CONTAMINANT HYDROGEOLOGY FETTER PDF
Page 1. "Fetter, C. W.() “Contaminant Hydrogeology”,. P Page 2. P Page 3. P Page 4. P Page 5. P Get this from a library! Contaminant hydrogeology. [C W Fetter]. Library of Congress Cataloging-in-Publication Data. Fetter, C. W. (Charles Willard). Applied hydrogeology / C.W. Fetterth ed. p. cm. Includes bibliographical.
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CONTAMINANT. HYDROCEOLOGY. C. IV. FETTER. Contaminant Hydrogeology. Contaminant Hydrogeology. C. W. Fetter Department of Geology University of. Request PDF on ResearchGate | Contaminant Hydrogeology - 3rd Edition of those contaminants while building on Fetter's original foundational work. Hydrogeology (2nd Edition) By C.W. Fetter Contaminant Hydrogeology - 3rd Edition | Request. PDF Contaminant Hydrogeology is intended as.
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Find a copy in the library Finding libraries that hold this item Details Additional Physical Format: Print version: Fetter, C. Charles Willard , Contaminant hydrogeology. If the hydraulic conductivity is the s a m e in all direc x. We first need to define s o m e terms relating to t e n s o r s. A zero-order tensor, also called a s c a l e r , is a quantity characterized only by its size or magnitude. E x a m p l e s in hydro geology include hydraulic head, chemical concentration, and temperature.
A first-order. Chapter One tensor, or v e c t o r , is a quantity that has both a magnitude and a direction. Vectors require three c o m p o n e n t s , each having a magnitude and direction. Velocity, specific discharge, mass flux, and heat flux are e x a m p l e s.
E x a m p l e s in hydrogeology are intrinsic permeability, hydraulic conductivity, thermal conductivity, and hydrodynamic dispersion. T h e hydraulic head is a scaler. However, the gradient of the head is a vector as it has b o t h a magnitude and a direction.
T h e gradient of h is designated as grad h:. An equivalent notation is the use of the vector differential operator, del, which has the symbol V. This operator is equivalent to.
Another vector is the specific discharge, q. Associated with any vector is a positive scaler with a value equal to the magnitude of the vector. If q is the magnitude of the vector q, this can be e x p r e s s e d as. A s e c o n d - o r d e r tensor, such as K, hydraulic conductivity, can be described by nine c o m p o n e n t s. If the c o o r d i n a t e system is oriented along the principal axes, the tensor b e c o m e s 0.
F o r the special c a s e w h e r e we orient the a x e s of the x, y, and z c o o r d i n a t e system with the three principal directions of anisotropy, K is the matrix shown in 1. If we multiply two vectors together and the result is a scaler, then the product is called a d o t p r o d u c t , o r i n n e r p r o d u c t. B a s e d on grad h, we can find a velocity vector v such that the magnitude and direction vary throughout the p o r o u s media.
If we apply the del o p e r a t o r to v, we obtain the following:. If we apply the del o p e r a t o r to grad h, the result is the s e c o n d derivative of head: In o t h e r words, the mass entering the REV less the mass leaving the REV is equal to the c h a n g e in mass storage with time.
T h e representative elementary volume is shown on Figure 1. T h e three sides have length dx, dy, and dz, respectively. T h e area of the two faces normal to the x axis is dy dz, the area of the faces normal to the y axis is dx dz, and the area of the faces normal to the z axis is dx dy. These are length, L mass, M, and time, T. T h e fundamental dimensions for density' are mass per unit volume. Volume is length cubed, so the shorthand expression for the fundamental dimensions of density is M t.
Specific discharge has the dimensions of velocity, so the fundamental dimensions are I. T, and area has fundamental dimensions 3. T h e net mass accumulation within the control volume due to the flow c o m p o n e n t parallel to the x axis is the mass inflow minus the mass outflow, or.
Similar terms exist for the net mass accumulation due to flow c o m p o n e n t s parallel to the y and z axes:.
T h e s e three terms can be s u m m e d to find the total net mass accumulation within the control volume. F r o m the law of conservation of mass, Equation 1. U n d e r this assumption Equation 1.
We may substitute Darcy's law for the specific discharge c o m p o n e n t s given on the left side. If the xyz c o o r d i n a t e system is aligned with the principal a x e s of anisotropy, then Equation 1.
T h u s the c h a n g e in the volume of water in storage is proportional to the c h a n g e in head with time. T h e right side of Equation. Another form of e x p r e s s i o n is called Einstein's summation notation.
For example. It is implied in the preceding equation that the specific discharge is parallel to the direction of dlj dl and that the medium is isotropic. Grad h is a vector that we will call h. Hydraulic conductivity. To d e s c r i b e Darcy's law in the most general form, we need three equations.
In Einstein's summation notation, the is dropped with the understanding that the summation is over the repeated indices: In general, we will use the standard form of differential equations rather than any of the shorthand notation. However, the literature cited in this text often uses the c o m p a c t forms and the reader should be aware of them. References American Water Resources Association. Status of waterb o r n e diseases in the U.
Journal of American Water Works Association 67, no. Ames, B. Ranking possible carcinogenic hazards. Bouwer, Herman. Groundwater hydrology. New York: McGraw-Hill B o o k Company, pp. Burmaster, D. Groundwater contamination, an emerging threat. Technology Review 85, no. Darcy, Henry. Les fontaines publiques de la ville de Dijon. Victor Dalmont, pp. Eckhardt, D. A, and E. Geological Survey Water Supply Paper , pp. Fetter, C. T h e c o n c e p t of safe groundwater yield in coastal aquifers.
