relationship
dF, dF, .^ where dA is an elemental cross-section area of the ~ (1) cubic element. Assuming that these two components may be superposed, the total amount of material Equation 1 is a mathematical statement of the law transported parallel to any given direction is obtained of conservation of mass under the conditions stipulated. by summing the convective and dispersive transports. Vectorially, the equation is written Thus, if Fx represents the total amount of mass per unit cross-sectional area transported in the x direction V-F=JdC/dt. per unit tune, then Substituting the expressions for Fx, Fv, and Ft and assuming that Dx=Dv=De=D= constant and since V-q=0, equation 1 becomes where / is the porosity of the medium. The negative (2) sign before the dispersive term indicates that the dt contaminant moves toward the zone of lower fluid j a a a a where-37is the operator-rr-fw-5 \-v-~ 1-10-5 Equation 2 concentration. dt r dt ox oy oz 370-837 O 70 2 16 FLUID MOVEMENT IN EAETH MATERIALS
applies to both diffusion and heat transfer within a that the dispersion coefficient is characterized by three moving medium. Note that the terms u, v, and w as independent components parallel to the chosen refer used in equation 2 now represent average velocities or ence axes. Under this assumption the dispersion tensor the Darcy velocity divided by the porosity. is a second-rank tensor consisting of nine components. Because the characteristics of the dispersion coeffi Jost (1960, p. 4), in discussing diffusion systems, cient other than its directional nature are not well argued that the existence of components other than understood, equation 2 was developed on the assump those making up the main diagonal of the matrix is tion that D was a constant. If D is a function of con doubtful. However, this argument cannot readily be centration, that is, if D=/(<7), equation 2 is written extended into the study of dispersion in granular media. Thus, for simplicity it is assumed that the dispersion coefficient is approximated by a tensor composed of, at most, nine components. Using the standard notation for second-order tensors, Equation 3 is nonlinear for D=/(<7) and can be solved the dispersion component of the transport equation only for simple expressions of j(O), generally by can be expressed numerical methods. These types of problems are en countered in the analysis of soil moisture and in some .Gi =DiJ i j=12tB. (4) other types of diffusive processes. Excellent discussions of the numerical methods applicable in obtaining solu tions for the one-dimensional equations are presented In other words, the three components of mass transport by Crank (1956) and also in many articles relating to are written soil-moisture movement.
ANISOTROPIC DISPERSION All experiments conducted in two dimensions indicate that the magnitudes of the dispersion coefficient de dC pend on the reference direction, with the larger value oriented in the direction parallel to the flow. To include this directional dependency in the equations requires and the dispersion tensor D tf can be represented by the that the coefficient D be represented by a more general matrix mathematical construct known as a tensor. The com plex properties of tensors are better described in texts on advanced mathematics and need not be included here. In generalizing the dispersion coefficient into a tensor, The advantage of the tensor notation is that it Bear (1961) assumed an analogy with a tensor repre provides a shorthand method of describing, in general, senting elastic deformation or tensor of the fourth the physical phenomena. Equation 4 describes the rank. In three dimensions this tensor has 81 compo most useful case of dispersion in a porous medium and nents, but because of symmetry the tensor has 21 indicates that the three components of mass transport nonzero components (Jeffreys, 1961, p. 78). Scheidegger are determined not only by the direction of the maxi (1961), in a verification of Bear's tensor, assumed dis mum concentration gradient but also by the gradients persion to be analogous to turbulence rather than to at right angles to this direction. In an anisotropic elasticity. Because turbulence is not symmetrical to medium, because the dispersion constant is dependent the same extent as elasticity, its tensor consists of on the fluid and medium properties, the elements 36 rather than 21 nonzero components. For unidirec forming the matrix can be both time and space de tional flow in an isotropic medium, however, the pendent. The dispersion tensor can be described by a results of both Bear's and Scheidegger's analyses show single number, as in isotropic dispersion, or by a that the dispersion coefficient is described completely matrix which reflects the complexity of the porous by longitudinal and transverse components in other medium itself. Only for isotropic media has the coeffi words, by a tensor composed of only two components. cient of dispersion been described successfully by The difficulties inherent in the application of the experimental methods. tensor to evaluate mass transport arise from difficulties If the medium is isotropic and the Darcy velocity is in measuring the various components. Thus, as hi heat constant, the transport equation in the x direction is flow or diffusion, it is generally necessary to -assume Fx=uC Qx; where Qx is the dispersion component. THEORY OF DISPERSION IN A GRANULAR MEDIUM 17
The substitution of the transport equation into the SOME USEFUL TRANSFORMATIONS continuity equation results in the field equation The diffusion, or heat-flow, equation where grad g<7=0 has been studied extensively and published in detail by Carslaw and Jaeger (1959), Crank (1956), and Jost (1960). For various types of bourdary- valued problems it is expedient to transform the field equation so that the convective terms do not appear explicitly. t d*C Two methods can be used to remove the explicit dxdy (5) convective terms for problems in which the boundary conditions involve media of infinite extent. These transformations require the introduction of a new Equation 5 describes the field distribution for a system coordinate system or a change in the independent of anisotropic mass transport. If Dn =DM=Da&= con variables. First consider the transformation of a stant and Z^=0 for i^j, equation 5 is identical with standard-rectilinear coordinate system into a new equation 2. This results from the fact that equation 2 system called moving coordinates. The dispersion equa was developed under the assumption that D was a tion including the directional properties of coefficient tensor of zero rank. D\a Equation 5 is known as a quadric, and by use of standard transformations it can be reduced to the form -£7-\-UdC . 3dC \-V. dC.3 \-W -~-=dC ^- 5-0 of equation 3. This transformation involves rotating dt dx dy dz dz2 oy (6) the coordinate axes so that the reference axes parallel the principal axes of dispersion. Recent experimental A transformation similar to one commonly used in the and analytical studies point to the important fact that study of wave equations is given by the expressions hi isotropic and homogeneous media the principal axes of dispersion are oriented parallel and transverse to £=x ut, r}=y vt, £=z wt, and r=t. the mean direction of regional flow. This indicates that Using the chain rule of partial differentiation, the for such media the mass transport system can be above expressions are substituted in equation 6. For defined by two characteristic dispersion components example, the transformation of dC/dx is given by that are specified when the mean direction of regional flow is known. Assuming that the principal axes can be defined, the dx dx dispersion tensor can be transformed so that only the elements of the major diagonal remain, all others being Two similar expressions for rj and £ coordinates can be zero. The matrix representation of the tensor then obtained in the same manner. The tune differentiation becomes follows the same rule and is given by 'DX Q 0 dC_dCd![ ,dCdr_dC_ dC 0 Dy 0 0 0 D. dt~d^drdrdt dr U dl-' A similar method is utilized to determine the second In unidirectional flow, symmetry about the mean flow differentiation with respect to the x, y, and z coordi line exists so that DV=DZ. nates. Substitution in equation 6 reduces it to the form The possible representation of the dispersion coeffi cient in the manner depicted by the matrix is significant dC (7) because of the internal complexity of the porous en 6T Q 2 vironment. Actually, the dispersion of a fluid flowing through a homogeneous porous medium is the result of which is analogous to the heat conduction or diffusion the averaging process necessary to describe the flow equation. The transformation of coordinates did not of ground water. In essence, this signifies that provided involve rotation of the coordinate axes; hence, the all the possible microscopic changes in the magnitude dispersion coefficient D< is still oriented in the original and the direction of the pore channels can be expressed x, y, and z or the new £, 77, and f directions. quantitatively, the components of the dispersion tensor Introduction of the above transformation in effect can be computed by statistical means. creates a system based on coordinates that move in 18 FLUID MOVEMENT IN EARTH MATERIALS space at a rate dependent on the magnitudes of the mation of equation 6 must be utilized to circumvent average fluid-flow velocities. Physically, this may be the specification of concentration at infinity. pictured as an observer moving along with a velocity ~g Usually, mathematical solutions are readily found while watching the process of dispersion. The choice of for the dispersion equation describing isotropic condi this new coordinate system is possible because the tions; that is, for Dx=Dy=Dg=D. Thus, introducing transport due to convection is a conservative process; the independent variable hence, stopping the fluid movement to simplify the analysis of the dispersion process is permissible. *'=*A/ . 