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V "4‘ . ”15¢“. 'Dfl-f "‘l"‘5 3‘4"" "Es‘dfi‘iflgg x 3113.5 9 " .m' if? ‘ . v u ”.313;- w‘ 3“ at"; a ‘ “bf .‘~ nan. . aw“, ‘ -Ih au" .. . u... m. u.‘ r. ,u- ia 7 Agfitiign . .1 I: 4.32;. 1 Jun- I NT rim. LJL "thym— ‘r'mz. “.5“ cal-.4 {9.0.51.1} flint. nu... 4.1-. «Sui . nu . k ‘1 '3» ' w, y .fvg‘pr . u\ «- C: ‘ t - _!'v p I. N v. . "K‘J- H ‘I «I. .. 1:.» -_..- Michigan Smu- University "1 LIBRA R Y i i Wm ’ Major professor “ate 2%”, 28:1 /?66 0-169 A B S T R A C T DISPERSION IN ANISOTROPIC POROUS MEDIA by John Rodger Adams The problem of dispersion in flow through porous media arises in coastal aquifers, underground waste disposal, chemical sepa- ration by filtration, and secondary recovery of petroleum. This study is motivated by the coastal aquifer problem in which the spread or dispersion of salt water into fresh water may limit the use of wells for drinking water. The dispersion process is described by the con- vective-diffusion equation with a coefficient depending on the flow and porous medium as well as on the solvent and solute. The nature and functional form of the coefficient is the topic of direct interest. Existing theories state that the dispersion coefficient is a second rank tensor formed by the contraction of a fourth rank tensor dependent 0n the porous medium with a second rank tensor dependent on the flow. This theory, which is presented for isotropic media, is adapted to anisotropic media by a transformation of coordinates involving the permeability tensor. Experiments are conducted in which the dispersion of tracer spots is measured during flow in porous beds packed with nylon filaments. The filaments are kept parallel to make the bed anisotropic . Two filament sizes and two orientations are used. John Rodger Adams The experimental results are in reasonable agreement with Vprevious experiments in isotropic media but are not compatible with the existing theory. A new relation is proposed in which the tensor dispersion coefficient is equal to the contraction of a second rank tensor depending on the fluid and solid media with a second rank tensor depending on the flow. The power to which the velocity appears in the flow-dependent tensor is expected to depend on the: (1) angle between the flow and major permeability, (2) flow channel geometry, (3) the component. DISPESION IN ANISOTROPIC POROUS MEDIA by . John Rodger Adana Amssrs ‘ .' Submitted to 7‘ Jun—“‘13 Michigan State University '3 in partial fulfillment of the requirements for the degree of G 1/3505 V/x9/(47 A C K N O W L E D G M E N T S The author wishes to express his gratitude to Dr. H. R. Henry who suggested the topic and provided the inspiration to undertake the work. The support, encouragement, and patience of Dr. C. E. Cutts, Chairman of the Department of Civil and Sanitary Engineering, who urged the author along during trying times, is greatly appreciated. The author wishes to thank the committee members for their constructive criticism. Especial thanks are due to Dr. J. S. Frame, Professor, Department of Mathematics, for his careful explanations. Financial support from the Institute of Water Research, Dr. L. L. Quill, Director; and the Division of Engineering Research, Mr. J. W. Hoffman, Director, is acknowledged. ii ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS INTRODUCTION THEORY 2.1 General 2.2 Flow through Anisotropic Porous Nadia 2.3 Dispersion -EXPIRIMENTAL APPARATUS AND PROCEDURE 3.1 Apparatus 3.2 Procedure 3.3 Discussion of Procedure :3.1. Discussion of Experiments .. mm ANALYSIS i-{iil' Data Reduction T A B L E 0 F C 0 N T E N T S vi viii 15 26 29 31 33 TABLE OF CONTENTS (continued! APPENDIX TABLES FIGURES BIBLIOGRAPHY iv 69 86 L I S T O F T A B L E S 4-1 Parameters in Least Square Analysis of Tracer ‘ 56 Distribution 4-2 Summary of Experimental Work 57 4-3 Data for Bed 1 58 4-4 Data for Bed 2 59 4—5 Data for Bed 3 60 4-6 Data for Bed 4 61 4-7 Equations of Lines Fit to data by Least Squares Method 62 4-8 Anisotropic Permeabilities 63 4-9 Transformed Dispersion Coefficients 64 A-l Summary of Least Squares Check Computation Bed 1 65 A-2 Summary of Least Squares Check Computation Bed 2 66 A-3 Summary of Least Squares Check Computation Bed 3 67 A-4 Summary of Least Squares Check Computation Bed 4 68' L I S T 0 F F I G U R E S aim has 2-1 Flow Region in Anisotropic Medium 12 2-2 Transformed Flow Region-Isotropic Medium 13 3-1 Schematic Drawing of Apparatus 69 3-2 Overall View of Apparatus 70 3-3 Porcus Bed and Electrode Group 70 3-4 Inlet Constant Head Tank 71 3-5 Outlet Constant Head Tank 71 3-6 Electrode GrOups 72 4-1 Concentration of NaCl in water as a function of 73 Specific Conductance 4-2 Molecular Diffusivity and Kinematic Viscosity 74 4-3a Bed 1 Dx/Dm versus Pe 75 4-3b Bed 1 Dy/Dm versus Pa 76 4-4a Bed 2 Dx/Dm versus Pa 77 4-4b Bed 2 Dy/Dm versus Pe 78 4-5a Bed 3 Dx/Dm versus Pe _ 79 4-5b Bed 3 Dy/Dm versus Fe 80 4-6a Bed 4 Dx/Dm versus Fe 81 4—6b Bed 4 Dy/Dm versus Fe 82 4-7 Composite Plot of Data 83 A-l Bed 1 R/RLS versus n/nLs 84 vi Figure LIST OF FIGURES (continued) Bed 3 R/RLS versus n/nLS Bed 2 R/RLS versus n/nLS Bed 4 [URLs versus n/nLS vii 84 85 85 L I S T O F S Y M B 0 L S Common symbols are listed here. Symbols which appear rarely are defined in the text. Symbol Description Area Media dispersivity tensor, isotropic media Media dispersivity tensor, anisotropic media Cell constant Cell constant for electrode Concentration Diameter of solid medium element Dispersion coefficient tensor, isotropic media Dispersion coefficient tensor, anisotropic media Molecular diffusion constant Flow dispersion factor tensor, isotropic media Flow dispersion factor tensor, anisotropic media Acceleration of gravity ‘ Metric tensor Head Subscripts indicating major and minor principal values of symmetric second order tensors Permeability tensor viii Dimensions L /T L /'1‘ l/L l/L M/L Lz/T L2/T L/T L/T Symbol L I S T 0 F S Y M B O L S (continued) Description Permeability ratio, kI/kII Porosity Pressure Peclet number, ud/Dm Reynolds number, ud/v Measured conductivity Measured conductivity of base fluid Electrode conductivity of base fluid Temperature Time Velocity component Volume Coordinates in isotropic medium Coordinates in anisotropic medium Vertical distance Angles between flow direction and kI direction Unit weight Increment or difference Coordinates parallel and perpendicular to flow Viscosity Mass density Specific conductance Specific conductance of base fluid ix Dimensions -M/LT2 micromhos micromhos micromhos °c T L/T M/LZTZ M/LT M/L3 micromhos/L micromhos/L L I S T O F S Y M B O L S (continued! Smbol Description Dimensions l/pc Specific