DIFFUSION IN SATURATED SOIL. I: BACKGROUND The design of ...

DIFFUSION IN SATURATED S O I L . I: BACKGROUND By Charles D. Shackelford, 1 Associate Member, ASCE, and David E. Daniel, 2 Member, ASCE ABSTRACT: Recent studies suggest that diffusion may be an important, if not dominant, mechanism of contaminant transport through waste containment barriers. This paper represents the first of two papers pertaining to the measurement of diffusion coefficients of inorganic chemicals diffusing in saturated soil. In this paper, both steady-state and transient equations describing the diffusive transport of inorganic chemicals are presented. Several factors affecting diffusion coefficients are identified. A method for measuring diffusion coefficients for compacted clay soil is described. The definition for the diffusion coefficient for diffusion in soil (known as the effective diffusion coefficient, D*) is shown to vary widely. In general, variations in the definition of D* result from consideration of the different factors that influence diffusion of solutes in soil and the different ways of including the volumetric water content in the governing equations. As a result of the variation in the definition of D*, errors in interpretation and comparison of D* values can result if the appropriate definition for D* is not used. INTRODUCTION

The design of earthen barriers for the containment of buried wastes traditionally has been based on the assumption that the hydraulic conductivity controls the rate of leachate migration. However, recent field studies have indicated that diffusion is the controlling mechanism of solute transport in many fine-grained soils [e.g., Goodall and Quigley (1977), Desaulniers et al. (1981, 1982, 1984, 1986), Crooks and Quigley (1984), Quigley and Rowe (1986), Quigley et al. (1987), and Johnson et al. (1989)]. As a result, it is becoming necessary to evaluate the migration of chemicals through earthen barriers due to diffusion. While the measurement of the hydraulic conductivity of fine-grained soils is relatively common practice for geotechnical engineers, the measurement of diffusion coefficients is not. In fact, the concept of diffusion may be unfamiliar to many geotechnical engineers. In addition, the literature abounds with a wide variation in the terminology associated with the study of diffusion in soils. Variable terminology can lead to considerable confusion, and an enormous amount of time can be spent in attempting to sort out the details. This paper is the first of two papers describing the process of diffusion in soils. The intent of this paper is to familiarize the geotechnical engineer with background information required for the measurement and evaluation of diffusion coefficients for use with the design of waste containment barriers. The specific objectives of this paper are to present the equations used to describe diffusion of solutes in soil, to discuss the factors affecting diffusion coefficients, to clarify some of the variability in the terminology as'Asst. Prof., Dept. of Civ. Engrg., Colorado State Univ., Fort Collins, CO 80523. Assoc. Prof., Dept. of Civ. Engrg., Univ. of Texas, Austin, TX 78712. Note. Discussion open until August 1, 1991. Separate discussions should be submitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 26, 1990. This paper is part of the Journal of Geotechnical Engineering, Vol. 117, No. 3, March, 1991. ©ASCE, ISSN 0733-9410/91/0003-0467/$1.00 + $.15 per page. Paper No. 25602. 2

467

sociated with the study of diffusion in soils, and to describe methods of measurement. STEADY-STATE DIFFUSION

Diffusion in Free Solution Diffusion of a chemical or chemical species in solution (i.e., a solute) typically is assumed to occur in response to a concentration gradient in accordance with Fick's first law which, for one dimension, may be written as: dc J = -D0 (1) dx where J = the mass flux, c = the concentration of the solute in the liquid phase, x = the direction of transport, and D0 = the "free-solution" diffusion coefficient. However, several investigators [e.g., Robinson and Stokes (1959), Quigley et al. (1987), Daniel and Shackelford (1987), and Shackelford (1988, 1989)] have noted that there is a more fundamental basis for diffusive transport than the empirical Fick's first law. This fundamental basis, which takes the driving force for the solute ions or molecules as the gradient in the chemical potential of the chemical species, results in a number of expressions that help to provide insight into the factors affecting the free-solution diffusion coefficient, D0. One of these expressions is the Nernst-Einstein equation (Jost 1960), or uRT dc

J=

(2) N 8x and, by comparison with Eq. 1, the expression for the free-solution diffusion coefficient at infinite dilution [i.e., sufficient dilution such that solutes (ions, molecules) do not interact with each other in solution] becomes uRT

D

(3)

^~w

where R = the universal gas constant (8.134 J mor'KT 1 ), T = the absolute temperature, N = Avogadro's number (6.022 x 1023 mol -1 ), and u = the absolute mobility of a particle. The absolute mobility of a particle is the limiting velocity attained under a unit force which, in the aforementioned case, is the gradient in the chemical potential of the diffusing chemical species (Robinson and Stokes 1959). When Eq. 3 is combined with expressions relating the absolute mobility to the limiting ionic equivalent conductivity (Robinson and Stokes 1959) and to the viscous resistance of the solvent molecules, i.e., Stokes Law (Bird et al. 1960), two additional expressions for D0 result: D