2/'=2/VrT> 2'=ZV7T However, the transformation does pose difficulties as, for example, hi problems where certain conditions are specified at an initial point. In such problems the and substituting these independent variables in equa transformation tends to increase the difficulty of tion 6 gives inserting the specified conditions in the solution of the differential equation. To illustrate, consider the problem where the concentration C is specified at the point where x=y=z=Q; say, C(Q,Q,Q,f) = C0, where <70 is a constant. It is noted that when x, y, z=Q; £, TJ, f= qt, where q=u, v, w, respectively. The problem now becomes one of describing the concentration for a system in which the concentration is specified at points hi space depend ing on the time. This tends to increase the complexity i, j, fc=unit vectors hi the x, y, and z directions. To of the mathematical analysis, unless there is sym reduce this relation to the form of equations 7 and 8, metrical distribution of concentration about the source the transformations previously discussed can be used. of fluid at 3=0. DISPERSION EQUATION IN CYLINDRICAL AND The second transformation method involves the SPHERICAL COORDINATES substitution of a new dependent variable. In this instance, it is assumed that the solution of the problem The field equation as previously written is in rectan can be expressed functionally by gular or Cartesian coordinates. For some problems the cylindrical or the spherical coordinates may be more u suitable for analysis. The fundamental equations governing the dispersion process are readily derived in much the same manner as before by stating the mass "2A_w_ balance within a reference space as defined by the given coordinate system. Similarly, coordinate transforma tions are readily accomplished because the inter where T(x, y, z, t) is an unspecified function. Assuming relationships between any given coordinate systems that, generally, the solution of the diffusion equation is are known. of this nature, the next step is to determine the func In cylindrical coordinates the isotropic dispersion tion T(x, y, z, t). The solution must satisfy equation 6; equation is hence, substitution gives the expression 32r dc BO ac do ~*\7=:L'f (8) which is the diffusion equation. The boundary condi tions, as given in the preceding paragraph, must be And in spherical coordinates the equation is written as a time-dependent variable at a specified point in space. Because F0 (say) is specified at a given dC . dC. /l\atf, / 1 \dC point in space, an analysis using equation 8 introduces de no further difficulties. In general, the transformations leading to equations 7 and 8 work well for dispersion in media of infinite ex tent. In the event of a finite boundary the concentra (10) tion is specified along x, y} z=L some other transfor THEORY OF DISPERSION IN A GRANULAR MEDIUM 19
FIELD EQUATION FOR TWO-FLUID SYSTEMS Land surface In the development of the dispersion equation the density and viscosity of the contaminant fluid was assumed to be the same as the surrounding original or uncontaminated fluid. In cases where high concentra Sea level tions of a contaminant occur, as in an aquifer subject to salt-water encroachment in coastal regions, this Fresh water assumption is no longer valid. In such an environment both the viscosity and the specific weight of the con taminated fluid change. Addition of the gravity force is now necessary for analyzing the concentration dis Zone of diffusion tribution along the interface between the contaminated and uncontaminated fluids. The gravity force exerts so Interface large an influence on the flow field that the position of a particle of contaminated fluid at any given time is largely predetermined by this force. That is, the description of the velocity field is dependent on the FIGURE 2. Schematic diagram of coastal aquifer system. dynamic equilibrium position of the interface between the two fluids. Thus, definition of the dispersion process in a two-fluid system requires complete knowledge of both the fluid properties and the flow field. When no chemical reaction between the fluids and where s and n denote the directions parallel and normal the solids take place, the dispersion equation is the to the interface, respectively, as shown in figure 2. The same as previously described with the provision that components of flow in the specified reference frame velocity is space dependent. Hence, the field equation (eq 12) are is written _- ds y ds dC (11) (13) Equation 11 indicates that the field equation is com where k= - Accordingly, substituting the stream pletely specified provided the flux g is determined. pg Although conditions depart somewhat from those function in equation 13, the continuity equation may studied by Darcy, it is assumed that the flow equation be written may be generalized such that
2=- K(V#- f*, (12) The terms in the brackets are included to show the The schematic diagram of a two-fluid model under effect of viscosity M which is dependent on the salt dynamic equilibrium conditions is shown in figure 2. concentration in the fluid. The right side of the equation The description of the transport of salt requires the describes the effects of changes in density and viscosity. complete solution of equation 11 subjected to equation Equation 14 is analogous in form to a vorticity equa 12. Inasmuch as equation 11 is nonlinear it is virtually tion that frequently appears in the potential theory in unsolvable without the use of numerical methods or hydrodynamics. This vorticity equation has been various simplifications. For example, Jong (1959) studied in classical hydrodynamics, and methods of defined the flow system in terms of curvilinear coordi solution for various boundary conditions are described nates oriented with respect to the assumed position of in a number of texts; for example, Milne-Thomson the interface, thereby enabling the use of a two- (1961, p. 564). Equation 14 is also substantially the dimensional system. Paralleling the classical hydro same equation derived by Henry (1960) in evaluating dynamics, a stream function is introduced into the the two-fluid system. Henry used the same basic flow analysis. The pertinent components of the stream equation, expressed the specific weight 7 in terms of the function are defined as salt concentration (7, and assumed that the viscosity no FLUID MOVEMENT IN EARTH MATERIALS
change due to that concentration is small. Hence, the specified, provided * is specified by equation 15. Henry equation he derived is in the form (1960) has developed solutions for two special instances of the transport equation and has evaluated them numerically. Q dz' (15) ADSORPTION where ki=k ( ), y0 is the aquifer thickness, Q is the In the discussion and development of dispersion \ Po / aquifer discharge, C' is the ratio between the concen equations to this point, no means of mass transport has tration of salts in the aquifer water and the sea water been considered other than mechanical dispersion and C/CB, and xf is the dimensionless length x/y0. Equations convective transport. In the isotropic systems usually 14 and 15 differ only in that the first specifies possible studied in the laboratory, the spread of tracers in the effects of the viscosity changes. fluid is generally observed to be small, which indicates that the magnitude of the dispersive term is small. All To simplify the analysis, Jong (1959) dealt with the vorticity equation (eq 14), excluding the terms in soil complexes display an additional component of mass brackets since the change in viscosity is small. He transfer generally termed "adsorption." Adsorption is showed that the analysis can be carried out by replacing a chemical reaction in which mass transfer occurs by real fluids with fictitious fluids and by introducing actual removal of the dissolved substance from the singularities parallel to the interface of the system. liquid phase owing to mutual attraction between the The characteristics of the singularities are determined substance and the solid phase of the porous medium. by the actual observed velocity distribution; in other Generally speaking, in natural flow systems this process is of major magnitude with the fate of any contaminant words, the singularities are chosen so that the original introduced into the soil or the ground-water system velocity distribution in the two fluids is reproduced. The dependent to a large extent on the capacity of the solid method, thus, makes possible computation of the posi matrix material to adsorb the dissolved component. tion of the interface between two fluids of different This is recognized by most workers in the field of radio- densities. Because the equations are