Water Resources Bulletin 8, no. Applied' Ijydrogeology.
Contaminant cittadelmonte.info | Environmental Remediation | Solvent
Columbus, Ohio: Merrill Publishing Company, 5 8 8 pp. Gordon, J a m e s. OMNI Engineers, personal communication. Hanmer, Rebecca. Environmental Protection Agency, Testimony in a hearing on the seriousness and extent of ground water contamination before the Senate Subcommittee on Superfund, Ocean and Water Protec tion of the Committee of Environment and Public Works, August 1, Hess, K. Hindall, S. M and Michale Eberle. National and regional trends in water well drilling in the United States, 1 9 6 4 - 8 4.
Geological Survey Circular , 15 pp. Hoffman, J. Water softener salt: Geological Society of America, Abstracts with Programs 10, no. Hughes, J. L Evaluation of ground-water degradation resulting from waste disposal to alluvium near Bar stow, California. Geological Survey Professional Paper 8 7 8.
Hult, M. In National Water Summary, Konikow, L F. Groundwater contamination and aquifer reclamation at the Rocky Mountain Arsenal, Colorado. In Groundwater Conlamination. Washington, D. National Academy Press.
Lehr, J. Toxicological risk assessment distortions: Part I. Ground Water 28, no. Miller, David. Waste disposal effects on ground water. Berkley, Calif.: Premier Press, pp.
Nash, G. Discussion of a paper by E. Proceedings of the Society of Water Treatment and Examination Office of Technology Assessment. Protecting the na lion's groundwater from contamination. Congress, OTA 0 - 2 7 6 , two volumes. Peterson, N. Waste Age March 1 9 8 3: Solley, W. Merk, and R. Estimated use of water in the United States in Geological Survey Circular , 82 pp.
Spanggord, R J. R Mabey. Department of Energy. Site characterization plan overview. Office of Civilian Radioactive Waste Management, pp. Surface impoundment assessment national report. Office of Drinking Water. National u aler quality inventory. Nuclear Regulatory Commission. Water News. At Jl billion, Colorado arsenal is costliest cleanup yet. L Troise, and D. The water encyclopedia, Chelsea, Mich: Lewis Publishers. Wilson, Richard, and E. Risk assessment and comparisons: An introduction.
Wollenhaupt, N. C, and R.
Atrazine in groundwater: A current perspective. Introduction In this c h a p t e r we will c o n s i d e r the transport of solutes dissolved in g r o u n d water. T h e s e equations are similar in form to the familiar partial differential equations for fluid flow.
In recent years much work has b e e n d o n e on the theories of mass transport in r e s p o n s e to the great interest in p r o b l e m s of ground-water contamination. O n e of the o u t c o m e s of this has b e e n the d e v e l o p m e n t of what is essentially a n e w b r a n c h of subsurface hydrology, w h e r e the flow of fluid and solutes is treated by statistical models; these m o d e l s can a c c o u n t for the role of varying hydraulic conductivity that a c c o m p a n i e s aquifer heterogeneity.
Very recently fractal g e o m e t r y has b e e n used to d e s c r i b e the solute transport b a s e d on the c o n c e p t that aquifer heterogeneities have repeating patterns. Diffusion will o c c u r as long as a concentration gradient exists, even if the fluid is not moving. T h e negative sign indicates that the movement is from areas of greater concentration to t h o s e of lesser concentration.
For systems w h e r e the concentrations are changing with time, F i c k ' s s e c o n d l a w applies. In o n e dimension this is 2. In p o r o u s media, diffusion c a n n o t p r o c e e d as fast as it can in water b e c a u s e the ions must follow longer pathways as they travel around mineral grains. T o r t u o s i t y is a measure of the effect of the shape of the flowpath followed by water molecules in a p o r o u s media. Tortuosity in a p o r o u s media e. Flowpaths across a representative sample of a well-sorted sediment would tend to be shorter than those a c r o s s a poorly sorted sediment in which the smaller grains were filling the voids between the larger grains.
Thus the well-sorted sediment would tend to have a lower value for tortuosity than the poorly sorted sediment. With this definition, e. This definition will not be used in this text. T h e value of to, which is always less than 1, can be found from diffusion experiments in which a solute is allowed to diffuse a c r o s s a volume of a p o r o u s medium.
According to F r e e z e and Cherry 1 9 7 9 , to ranges from 0. Diffusion will c a u s e a solute to spread away from the place where it is introduced into a p o r o u s medium, even in the a b s e n c e of ground-water flow.
Figure 2. T h e solute concentration follows a normal, or Gaussian, distribution and can be 2. T h i s is an alternative definition of effective diffusion coefficient to the o n e given in Equation 2. T h e p r o c e s s of diffusion is c o m p l i c a t e d by the fact that the ions must maintain electrical neutrality as they diffuse. If we have a solution of NaCl, the N a. F o r e x a m p l e , solid waste containing a high c o n c e n t r a t i o n of chloride ion may be placed directly on the clay liner of a landfill.
T h e c o n c e n t r a t i o n of chloride in the leachate contained in the solid waste is so much greater than the c o n c e n t r a t i o n of chloride in the p o r e water of the clay liner that the latter may be c o n s i d e r e d to be z e r o as a simplifying assumption in determining a conservative estimate of the m a x i m u m diffusion rate.