conductance of solution (salt only) micromhos/L a Standard deviation L l. I N T R 0 D U C T I 0 N The problem of the dispersion or spread of a fluid or solution imbedded in another fluid flowing through a porous medium has been of interest to engineers for many years. One of the early problems which has taken on even greater importance is the fresh water- salt water interaction in groundwater aquifers along sea coasts. Knowledge of the location and mobility of the interface (or mixing zone) is critical to the location and operation of wells supplying water for domestic use in coastal regions. A problem of more recent concern is the contamination of groundwater sources by industrial or municipal wastes. These wastes may enter aquifers either by infil- tration or through the use of subsurface formations for the disposal of concentrated or hard to treat liquid wastes. Chemical and petro- leum engineers also have definite interests in dispersion during the flow through porous media. Chemicals may be separated by use of the different dispersion of each material during the flow through filter columns. The success of secondary recovery of oil by water or hydro- carbon displacement depends on the ability to avoid excessive mixing. The initial efforts at analysis of the fresh-salt water interface assumed that both fluids were static and that they were immiscible. These assumptions lead to the well known Gyben-Herzberg —2 relation between water table and interface elevations, (Todd, 1959)1. The next degree of approximation is to consider flow in the fresh water only and to retain the immiscible assumption. Henry (1959) obtained solutions for fresh water outflow through a vertical surface by means of the hodograph and complex mapping. De Josselin de Jong (1959) avoided the difficulty imposed by the need for two potential functions when motion is allowed in the salt water by representing the sharp interface by a distribution of vortices. As a follow-up to Henry's work, Lin (1964) used singularities to describe the interfacial discon- tinuities and used numerical integration to obtain a solution for the movement of the interface after a sudden change in fresh water flow rate. Somewhat parallel to the studies above, the effect of miscibility was included in efforts to describe the actual mixing zone between fresh and salt water. Wentworth (1948) and Carrier (1958) showed that molecular diffusion was unimportant compared with the macroscopic dispersion in the development of transition zones. Henry (1960) solved the problem including dispersion by an approximate method and obtained streamlines and isochlors which agree with field data (Kohout, 1960). The success of these calculations of the mixing region obviously depends on the accuracy of the description of the dispersion process. The simplest approach is to use a scalar dispersion coefficient. However, reference to Ogata and Banks (1961) and Ogata (1961) shows that the dispersion in the flow direction is several times the dispersion in the transverse direction. One approach is exemplified by 1Dates in parentheses refer to references in the bibliography. A -3 von Rosenberg's (1956) combination of the capillary model of porous medium (Scheidegger, 1960) and Taylor's (1953) theory of dispersion in circular tubes. Another approach is the use of statistics to describe the porous medium or the flow pattern. Scheidegger (1954) considered the dispersive nature of the flow of a single fluid through a porous medium. De Jong (1958) and Saffman (1959) applied stochastic processes in order to obtain dispersion coefficients. Saffman and De Jong arrived at similar results for longitudinal dispersion including the influence of molecular diffusion, but while De Jong predicted a ratio of lateral to longitudinal dispersion dependent on the distance traveled, Saffman predicted that lateral dispersion was entirely independent of molecular diffusion. This last statement is in opposition to that of Simpson (1962) who states that molecular diffusion is the primary mechanism of lateral dispersion. In a second paper (1960), Saffman obtains values of dispersion coefficients which are valid for all values of the Peclet number. Still another view of the dependence of the dispersion coef- ficient on the fluid, the porous medium, and the macroscopic flow pattern has been developed recently. Papers by Bear (1961) and Scheidegger (1961) propose that the dispersion coefficient is a symmetric second order tensor formed by the contraction of a fourth order tensor which depends on the porous medium and a second order tensor which is a function of the flow. Harleman and Rumer (1962) suggest that the manner of dependence on the flow is non-linear with the power of the velocity variable from component to component. -4 The merit of the last approach will be studied experimentally. The ultimate success would be to determine the relations for the tensor dispersion coefficient and flow function. However, the experimental results raise severe doubts about the validity of this theory. Convention- ally plotted results lead to expressions which are not compatible with the theoretical requirements. A possible extension of the original theory is proposed as is an alternative theory involving a second rank tensor for the media factor. The theoretical development will not include a solution of the convective dispersion equation but will attempt to present the dispersion process in anisotropic porous media in terms of the measured quantities. The anisotropic media is vital to the study as it should aid in exposing the effect of the solid medium, which is considered to be more significant than the fluid part of the system. -5 2. T H E 0 R Y 2.1 General The theory of flow through porous media, including the transformation between anisotropic media and isotropic media, will be treated first. The flow will be assumed to be slow, and laminar so that Darcy's law will apply. The transformation is used to replace an anisotropic medium in which the flow is not described by the Laplace equation with an isotropic medium in which the flow is described by the Laplace equation, but the domain is distorted. After this, the dispersion equation will be treated. The transformation will be used to aid in understanding the dispersion process in anisotropic media by applying it to what is known about dispersion in isotropic media. The treatment will center around the tensor dispersion coefficient and will essentially assume that the region is infinite in extent. As the nature of the transformation is not proper to Cartesian tensorsl, general tensor notation must be used in that part of the development. For continuity,general tensor notation will be used through- out. The operations concerning tensors generally follow the book by Aria (1962). A vector may be represented by its contravariant components which transform according to: -1 , 51 3 £1 8'1 (1) 3x 1Aris (1962) states that "The group of rotations ... is the only group of transformations considered in constructing Cartesian tensors." A vector may also be represented by its covariant components which trans- form according to: aj (2) Summation is implied by the appearance of the same index as both an upper and lower suffix in the same term. Note that an upper index in the denominator counts as a lower index in the equation. If an index appears only once in each term, it may assume any value in the range of the indices. Where details are required, the treatment will be two dimensional to save space and to prepare for the two dimensional experb mental work. Thus the indices range over the values 1 and 2. 