RTk0

° = J^

W

and

°o =

RT T^T

(5) 468

TABLE 1. Self-Diffusion Coefficients for Representative Ions at Infinite Dilution in Water at 25° C Anion (1)

oir F~

cr

Br

rHCO _ 3

NO3_"

sor cof

— — — — ^ — — — — — —

A> x 1010 (m 2 /s) (2)

Cation (3)

D 0 x 10 10 (m 2 /s) (4)

52.8 14.7 20.3 20.8 20.4 11.8 19.0 10.6 9.22

H+ Li + Na + K+ Rb + Cs + Be 2+ Mg 2+ Ca2+ Sx2* Ba 2+ Pb 2+ Cu 2+ Fe2+» Cd2+* Zn 2+ Ni 2+ * Fe 3+ * Cr3^ Al 3+ "

93.1 10.3 13.3 19.6 20.7 20.5 5.98 7.05 7.92 7.90 8.46 9.25 7.13 7.19 7.17 7.02 6.79 6.07 5.94 5.95

— — — — — — — .— — .— —

"Values from Li and Gregory (1974).

where F = the Faraday (96,490 Coulombs/equivalent), \z\ = the absolute value of the ionic valence, \ 0 = the limiting ionic conductivity, T| = the absolute viscosity of the solution, and r = the molecular or hydrated ionic radius. The limiting ionic conductivity is the conductivity of an aqueous solution containing the specified ion at infinite dilution. Eqs. 4 and 5 commonly are referred to as the Nernst and the Einstein-Stokes equations, respectively, and indicate that D0 is affected by several factors, including the temperature and viscosity of the solution, and the radius and valence of the diffusing chemical species. Based on X,0 values from Robinson and Stokes (1959), the D0 values for several ions have been calculated using Eq. 4 and the results are shown in Table 1. Similar tables can be found in Li and Gregory (1974), Lerman (1979), and Quigley et al. (1987). The values of D0 reported in Table 1 should be considered to be the maximum values attainable under ideal conditions (i.e., molecular scale, infinite dilution). Under nonideal conditions (e.g., macroscopic scale, concentrated solutions), a number of effects, negligible for ideal conditions, become important. For example, when two oppositely charged ions are diffusing in solution, an electrical potential gradient is set up between the ions (Robinson and Stokes 1959). Due to this electrical potential gradient, the slower moving ion speeds up while the faster-moving ion slows down, the overall result being that both ions migrate at the same speed. This electrical potential effect is responsible, in part, for the differences between the simple electrolyte diffusion values shown in Table 2 and their respective self-diffusion coefficients given in Table 1. Other effects responsible for the difference in DQ 469

TABLE 2. Limiting Free-Solution Diffusion Coefficients for Representative Simple Electrolytes at 25° C [after Robinson and Stokes (1959)] D 0 x 10 10 (m a /s) (2)

Electrolyte (1) HC1 HBr LiCl LiBr NaCl NaBr Nal KC1 KBr KI CsCl CaCl2 BaCl2

33.36 34.00 13.66 13.77 16.10 16.25 16.14 19.93 20.16 19.99 20.44 13.35 13.85

values under nonideal conditions include solute-solute and solute-solvent interactions [e.g., see Robinson and Stokes (1959)]. Types of Diffusion In addition to the previously described factors, the value of D0 depends on the type of diffusion. There are essentially four different types of diffusion (Robinson and Stokes 1959; Li and Gregory 1974; Lerman 1979; Shackelford 1988, 1989): (1) Self-diffusion; (2) tracer diffusion; (3) salt diffusion; and (4) counterdiffusion or interdiffusion. The four different types of diffusion are represented schematically in Fig. 1. In the illustration, sodium chloride (NaCl) and/or potassium chloride (KC1) are assumed to be the diffusion compounds. In true self-diffusion, the initial system would contain two half-cells, each with equal concentrations of NaCl, but without any isotopically different species. In such a system, the movement of the molecules would truly be random, but the motion of the molecules could never be traced. Therefore, the true self-diffusion system is approximated by the introduction of the isotopic (tracer) species, depicted schematically in Fig. 1(a). In this case, each half-cell of the system initially contains an equal concentration of sodium chloride (NaCl). However, in one half-cell, a small amount of the sodium, Na + , has been replaced by its isotope, 22Na+. When the two half-cells are connected, diffusion of both Na+ and its isotope, 22Na+, occurs, but in opposite directions, owing to the small concentration gradients of each species. Since the concentration gradient is extremely small, the movements of the radioactive "tracer" ions (22Na+) and the Na+ ions are not tied to that of the ions of opposite signs (i.e., Cl~), and the tracer ions may be considered to be moving relative to a stationary background of nondiffusing ions (Robinson and Stokes 1959). This movement of the tracer ions is termed "selfdiffusion," and the diffusion coefficient describing it is called the "self-diffusion coefficient.") Tracer diffusion is the same as self-diffusion except the isotopic species is of a different element. For example, consider a system of two half-cells, 470