If the solid waste and the clay are b o t h saturated, the chloride ion will diffuse from the solid waste, w h e r e its c o n c e n t r a t i o n is greater, into tlie clay liner, even if there is no fluid flow. T h e c o n c e n t r a t i o n of chloride in the solid waste, CQ, will be assumed to be a constant with time, as it can be replaced by dissolution of additional chloride. T h e c o n c e n t r a t i o n of chloride in the clay liner, Q x, t , at s o m e distance x from the solid waste interface and s o m e time t after the waste was placed, can be determined from Equation 2.
This is a solution to Equation 2. This m e a n s that the solution described by Equation 2. T h e standard deviation is the square root of the variance. T h e complementary error function is tabulated in Appendix A. F o r s o m e applications it may be necessary to find erfc of a negative number. Appendix A d o e s not give values for e r f c B for negative values of B. O for 3.
Insert values into Equation 2. In yr, diffusion over a 5-m distance would yield a c o n c e n t r a t i o n that is 0. F r o m the p r e c e d i n g e x a m p l e p r o b l e m it is obvious that diffusion is not a particularly rapid m e a n s of transporting dissolved solutes.
Diffusion is the p r e d o m i n a n t m e c h a n i s m of transport only in low-permeability hydrogeologic regimes. However, it is p o s s i b l e for solutes to m o v e through a p o r o u s or a fractured medium by diffusion even if the ground water is not flowing. Transport by Advection Dissolved solids are carried along with the flowing g r o u n d water.
F o r one-dimensional flow normal to a unit cross-sectional area of the p o r o u s media, the quantity o f water flowing is equal to the the. It is n o t the average rate at which the water m o l e c u l e s are moving along individual Ilowpaths, which is greater than the average linear velocity due to tortuosity.
T h e one-dimensional mass Bux, l ' , d u e to advection is equal o t h e quantity o f x. T h e derivation of this equation is given in Section 2.
Si ilution of the advective transport equation yields a sharp concentration front. On the advancing side o f the front, the concentration is equal t o that o f the invading ground water, whereas on the o t h e r side of the front it is unchanged from the background value I hi-, is known as p l u g H o w , with ill the x i r e fluid being replaced by the invading solute front.
The sharp interface that results from plug flow is shown in Figure 2. If o n e obtains a sample of water for p u r p o s e s of monitoring the spread of a dissolved contaminant from a b o r e h o l e that penetrates several strata, the water sample will be a c o m p o s i t e of the water from e a c h strata.
Due to the fact that advection will transport solutes at different rales in e a c h stratum, the c o m p o s i t e sample may be a mixture of water containing the transported solute c o m i n g from o n e stratum and uncontaminated ground water c o m i n g from a different stratum w h e r e t h e average linear velocity is lower. T h e concentration of the contaminant in the c o m p o s i t e sample would thus be less than in the source.
Fetter, Applied Hydrogeology, 2d ed. Macmillan Publishing Company, 1 9 8 8. Mechanical Dispersion G r o u n d water is moving at rates that are b o t h greater and less than the average linear velocity. T h e s e factors are illustrated in Figure 2. If all g r o u n d water containing a solute were to travel at exactly the s a m e rate, it would displace water that d o e s n o t contain the solute and create an abrupt interface b e t w e e n the two waters. However, b e c a u s e the invading solute-containing water is n o t all traveling at the s a m e velocity, mixing o c c u r s along the flowpafh.
T h e mixing that o c c u r s along the direction of the flowpath is called longitudinal dispersion. An advancing solute front will also tend to spread in directions normal to the direction of flow b e c a u s e at the p o r e scale the flowpaths can diverge, as shown in Figure 2.
T h e result of this is mixing in directions normal to the flow path called t r a n s v e r s e dispersion. If we assume that mechanical dispersion can be described by Fick's law for diffusion Equations 2. T h i s is equal to a property of the medium called dynamic dispersivity, or simply dispersivity, a, times the average linear velocity. If i is the principle direction of flow, the following definitions apply: Fetter, Applied Hydrogeology.
Macmillan Publishing Company,! Hydrodynamic Dispersion T h e p r o c e s s o f m o l e c u l a r diffusion c a n n o t b e separated from mechanical dispersion i n flowing g r o u n d water.
It is represented by the following formulas: T h e vertical line at V represents the advective transport without dispersion. Effects of diffusion and mechanical dispersion are shown. T h e p r o c e s s of hydrodynamic dispersion can be illustrated by Figure 2. T h e advecting g r o u n d water carries the mass of solute with it.
With this distribution the. The ground-water flow 0. Derivation of the Advection-Dispersion Equation for Solute Transport This derivation of the advection dispersion equation is based on work by Freeze and Cherry 1 9 7 9 , B e a r 1 9 7 2 , and Ogata 1 9 7 0 Working assumptions are that the p o r o u s medium is h o m o g e n e o u s , isotropic, and saturated with fluid and that flow conditions are such that Darcy's law is valid.
T h e derivation is based on the conservation of mass of solute flux into and out of a small representative elementary volume R E V of the p o r o u s media. A representative elementary volume is illustrated in Figure 1. Mass of solute per unit volume of aquifer is the product of the porosity, n , and C. Porosity is considered to be a constant e. T h e solute will be transported by advection and hydrodynamic dispersion. T h e negative sign indicates that the dispersive flux is from areas of greater to areas of lesser concentration.