2.2 Flow through anisotropic porous media The flow of an incompressible fluid can be described by two differential equations and appropriate boundary conditions. The conservation of mass is given by the incompressible continuity equation as: (3) gri- Igna- II where: u1 = velocity in the x1 direction The equation of motion is the Navier-Stokes equation, which is given by Aria (1962) as: 1 2 1 Du _ 11 82 fig jk a u p—--s (v—+ +ug -— (4) ”t axj 3x1) axjaxk where: z is measured vertically upward p = pressure p = mass density y = unit weight u = viscosity ij = metric tensor g? = g; + uJ i;{ This equation (4) is valid for an incompressible fluid in a gravitational field. Note that for Cartesian coordinates gij = 0 if i = j, and g11 = 1. For steady motion the first term in the acceleration is zero. For sufficiently slow motion the remaining terms in the acceleration are small, so that, for slow steady flow Eq. 4 may be written as: 2 i sij (y m3. 92f u ng a f’ k (5) 3x ax axJax It is convenient to replace the lefthand side of Eq. 5 by y 92; , with ax h = p/y + 2, so that the equation of motion becomes: 2 111—:Ejka‘ui (6) 1 yg jk Bx ax 3K However, this equation applies to each particle of fluid flowing through the flow channels between the solid particles. The complex and unknown geometry of the pore spaces precludes the formulation of the boundary value problem, let alone the solution of it. Darcy's law for the bulk velocities is used in porous media problems and will be used here. A generalized form of Darcy's law (Collins, 1961) is: i x_ ij ah = - k —. . u PH ax] (7) -8 where: P = porosity k1] = permeability tensor This equation (7) is valid for a homogeneous medium, in which the per- meability is not a function of location. Note that the velocity vector and the head gradient are not necessarily parallel. The relation between Eqs. 6 and 7 must be made very clear. The velocity in Eq. 6 is a point function defined in the region consisb ing of the pore space in the porous medium. The velocity in Eq. 7 is an artificial average velocity equal to the volume flow rate divided by the area of the pore space in the appropriate direction. The same sort of thing is true of the head gradients. The head gradient in Eq. 6 is evaluated at a point in the actual flow while the head gradient in Eq. 7 is again an average over a portion of the bed. Thus the Navier- Stokes equation and the complex geometry are replaced by the Darcy equation and an homogeneous continuum. The complete derivation of Darcy's law from the Navier- Stokes equation is practically impossible. Any geometric model of a porous medium which would permit solution of the Navier-Stokes equations would be too regular to be a good model of a real porous medium. However, the concept of the "capillary bundle" theory (Collins, 1961) can help one to understand the interrelation. The porous medium is considered to be a collection of parallel tubes of various cross sections. The solution of the Navier-Stokes equation for steady flow -9 through a straight tube of uniform cross section yields the following expression for the average velocity in the tube: 0’ h 1 (a) < I I tlm %“ l . . l . . where: the tube axis is in the x direction S is the constant dependent on the shape of the cross section V1 is the average velocity in the tube' There is a strong similarity in form between Eq. 8 and Eq. 7. There are still the undetermined relations between the velocities and between the head gradients in the two equations, as well as the effect of geometry on S and kij. The similarity is significant however, for if the geometric difficulties could be resolved,or if Eq. 6 could be solved for irregular geometries the derivation could be completed. Such efforts are not of interest in this project. For the purpose at hand the flow of an incompressible fluid through a porous medium will be described by Eq. 3, Eq. 7 and the appropriate boundary conditions, which are: 1) h = constant (or function of time) on boundaries open to a reservoir 2) uh = 0 on impermeable boundaries where: un is the velocity component normal to the boundary. -10 If Eq. 7 is substituted into Eq. 3 the following equation is obtained: 3 13 I. as \ _ _(_k )—0 (9) 1< P axj/ Since the medium has been assumed to be homogeneous this may be written: 2 kij a h = 0 (9a) axiaxj Furthermore, if the medium is isotropic, Laplace's eqdation is obtained: 2 2 _a_IET+3—hz=° (10) 5X 3X Bx BX Standard mathematical techniques yield solution to flow problems in most cases if the medium is isotropic. Collections of solutions to Laplace's equation with reference to groundwater problems can be found in several books, including those by Harr (1962), Muskat (1946), and Polubarinova- Kochina (1962). The solution of Eq. 9, however, is more difficult But a change of variable involving a scale change will convert the anisotropic medium into an isotropic one. Tliough several variations are possible, the transformation used here will be a scale change in the direction of the minor permeability. The transformation is carried out after the flow region in the anisotropic medium has been expressed in the coordi- nate system which coincides with the principal axes of the permeability tensor. The existence of these axes depends on the assumption that the Permeability tensor is symmetric. The relation between the coordinates -11 convenient for the flow region and the principal axes of the permeability are given by the rotation: E = cosa sina x (11) n = -sina coso y The angle is determined by the relation: k11 k12 = cosa -sina kI 0 cosa sina (12) k21 k22 = sina cosa O k sina cosa I where kI and kII are the major and minor principal values of the permeability. In the x, y coordinate system, Eq. 9a becomes: 2 2 13% + knM =0 (13) BK BY - k _ _ . Now let Kr — I/kII, X - x, and Y — nyr in Eq. 13. 