derived for steady- isotope disposal; however, the major research is con state conditions, the final position of the interface can fined to surface chemistry which takes into account the be computed for any given flow conditions. Equation 15 describes the flow conditions, in terms microscopic aspects of the adsorption process. Vermeulen and Hiester (1952), in a macroscopic in of a stream function, for the change in specific weight vestigation of ion-exchange columns, specified that the of the fluid with the change in salt content. However, ratio of the tracer velocity to the main body of fluid the transport of the salt is also dependent on mechanical velocity under equilibrium conditions may be expressed dispersion; thus, equation 15 specifies only a part of the problem. In other words, equation 1.5 describes the by flow field within the transition zone containing fluids of UT 1 iLf Sp. U different densities, and the transport of mass within where this zone needs to be described by the dispersion S= trace component adsorbed per gram of solid equation. The solution of the vorticity equation given matrix, by Jong makes it possible to compute the location of p=bulk density of solid matrix, and the interface for any specified time, from which the UT= velocity of tracer. stream function can be determined for the whole flow This expression was established empirically and has field. When unsteady conditions prevail, these solutions been used to some extent to determine rate of move may still be used as long as the changes in the flow ment of radioactive isotopes. field are small. The transport system, however, must To develop the transport equation which includes be described subsequent to description of flow in order adsorption, it is assumed that the rate of adsorption to determine the dispersion effects on the frontal zone. is the third component in the dispersion equation. This vSince by definition q*=6V/dy and qv= 6V/dx, sub assumption simply implies that the chemical reaction stitution into the two-dimensional steady-state trans or the rate of adsorption can be superimposed on the port or dispersion equation leads to dispersion and convection system previously described by equation 3. The adsorption component in the physi (16) Q dy' 6V cal model can be pictured as a mathematical sink or source with its strength dependent on the difference of Equation 16 is thus an approximation of the steady- concentration of the liquid phase and the solid phase. state mass transport process which can be completely A simplified model is illustrated in figure 3. Here, it is THEORY OF DISPERSION IN A GRANULAR MEDIUM 111
Total inflow esses in fluids flowing in porous substances. Note that, when the term mS is much smaller than C, the expres sion reduces to a form similar to Langmuir's equilibrium isotherm for low concentrations or
Liquid phase (19)
Solid dx Equation 19 is applicable to a first-order irreversible phase reaction wherein the soil particle may be thought to act as a mathematical sink. Extending the expression given as equation 18 a little further, it may be written
Convection and dispersion components where n< 1. Note that for the special case of n= 1, this (From Bird and others, 1960) is equation 18. The need to determine three unknown parameters makes the above equation difficult to FIGURE 3. Schematic diagram of transport In porous medium. evaluate even under the rather idealized conditions encountered in a laboratory. Again assuming that assumed that the adsorption component is in one direc mS^C, the equation may be written in the simpler tion only. form Superposition of the three components adsorption, %j-=bCn > (20) dispersion, and convection allows the writing of the d\ continuity or mass-balance equation as follows: which is virtually the expression for Freundlich's (17) equilibrium isotherm. The three types of reaction rates represented by equations 19-21 are discussed by Crank (1956, p. 121) where S is defined simply as the concentration of the in reference to a diffusion system involving various solid phase. The function 8 has the same dimension boundary conditions. The solution of the simultaneous as C. equations represented by equations 3 (for a constant The analysis of dispersion in adsorbing media that D), 18, and 19 can be determined by operational is, the use of equation 17 depends on the representa methods. A numerical method, however, must be tion of S in terms of C, where it is assumed that the used to solve equation 20. When equation 19 is used relationship £)$/()<=/(
most cases the fate of the contaminant in a ground- Numerous solutions available in the published lit water reservoir is largely dependent on the amount erature are presented in the following pages. Whenever transferred between the liquid and solid phases. possible, a reference is given so that the complete Perhaps the most realistic macroscopic appraisal development can be consulted by those who are available is in the results of reaction-tower experiments. interested. In addition a limited number of functions, The analytical difficulties that appear whenever the which the author arranged to have computed, are adsorption isotherm is nonlinear are formidable, presented in graphical form as appropriate. however, not insurmountable. The first two examples considered are for dispersion within.a semi-infinite medium in a unidirectional flow MATHEMATICAL TREATMENT OF THE field. It is postulated that the concentration is specified DISPERSION EQUATION at z=0 and at the initial time f=0. The system is The analytical treatment of dispersion phenomena defined schematically in figure 4. For the specified field within a porous matrix, based on the preceding theoreti conditions, the field equation (eq 6), omitting the cal development, presupposes that the dispersion subscript x, becomes simply coefficient is a known parameter. Thus, the full defini n 620 dC.dC tion of the physical system depends on accumulation D 5-5-='^ 5 T-5T' (21) dx2 dx ot of data concerning this specific parameter. It will become apparent in the following discussion that only The subscript x has been dropped in equation 21, since the simpler dispersion systems are treated primarily dispersion occurs only in the x direction. because of the need for laboratory confirmation. The discussion is, therefore, not intended to be a complete Example 1. The plane a*=0 is maintained at a con presentation of all known solutions of the differential centration C= CQ. equation. Also, this section presents methodology The boundary conditions are given by considered useful in the analysis of the dispersion equation. 0z0=0, x>Q The analytical methods, namely, Laplace transforms, the product solution, and the point source solution are 0(00,0=0, the principal tools used in the study of the differential equation. These techniques are discussed to some The method best suited is the Laplace transfonna- extent in the work of Carslaw and Jaeger (1959) and in many texts in advanced calculus. All the examples to follow are limited to flow in a semi-infinite medium. The use of more complex bound t aries adds little to the study of the fundamental characteristics of the mass transport system because of the complexity of the final expression. The principal u average areal flow objective is to investigate initially the nature of dis persion. Hence, the convective term div ~^C will be assumed to be given by u dC/dx-, in other words, the flow field is assumed constant and unidirectional. Space dependency of the dispersion coefficient is not considered; variation is limited to directional depend ency only. In systems involving an adsorptive solid medium, the problem becomes highly complex and thus no analysis has yet been attempted for anything more complex than one-dimensional mass transport. The examples given are intended only as samples of solutions available and may not be physically realistic. This is especially true in examples 3 and 4 where a system involving a disk source was chosen to depict a two-dimensional case. Example 3 actually may not be physically realistic; however, it may possibly occur when the velocity u is small. That is, example 3 implies C specified at x = 0 that a concentration gradient occurs at z=0. FIGURE 4. Definition sketch of one-dimensional system. THEORY OF DISPERSION IN A GRANULAR MEDIUM 113
tion; substitution of the transform leads to the Consider now, the system that is described within a subsidiary system given by the equation two-dimensional frame of reference because of sym metry about a given axis. The definition plane is given dC. - in figure 4. Because of the mathematical complexities, this system is illustrated using circular surface sources maintained at constant strength and constant concen The solution of this subsidiary equation subjected tration. The results are sufficiently complex that in a to the restraints due to the boundary condition is discussion on evaluation of the dispersion coefficient ux (p. 130-132) an approximation is used to simplify the computations. Assuming again a unidirectional ground-water flow The inverse transform as given by Erdelyi and others field with parameters Dx and u oriented in the x direc (1954,p.246) is readily determined as tion and DT in the r direction normal to x, the field equation for the radially symmetrical system becomes (21a)/ot N