T h e total a m o u n t of solute entering the representative elementary volume is F dzdy-Y. T h e total a m o u n t of solute leaving the representative elementary volume is SP F. T h e rate of mass c h a n g e in the representative elementary volume is BC n, Bt. By the law of mass conservation, the rate of mass c h a n g e in the representative elementary volume must be equal to the difference in the mass of the solute entering and the mass leaving.
Equation 2. In a h o m o g e n e o u s medium with a uniform velocity field, Equation 2. Diffusion versus Dispersion In the previous section the mass transport equation was derived on the basis of hydrodynamic dispersion, which is the sum of mechanical dispersion and diffusion.
It would have b e e n p o s s i b l e to separate the hydrodynamic dispersion term into the two c o m ponents and have separate terms in the equation for them. However, as a practical matter, under most conditions of ground-water flow, diffusion is insignificant and is neglected. It is p o s s i b l e to evaluate the relative contribution of mechanical dispersion and diffusion to solute transport. A P e c l e t n u m b e r is a dimensionless n u m b e r that can relate the effectiveness of mass transport by advection to the effectiveness of mass transport by either dispersion or diffusion.
Shown on this figure are the results of a n u m b e r of experimental measurements using sand c o l u m n s and tracers as well as s o m e experimental curves from several investigators Perkins and J o h n s o n In this manner the L. At very d. This shows up on the left side of Figure 2. In this z o n e diffusion is the predominant force, and dispersion can be neglected. Perkins and O.
Johnson, Society of Petroleum Engineers r. Although the curve has the s a m e shape as in a , it o c c u r s at Peclet n u m b e r s roughly 1 0 0 times greater. T h i s m e a n s that diffusion has m o r e c o n t r o l over transverse dispersion at higher Peclet n u m b e r s than it d o e s for longitudinal dispersion.
At higher Peclet n u m b e r s mechanical dispersion is the predominant cause of mixing of the contaminant plume Perkins and T. J o h n s o n ; B e a r ; B e a r and Verruijt 1 9 8 7 and the effects of diffusion can be ignored. Under these conditions D , can b e replaced with v , i n the advection-dispersion equations. T h e advection-dispersion equations can be solved by either numerical or analytical methods.
Analytical m e t h o d s involve the solution of the partial differential equations using calculus based on the initial and boundary value conditions. They are limited to simple geometry and in general require that the aquifer be h o m o g e n e o u s.
A n u m b e r of analytical solutions are presented in this chapter. They are useful in that they can be solved with a calculator and a table of e r r o r functions or even a pencil and paper, if o n e is so inclined Numerical m e t h o d s involve the solution of the partial differential equation by numerical m e t h o d s of analysis.
T h e y are m o r e powerful than analytical solutions in the s e n s e that aquifers of any geometry can be analyzed and aquifer heterogeneities can be a c c o m m o d a t e d. However, there can be other p r o b l e m s with numerical models, such as numerical errors, which can cause solutions to s h o w e x c e s s spreading of solute fronts or plumes that are n o t related to the dispersion of the tracer that is the subject of the modeling.
B e a r and Verruijt 1 9 8 7 present a g o o d introduction to the use of numerical m o d e l s to solve mass transport equations. In o r d e r to obtain a unique solution to a differential equation it is necessary to specify the initial and the boundary conditions that apply. T h e i n i t i a l c o n d i t i o n s describe the values of the variable under consideration, in this case concentration, at s o m e initial time equal to 0.
T h e b o u n d a r y c o n d i t i o n s specify the interaction between the area under investigation and its external environment. T h e r e are three types of boundary conditions for mass transport. A v a r i a b l e flux boundary constitutes the boundary condition of the third type. B o u n d a r y and initial conditions are shown in a shorthand form. For one-dimensional flow we n e e d to specify the conditions relative to the location, x, and the time, I.
By convention this is shown in the form. We could also have specified an initial condition that within the domain the initial solute concentration was q. This would be written as C x, 0. O t h e r e x a m p l e s of concentration first type boundary conditions are exponential decay of the s o u r c e term and pulse loading at a constant concentration for a period of time followed by another period of time with a different constant concentration.
Pulse loading w h e r e the concentration is q, for times from 0 to tg and then is 0 for all time m o r e than tg is e x p r e s s e d as.
T h e variable-flux boundary, a third type, is given as dC D ox. Sand c o l u m n e x p e r i m e n t s have b e e n used to evaluate b o t h the coefficients of diffusion and dispersion at the laboratory scale. A rube is filled with sand and then saturated with water. Water is m a d e to flow through the tube at a steady rate, creating, in effect, a permeameter.
A solution containing a tracer is then introduced into the sand c o l u m n. T h e initial concentration of the solute in the column is zero, and the concentration of the tracer solution is Q. This is called a fixed-step function. T h e solution to Equation 2. In nature there are not many situations w h e r e there would be a sudden change in the quality of the water entering an aquifer.
A much m o r e likely condition is that there would be leakage of contaminated water into the ground water flowing in an aquifer. F o r the one-dimensional case, this might be a canal that is discharging contaminated water into an aquifer as a line s o u r c e Figure 2.
T h e rate of injection is c o n s i d e r e d to be constant, with the injected mass of the solute proportional to the duration of the injection. T h e initial concentration of the solute in the aquifer is zero, and the concentration of the solute being injected is CoT h e solute is free to disperse both up gradient and down gradient.