2 2 kI§—: + kn Kr fl =0 (14) BK BY This is of course Laplace's equation. The medium in the X, Y plane is isotropic with permeability kI. Now consider the transformation in some detail by applying it to a rectangular region in the g, n plane, with the permeability directions as shown in Fig. 2-1. The n axis is vertical. The components of the permeability tensor in the g, n plane are given by Eq. 12. The boundary conditions are: h = 0, g = L; h = ho’ g = 0; %% = 0, n = O, and n = H. 4:; x Fig. 2-1 Flow Region in Anisotropic Medium ‘ ,Tho transformation to the isotropic medium is now: i .. X - x - g cosa - n sina . - q (15) 0.7. Y - y/Rr - Jkr (g sina + n cosa) A further notation into a g', n' system, in which.g' is parallel to the .long side of the region may be defined by the appropriate substitutions ." « x'gqi"into Eq. 11. An angle 3 may be found in terms of the angle a and the 2 ":;fi permeability ratio as follows. The g' axis is given by the equation: -X sing + Y cosa - 0 , ‘2‘ p (16) ;i 3 .92* Y - X tans A .-, .sBut'in the anisotropic medium.the g axis is given by the equation: -x sina + y cosa - 0 . (I7) .y - x tana “by.use of Eq. 15 in Eq. 17 -13 And equating Eqs. 18 and 16 it is fOund that: tans = JRr tana (19) The region in the isotropic medium is shown in Fig. 2-2. The boundary conditions are essentially unchanged. n: Fig. 2-2 Transformed Flow Region -- Isotropic Medium In this plane there must be end regions of curvilinear flow. In the center of the region the long boundaries will force the velocity to be parallel to the 5' direction. However, the velocity will be perpendicular to the permeable ends which are also lines of constant head. This follows immediately from the fact that the solutions of Laplace's equation in terms of stream function or a potential function are orthogonal. This streamline curvature will remain after trans- forming back into the rectangular domain in the anisotropic medium. Curvilinear flow will affect the dispersion and this is an unnecessary complication. If the medium were sufficiently long itfwould be possible to use the central section only for the dispersion tests. This would probably require excessively long apparatus since the ends can be -14 expected to influence the flow pattern within the bed to a distance the same size as the height of the end. The transformation itself suggests another way to avoid this trouble. A rectangular region in the isotropic ' medium would have parallel flow throughout, and this could be trans- i formed into a parallelogram shaped region in the anisotropic region, also with parallel flow. The slope of lines of constant head for flow in a given direction in the anisotropic medium is the important thing to be ‘ obtained from this inverse transformation. This slope can be obtained by starting with Eq. 9,by making use of the transformation, or by 9 inspecting the condition for flow parallel to the axis. The last approach is the easiest since, for this flow the velocity in the n ,3... direction is zero and Eq. 7 yields: u1 = _ x. (kll gg + k12 g3) Pu a: BB (20) 2 _ x_ 21 g3 22 QB u - - PH (k BS + k 61") I These equations may be solved to yield: DE - ulk22 ag k11k22 _ k12k21 (21) DE u1k21 an k11k22_ k12k21 Now along lines of constant head, the total differential of the head is zero, or: o:%dx+%dy (22) -15 This may be solved for: 3_h 111=§§ ‘ dg QB (23) all Now substituting from Eqs. 20 and 13 the slope of the lines of constant head becomes: E3 coszo+l(r sinzoz at; = (Kr - l)sin01 cosa (24) The apparent difficulty which is caused by a = 0, a = n/2, or kI = kII’ for which values the denominator of Eq. 21 becomes zero, may be resolved in each case by returning to the original statement of the problem. In each of these cases k12 = k21 = 0 so the rotation to principal axes is not necessary, and the velocity vector is perpendicular to the lines of constant head. 2.3 Dispersion A brief dimensional discussion is of interest before proceeding with the treatment of the dispersion coefficient. The convective- diffusion equation which is a mass conservation statement for the tracer is (Bear, 1961): §g+uiaC___a.(DiJB¢_,)=o (25) at 6x1 1 J where: c = tracer concentration t = time Dij = dispersion coefficient -16 This is valid for a substance of concentration c embedded in an incom— pressible fluid (Henry, 1960). Though a dimensionlal analysis of the combined problem of flow with dispersion does not result in a useful basis for analysis, some discussion of the dimensional nature of the dispersion coefficient is helpful. The convective-diffusion equation may be put into dimension- less form by the following substitution: i i E = x /d E = c/co E i = ui/uo 3 ij = Dij/D m E = tuo/d where: The overscored quantities are dimensionless, d is diameter of solid material and Dm is the effective molecular diffusivity. If this substitution is made into a one-dimensional form of the con- vective-diffusion equation the following expression results: u c — u c — D c 2- 5:: d axlax Now divide by uoco/d to obtain ; - — D 2- eel— = .—'". n“ 5.: -1 ....) ' 3K 0 3X 6X tv_—v_wv -v—pf v '7 -17 The parameter Dm/uod is the reciprocal of the Peclet number and appears in the same manner as the reciprocal of the Reynolds number appears in the Navier-Stokes equation. For a fluid-tracer mixture in a porous medium the effective molecular diffusivity is a function of the porosity as well as the molecular diffusivity of the tracer in fluid. If there is no flow, molecular diffusion will take place but the rate is reduced/by the interference of the solid medium. On the other hand, if even slow flow occurs, the porcus material causes larger scale mixing and the dispersion is much more than could be caused by molecular diffusion alone. Of course the actual transfer of tracer material across streamlines is always due to molecular action in laminar flow. So long as the flow within the pore spaces remains laminar, the Reynolds number should not be used as the independent variable. Instead the ratio of D to Dm should be a function of the Peclet number. Unfortunately, dimensional reasoning does not help much when an effort is made to determine the functional relation. Scheidegger (1961) and Bear (1961) propose that the disbeSsion coefficient is given by the contraction of a fourth order tensor dependent on the porous medium with a second order tensor dependent on the flow. ij 3 ij kl D A le (27) This is only an hypothesis and the results of this study do not support it, at least when the usual analysis is carried out. The medium factor has dimensions of length and the flow factor has