Diffusion

Diffusion

Prevented

Allowed

22

NaCl + NaCl

NaCl

(a)

NaCl

(b)

NaCl

Water

(c)

NaCl

KCI

(d)

42

KCI 4-

NaCl

FIG. 1. Diffusion Cells for Different Diffusion Systems: (a) Self-Diffusion; (h) Tracer Diffusion; (c) Salt Diffusion; (d) Counterdiffusion [after Shackelford (1988)]

each containing equal concentrations of NaCl. If a small amount of Na+ in one of the half-cells is replaced by an equal amount of a radioisotope of a different element, say 42K+, and the two half-cells are connected, the diffusion of the 42K+ may be traced [Fig. 1(b)]. In this case, the diffusion of 42 + K is termed "tracer diffusion" to distinguish it from self-diffusion. At infinite dilution, the tracer diffusion and self-diffusion coefficients are the same. Salt diffusion is illustrated by Fig. 1(c). In this case, one half-cell contains a sodium chloride solution whereas the other half-cell contains only the solvent. When diffusion is allowed, both the Na+ and the Cl~ ions diffuse in the same direction. Counterdiffusion or interdiffusion describes the process whereby different ions are diffusing against, or in opposite directions to, each other. A system describing such a process is shown in Fig. 1(d). In this system, two halfcells with equal concentrations of sodium chloride (NaCl) and potassium chloride (KCI) are joined together resulting in the diffusion of Na+ and K+ ions in opposite directions. This same process applies to any system in which concentration gradients are established in opposite directions. Equations for counter diffusion coefficients can be found in Robinson and Stokes (1959), Jost (1960), Helfferich (1962), Olsen et al. (1965), Li and Gregory (1974), Lerman (1979), Low (1981), and Shackelford (1988, 1989). In reality, both self- and tracer diffusion are iimiting cases of counterdiffusion, and salt diffusion and counterdiffusion usually occur simultaneously 471

Effective

Length, L

FIG. 2. Concept of Effective Length in Transport through Soil [after Shackelford (1988)] in most systems. The case of salt diffusion [Fig. 1(c)] best represents most practical field problems involving containment of waste by earthen barriers. Diffusion in Soil Solutes diffuse at slower rates in soil than in free solution because the pathways for migration are more tortuous in soil. Also, diffusive mass fluxes are less in soil than in free solution because solid particles in soil occupy some of the cross-sectional area. These effects are illustrated schematically in Fig. 2. Effect of Reduction in Cross-Sectional Area of Flow Due to the reduced cross-sectional area of flow in soil, the concentration of the diffusing species, c, is the concentration in the liquid phase of the pore space. Since fluxes are defined with respect to the total cross-sectional area, Eq. 1 must be modified for diffusion in soil as follows dc J = -D 0 9 — dx

(6)

where 6 = the volumetric water content defined as follows B = nSr

(7)

where n = the total soil porosity and Sr = the degree of saturation of the soil, expressed as a decimal. Therefore, the maximum flux for liquid phase diffusion will occur when the soil is saturated (Sr= 1.0), all other conditions in Eq. 6 being equal. Effect of Tortuous Pathway The tortuosity of the soil usually is accounted for by including a tortuosity factor, T, in Eq. 6 as follows [e.g., Porter et al. (1960), Olsen and Kemper (1968), and Bear (1972)] dc

(8)

J = -D0T6 —

dx

Typical values of T (discussed later) are < 1. 472

Other Effects Additional factors, not included in Eq. 8, tend to reduce the rate of diffusive transport of solutes in soil. Kemper et al. (1964) incorporated a "fluidity" or "mobility" factor, a, into Eq. 8 to account for the increased viscosity of the water adjacent to the clay mineral surfaces relative to that of the bulk water. In addition, Porter et al. (1960) and van Schaik and Kemper (1966) added a factor, 7, to account for exclusion of anions from the smaller pores of the soil. Anion exclusion can result in compacted clays and shales when clay particles are squeezed so close together that the diffuse double layer of ions associated with the particles occupies much of the remaining pore space (Freeze and Cherry 1979; Drever 1982). The process is also known as salt filtering, ultrafiltration, or membrane filtration (McKelvey and Milne 1962; Hanshaw and Coplen 1973; Mitchell 1976; Freeze and Cherry 1979; Drever 1982). Berner (1971) states that anion exclusion may occur in natural deposits when the average porosity of the soil has been reduced to 0.3. Anion exclusion may also be operative in highly unsaturated soils and in relatively small pores where the available cross-sectional area of flow is reduced (Olsen and Kemper 1968). Eq. 8 can be modified to account for these additional effects, or dc J = -DOTOCYG —