Saufy, Wafer Resources Research 16, no. Copyright by the American Geophysical Union. Saury 1 9 8 0 gives an a p p r o x i m a t i o n for the o n e dimensional dispersion equation as C. Dimensionless-type curves for the continuous injection of a tracer into a one-dimensional.
Sauty, Wafer Resources Research 16, no. This approximation c o m e s about b e c a u s e for large Peclet numbers, the s e c o n d term of Equations 2. In Figure 2. Curves are plotted for three Peclet numbers, 1, 10, and This Peclet n u m b e r defines the rate of transport by advection to the rate of transport by hydrodynamic dispersion.
For Peclet n u m b e r 1, the fixed step function and the continuous-injection function give quite different results, whereas for Peclet n u m b e r 1 0 0 they are almost identical. T h e approximate solution lies midway between the other two.
This figure suggests that for Peclet numbers less than about 10, the exact solutions n e e d to be considered, whereas for Peclet numbers greater than 10, the approximate solution is probably acceptable, especially as the Peclet n u m b e r approaches This Peclet n u m b e r increases with flow path length as advective transport b e c o m e s m o r e dominant over dispersive transport.
Thus for mass transport near the inlet boundary, it is important to use the correct equation, but as o n e g o e s away from the inlet boundary, it is less important that the correct form of the equation is employed.
A solution for Equation 2. T h e third condition specifies that as A' a p p r o a c h e s infinity, the concentration gra dient will still be finite. Under these c o n d i t i o n s the solution to Equation 2. T h i s equation also reduces to the a p p r o x i m a t e solution, Equation 2. Breakthrough b e c o m e s m o r e symmetric with increasing.
If a tracer is continuously injected into a uniform flow field from a single point that fully penetrates the aquifer, a two-dimensional plume will form that l o o k s similar to Figure.
Saury, Wafer Resources Research 16, no. It will spread along the axis of flow due to longitudinal dispersion and normal to the axis of flow due to transverse dispersion. This is the type of contamination that would o c c u r due to leakage of liquids from a landfill or lagoon.
T h e mass transport equation for two-dimensional flow, Equation 2. B e a r 1 9 7 2 gives the solution to Equation 2.
T h e two-dimensional growth of a p l u m e from a c o n t i n u o u s s o u r c e can be tracked through time using a solution to Equation 2. Emsellem s e e Fried 1 9 7 5. If a slug of contamination is injected over the full thickness of a two-dimensional uniform flow field in a short period of time, it will move in the direction of flow and spread with time.
T h i s result is illustrated by Figure 2. T h e p l u m e that resulted from the field test is m o r e c o m p l e x than the laboratory p l u m e due to the heterogeneities e n c o u n t e r e d in the real world and the fact the p l u m e may not be following the diffusional model of dispersion. De J o s s e l i n De J o n g 1 9 5 8 derived a solution to this p r o b l e m on the basis of a statistical treatment of lateral and transverse dispersivities.
B e a r 1 9 6 1 later verified it experimentally. Experimental results from J. Bear, Journal of Geophysical Research 66, no. Vertically averaged chloride concentration at 1 day, 8 5 days, 4 6 2 days, and 6 4 7 days.
Mackay et al. Wafer Resources Research 2 2 , no. T h e lower the ratio, the b r o a d e r the shape of the resulting plume will be.
T h i s illustrates the fact that it is important to have s o m e k n o w l e d g e of the transverse dispersivity in addition to the. T h e r e is a paucity of data in the literature on the relationships of longitudinal to transverse dispersivities. In addition, dispersivity ratios based on field studies are b a s e d on fitting the diffusional model of dispersion to cases where it might not be applicable. Diffusion and dispersivity can be determined in the laboratory- using c o l u m n s packed with the p o r o u s media under investigation.
T h e results of column studies are often reported in terms of p o r e volumes of fluid that is eluted. O n e p o r e volume is the crosssectional area of the c o l u m n times the length times the porosity ALn. T h e unit discharge rate from the column is the linear velocity times the porosity times the cross-sectional area v nA. T h e total n u m b e r of p o r e volumes, U, is the total discharge divided by the volume of a single p o r e volume: I'nAI ".
It can be s e e n that the n u m b e r of p o r e volumes is equivalent to a dimensionless time, Ir. With this equivalency Equation 2.
If the data plot as a straight line, they are normally distributed, the diffusive form of the advection-dispersion equation is valid, and the slope of the line Is related to the longitudinal hydrodynamic dispersion. T h e value of D, can be found from 2. T h e average linear velocity in the c o l u m n can be found from the quantity of water discharging per unit time divided by the product of the cross-sectional area and the porosity.
T h e effective diffusion coefficient can either be m e a s u r e d in a c o l u m n test or estimated. Pickens and Grisak 1 9 8 1 c o n d u c t e d a laboratory study o f dispersion in sand c o l u m n s. T h e results of the three tests are plotted in Figure 2.
T h e results of test R2 have a reverse s l o p e as deionized water replaced the saline solution. It can be s e e n that the results form a straight line. F o r chloride in water at 25C, the molecular diffusion coefficient is 2.
B a s e d o n this, Pickens and Grisak estimated the effective diffusion c o e f 5. T h e hydrodynamic dispersion coefficients are b a s e d on the s l o p e of the straight lines. T h e following values were o b t a i n e d for the three tests: Pickens and G.
Grisak, Water Resources Research 17, no. Dispersivity can he determined in the field bv two means II there is a contaminated aquifer, the plume of known contamination can be mapped and the advection dispersion equation solved with dispersivity as the unknown.