dimensions of velocity. But, as Harleman and Rumer (1962) point out, the experimental evidence -18 does not support the requirement that the velocity appear to the first power. In fact, they suggest that the exponent may vary from component to component.' This requires that the dimensional nature of Aijkl and Fk1 vary from element to element. In a later publication by Harleman, Mehlhorn, and Rumer (1963), all the dimensions are assigned to the medium factor and the flow factor is made dimensionless. This means that the medium factor depends on the fluid as well as the solid phase, since the time dimension can enter only with a fluid property such as the kinematic viscosity or the molecular diffusivity of the tracer. This in only indicated, not directly stated, in the 1963 paper by Harleman and associates. Keeping equation 27 at hand, consider the properties of Dij and Aijk1 in isotropic media, and then use the transformation of the previous section to obtain some knowledge about their properties in the anisotropic media. However, since Fk1 is expected to depend primarily on the velocity it is reasonable to expect that Fk1 will have contravariant components. This agrees with the proposal of Scheidegger (1961), who assumed that Fk1 = jkul/Iul , which is symmetric. Scheidegger is concerned primarily with Aijkl, and by symmetry arguments similar to those used in crystal structure shows that for an isotropic medium there are only twenty-one non-zero elements in the three dimensional case. The number of non-zero elements in Aijk1 reduces to eight (of sixteen) in two dimensions. He also shows that there are only two independent factors involved in the non-zero terms. Thus, in two dimensions: 11 _ 22 A ll-A 22=AI (28) (28 can't) 12 12 _ 21 _ A 12 ’ A 21 ‘ A 21 ‘ (AI ‘ A11) /2 and the other eight terms are zero. In the isotropic medium the only preferred direction is the direction.of flow. Therefore Dij can be written as follows in the coordinate system pertinent to the domain and velocity: D = D (29) The contraction expressed in Eq. 26 is written out with Aijkl given by , Eq. 28, D11 given by Eq. 29 and FR1 left undetermined. ll 11 22 D = DI AIF + A F 12 D l o - < (A, - An)/2 ) (1,12 + r2 ) (so) 21 . n - o - ( (AI -IAII) /2) (r12 + r21 ) . 22 1 11 22 n .. 1111 = AIIF + AIF The expressions for-D12 and D21 imply that either AI - All or 'i71F13 + lei! 0. But the experimental evidence that DI and DII'and 5’ the relations for D11_and D22 require the second condition. A well --fl. ‘ . fis—s- fim‘ I -20 symmetry property is unchanged by a rotation. Also, the fourth rank tensor, Aijkl, is isotropic, and its components are not changed by a rotation. Thus a skew-symmetric portion of Fk1 will never appear in .D1] and will be assumed to be zero for simplicity. Consequently, Fk1 is also symmetric and may be written as: 11 _ F — FI F12 = F21 = o (31) 22 F - FII . kl Now Eq. 30 may be rewritten by substituting Eq. 31 for F : D11 = D = F + A F I AI I II II D12 = D21 = o (32) D22 = DII = AIIFI + AIFII Though written in general tensor notation, equations 28 to 32 and the associated discussion apply to an isotropic porous medium. It is now possible to apply the transformation presented earlier in order to learn something about the dispersion coefficient in an anisotropic medium. It is necessary to keep very close tabs on the transformations because the inverse contravariant transformation is the same as the direct covariant transformation for this particular transformation. It is convenient to write second order tensors as matrices, and it will be helpful to write out both contravariant and covariant, direct and inverse transformations. -21 Direct: from anisotropic to isotropic Contravariant: X 1 0 x == (33a) Y 0 Jkr y Covariant: X 1 0 x = (331:) ,/1 Y 0 K— y r Inverse: from isotropic to anisotropic x 1 '0 ix = (348) y 0 /1_ Y K Covariant: x l O ‘ X = (34b) y 0 JR Y These transformations may be applied to the three tensors given in Eqs. 28, 29, and 31. Eq. 12 indicates the appropriate form to carry out the transformation by matrix multiplication. The transformation may also be expressed by a contraction such as, dij = TiTiDmn. The fourth rank tensor is best handled by writing out the contraction, since appropriate and conformable matrix representations would have to be figured out first. Now the dispersion coefficient in the anisotropic medium in terms of the dispersion coefficient in the isotropic medium is given by: 11 11 d = D = DI 22 1 22 1 d =—-—D =———’D (35) Kt Kr II d12 = d21 = 0 ”@313, the flow factor becomes: I ‘11‘ 11 f ‘F =FI 22 . 1 22_ 1 .f ink-:1? — R-t-FII (36) f12 _ f21 _ 0 The fourth. order tensor is best handled in several groups of components which transform similarly. 1) Components in which each index value occurs the same umber of times as a lower suffix as it does as an upper suffix. 11 _ 11 _ A 11 ' A 11 ‘ AI 22 22 a 22 ' A 22 ‘ A1 812 _ 12 21 21 a (37’) 12 A 21 ‘ a 12 = a 21 12 12 21 21 A 12 ‘ A 21 ' A ,12 ‘ A 21" (A1 A11) ’2 2) Components in which both upper siffixes have the same , timelue and both lower suffixes have the sane value. \ <. ,L' . 11 11 a 22 ' Kr A 22 ‘ Kr A11 22 a 1 2 (37b) 2 1 a—A .— 11 K.r 11 Kt A11 's 3) Components in which one index value occurs only. once. ‘ 11' ' 11.. .11. 11 I . ";12; .aflImAu ‘2NAH21 WW5 \ 12 11 21 21 zz‘fKA 12 azz'flrA 22 22 _ f1 22 22 _ /1_ 22 , . 12' K A 12 a 21‘ K A 21 (”cm”) I: r 12 _ f1 12 21 _ /'1_ 21 El11'1 - (:x2) n: This requires that A1 = 0, so A; is also symmetric with the same 2 1 principal axes as Dij and F3. If the flow in the isotropic medium is -47 not aligned with apermeability axis from the anisotropic medium, then all three second rank tensors must be rotated into permeability coordinates before the transformation to the anisotropic medium may be applied. When this transformation is carried out, the mixed tensor, A; transforms as follows: 1 1 a1 ' A1 1 1 a2 - /K A2 ' (65) 2 _ f1 2 81 K— A1 r 2 2 a2 ‘ A2 Now dlJ may be written in terms of A; and FJm as: dll = Dll = AlFll + A1F12 1 2 d12 = /1' D12 = /l—-(A1F21 + A1F22) r ‘ Kr l 2 (66) d21 = /l__D21 = J1—-(A2F11 + A2F12) K 1 2 r r 22 l 22 1 2 21 2 22 d - R;' D - E; (AID + AZF ) For the case in which the flow and major permeability are parallel in the anisotropic medium this simplifies to: 11 _ _ _ 1 d — dI - DI - AlFI d22 = d ='l— D =-l— AZF (67) II Kr 11 Kr 2 II 12 21 -48 For the case in which the flow is perpendicular to the major permeability in the anisotropic medium this simplifies to: 11 _ l. _ 1_ 1 d ‘ d1 ‘ K D1 ’ K A1FI r r 22 2 d ' d11 ' D11 ‘ A2F11 (68) d12 = d21 = 0 These equations are compatible with the experimental results. However, little information can be gained by