Example 2. The region z<0, initially at concentration Equation 22 is equation 9 when symmetry exist about The boundary conditions of interest are specified the axis x; hi other words, the variation in concentra by tion with respect to the angle 6 is zero. C(x, 0) = C0, z<0, C(x, 0)=0,x Example 3. A disk source, 0 Again, various analytical methods are available; Fo at x=Q, Q = erfc (21b) where £=x ut. For the special case DT =DZ=D and u=0, the expression is the same as equation 9, page where £=x ut. 260, in Carslaw and Jaeger (1959). To obtain an expression for the continuous disk The graphical representation of the solutions given source, the above equation is integrated between the for examples 1 and 2 appears in figures 5 and 6. Note limits 0 to some time t. After some algebraic manipu that because a constant concentration is maintained at lation the integral equation is written x==0, example 1 exhibits an asymmetrical concentra tion distribution about x ut. However, it has been C^ f shown that the conditions of flow through porous f o Un Jo media are such that the second term in equation 21 a is negligible (Ogata and Banks, 1961). Hence, the two where systems exhibit no difference at some distance from the u ux 4DXDT source. 370-837 O .70 3 114 FLUID MOVEMENT IN EARTH MATERIALS 0.001 FIGURE 5. Concentration distribution; plane source at i=0 maintained at constant concentration. THEORY OF DISPERSION IN A GRANULAR MEDIUM 115 -2.0 (x-ut) FIQUEE 6. Concentration distribution; region x<0 initially at constant concentration. The above expression may be written in two The steady-state solution, although simpler, cannot alternative ways, or be integrated and must be evaluated numerically. Along the axis r=0, the integral equation can be C expressed in terms of an elementary function or, uC , where -\dr l=_ *1/21 Jofe Example 4- A disk surface source 0 -exp C(Q, r)=0, r>a C(a>,r)=0, r>0. C(x, co)=0, z>0. For the steady-state condition, that is, letting t >°° in the above expression, the equation obtained To analyze this problem, it is assumed that the is solution is represented by the product of two func tions of single variables, that is, C=R(r)X(x), 116 FLUID MOVEMENT IN EARTH MATERIALS where R(r) is a function of r only, and X(x) is a A similar solution for DT=DX D and u=0 is function of x only. Substitution for C in equation 22, given by Carslaw and Jaeger (1959, p. 214). noting that for steady state dC/dt=Q, the partial The steady-state solutions to examples 3 and 4 are differential equation is reduced to the subsidiary almost identical, except for the factor l/yT+VX2- The equations two solutions, presented as examples 3 and 4, are repre sented .graphically in figures 7 and 8. The numerical computations for the integrals in examples 3 and 4 were accomplished by integrating under a curve with a polar planimeter. In both examples the function could not be integrated into a readily calculable form. However, the where f2 is an unspecified constant. integrals can be evaluated by numerical methods for The solutions of the subsidiary equation are readily various values of rj=r/a, which may be used in evalu obtained. The solutions are ating experimental data. The previous examples considered dispersion in flow systems described by a single-space variable and by )=_42 exp [2a two-space variables. Examples 3 and 4 considered a radially symmetrical system with flow occurring along Because of the condition specifying C be zero as the cylindrical axis. Consider now, a line sink or a line x »oo, the positive sign before the radical in the source at r=0 in a cylindrical reference frame. This is second equation is discarded. The solution is thus generally termed "a radial flow system." The definition of the geometrical configuration is presented in figure 9, =r exp and the applicable differential equation follows. Writing the expression for the conservation of mass, The method of determining ^l(f) is described fully with reference to the definition sketch (fig. 9), the by Sneddon (1951, p. 59), the result is differential equation is d /n ac\.z>ao do dc ,OQ, -£=a f° exp [2a (l- a-- ( D --5- - --=-' (23) ko Jo dr\ or/ or or dt i .i i i i i i i___ i ___i ___i ___i _____i___ i 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 FIQUBE 7. Steady-state concentration distribution; circular surface source of constant strength. THEORY OF DISPERSION IN A GRANULAR MEDIUM 117 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 FIGURE 8. Steady-state concentration distribution; circular surface source maintained at constant concentration. Here, the concentration and the dispersion coefficient inversely proportional to the distance from the source do not vary with the angle 8 nor with the distance z at f=0, this same relationship is assumed to hold for which is normal to r. Equation 23 is written to allow for the dispersion coefficient. a possible space dependency of the dispersion coefficient The expression for fluid velocity may be formulated D in the direction r. The basis of this space dependency provided the quantity of flow or flux is specified at some comes from experimental and statistical findings that value of r=r0) generally near the source. Since flow is D is directly proportional to the velocity of the fluid assumed steady, continuity requires that the velocity stream. Since it is readily seen that the velocity is at any point is given by the expression u u<,rQfr, where u0 is the average velocity at r r0. Substituting the variable velocity into equation 23 leads to the expression _ (24) dr dr~ dt Experimental evidence indicates that the dispersion coefficient D in a homogeneous medium varies very nearly with the first power of velocity; thus, D can be written where Dm is a constant related to the solid and liquid properties. Substituting the preceding expression into equation 24 gives D/dC l_dC} FIGURE 9. Definition sketch of radial flow. 6V V r2 dr DQ dt' 118 FLUID MOVEMENT IN EARTH MATERIALS where a=u0r0. Accordingly, collecting the terms leads and denominator. Shamir and Harleman (1967) have to the equation obtained an approximate solution of equation 25, numerically. Experimental analysis of this problem _a_ dC=j^ dC (25) has been done by Lau, Kaufman, and Todd (1959); 6V2 D0 6V DQ dt' however, certain fundamental simplifications were re quired for the comparison of experimental with theo Example 5. Dispersion in a fluid flowing radially from retical results. A result of the finite difference computa a line source at r=0. tion for the value D0/u0r0=l'XlQ~2 is presented in The boundary conditions to be satisfied are figure 10. The solutions for dispersion hi one-dimensional, two-dimensional, and the special case of radial-flow tf(r,0)=0, r>rQ systems have been presented. Continuation of this #(00, 0=0, t>0. analytical trend would suggest that dispersion in a three-dimensional system should be considered next. The method of analysis because of convenience and However, the relative difficulties in coping with a simplicity in determining the coefficient of the solu three-dimensional ground-water system renders the tion is the Laplace transform. Equation 25 trans attempt hardly worthwhile. Thus, it seems more useful forms to in the final two examples of dispersion without chemical reaction to consider approximate solutions for two- _ == directional dispersion. In the two-dimensional field dr2 D0 dr D0 ' equations (eq 22) it is assumed that, if the term Dxd?C/dx2 is small compared to the other terms, the sys The solution of the transformed problem is tem can be described as consisting of flow and dispersion that are orthogonal to each other. This is signified in the definition sketch presented as figure 11. For dispersion process from a disk source, the field equation is £ (26) where z=r a/4D0p, z0=r0 a/4D0p and K^(x) is the modified Bessel function of the second kind of For the plane source hi which y>0 initially at a con order %. stant concentration, the appropriate field equation The inversion of the Laplace transformed system is has been obtained by the use of the hi version theorem dC dCdx' (Ogata, 1958, p. 88). The final expression is given by (27) Equation 27 is identical with the diffusion through a wall, and equation 26 is identical with diffusion from a cylinder moving hi the direction x at the rate u. where Example 6. Approximate radially symmetrical model of lateral dispersion. The approximation of lateral dispersion, equation 27, in which the justification of its use is given, was treated by Ogata (1961). In previous discussion on transformation of the coordinates, it was pointed out that the coordinate transformation £=x utm&y be used to eliminate the convective term. Equation and JK(X), YX(X) are Bessel functions of the first and 26 may thus be written second kind of order %, respectively. The solution hi its present form has not been evalu ated; however, a finite difference scheme has been ap ^VoWrdr/ dt plied to the differential equation. The integral in its present form is cumbersome to evaluate because of the Although C depends on x, this need not be specified appearance of the Bessel function in both numerator in the boundary conditions since x does not appear THEORY OF DISPERSION IN A GRANULAR MEDIUM 119 1.0 0.9 EXPLANATION 1 (/ -/ )= 1.75 feet 0.8 2 (r-y0 )=2.75feet 3 (f-f ) = 3.50 feet 4 ( f-r0) = 4.00 feet 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.68 0.76 0.84 0.92 1.00 1.08 1.16 1.24 FIGURE 10. Concentration distribution for radial-flow system. explicitly in the differential equation. Hence, the Particular solutions valid for the region of interest boundary conditions are stated with the specified constraints are "1 -Q fl ^(qa^gr) PL C(r,0)=0,r>a, c=^ /i(go)g0(gr) (d<7/dr)r=0=0, t>o. p The inverse of the above expression is given by The solution of the above differential equation for Carslaw and Jaeger (1959, p. 346). The solution for the specified boundary condition is obtained by all values of r is given by Laplace transformation. The subsidiary equations for r>a and ra' ' thus computed from the same expression. - C where \=C-\ - and g*=p/D. The two regions are Example 7. Approximate steady-state model of lateral described by identical equations when X is used. dispersion in a unidirectional flow field. 