Pinder 1 9 7 3 used this approach in a groundwater modeling study of a plume of dissolved chromium in a sand and gravel aquifer on k i n g Island, New York. He started with initial guesses of a and a and then varied them during successive model runs until the c o m p u t e r model yielded a reasonable reproduction of the o b s e r v e d contaminant plume.
O n e of the difficulties of this approach is that the concentration and volume of the contaminant s o u r c e are often not known. T h e r e are a variety of variations to this approach.
Natural gradient tests involve the injection of a tracer into an aquifer, followed by the m e a s u r e m e n t of the p l u m e that d e v e l o p e d under the prevailing water table gradient e. A single-well tracer test involves the injection of water containing a conservative tracer into an aquifer via an injection well and then the s u b s e q u e n t pumping of that well to recover the injected fluid.
T h e Huid velocities of the water b e i n g p u m p e d and injected are m u c h greater than the natural ground-water gradients.
Gelhar and Collins 1 9 7 1 derived a solution to Equation 2. T h e relative c o n c e n t r a t i o n of the water b e i n g withdrawn from the injection well is. Pickens and Grisak 1 9 8 1 performed a single-well injection-withdrawal tracer test into a confined sand aquifer a b o u t 8.
T h e injection well was 5. Clear water was injected at a constant rate for 24 hr prior to the stan of the test to establish steady-state conditions. T h e tracer used during the tests was I, a radioactive iodine, which was added to the injected water. T w o tests w e r e performed on the well. T h e first test, S W 1 , had an injection rate of 0. A total volume of Water with the tracer was added at a rate of 0. A total of 2 4 4 m of water was added, and the average position of the injection front reached to 4.
During the withdrawal phase a total of 8 8 6 m of water was p u m p e d over a period of T h e results of the test are shown in Figure 2. T h e dots represent field values and the solid lines are curves, which w e r e c o m p u t e d using Equation 2.
Various curves were c o m p u t e d for different values of a , and the curves with the b e s t fit to the field data w e r e plotted on the graphs. This test illustrates the scale d e p e n d e n t nature of dispersivity. T h e s e c o n d test, in which a larger volume of water was injected, tested a larger volume of the aquifer than the first test and yielded a higher dispersivity value. A t the laboratory scale the m e a n value of a was determined to be 0. With the single well injection withdrawal test, a was 3 cm w h e n the solute front traveled 3.
In a two-well recirculating withdrawal-injection tracer test with wells located 8 m apart, x was determined to be 50 cm. T h e greater the flow length, the larger the value of longitudinal dispersivity n e e d e d to fit the data to the advection-dispersion equation.
Lallemand-Barres and P e a u d e c e r f 1 9 7 8 published a graph on which dispersivity, as measured in the field, was plotted against flow length on log log paper Figure 2. This graph suggested that the longitudinal dispersivity could be estimated to be about.
Field-measured values of longitudinal dispersivity as a function of the scale of measurement. Lallemand-Barres and P. The largest circles represent the most reliable data. Gelhar, Water Resources Research 2 2 , no. Gelhar 1 9 8 6 published a similar graph Figure 2.
T h e additional data on the Gelhar graph suggest that the relationship b e t w e e n a and flow length is m o r e L. T h e longitudinal dispersivity that o c c u r s at field-scale flow lengths can be called m a c r o d i s p e r s i o n.
In a flow domain that e n c o m p a s s e s a few p o r e lengths, mechanical dispersion is caused by differences in the fluid velocities within a pore, b e t w e e n p o r e s of slightly different size, and b e c a u s e different flow paths have slightly different lengths. However, at the field scale, even aquifers that are c o n s i d e r e d to be h o m o g e n e o u s will have layers and z o n e s of somewhat different hydraulic conductivity.
If mechanical dis persion can be caused by slight differences in the fluid velocity within a single pore,. Hydraulic conductivity is frequently determined on the basis of a pumping test, w h e r e water is removed from a large volume of the aquifer. As a result, the hydraulic conductivity that is obtained is an average value over the entire region of the aquifer contributing water to the well.
This averaging will c o n c e a l real differences in hydraulic conductivity across the aquifer. T h e s e differences exist in both vertical and longitudinal sections.
Simon aquifer in Illinois. Gelhar, Water Resources Research. The borings from which the cores were obtained are separated by one meter horizontally. Sudicky, Wafer Resources Research 2 2 , no. Figures 2. Even aquifers that are usually c o n s i d e r e d to be h o m o g e n e o u s still have variations in porosity and hydraulic conductivity. Hydraulic conductivity of g e o l o g i c materials varies over a very wide range of values, up to nine orders of magnitude.
Porosity varies over a much, m u c h smaller range: From the standpoint of describing aquifers mathematically, it is s o m e t i m e s useful to a s s u m e that hydraulic conductivity follows a lognormal distribution, which m e a n s that the logarithms of the conductivity values are normally distributed, whereas porosity is normally distributed F r e e z e 1 9 7 5. However, s i n c e hydraulic conductivity varies over a much larger range, it is the m o r e i m p o r t a n t.
Sample locations are every 5 cm vertically and every 1 m horizontally. Hydraulic conductivity was less than This leads us to an explanation for the scale factor. As the flow path gets longer, ground water will have an opportunity to e n c o u n t e r greater and greater variations in hydraulic conductivity and porosity.