use of Eqs. 67 and 68. The experimental results are presented so as to make ij appear to be equal to (Pen)jm, where the exponent varies from component to component but the argument does not change. The coefficient A; depends on the solid and fluid phases. The exponent depends in some complex way on the pore geometry. As previously noted, theoretical studies of dispersion have resulted in exponents of either 1 or 2. It is proposed that the exponent is a function of the angle between the flow and major principal axis of the permeability tensor, of the directness or tortuosity of the flow channels, and of the component itself. A porous medium with fairly straight flow channels and the flow and major permeability parallel would have an exponent in the expression for the longitudinal component of the dispersion coefficient which would be near 2. The lateral component in the same medium would have a smaller exponent. On the other extreme, a perfectly random packing should have an exponent near 1 in the longitudinal component and have an exponent less than 1 in the lateral component. -49 5. C O N C L U S I O N S The original plan of this study had been to take the existing theory of dispersion in isotropic porous media and adapt it to aniso- tropic porous media by use of the transformation commonly applied in flow problems. The statement that Dij = Aijlekl was recognized as an hypothesis, not as an established fact. The early efforts at analyzing the experimental results pointed to severe difficulties with this relation. After some reconsideration of the proper parameters to use in analyzing the data, the contradiction between this theory and the experimental evidence became more apparent, as is described in section 4.3. Thus the first conclusion is that the expression above, combined with an exponential plot of experimental results is not correct. The theory might be saved by using a polynomial relation for Fk1 in terms of the velocity. This is compatible with the theories for one- dimensional dispersion, but it is contrary to the work of other experimenters in this area. A more positive conclusion is the verification of the assertion by Harleman and Rumer that the exponents in the flow factor are different for different components. This is, in fact, the main reason the original theory fails. An effort to devise a combination of two tensors that would i . . represent D j as given by experiment was then necessary. This resulted in Eq. 63: Dij = Aéij The relation between these three tensors is quite different from the relation of Eq. 27. Now the flow factor is directly identified with the dispersion coefficient in one index and is identified with only one of two indices on the medium tensor. The medium tensor is symmetric in an isotropic medium, which is less restrictive than the requirement in the previous theory that the fourth rank tensor be isotropic. Several statements about the factors affecting the exponent on the flow factor can be made from the experimental results. In a: power relation like the present one, the exponent is the dominating term, especially since the argument is the same in all components. It is postulated that the exponent depends on three things: (1) angle between the flow and major permeability, (2) flow path geometry through the pore space, (3) the component. More conclusive results must wait upon further research. The problem is very complex and many ways are open to examination. Several possibilities offer some chance of success, including: (1) experiments similar to the ones in this study in different media, either isotropic or anisotropic, (2) dispersion experiments using the Hele-Shaw model of the porous medium, (3) mathematical investigation of the nature of the dispersion process. Whatever the direction of experimental work, the use of more precise instrumentation is required. This is particularly true when physical properties other than diffusivity are excluded since this limits the tracer concentrations to very small values. Quantitative practical application of the theoretical and empirical studies of the fundamental nature of the dispersion process in flow through porous media is still in the future. The immediate need -51 is for research to provide a definite indication of the proper theory to describe the dispersion process. After the correct theory is determined, emphasis can be placed on the properties of porous media; especially of natural deposits where the significant dispersion related problems occur. APPENDICES -52 -53 A P P E N D I X Least Squares Analysis of Data The existing theories postulate and the available experiments support the correctness of a relation such as D = AV“. Following this trend, the present data is presented on log-log plots and a straight line is fitted through the data. Due to the spread of the data, the least squares method was selected to determine the equation of the plotted lines. The range of validity of these lines should be noted carefully. If the Peclet number is less than about 0.1 (Collins, 1961) the ratio of D/Dm is constant. Consequently for Peelet numbers between 0.1 and about 10 the relation is non-linear as the slope increases from zero to a value between 1 and 2 as the Peclet number increases. The upper limit for these lines is not clearly defined, but will depend on a transition from laminar to turbulent diffusion. The least squares analysis of each set of data is carried out on the logarithmic form of the exponential relation: .1 ,1 log D/Dm = logA + nLogPe (69) The simultaneous equations for logA and n are: NlogA + nilogPe = Zlog(D/Dh) (70) log AIflogPe-+ n2(logPe)2 = 2KlogPelogD/Dm) -54 Where: A = coefficient of Pen n = exponent on Pe N = number of data points The lines determined by these equations will be the best fit in a least squares sense; that is, the sum of the squared differences between the data and the line is minimized. However, for some of the graphs, especially Figs. 4—3b, 4-5a, and 4-6a, inspection would suggest that a steeper slope would provide a better fit than the least squares line. In order to verify the least squares lines and check the visually fitted lines, the sum of the squared differences were calculated. For each graph the least squares line, the visual line and at least one other line were selected for this computation. All lines pass through the centroid of the data so selection of the slope determined each line. The results are presented in terms of the root mean square difference, which is simply the square root of the mean of the squared differences. Tables A-l to A-4 list the slope of each line, the actual value of the root mean square difference (R), and the ratio