120 FLUID MOVEMENT IN EARTH MATERIALS r or y The application of Laplace transform- in equation 27 leads to the subsidiary equation Fr = Drb C/6r where C'=C/C0. The solution of this subsidiary equation is readily determined as Hence, ushig the complex inversion theorem, the solution obtained is, FIGURE 11. Definition sketch of system depicting no dispersion in X direction. The model considered is dispersion in a fluid moving in the direction x with the plane ?/=0 initially at a The expression for example 6 has been computed constant concentration C0. This special case was a'u numerically for various values of The expression treated by Harleman and Rumer (1962, p. 24). 4DTx Equation 27 is the appropriate differential equation for example 7, on the other hand, is a standard tabu and the boundary conditions are lated function; hence, numerical values are available. These two expressions are presented graphically as figures 12 and 13. Because of the simpler nature of ,y)=0, <=°<2/<0 these expressions, as compared with those obtained in examples 3 and 4, their use, whenever applicable, is =0, x>Q- favored for computing-numerical values of the disper sion coefficient for any given experiment. Figure 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1-1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FIGURE 12. Approximate concentration profile due to lateral dispersion; radially symmetrical model. THEORY OF DISPERSION IN A GRANULAR MEDIUM 121 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1,5 1.6 1.7 FIGURE 13. Approximate concentration profile due to lateral dispersion; two-dimensional model. shows the curves for examples 3, 4, and 6 for two In the examples to follow, equations 28 and 29, with values of 4Drx/ua?. As expected for higher values of suitable modifications, are used to describe the concen iDrX/ua2, the difference between the three functions tration distribution within an adsorbing medium. The become appreciable. For a first approximation, how flow occurs in a semi-infinite medium; hence, for all ever, the use of the solution obtained in examples 6 following examples the boundary conditions are or 7 would be simpler and of sufficient accuracy. The following examples of mathematical analysis of C(Q,t) =, *>0, dispersion consider an adsorbing medium. The system <7(z,0) =0, x>0, will be limited to a semi-infinite medium. The adsorp tion rate is assumed to be linear or more specifically 0(00, £)=0, t>Q, of the form dS/dt=b(C mS). For flow in a medium S(z,0) =0, z>0. with finite boundaries, the reader is referred to several examples given by Crank (1956) as the complete solu Example 8. Dispersion in an adsorbing medium; tions are too complex to be reproduced here. adsorption rate given by dS/dt=bC, (m=O). When mass transfer due to adsorption is superimposed The solution of simultaneous equations 28 and 29 on the convective and dispersive transport, the con (with m=0) for the prescribed conditions can be tinuity consideration in a one-dimensional flow system determined by assuming that it may be written in the leads to the differential equation form (28) C(x,t)=T(x,t) exp - The tune rate of change in the solid-phase concentration Substituting the above expression into equation 33 was previously indicated to be approximated by the leads to an equation with T as the dependent variable, linear equation or ~=b(C-mS). (29) 122 FLUID MOVEMENT IN EARTH MATERIALS exp(Dr x\2/ua2) F(\)d\ 4Dr x/ua =2.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 FIGURE 14. Comparison of examples 3, 4, and 6. For the function T, the condition at z=0 is given by The differential equation for the specified condition is reduced to udC/dx dS/dt. Using the Laplace transform, the subsidiary equations obtained are The general solution of the equation above is given u =o exp The solution satisfying these simultaneous equations is TT=-C 1 exp Lr~ bx Co p \p+km/_\ The integral above is a well-tabulated function called the error function, or The inverse of the above expression is available in tables of the inverse transform. The solution ob tained by Hougen and Marshall (1947) is -= exp [(l-M)ux/2D] erfc l- fb 7~-=lQ exp ( bmt) I exp exp [(l+M)ux/2D] erfc Jo where Example 10. Dispersion in an adsorbing medium; adsorption rate given by dS/dt=b(C mS), with dS/dt»dC/dt. Example 9. Dispersion in an adsorbing medium; The postulated conditions reflect primarily the adsorption rate given by dS/dt=k(C mS) with mathematical viewpoint of looking toward a simpli ^0 and dC/dt«dS/dt. fied equation. Again, the use of the Laplace integral THEORY OF DISPERSION IN A GRANULAR MEDIUM 123 transform reduces the partial differential equations Note that the preceding solution is the same as that to the subsidiary expressions for example 9 when (t x/u) is replaced by t. dC Example 12. Dispersion in an adsorbing medium; adsorption rate given by dS/dt=b(C-mS). This final example employs equations 29 and 30. p'S=b(C-mN}. The subsidiary equations, obtained by use of the Laplace transform, are The solution of these simultaneous equations, obtained by classical methods, is 6XP pS=b(C-mN). The inverse is obtained by standard means and is The particular solution of the transformed equations described by Ogata (1964b). The expression obtained for the specified boundary conditions is given by the is the expression, O 2i& I exp (£2-< Co -y/TT JO exp -£- The process of inversion is described by Ogata \TftJ 0 (19646), and the final expression is written where a=ux/2~D, p=Dba2/u2, and 3(x,y) is the -= |_,_ exp (-r- Goldstein J function which has been defined in Jl example 3. 2V/W Example 11. Dispersion in an adsorbing medium; where c=6 72/£2, T=bm(t-y2/?), a=ux/2D, y2=x2/4D adsorption rate given by dS/dt=b(C mS), with and J(c, T) is the Goldstein J function as defined previously. The equation considered is again a vjariation of Note that this solution is identical with the solution the complete field equations 28 and 29 with the in example 10 when the term (t y2/^2) is replaced by t. condition that dispersion in the direction of flow is Because the development of the solutions for exam small. The method of analysis again involves appli ples 10-12 is relatively recent, their numerical evalua cation of the Laplace integral transform. The tion has not been completed. The solution for example subsidiary equations for this case are 8 can be readily computed, whereas the solution for 9 has been used in analyzing reactor columns. Figures 15 and 16 are graphical representations of the ex u pressions for these two examples. The number of conditions, other than the one- pS=b(C-mS). dimensional system, that have been investigated are relatively few. Because of the similarity between equa The solution of these simultaneous equations tions describing dispersion in ground water and those can be determined by elementary means, and the describing diffusion and heat-conduction systems, resulting expression is numerous analytical solutions are available. And with the increased use of digital computers, a variety of more JLl f f» complex mathematical problems can now be solved Co p L (p+bm) numerically. The solution as given by Ogata (1964b) is NATURE OF THE DISPERSION COEFFICIENT a r / r\~\ rte/w r / 7 r\ I In the analytical development of the macroscopic £=l-exp -bmtt--) e-*I0\ Jbm(t--)\ \d\ Co *L \ u/JJo UL\ \ uj J theory of dispersion it is assumed that the characteristics of the dispersion coefficient are known, and that it is virtually a constant. In reality, the nature of this coefficient is unknown and is the subject of considerable 124 FLUID MOVEMENT IN EARTH MATERIALS 0.5 10.0 FIOUBE 15. Concentration profile of dispersion in an adsorbing medium with dS/dt=bC. investigation. Extension of the theory describing dis realized unless a porous matrix can be described in its persion in a porous medium, or the ability to use generality, including parameters such as the grain dispersion equations for systems other than isotropy, is distribution and degree of cementation. This need is entirely dependent on the evaluation of this coefficient reflected in all problems involving the granular medium by either analytical or laboratory means. Further, this and in some quarters gave rise to such correction terms critical evaluation depends entirely on the ability to as "tortuosity." However, the only route opened at describe analytically the physical properties of the present is to assume that the Reynolds number estab porous medium. This stage of investigation has not been lishes flow similitude and the