Even if the average linear velocity remains the same, the deviations from the average will increase, and h e n c e the mechanical dispersion will also increase. It is logical that the flow path will eventually b e c o m e long e n o u g h that all p o s s i b l e variations in hydraulic conductivity will have b e e n e n c o u n t e r e d and that the value of mechanical dispersion will reach a maximum.
If o n e assumes that the distribution of hydraulic conductivity has s o m e definable distribution, such as normal or lognormal, and that transverse dispersion is occurring, it c a n be s h o w n that apparent macrodispersivity will a p p r o a c h an asymptotic limit at long travel distances and large travel times M a t h e r o n and de Marsily 1 9 8 0 ; Molz, Guven, and Melville 1 9 8 3 ; Gelhar and A x n e s s 1 9 8 3 ; Dagan 1 9 8 8.
W h e n the asymptotic limit is reached, the p l u m e will c o n t i n u e to spread. In this region the variance of the p l u m e will g r o w proportionally to the time or m e a n travel distance, as it d o e s at the laboratory c o l u m n scale.
T h e advective-dispersion m o d e l is b a s e d on the assumption that dispersion follows Fick's law. S o m e authors c o n t e n d that dispersion follows F i c k ' s law only at the laboratory scale, w h e r e it is c a u s e d by local mechanical dispersion, and for very long flow paths, w h e r e the effects of advection through h e t e r o g e n e o u s materials and local transverse dispersion create m a c r o s c a l e dispersion that follows F i c k ' s law e.
T h e normal m a n n e r of determining a field-scale dispersion coefficient is to l o o k for a natural tracer or inject a tracer into an aquifer and o b s e r v e the resulting development of a plume.
A solute-transport m o d e l is then c o n s t r u c t e d and the c o m p u t e d solute distribution is fitted to the o b s e r v e d field data by adjusting the dispersion coefficients. Dispersion coefficients o b t a i n e d in this m a n n e r are fitted curve parameters and do n o t represent an intrinsic property of the aquifer. It is apparent that flow and transport modeling b a s e d on a single value for porosity and hydraulic conductivity is a g r o s s simplification of the c o m p l e x i t y of nature.
F o r analytical solutions, we are constrained to u s e of a single value for average linear velocity, and for numerical m o d e l s we often use a single value b e c a u s e that is all we have.
A d e t e r m i n i s t i c m o d e l is o n e w h e r e a partial differential equation is solved, either numerically or analytically, for a given set of input values, aquifer parameters, and boundary' conditions. T h e resulting output variable has a specific value at a given place in the aquifer.
It is assumed that the distribution of aquifer parameters is known. T h e equations given earlier in this c h a p t e r are e x a m p l e s of deterministic models. A s t o c h a s t i c m o d e l is a model in which there is a statistical uncertainty in the value of the output variables, such as solute distribution. T h e probabilistic nature of this o u t c o m e is due to the fact that there is uncertainty in the value and distribution of the underlying aquifer parameters, such as the distribution and value of hydraulic conductivity and porosity F r e e z e 1 9 7 5 ; Dagan 1 9 8 8.
T h e idea behind stochastic modeling is very attractive. It is obvious that it takes a great effort to determine hydraulic conductivity and porosity at m o r e than a few locations in an aquifer system. If we could determine the distribution of aquifer properties with a high d e g r e e of detail, then a numerical solution of a deterministic model would yield results with a high degree of reliability. However, with limited knowledge of aquifer parameters, a deterministic model makes only a prediction of the value of an output variable at a given point and time in the aquifer.
T h e stochastic model is based on a probabilistic distribution of aquifer parameters. At the outset it is recognized in the stochastic m o d e l that the result will be only s o m e range of possible o u t c o m e s. It cannot tell us what the concentration of a solute will be at a particular point in the aquifer at a given time.
T h e stochastic model thus recognizes the probabilistic nature of the answer, whereas the deterministic model suggests that there is only o n e " c o r r e c t " answer. Of course, the e x p e r i e n c e d hydrogeologist recognizes the uncertainty even in the deterministic answer.
T h e r e have b e e n literally hundreds of papers written since 1 9 7 5 on various aspects of stochastic modeling of ground-water flow and solute transport e. Stochastic m o d e l s have reached the stage of development where their accuracy has b e e n tested by c o m p a r i s o n of model-predicted results with the movement of a tracer in field tests S p o s i t o and Barry ; Barry, Coves and Sposito 1 9 8 8.
T h e greatest uncertainty in the input parameters of a model is the value of hydraulic conductivity, b e c a u s e it varies over such a wide range for geologic materials, if we make a measurement of hydraulic conductivity at a given location, the only uncertainty in its value at that location is d u e to errors in measuring its value.
However, at all locations w h e r e hydraulic conductivity is not measured, additional uncertainty exists. If we make a n u m b e r of m e a s u r e m e n t s of the value of hydraulic conductivity, we can estimate this uncertainty using certain statistical techniques. Let us define Y as the log of the hydraulic conductivity, K, and assume that the log value Y is normally distributed. Freeze et al. Used with permission. F o r a normally distributed population, the probabilistic value is called a probability density function P D F and is d e s c r i b e d by the mean and the variance.
T h e variance is a measure of the degree of heterogeneity of the aquifer. T h e PDF can be represented as a bell-shaped curve with the peak equal to the mean, as in Figure 2.