of each of these to the respective value for the least squares line. The relative results are presented graphically in Figs. A-l to A-4. As an indication of the shape of the curves in these figures, the difference between the high and low slope ratio at a root mean square difference 5 percent higher than the minimum is also given in Tables A-l to A-4. From this computation, the lines in Figs. 4-3a, 4-4b, 4-5b, and 4-6b are relatively good fits as a moderate change in slope causes a significant increase in the difference. The reason for the steeper visually chosen slopes in several cases is also evident. -55 For instance, in Fig. 4-3b, the visually selected slope of 1.85 is 1.37 times the least squares line slope of 1.35 but the root mean square difference is increased by only 2 percent. In Fig. 445a,'the visually selected line has a slope of 1.30 times the least squares slope and increases the difference by 5 percent, and in Fig. 4-6a a slape of 1.52 times the least square slope is needed to increase the difference by 5 percent and the visual slope of 2.15 is 1.89 times the least square slope yet has a difference just 15 percent larger. However, from Fig. A-3, the line chosen for Fig. 4-4a by comparison with the other fitted lines is not a good fit in the least squares sense. This completes the discussion of the quality of the fitted lines. There are no gross errors in the least squares analysis though the minimum is not particularly sharp for half of the graphs. The difference between the least squares lines and visually selected lines points out a difference between the methods. The least squares method satisfies a mathematical condition while the visually method tries to put the line as close to as many points as possible, especially near the ends of the range. The significance of the lines, especially of the slopes, is discussed in the section 4.3. Even if the spread of the data raises some question about the numerical value of some of these slopes, the fact remains that they differ from bed to bed and from orientation to orientation. T A B L E 4 - l P A R A M E T E R S I N L E A S T S Q U A R E S ANALYSIS OF TRACER DISTRIBUTION 2 4 N EX IX D 3 2 2 2 5 10 34 70 7 28 196 588 -56 -57 T A B L E 4 - 2 S U'M M A R Y O F E X P E R I M.E N T A L W 0 R K Bed Number 1 2 3 4 d, cm 0.0508 0.0102 0.0509 0.0102 Orientation H H V V P 0.30 0.35 0.35 0.41 ‘55? 0.22 0.26 0.26 0.32 k, cm2 1.01(10)‘5 0.18(10)"5 0.76(10)'5 0.058(10)'5 Re, min 0.15 0.008 0.16 0.014 Re, max 1.30 0.20 1.54 0.085 Pe, min 426 15 370 27 Fe, max 3550 397— 3000 151 T, °c, min 22.5 24.4 23.4 23.1 T, °c, max 23.4 27.5 27.0 24.6 T A B L E 4 - 3 D A T A F O R B E D 1. Run oT, u Re t Pe 25' By No. C cm/sec sec Dm Dm DT-7 23.0 0.184 0.96 515 2685 2310 824 DT-lO 23.0 0.138 0.75 510 2100 4060 960 DT-ll 23.0 0.136 0.74 690 2070 1060 1020 DT-12 23.0 0.135 0.74 737 2070 4070 802 D-3 22.5 0.050 0.27 2070 '774 308 ~231 D-5 22.5 0.087 0.46 1165 1320 625 1210 D-6 22.9 0.118 0.64 895 1790 2020 4800 D-7 23.1 0.174 0.95 605 2680 2500 5030 D-8 23.4 0.233 1.30 445 3550 9690 2830 D-9 23.3 0.199 1.09 505 2980 4760 4660 D-ll 23.0 0.095 0.52 1060 1450 1750 3170 D-12 23.0 0.055 0.30 1810 838 263 317 D-13 23.7 0.027 0.15 3865 426 227 136 -58 TABLE 4-4 0 A T A F 0 R B E D 2. 3:“ 0T cm/Zec Re 52c Pe g_:- %% 0-1 24.4 0.0576 0.065 1800 136 109 167 0-2 27.5 0.0065 0.008 10J80 15 72 15 D-3 27.1 0.0113 0.014 9120 27 327 73 0-5 25.7 0.0287 0.034 3600 70 342 109 D-6 26.0 0.0612 0.072 1675 148 494 148 D-8 26.1 0.1096 0.13 932 264 483 408 0-9 26.6 0.1698 0.20 607 397 1280 638 0-11 25.4 0.0696 0.080 1485 167 606 650 0-12 25.5 0.0387 0.045 2683 93 206 76 -59 T A B L E 4 - 5 -60 D A T A F O R B E D 3; Run u Re t Pe Ex By No. cm/sec sec Dm Dm DT-l 24.0 0.131 0.72 790 1630 1850 1440 DT-2 24.0 0.130 0.72 805 1630 222 658 D-l 27.0 0.070 0.42 1495 817 362 264 D-3 27.0 0.162 0.97 636 1890 3870 506 D-4 27.0 0.210 1.26 487 2450 1090 732 D-S 27.0 0.258 1.54 400 3000 2910 722 D-6 27.0 0.177 1.06 580 2060 1890 1470 D-7 23.9 0.151 0.84 680 1920 12,250 496 D-9 23.4 0.057 0.31 1820 735 366 171 D-ll 24.8 0.224 1.27 460 2750 782 875 D-12 23.7 0.029 0.16 3526 370 70 93 T A B L E 4 - 6 D A T A F 0 R E D Run 0 u Re t Pe 2§_ 2y No. cm/sec sec Dm Dm D-l 23.6 0.0662 0.072 1555 133 4400 178 D-2 24.2 0.0532 0.058 1930 108 760 117 D-3 23.4 0.0200 0.022 5152 42 82 49 D-4 23.8 0.0429 0.047 2400 88 279 88 D-5 23.1 0.0312 0.134 3310 66 133 79 D-6 23.2 0.0127 0.14 8210 27 170 39 D-7 23.9 0.0652 0.072 1569 133 161 113 D-8 23.3 0.0355 0.038 2891 73 459 64 D-lO 24.6 0.0511 0.057 2005 102 294 120 D-ll 24.6 0.0758 0.085 1339 151 1480 205 D-12 23.1 0.0224 0.024 4623 47 775 88 -61 -62 T A B L E 4 - 7 E49 U A T I O N S O F L I N E S F I T T O D A T A B Y L E A S T S Q,U A R E S M E T H O D Bed and Direction Equation 1-x a3 = 0.00171361'85 l-y 3% = 0.058Pe1'35 2-x % = 0.647Pe1'35 2-y %%-= 1.19Pe1°05 3-x %§'= 0.0074Pe1°62 3-y % = 0.127Pe1'14 4-x %§-= 2.74Pe1'14 4-y 3% = 2.54peo'83 £35312: x 0.191Pe1'18 0 65 y 0.55Pe ' TABLE 4-8 ANISOTROPIC PERMEABILITIES - .1 Bed kI, cm2 kII,cm2 Kr ’ k 11 -5 -5 1 1.014(10) 0.476(10) 2.14 2 0.178(10)‘5 0.036(10)’5 4.95 3 1.613(10)’5 0.757(10)’5 2.14 4 0.28700)“5 0.058(10)‘5 4.95 TRANSFORMED TABLE 4-9 DISPERSION -64 COEFFICIENTS 95 BI 21 2;; Bed Dm Dm Dm Dm 1 0.00171361'85 0.0017861'85 0.058Pe1'35 0.124Pe1°35 3 0-0074Pel'62 0.162Pe1'62 0.127Pe1'14 0.1271>e1'14 2 “647139135 “647139135 1.19Pe1'05 5.89Pe1'05 4 2.74p61'14 13.57pe1°14 2.541,e0.83 2541,8083 T A B L E A - 1 SUMMARY OF LEAST SQUARES CHECK COMPUTATION -65 Bed 1 Orientation X Quantity Value Ratio Value Ratio 1.35 0.73 0.85 0.63 310 e S 1.85 L.S. 1.00 1.35 L.S. 1.00 P ’ 2.34 eye 1.27 1.85 eye 1.37 2.00 1.48 Root mean 0.54 1.15 0.82 1.13 square 0.47 1.00 0.73 1.00 difference, R 0.57 1.21 0.74 1.02 0.76 1.05 Spread of slope ratio 0.40 0.78 at R/RLS = 1.05 T A B L E A - 2 SUMMARY OF LEAST SQUARES CHECK COMPUTATION -66 Bed 2 Orientation X Quantity Value Ratio Value Ratio 0.50 0.83 0.87 0.83 810 e S 0.60 L.S. 1.00 1.05 L.S. 1.00 P ’ ' 0.96 1.60 1.24 eye 1.18 1.35 eye 2.25 Root 0.60 1.02 0.50 1.07 mean 0.59 1.00 0.47 1.00 square 0.68 1.15 0.49 1.03 difference, R 0.90 1.52 Spread of slope ratio 0.58 0.36 at R/RLS = 1.05 T A B L E A - 3 SUMMARY OF LEAST SQUARES CHECK COMPUTATION Bed 3 -67 Orientation Quantity Value Ratio Value Ratio 1.28 0.79 0.93 0.82 Slope, S 1.62 L.S. 1.00 1.14 L.S. 1.00 2.10 eye 1.30 1.41 eye 1.24 Root mean 1.01 1.02 0.49 1.20 square difference, 0.98 1.00 0.41 1.00 R 1.03 1.04 0.44 1.08 Spread of lepe ratio 0.60 0.36 at R/RLS = 1.05 T A B L E A - 4 SUMMARY OF LEAST SQUARES CHECK COMPUTATION -68 Bed 4 Orientation X Quantity Value Ratio Value Ratio 1.00 0.88 0.66 .80 810 e S 1.14 L.S. 1.00 0.83 L.S. .00 P ’ 1.50 1.32 1.00 eye .20 2.15 eye 1.89 Root 0.99 1.04 0.22 .09 mean 0.95 1.00 0.20 .00 square 0.97 1.02 0.22 .08 difference, R 1.09 1.15 Spread of slope ratio 0.65 .34 at R/RLS = 1.05 -69 com woouom mo unsung Hopscoo Hun .wgh mama owmuoum muouoEonmfla mummz x a . . _ _ _ . _ _ _ // _ . _\ _ _ o>Hm> mwcoauomao xcmH new: ucmuwcoo 3onuso Jena noon unnumcoo BOHMCH Fig. 3-3 Porous Bed and Electrode Group -71 0.