Schmidt number estab reached as yet; thus, the discussion to follow is limited lishes similitude in the diffusion. Then, the correlation to evaluation of the dispersion process in isotropic of these two similarity criteria makes it possible to write material. an analytical expression for the functional relationship Many experimental and analytical papers published between the flow and diffusion processes. in journals show the dependency of the magnitude of Characteristics of the dispersion coefficient for con the dispersion coefficient on the flow rate and the effec ditions encountered in the laboratory are generally tive gram size diameter of the porous medium. This studied through statistical evaluation, or by correlation dependency comes about because, provided the sand is of the experimental data with results of a mathematical uniform, the geometric characteristics are described by model. The statistical model is important in that to a a single parameter the sand-grain diameter. But, limited extent it does point out the parameters that whenever the materials that make up the matrix depart control the magnitude of the coefficient. Its weakness from isotropic conditions, the representation of the lies in the need to reduce the porous matrix to an ex medium by a so-called characteristic length becomes a tremely simplified structure in order to render the difficult task, as experienced in attempts to generalize problem mathematically tractable. The use of the the concept of a friction factor in fluid flow through macroscopic model is, however, a little more difficult porous media (Carman, 1956). Generalization of the since the nature of the coefficient is only specified as a results of laboratory and analytical studies cannot be constant. Thus, any correlation of these coefficients with THEORY OF DISPERSION IN A GRANULAR MEDIUM 125 FIQTIBK 16.- Concentration profile of dispersion in an adsorbing medium with dS/df ^(C-mS), and T^'r various parameters must be accomplished by methods to the actual porous matrix. The basic theory of the of regression, unless similarity can be established be statistical analysis is described in four papers published tween the various flow conditions in the laboratory and in separate scientific journals. These papers were written the field. by Scheidegger (1954), Jong (1958), and Saffman (1959, Numerous experiments indicate that the use of the 1960). Saffman's and Jong's models are similar in Reynolds number and Schmidt number are useful in nature; however, Scheidegger, who introduced the con establishing results comparable to those obtained by cept of statistical hydrodynamics, chose a mathe statistical analysis, provided the porous matrix is iso- matical model that assumed random motion of the fluid tropic. However, in natural soils, or when air entrap particles. ment occurs in flows, the dispersion coefficient cannot The results of both Jong and Saffman are reproduced. be readily correlated to an effective diameter or a The porous medium model of Jong is reproduced as characteristic length. Thus, the expression for the D is figure 17. The pore channels are assumed to be capillary written Dt=iDmu, where Dm is an experimental constant tubes multiply connected at some end points and uni that depends on characteristics of the porous medium. formly distributed in all directions in space. The medium is assumed to be made up of spherical particles that form polyhedral cavities connected by triangular capil lary tubes. The velocity within a tube varies according The statistical development of the dispersion theory to the position of the fluid particle within the canal; is of interest because it reveals the dependence of the however, for simplicity it is postulated that the mean magnitude of the dispersion coefficient on the charac velocity is represented by the rate at which the fluid teristics of the flow and the porous medium. Stringent particle containing, say, a contaminant or tracer, passes conditions, however, must be postulated to simplify the from point to point. The multiple connections at the model so that the mathematical concept can be utilized. tube-end points represent a limited choice of direction As is generally true in most mathematical development, confronting an arriving particle of fluid containing the the mathematical model adopted has little resemblance contaminant or tracer. This postulate, to some degree, 126 FLUID MOVEMENT IN EARTH MATERIALS length I. The final results of Jong's analysis are the statements that the dispersion coefficients are related to the flow and media parameters according to the expressions DT =Bu Z/16 Dx =u I -ln 7)/6. (30) In equation 30, the symbols are defined as follows: u= average velocity, /=length of elementary canal, A =f unction of Z0 where Z0 is the distance along the z direction at which the maxi mum number of particles are located at time T0, In 7=Eulers constant = 0.577. The elementary length of the porous medium, and the residence time of the tagged fluid therein, constitute the variables of the statistical analysis. Although the elementary length may be used to describe the medium, the magnitudes of the dispersion coefficient cannot be computed directly unless there is a relating measurable parameter associated with the characteristic length. Jong reports that for his experimental model, in which the medium was composed of glass beads, the calculated value of I was one-third of the mean grain diameter. For conditions other than isotropy, it is expected that the value of I will depend on the grain-size distribution and the mode of packing. Staffman (1959), in his first paper on the theory of dispersion, used a statistical model identical with that of Jong's. The basis of his analysis is the visualization of the fluid path as a random walk process where the (From Jong, 1958) length, the direction, and the duration of each step FIGURE 17. Random path chosen by a fluid particle moving are random variables. The expressions for radial and through the canal system. axial dispersion derived from this mathematical model are indicates randomness of the particle path, and together with the fact that the pressure gradient is proportional to cos 6 (fig. 17), makes it possible to trace or determine (31) the probable location of the fluid particle at any given time. where The complete analysis presented by Jong (1958) is not repeated here inasmuch as it is basically a justification for the use of a given probability function. Of immediate interest, however, is the end result which follows and or indicates the functional dependency of the dispersion coefficient on the flow field and the pore characteristics. The use of average velocity to characterize the flow makes it possible to characterize the pore system by a The symbols in equation 31 and the related expressions length parameter defined here as the elementary are defined as follows: THEORY OF DISPERSION IN A GRANULAR MEDIUM 127 £=time, the mathematical difficulties, and thus cannot in any Urn G(x,t)=Q,t>0. X »oo The first statement indicates that over a defined plane, say z=0, the concentration is to be maintained at a known and feasible value. The second condition is merely a statement of the concentration of the fluid which initially fills the porous matrix. This condition can be established to be either zero or any known value <70. For the conditions depicted, the solution of the differ ential equation is as given in example 1. Most studies L-B- =P indicate that the second term in that equation is gener Screen ally small, so that the solution presented as example 2 gives a good approximation (Ogata and Banks, 1961). Thus, for purpose of correlation, the expression to be used (see eq. 26b) is ft _ 2 erfc.e where T=ut/x and X=Dfux. The laboratory model constructed to conform with the specified boundary conditions consists of a tubular 1 Tracer reservoir column filled with unconsolidated sand. Except for 2 Solute mixing and constant head tank the contact at the wall of the tube, where some flow 3 Tap water constant head tank 4 Overflow tanks channeling may occur, the model satisfies the condi 5 Petcocks tions specified for a semi-infinite porous medium. 6 Sand column 7 Downstream overflow The tracer chosen depends on the nature of the in 8 Overflow control valve vestigation. Common salt has proven the easiest to 9 Sampler tubes measure and is used whenever density is not a critical 6 factor. Because of its electrolytic characteristic, in-place measurement of the salt concentration can be obtained by means of an electric probe. At points where large enough samples of fluid may be taken, however, chem ical methods represent an economical and rapid means of determining the concentration at any specified tune. In either event the requirement is that measurements be taken at some distance from the tracer source, at a location and in a manner which does not disturb the flow. A typical setup for a columnar model is presented 7 in fibure 18. 