(9)

dx

Since in most cases it is difficult, if not impossible, to separate the effects of geometry (T), fluidity (a), and anion exclusion (7) in soil diffusion studies, it seems best to define a single factor that accounts for all of them. Nye (1979) has done this by defining the "impedance f a c t o r , " / , as ft = Ta7

(10)

Olsen et al. (1965) included the volumetric moisture content, 6, into the definition of the "tortuosity factor" and called it the "transmission factor," t„ or tr = T0178

(11)

When Eq. 9 is written in terms of an impedance factor (Eq. 10) or a transmission factor (Eq. 11), the following equations result, respectively J = -D0ffi

dc dx

(12)

dc

J = -D0t, — (13) dx The similarity in the forms of Eqs. 8, 12, and 13 should be noted. Due to this similarity, many researchers report tortuosity factors, T, when they may in effect be measuring impedance factors,/-, or transmission factors, tr. For this reason, it seems more appropriate to define an "apparent tortuosity factor," Ta, in which is included not only the actual, geometric tortuosity, T, but also all other factors, which may be inherent in its measurement, in473

eluding solute-solute and solute-solvent interactions. Since the volumetric water content, 6, can be determined independently of all other factors, Fick's first law describing the diffusion of a chemical species in soil is more conveniently expressed as: dc

J = - £>0T„e — dx

(14)

Effective Diffusion Coefficient At present, tortuosity factors cannot be measured independently. Therefore, it is convenient to define an effective diffusion coefficient, D*, as follows D*=D0Ta

(15)

When Eq. 15 is substituted into Eq. 14, Fick's first law for diffusion in soil becomes J = -D*9 — dx

(16)

Eq. 16 can be utilized to determine effective diffusion coefficients of chemical species, D*, diffusing in soil from experimental results. After D* is determined, the apparent tortuosity factor can be calculated from Eq. 15 using an appropriate value for the free-solution diffusion coefficient. Some typical values for Ta reported in the literature are presented in Table 3. There are several definitions for the effective diffusion coefficient, D*, besides that of Eq. 15. Some of these definitions for D* are reported in Table 4. Of particular importance is the fact that some investigators have included 6 in the definition of D* while others (including the writers) have not. Caution should be exercised when interpreting the effective diffusion coefficient data of various researchers. Errors in interpretation of 50% or more can result if the appropriate definition of D* is not used. Definition of Concentration A few investigators [e.g., Porter et al. (I960)] have defined the solute concentration in terms of the total volume of soil (i.e., c' = 9c) and rewritten Eq. 16 in terms of this modified concentration as follows J = -D* — dx

(17)

where D* is as defined in Eq. 15. When the definition of D* includes the volumetric water content (i.e., D* = Z>oT„6), Eq. 17 is written as follows [e.g., Porter et al. (I960)] J= -Df—

(18) dx

where „. D* D$ = —

(19) 474

TABLE 3.

Representative Apparent Tortuosity Factors Taken from Literature Saturated or unsaturated (2)

Soil(s) (1)

T„ Values8 (3)

Reference (4)

(a) MC1 Tracer Bentonite: sand mixtures 50% sand:bentonite mixture Bentonite .sand mixtures

Saturated Saturated Saturated

Sandy loam Sand Silty clay loam Clay Silt loam Silty clay loam; sandy loam Silty clay Clay

Unsaturated Unsaturated Unsaturated Unsaturated Unsaturated Saturated Saturated Saturated

Silty clay loam; sandy loam Sandy loam

Saturated Saturated

0.59-0.84 0.08-0.12 0.04-0.49

Gillham et al. (1984) Gillham et al. (1985) Johnston et al. (1984)

(6) Cl~ Tracer 0.21-0.35* 0.025-0.29* 0.064-0.26* 0.091-0.28* 0.031-0.57* 0.08-0.22* 0.13-0.30* 0.28-0.31*

Barraclough and Nye (1979) Porter et al. (1960) Porter et al. (1960) Porter et al. (1960) Warncke and Barber (1972) Barraclough and Tinker (1981) Crooks and Quigley (1984) Rowe et al. (1988)

(e) Br Tracer 0.19-0.30* 0.25-0.35*

Barraclough and Tinker (1981) Barraclough and Tinker (1982)

(