‘ q 4.. x04. . IV‘ . IV. .0 ..Lu- on. a! an O I a E. O at‘“. .vasoo Fig. 3-4 Inlet Constant Head Tank Fig. 3-5 Outlet Constant Head Tank -72 umm Bonuoo no Emmuumc3on O O O O O 3 maoouo onouuooam cum .me uom soHMCH no Emmuumms Am 0 so 24 EU NN.H I O O O O O Eu moH -73 101 100 10'1 1/' c, 619 10’2 g’ l, 10'3 10'1 100 101 102 10 -l-'micromhos/cm p25 Fig. 4-1 Concentration of NaCl in water is a function of specific conductance T = 25°C 1.9 1.7 (10)°Dm ,r””””" 2 C111 SEC 1.5 1.3 20° 22° 24° 0 26° 28° 30 T, c 1200 5 (10) 0 1000 1..\ 2 h‘nun...-~ fl. sec -““--—~n 800 O O O O O 20 22 24 26 28 30 T, °c Fig. 4-2 Molecular Diffusivity and Kinematic Viscosity 515’ -75 10S 104 .. U .37 0 0 3 10 + / ,g9 (21 C) 0 102 102 103 10 Pe Fig. 4-3a Bed 1 gar versus Pe , 5'1? -76 10 10 G 10 [/7 <9 10 102 103 Fe Fig. 4-3b Bed 1 versus Pe SIS 10 SI? 10 10 10 10 -77 10 1 102 Fe Fig. 4-4a Bed 2 g—E versus Fe 10 SF? -78 Pe Fig. 4-4b Bed 2 13% versus Fe 104 .1..- ...- 103 G /9 / o 104 {9’ I ‘ O / H O 101 101 102 10 10 10 C X 10 10 -79 II o // f o 10 3 10 Pe . 4-Sa Bed 3 P1 versus Pe 1h 10 10 E13 10 10 -80 ./ 10 Fig. 4-5b Bed 3 21 Dm 10 Fe versus Fe 10 10 10 El? 10 10 0 \° 10 10 Pe Fig. 4-6a Bed 4 % versus Fe 10 -82 104 ._ 103 22 on Q 0 0 102 07‘! 0 O / fi/3 10 1 10 1 102 10 Fe Fig. 4-6b Bed 4 1% versus Pe 10 10 Bed 1 Note: In all cases 2 3 the upper line is D /D , 4 x m Harleman &—— — — — — — — the lower line is D /Dm' Rumer Y 10 103 D— D11! 102 101 10° Pe Fig. 4-7 Composite Plot of Data 1.30 1.20 R/R LS 1.10 1.00 1.20 11/11LS 1.10 1.00 s19 ° )1 /E] I .5 1.0 1 n/°1.s .5 Fig. A-l Bed 1 R/RLS versus n/nLS \L/ ' 7 / SI? 1. 1 n/“Ls .5 Fig. A-I Bed 3 R/KLS versus n/nLS -85 1.50 19 135 1.40 /\/L D‘“ \ 1.20 —°\/\/\ R/RLS 1.10 Q 22 Dm \ 1.00 0 0.5 1.0 1.5 2.0 “”15 Fig. A-3 Bed 2 R/RLS versus n/nLS 1.10 _ ' - f) ‘1 59% 3%.?- MLS \M / 1.00 0 0.5 1.0 “”15 1.5 2.0 Fig. A-4 Bed 4 R/RLS versus n/nLS -86 B I B L I O G R A P H Y Aris, R., Vectors, Tensors, and the Basic Equation of Fluid Mechanics, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. Bear, J., "On the tensor form of dispersion in porous media," Journal of Geophysical Research, Vol. 66, No. 10, 1961. Bear, J., and Todd, D. K., "The transition zone between fresh and salt waters in coastal aquifers," Contribution No. 29, Water Resources Center, University of California, 1960. Bell, J. R. and Grosberg, P., "Diffusion through porous materials," Nature, Vol. 198, No. 4769, March, 1961. Benson, M. A., "Spurious correlations in hydraulics and hydrology," Journal of the Hydraulics Division, Proceedings ASCE, Vol. 91, No. HY 4, July, 1965. Carrier, G. F., "The mixing of ground water and sea water in permeable subsoils," Journal of Fluid Mechanics, Vol. 4, No. 5, 1958. Collins, R. E., Flow of Fluids Through Porous Materials, Reinhold Publishing Corporation, New York, 1961. Harleman, D. R. F., Mehlhorn, P. F., and Rumer, R. R. Jr., "Dispersion permeability correlation in porous media," Journal of the Hydraulics Division, Proceedings ASCE, Vol. 89, No. HY2, May, 1963. Harleman, D. R. F., and Rumer, R. R. Jr., "The dynamics of salt-water intrusion in porous media," Report No. 55, Hydromechanics Laboratory Massachusetts Institute of Technology, August, 1962. Harr, M. E., Groundwater and seepage, McGraw-Hill Book Co., Inc. New York, 1962. Hawley, M. C., "Solute dispersion in liquid-solid chromatographic columns," Thesis for the degree of Ph.D., Michigan State University, 1964. Henry, H. R., "Salt water intrusion into fresh water aquifers," Journal of Geophysical Research, Vol. 64, No. 11, 1959. Henry, H. R., "Salt Intrusion into Coastal Aquifers," Commission of subteranean water, International Association for Scientific Hydrology, Publication No. 52, 1960. A -87 B I B L I O G R A P H Y (continued) Hildebrand, F. B., Advanced Calculus for Engineers, Prentice-Hall, Inc., New York, 1949. De Josselin de Jong, G., "Longitudinal and transverse diffusion in granular deposits," Transactions of the American Geophysical Union, V0. 39, 1958. De Josselin de Jong, G., "Vortex theory for multiple phase flow through porous media," Contribution No. 23, Water Resources Center, University of California, 1959. Kohout, F. A., "Cyclic flow of fresh water in the Biscayne aquifer of northeastern Florida," Journal of Geophysical Research, Vol. 65, 1960. Lin, K. M., "Transient tw0phase flow through porous media," Thesis for degree of Ph.D., Michigan State University, 1964. Muskat, M., Flow of Homogeneous Fluids through Porous Media, J. W. Edwards, Inc., Ann Arbor, Michigan, 1946. Ogata, A., "Transverse diffusion in saturated isotropic granular media," Professional Paper No. 411-B, U. S. Geological Survey, 1961. Ogata, A. and Banks, R. B., "A solution of the differential equation of longitudinal dispersion in porous media," Professional Paper No. 411-A, U. S. Geological Survey, 1961. Polubarinova-Kochina, P. Ya., Theory of Groundwater Movement, Translated from Russian by J. M. R. De Wiest, Princeton University Press, 1962. von Rosenberg, D. U., ”Mechanics of steady state single-phase fluid dis- placement from porous media, Journal, AIChE, Vol. 2, No. l, 1956. Saffman, P. G., "A Theory of dispersion in porous media," Journal of Fluid Mechanics, Vol. 6, 1959. ‘ Saffman, P. G., "Dispersion due to Molecular diffusion and macroscopic mixing in flow through a network of capillaries," Journal of Fluid Mechanics, Vo.. 7, 1960. Scheidegger, A. E., "Statistical hydrodynamics in porous media," Journal of Applied Physics, Vol. 25, No. 8, 1954. Scheidegger, A. B., The Physics of Flow Through Porous Media, The Mac- Millan Company, New York, 1960. Scheidegger, A. B., "General theory of dispersion in porous media," Journal of Geophysical Research, Vol. 66, No. 4, 1961. -88 B I B L I 0 G R A P H Y (continued) Simpson, E. 8., ”Transverse dispersion in liquid flow through porous media," Professional Paper No. 4ll-C, U. 8. Geological Survey, 1962. Taylor, G. I., "Dispersion of soluble matter in solvent flowing slowly through a tube," Proceedings, Royal Society of London, Vol. 219, Series A, 1953. Todd, D. K., Groundwater Hydrology, John Wiley & Sons, New York, 1959. Wentworth, C. K., "Growth of the Gyben-Herzberg transition zone under a rinsing hypothesis," Transactions American Geophysical Union, Vol. 29, No. 1, 1948.