1 Since the solution is available, the magnitude of D -^ -Perforated steel plate can be rapidly computed. The simplest means of calcu 9 r^ lating this result is to establish the value of the ratio - O-~ Vnlves X=8/ 10 EXPLANATION O Sand 1, d = 0.00371 ft O Sand 2, d = 0.00199 ft C Sand 3, d = 0.00135 ft -O- Glass beads, d = 0.0116 ft 1.0 (j) Glass beads, d = 0.00306 ft Data from- t^ Dankwerts (1953) Sc =0.08/Re - Day (1956) | Jong (1958) -%-Kramers and Alberda (1953) 0.1 J^f Banks (personal commun., 1958) Sc --= 1 /Re 0.01 -o 0.001 I I I I I I___I I I I I I I I 0.01 0.1 1-° 10 100 1000 Re = ud/v FIGURE 19. Plot of the Schmidt number versus the Reynolds number. In the section headed "Field equation for anisotropic the axially symmetrical system described by the dispersion," the directional properties of the dispersion equation coefficient were discussed. Laboratory observations and 1 dC\ . d2C dC dC analytical studies, like those of Bear (1961), indicate that the magnitude of the coefficient is different in at least two directions, parallel to and transverse to the Again, DT and Dx represent the radial and longitudinal direction of flow. dispersion coefficients, respectively, and the fluid flow The few available laboratory data for transverse is in the direction parallel to the x axis. diffusion are generalized by a plot of the Peclet number To determine the grouping of the parameters involved versus the Reynolds number. A general review of the in the process of mass transport, the above equation is published data relating to oil production is presented rewritten in dimensionless variables. To accomplish this, by Perkins and Johnston (1963). However, the specific put x=xd, r=pd, and t r d/u where d is a representa area of interest, as displayed by the analytical and tive or characteristic length describing the porous experimental data, is that related to flow conditions for medium. Substitution of these dimensionless variables which the Reynolds number >1. In a ground-water leads to the expression regime, this represents a range of flow that is not nor dC' dCf mally observed. _ _ Pe dp2 P dp dx2 dx ~ dr For the lower range of the Reynolds number, repre sentative of ground-water flow conditions, the analysis where PeT=ud/DT is the radial Peclet number and uses the two-dimensional dispersion equation with the ¥ex=ud/Dx is the axial Peclet number. constant dispersion coefficient developed previously. The Schmidt number, for directions r and x, and the The most practical flow system for studying dispersion, Reynolds number characterize the dispersion and con because of ready representation in the laboratory, is vection, respectively, for a radially symmetrical system. THEORY OF DISPERSION IN A GRANULAR MEDIUM 131 1.0 -O- -o 0.11 C EXPLANATION f f) II 0.1 o 0.08 O Sand 1 3 Sand 2 O O Sand 3 O -O- Glass beads, large (J) Glass beads, small Data from - ^ Dankwerts (1953) - - Kramers and Alberda (1953) P Banks (personal commun.,1958) 0.01 i i i i 1.0 1.0 10 100 Re = ud/v FIGURE 20. Plot of the Pedet number versus the Reynolds number. Published articles, for example, Kramers and Alberda 1961) was based on the preceding expression. Because (1953), indicate that the longitudinal and radial Peclet of some variation in the experimental data, only the numbers are constant. Therefore, both radial and average value of I>r=lX10~5 cm2/sec was determined longitudinal dispersion coefficients may be expressed to be significant. The velocity of the fluid in the experi explicitly by the relationship D=Dmu, where Dm is a mental model was approximately 4X10"4 cm/sec, and constant for any given granular material. the computed value of D compares well with results Because of difficulties in building a two-dimensional obtained by others and summarized by Perkins and physical model, the radial dispersion is investigated by Johnston (1963). It is also interesting to note that this considering a system where the longitudinal dispersion value of the dispersion coefficient is near that of is small. One approximate analysis for steady-state molecular diffusion of salt in water. condition is described by Ogata (1961) and may be A second model of interest is the one used by Harle- used to calculate the magnitude of the radial dispersion man and Rumer (1962). A two-dimensional dispersion coefficient from any suite of experimental data. This model was analyzed both theoretically and experi approximate equation is given in example 6. The com mentally. The mathematics of the mass transport plete solution, however, as previously stated, is too system is described in example 7 where the solution was complex for rapid computation of the dispersion presented as coefficient; hence, the expression T^ofl+erf C_=1i ~exp CQ Computed values of the dispersion coefficient, applicable for r=0 may be used. based on the observed and the approximate mathe Evaluation of the experiment conducted in Phoenix matical models, show a range of Dv from 9X10~5 to by the Geological Survey (Skibitzke and others, 6.5X10"4 cm2/sec for values of the fluid velocity rang- 132 FLUID MOVEMENT IN EARTH MATERIALS ing from 0.012 to 0.28 cm/sec. By correlation of the is simply a perturbation from the described, average dispersion coefficient with the Reynolds number, the flow. There are other mechanisms involved hi the trans proposed expression for the dispersion coefficient port of the dissolved component, but their magnitude transverse to the direction of flow was is small for normal ground-water flows, excepting adsorption. This mechanism of dispersion is treated to be analogous to eddy diffusion; thus, an expression similar to Fickian law of diffusion is used in the Using this expression with the results of the Phoenix mathematical model to describe the dispersion experiment, a value of 0.96 X10~5 cm2/sec was obtained component. for Dv, which shows good agreement. Evaluation of dispersion theory, based on the stipu lated assumptions, entails laboratory investigations SUMMARY that are necessarily simple. Laboratory analysis of a The theory of dispersion of miscible fluids has been linear flow system can be readily carried out and is developed extensively for fluid flow within a homoge especially useful in evaluating the one-dimensional prog neous porous medium. In the development, a fluid ress of a slug of tracer. Such laboratory investigation containing a given dissolved substance is described is described in the section on determination of the as moving according to Darcy's law, provided density dispersion coefficient. The results comparing theoreti differences between that fluid and the natural host cal and actual concentration distributions in the frontal fluid are negligible. However, the transportation of zone of a linear flow system are typified in figure 21. the dissolved substance has both a convection com For such a flow system there is ample evidence of this ponent, as described by Darcy, and a dispersion com nature which indicates that the theory of dispersion ponent. In all investigations the dispersion component hi a homogenous medium is a good first approximation. is considered to be primarily an effect of the microscopic On the basis of findings from experimental evaluation velocity variation within the pore channel and, hence, of the theory of dispersion, the application of the Experimental - glass beads -1.5 -2.0 (From Ogata, 1958) FIGURE 21. Example of comparison between theoretical and experimental results. From Ogata, 1958. THEORY OF DISPERSION IN A GRANULAR MEDIUM 133 theory is progressing to more complex flow systems. persion are apparent. The generalized description of the A good example is the derivation of an analytical ex transport of mass within any heterogeneous porous pression representing the mixing of two miscible fluids medium is unattainable by any presently known of different densities. This effort is directed particularly methods. However, the probelms encountered in a toward analyzing the conditions encountered in coastal real situation do not require complete and rigorous aquifers where observations reveal the development of formulation; usually, an engineering estimate of the a diffused zone between the fresh- and salt-water zones. maximum possible effect under the observed conditions The theoretical treatment is basically that of a one- is acceptable. dimensional dispersion system in which the coordinates The investigations of mass-transport phenomena have are oriented with the position of the interface between proceeded along both analytical and experimental lines. the two fluids. The experimental methods have been used to verify the The field applicability of the analysis for a unidirec analytical expressions obtained by solving the differ tional transport system is limited. For problems in ential equation of mass transport. Qualitative data volving any injection of contaminant, the dispersion defining the phenomenological process have also been phenomena take place in at least two-dimensional obtained in the laboratory to demonstrate the nature space. Accordingly, a description of the transport com of the flow. The analytical developments have proceeded ponent perpendicular to the direction of flow is impera along established lines to obtain expressions for pre tive. The approach to this description is to assume that dicting the possible concentration of a given dissolved the generalization can be accomplished by assigning to substance under various specified conditions. As with the dispersion coefficient the properties of a tensor. all analyses, their applicability is limited by the ability Justification of this method is based on experimental to describe the parameters that control the behavior evidence which indicates that the magniture of the mechanisms; hence, only the most simple