If we have measured the value of Y at a n u m b e r of locations and wish to estimate t. O n e approach is to say that the most likely estimate of Yj is the mean of the measured values of Y , and the uncertainty in this value is t. This t. Hydraulic conductivity values measured at locations c l o s e to each other are likely to be somewhat similar. T h e farther apart the measurements, the less likely that the values will be similar.
This is due to the fact that as distances b e c o m e greater, the c h a n c e that there will be a c h a n g e in g e o l o g i c formation increases.
T h e value of the autocorrelation function Y. An estimate of the autocorrelation function, r , can be obtained from the measured sample values by the following r.
T h e autocorrelation factor can be e x p r e s s e d in terms of either lag, p , or distance, Yk. If the autocorrelation function has an exponential form, then it can be e x p r e s s e d as p.
W e can d e s c r i b e the distribution o f heterogeneity o f Y b y the u s e o f three stochastic functions, p , r r o r r j Y. If we a c c e p t the idea that we don't k n o w the value of the hydraulic conductivity and the porosity everywhere, then we must a c c e p t the idea that it is not p o s s i b l e to predict the actual c o n c e n t r a t i o n of a solute that has u n d e r g o n e transport through an aquifer. However, it is important to n o t e that the p r o c e s s of advective transport d o m i n a t e s macrodispersion.
T h i s m e a n s that w h e t h e r o n e uses a deterministic m o d e l or a stochastic m o d e l , the large picture of solute transport will emerge, since b o t h a c c o u n t primarily for advective transport, with the dispersion factor tending to s m e a r the leading e d g e of the plume. Dagan 1 9 8 7 , 1 9 8 8 has derived a linear m o d e l o f stochastic transport.
In o r d e r to d o s o , h e neglected all nonlinear terms, such as t h o s e arising from the deviation of solute particles from their mean trajectory. Neuman and Zhang 1 9 9 0 and Zhang and Neuman 1 9 9 0 have derived a quasilinear stochastic m o d e l that is m o r e general than Dagan's linear model, which is applicable only for solute transport domains with a large Peclet n u m b e r t h a t is, t h o s e with a long flow distance. However, as solute transport over long flow distances represents a practical p r o b l e m and Dagan 1 9 8 8 presents s o m e useful closed-form analytical solutions, we will e x a m i n e his results.
F o r a conservative solute the position of the center of mass of the solute slug can be obtained from the advective equation. T h e geometric mean of K is found by taking the natural log of each value, finding the mean of the natural logs, and then finding the exponential of the m e a n of the natural logs.
X' is the residual of the d. T h i s equation has b e e n solved for b o t h longitudinal and transverse dispersion. Solutions of 2. C l o s e d forms of these equations are available for two conditions: T h e anisotrophy ratio, je , is indicated by fi. In Equations 2. Dependence of longitudinal spatial moment, X , on travel time, t. Dagon, Wafer Resources Research 24, no.
Dagan 1 9 8 8 defined an apparent longitudinal macrodispersivity coefficient as "the value of the constant dispersivity that would lead to the solution of the convection dispersion equation for the s a m e X , as the actual, time-dependent o n e.
It can be seen that this parameter has limited d e p e n d e n c e on the anisotropy ratio. T h e apparent macrodispersivity a p p r o a c h e s an asymptotic value with increasing travel time, which c o r r e s p o n d s to increasing travel distance.
This figure demonstrates that at all times longitudinal macrodispersion is dominated by advective transport. T h e asymptotic value h.
It can be seen that the effect of the anisotropy ratio is significant for both these moments. Dependence of the apparent longitudinal macrodispersivity upon travel time. Dagan, Water Resources Research 2 4 , no. Fractal geometry is a way of looking at irregular o b j e c t s , such as coastlines or aquifers.
O n e of the p r e c e p t s of fractal g e o m e t r y is that irregular o b j e c t s in nature tend to have patterns that repeat themselves at different scales, a p h e n o m e n o n k n o w n as self-similarity.
F o r e x a m p l e , in a sedimentary aquifer the relationship of individual p o r e s to e a c h o t h e r may be similar to the relationship of laminae to e a c h other, which may be similar to the relationship of b e d s to e a c h other, which may be similar to the relationship of g e o l o g i c formations to each other. In a classic paper Mandelbrot 1 9 6 7 d e m o n s t r a t e d that the measured length of an irregularly shaped o b j e c t , in that c a s e the coastline of Great Britain, d e p e n d s upon the.
Dependence of the horizontal plane transverse spatial moment, X , on travel time. T h e degree o f irregularity o f the coastline i s independent o f the scale at which the measurement takes place. A coastline has a similar irregular shape if viewed from the b e a c h , an airplane, or a space station.
If we measure a straight line with a ruler, the length is constant and equal to the n u m b e r of units times the unit length in which the measurement is made. If the unit length is halved, the n u m b e r of units is doubled, but the overall length remains the same. If we measure an irregular line, the accuracy of the measurement is a function of the scale of the measuring device. In the c a s e of our coastline, if we used a ruler that had a minimum scale of 1 0 0 km, we would get a certain approximate length.
If we then used a ruler that had a minimum scale of 1 km, we would get a different, m o r e accurate measurement, which would be longer b e c a u s e we could m o r e accurately trace the irregularities of the coast. If we made yet another measurement using a ruler with a 1 m scale, we would obtain a third, even m o r e accurate length, that was longer still. Even so, we have yet to measure the curve of the coastline as it b e n d s around individual rocks, much less the curve of the coastline around grains of sand.
Using conventional geometry, the measured length is a function of the minimum scale of measurement.