linear flow dispersion coefficient for a homogeneous medium is systems are described. different in the directions transverse and parallel to Within a limited range, the theories of dispersion the line of flow. Because the generalization depends on have been verified by the results of experimental the ability to identify separately all given causes of analysis. However, these expressions are only useful spreading of the contaminant, such as the effect of whenever ground-water investigations can be brought the injection source, the tensor components are not yet into the laboratory; for example, when the use of tracers quantified. is needed in establishing certain details of the fluid flow. Additional mechanisms, such as mass transfer from Extension of such investigations into more complex the liquid to the solid phase due to adsorption, are of porous media depends on successfully directing the fundamental importance whenever it is required that research in dispersion theory so that the requisite the fate of a given substance in a ground-water body studies of the controlling parameters are possible. be determined. The adsorption process may be de scribed by the historical evaluation of the mechanism REFERENCES CITED involved, and its effect on the transport of the given Adamson, A. W., 1960, Physical chemistry of surfaces: New substance through the field of flow. In many instances York, Interscience Publishers, 629 p. of transport through a porous medium, the large solid Aris, R., and Amundson, N. R., 1957, Some remarks on logitu- surface area available makes the adsorption phenom dinal mixing or diffusion in fixed beds: Am. Inst. Chem. enon the most important feature. Description of this Eng. Jour., v. 3, no. 2. p. 280-281. Bachmat, Y., 1967, On the similitude of dispersion phenomena phenomenon involves a chemical process; hence, the in homogeneous and isotropic porous medium: Water thermodynamics of the mass transfer must be deter Resources Research, v. 3, no. 4, p. 1079-1083. mined for detailed analysis. At present, for lack of Bear, Jacob, 1961, On the tensor form of dispersion in porous better data, approximation of the adsorption rate is media: Jour. Geophys. Research, v. 66, no. 4, p. 1185-1198. used in the mathematical analysis. There is, as yet, no Bear, Jacob, and Todd, D. K., 1960, The transition zone between experimental evidence to indicate the validity of these fresh and salt waters in coastal aquifers: California Univ., Water Resource Center Contribution 29, 156 p. expressions used in the mathematical model. Beran, M. J., 1957, Dispersion of soluble matter in flow through In all attempts to describe the flow conditions in a granular media: Jour. Chem. Physics, v. 27, no. 1, p. porous matrix, it has been found that the generalization 270-274. of the expression for flow is formidable piimarily Biggar, J. W., and Nielsen, D. R., 1960, Diffusion effects in because of the inability to represent statistically the miscible displacement occurring in saturated and unsat- urated porous materials: Jour. Geophys. Research, v. 65, physical properties of the given medium in relation to no. 9, p. 2887-2896. the fluid flow (Skibitzke and others, 1961). Thus, the Bird, R. B., Stewart, W. E., and Lighfoot, E. N., 1960, Transport limitations placed on the analytical theories of dis phenomena: New York, John Wiley & Sons, 780 p. 134 FLUID MOVEMENT IN EARTH MATERIALS Carman, P. C., 1956, Flow of gases through porous media: New Muskat, Morris, 1946, The flow of homogeneous fluids through York, Academic Press Inc. porous media: Ann Arbor, Michigan, J. W. Edwards, Inc., Carrier, G. F., 1958, The mixing of ground water and sea water 749 p. in permeable subsoils: Jour. Fluid Mech., v. 4, no. 5, p. Ogata, Akio, 1958, Dispersion in porous media: Evanston, 111., 479-488. Northwestern Univ., Ph. D. dissert. Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of heat in 1961, Transverse diffusion in saturated isotropic granular solids: London, Oxford Univ. Press, 510 p. media: U.S. Geol. Survey Prof. Paper 411-B, 8 p. Cooper, H. H., Jr., 1959, A hypothesis concerning the dynamic 1964a, The spread of a dye stream in an isotropic granular balance of fresh water and salt water in a coastal aquifer: medium: U.S. Geol. Survey Prof. Paper 411-G, 11 p. Jour. Geophys. Research, v. 64, no. 4 p. 461-467. 1964b, Mathematics of dispersion with linear adsorption Crank, J., 1956, The mathematics of diffusion: London, Oxford isotherm: U.S. Geol. Survey Prof. Paper 411-H, 9 p. Univ. Press, 347 p. Ogata, Akio, and Banks, R. B., 1961, A solution of the differen Dankwerts, P. V., 1953, Continuous flow system distribution of tial equation of longitudinal dispersion in porous media: residence times: Chem. Eng. Sci., v. 2, no. 1, p. 1-13. U.S. Geol. Survey Prof. Paper 411-A, 7 p. Day, P. R., 195(5, Dispersion of moving salt-water boundary Orlob, G. T., and Radhakrishna, G. N., 1958, The effects of advancing through saturated sand: Am. Geophys. Union entrapped gases on the hydraulic characteristics of porous Trans., v. 37, no. 5, p. 595-601. media: Am. Geophys. Union Trans., v. 39, no. 4, p. 648-659. ErdeUyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Perkins, T. K., and Johnston, O. C., 1963, A review of diffusion 1954, Tables of integral transforms, v. 1: New York, and dispersion in porous media: Soc. Petroleum Engineers McGraw-Hill Book Co., 391 p. Jour., v. 3, no. 1, p. 70-84. Glasstone, Samuel, 1961, Textbook of physical chemistry: Rifai, M. N. E., Kaufman, W. J., and Todd, D. K., 1956, Princeton, N. J., D. Van Nostrand Co., 1320 p. Dispersion phenomena in laminar flow through porous Harleman, D. R. F., and Rumer, R. R., Jr., 1962, The dynamics media: Calif. Univ., Inst. Eng. Research Ser. 93, Issue of salt-water intrusion in porous media: Massachussetts No. 2, 157 p. Inst. Technology Hydrodynamics Lab. Rept. 55, 125 p. Saflfman, P. G., 1959, A theory of dispersion in a porous medium: Henry, H. R., 1960, Salt intrusion into coastal aquifers: Internat. Jour. Fluid Mech., v. 6, p. 321-349. Assoc. Sci. Hydrology, Comm. Ground Waters Pub. 52, p. 1960, Dispersion due to molecular diffusion and macro 478-487. scopic mixing in flow through a network of capillaries: Jour. Hougen, 0. A., and Marshall, W. R., Jr., 1947, Adsorption from Fluid Mech., v. 7, p. 194-208. a fluid stream flowing through a stationary granular bed: Scheidegger, A. E., 1954, Statistical hydrodynamics in porous Chem. Eng. Process, v. 43, no. 4, p. 197. media: Appl. Physics Jour., v. 25, p. 994-1001. Jeffreys, Harold, 1961, Cartesian tensors: London, Cambridge 1960, The physics of flow through porous media: Canada, Univ. Press, 92 p. Univ. Toronto Press, 313 p. Jong, G. de Josselin de, 1958, Longitudinal and transverse dif 1961, General theory of dispersion in porous media: Jour. fusion in granular deposits: Am. Geophys. Union Trans., v. Geophys. Research, v. 66, no. 10, p. 3273-3278. 39, no. 1, p. 67-74. Shamir, U. Y., and Harleman, D. R. F., 1967, Numerical solutions for dispersion in porous media: Water Resources 1959, Vortex theory of multiple phase flow through porous media: California Univ. Water Resource Center Contri Research, v. 3, no. 2, p. 557-581. bution 23, 80 p. / Skibitzke, H. E., 1964, Extending Darcy's concept of ground- water motion: U.S. Geol. Survey Prof. Paper 411-F, 6 p. Jost, W., 1960, Diffusion in/solids, liquids, gases: New York, Academic Press Inc., 55o p. Skibitzke, H. E., Chapman, H. T., Robinson, G. M., and McCullough, R. A., 1961, Radio-tracer techniques for the Kitagawa, K., 1934, Sur le dispersement et l'e"cart moyen de study of flow in saturated porous material: Internat. Jour. l'6conlement des eaux sonterraines, I. Experiences avec un Appl. Radiation and Isotopes, v. 10, no. 1, p. 38-46. modele le laboratoire: Kyoto Imperial Univ., Coll. Sci. Mem. Skibitzke, H. E., and Robinson, G. M., 1963, Dispersion of Ser. A, v. 17, p. 37-42. ground water flowing through heterogeneous materials: U.S. Kohout, F. A., 1960, Cyclic flow of salt water in the Biscayne Geol. Survey Prof. Paper 386-B, 3 p. aquifer of southeastern Florida: Jour. Geophys. Research, v. Slichter, C. S., 1905, Field measurements of the rate of movement 65, no. 7, p. 2133-2141. of underground water: U.S. Geol. Survey Water-Supply 1961, Fluctuation of groundwater levels caused by dis Paper 140, p. 9-85. persion of salts: Jour. Geophys. Research, v. 66, no. 8, p. Sneddon, I. N., 1951, Fourier Transforms: New York, McGraw- 2429-2434. Hill Book Co., 542 p. Kramers, H., and Alberda, G., 1953, Frequency response analysis Taylor, Geoffrey, 1953, Dispersion of soluble matter in solvent of continuous flow system: Chem. Eng. Sci., v. 2, no. 4, p. flowing slowly through a tube: Royal Soc. London Proc., 173-181. ser. A, v. 219, p. 187-203. Lau, Leung-Kai, Kaufman, W. J., and Todd, D. K., 1959, Vermeulen, Theodore, and Hiester, N. K., 1952, Ion exchange Dispersion of a water tracer in radial laminar flow through chromatography of trace elements: Ind. Eng. Chemistry, homogeneous porous media: Calif. Univ., Inst. Eng. v. 44, p. 636-651. Research Ser. 93, Issue No. 5, 78 p. Wentworth, C. K., 1948, Growth of the Ghyben-Herzberg Mime-Thomson, L. M., 1961, Theoretical hydrodynamics: New transition zone under a rinsing hyopthesis: Am. Geophys. York, MacMillan Co., 660 p. Union Trans., v. 29, p. 97-98. U. S. GOVERNMENT PRINTING OFFICE : 1970 O - 370-837