Evaporation of pendant water droplets in fractures

Transcription

1 WATER RESOURCES RESEARCH, VOL. 33, NO. 12, PAGES , DECEMBER 1997 Evaporation of pendant water droplets in fractures Clifford K. Ho Sandia National Laboratories, Albuquerque, New Mexico Abstract. This paper investigates the evaporation of pendant water droplets in fractured tuffaceous rock. The droplets are assumed to exist near asperities and cavities as a result of gravity-dominated flow through fractures. Because of the curvature of the droplet, the vapor pressure near the liquid-vapor interface is greater than the saturated vapor pressure of water held in the matrix at the system temperature. Expressions for the vapor pressure and evaporation rate of water droplets are developed as a function of droplet radius and relative humidity of the surroundings. The relative humidity is calculated using Kelvin's equation to account for vapor pressure lowering in the surrounding tuffaceous rock. Results indicate that the vapor pressure of water droplets is not significantly increased above the saturated vapor pressure of freestanding water when the droplet size is greater than 100/xm. However, if the relative humidity of the surroundings is exactly equal to 1 or if the droplet radius is very small (<<1 /xm), the effects of interface curvature will enhance the evaporation of water droplets. Finally, expressions are developed to determine the minimum liquid flux required to propagate flow through either a single fracture or a network of fractures without being evaporated. 1. Introduction Performance assessments of the potential high-level nuclear waste repository at Yucca Mountain, Nevada, require understanding of processes that govern the flow of water through fractured rock. While the importance of hydrologic processes in fractures have at least been acknowledged (albeit not well understood), the evaporation of water in fractures under ambient conditions has received little attention. Evaporation of water in fractures surrounded by partially saturated rock can occur if a vapor pressure gradient exists between the water in the fractures and the water held in the surrounding rock pores. High capillarity in the rock matrix tends to lower the vapor pressure of trapped pore water to a value below the saturated vapor pressure. In contrast, pendant droplet formation in fractures yields vapor pressures that are greater than the saturated vapor pressure. Thus vapor pressure gradients can exist between water flowing in fractures and surrounding pore water in the matrix. Figure 1 shows two conditions that could lead to the formation of pendant water droplets in fractures: gravityinduced pendant droplets at asperities and cavities during nonequilibrium flow through the fractures. These conditions and the potential for droplet evaporation are important to performance assessments in nuclear waste management that determine the rate and likelihood of waste package corrosion caused by water dripping into repository drifts. The remainder of this paper develops mathematical relations that govern the evaporation rates of water droplets. The sensitivity of the evaporation rate on droplet radius and relative humidity of the surroundings (and hence liquid saturation in the host rock) is investigated. The rate of evaporation is then used in a mass balance to estimate the minimum amount of liquid flow that must be applied to either a single fracture or a network of fractures to propagate flow through the fractures without being evaporated. Copyright 1997 by the American Geophysical Union. Paper number 97WR /97/97WR Evaporation Rates of Pendant Droplets in Fractures The rate of evaporation of a water droplet (Figure 2) can be determined using Fick's law assuming steady state, isothermal diffusion: m e -- -DA - - (1) r=rb where /'/1 e is mass rate of evaporation (kg/s), D is the binary diffusion coefficient of water vapor in air (m2/s), 1 is the surface area of the water droplet (m2), C is water vapor concentration (kg/m3), r is radial distance (m), and r, is the radius of the water droplet (m). The binary diffusion coefficient [Vargafiik, 1975] and the surface area can be calculated as follows: A = 4,rr 2 (3) where D ø is the binary diffusion coefficient of water vapor in air at standard conditions (=2.13 x 10 -s m2/s at P = i x 10 s Pa, T = 0øC), P is pressure (Pa), T is the system temperature (K), and 0 is a material parameter (= 1.8 for vapor-air mixtures). The actual surface area of an evaporating pendant droplet as shown in Figure i is more complex and potentially different than that for a sphere, but for simplicity, evaporation from a spherical droplet with a radius equal to rt, is assumed in this study. For nonspherical evaporating surfaces, rt, can represent a mean radius of curvature, and the constant value of 4 in (3) can be replaced by a constant dependent on drop size, surface tension, density, and other system parameters. It should be noted that the error in assuming a spherical shape grows as the droplet size increases due to increasing effects of gravity. The water vapor concentration, C, in (1) can be determined by solving the steady state, radial diffusion equation with appropriate boundary conditions:

2 2666 HO: EVAPORATION OF PENDANT WATER DROPLETS Table 1. Model Input Parameters Used in This Study..,..:: ii:ii ::iiii... ' - " /...:1111iii'"'"" "' -"'. ' ' --'- Droplet Formation '..., : ii.: %' x- ' - ' Gr ity ß ::::::::::::::::::::::::::. - _R øck :m' ' ' '""i :: :: :: ::::'"' " --- tu;ed by..... '..t '!?, : -i :: ::: '* '---'... :;i ::. ' '.....?...?" : " ' x"½--- ' '... For marion v ' i i't ' '" " ' Open Ca9ity I -,½-,0'.m Figure 1. Pendant water droplets formed in fractures as a result of asperities or cavities. Parameter Temperature, T (K) Saturated vapor pressure, Psat (Pa) Water vapor gas constant, R (J/kg-K) Water liquid density, p (kg/m 3) Total gas pressure, P (Pa) Water surface tension, rr (N/m) Binary diffusion coefficient, D ø (m2/s) Material parameter for diffusion, 0 Matrix van Genuchten parameter, a (1/Pa) Matrix van Genuchten parameter,/3 Matrix residual liquid saturation, Sr Matrix full liquid saturation, S Bulk fracture permeability, kf, t (m 2) Value x x X O.O x 10-3 where dr d r2 (do) rr : 0 (4) C(r = rt ) = Ct (5) C(r --> oo) = Coo (6) The solution to (4) with boundary conditions (5) and (6) is given as follows: rb C(r) = (Ct - Coo) --+ Coo (7) Equations (2), (3), and (7) are used in (1) to derive an expression for the evaporation rate of a water droplet with radius, r. 4,rDrt,(Pt, - Poo) me ' gr (8) where the ideal gas law (C = P/RT) has been used to express the water vapor concentration, C, in terms of water vapor pressure. P is the water vapor pressure at the surface of the water droplet, Poe is the water vapor pressure of the surroundings, and R is the ideal gas constant for water vapor (=462 J/kg-K). Table 1 provides values for the model input parameters that are used in this study. It is important to note that in (8), the vapor pressure at the surface of the water droplet, P, is greater than the saturated vapor pressure of freestanding water. In contrast, the surrounding vapor pressure, Poe, is less than the saturated vapor pressure. In both cases, the deviations in vapor pressure are caused by the curvature of the interface between the liquid and gas. The curvature is convex in the case of a droplet, and the capillary pressure causes an increase in the liquid pressure over that of the surrounding gas pressure. However, the curvature of the water in the surrounding rock pores is concave because of strong matric potentials, and the liquid pressure of the water in the pores is less than the gas pressure. Assuming negligible solute effects, the vapor pressure at the surface of the water droplet, Po, is given as follows [Carey, 1992]: (9) Pb = Psat(T) exp (2a/r where Psat(T) is the saturated water vapor pressure of freestanding water at the system temperature, o-is the surface tension of liquid water in contact with air or its own vapor (=0.073 N/m), and p is the liquid density. Note that the numerator of the exponent in (9) is just the capillary pressure of the water droplet as given by the Young-Laplace equation. Figure 3 shows a plot of the ratio of the vapor pressure at the surface, Pb, to the saturated vapor pressure, Psat, as a function of droplet radius assuming a temperature of 20øC (see Table 1 for parameters). The vapor pressure ratio increases as the droplet radius decreases. Note that for droplet radii greater than or equal to --1 mm, the vapor pressure ratio is close to 1 and the effects of interface curvature on vapor pressure are negligible. The surrounding vapor pressure, Poe, is assumed to be constant so that the matrix provides an infinite sink for vapor diffusion from the fractures. The surrounding vapor pressure is Co = P(r)/RT Coo Figure 2. Water vapor concentration profile of an evaporating droplet Figure 3. Ratio of vapor pressure of a spherical water drop to the saturated vapor pressure of freestanding water as a function of droplet radius at 20øC.

3 HO: EVAPORATION OF PENDANT WATER DROPLETS 2667 assumed to be dependent on the matrix capillary pressure of the host rock and can be calculated using Kelvin's equation [Carey, 1992; Bear, 1972]' where 1.2. Pc =- - 1 ol Ss-S r X = 1-1//3 (10) (11) 1... i Liquid Saturation Figure 4. Relative humidity (vapor pressure divided by saturated vapor pressure) in the host rock as a function of matrix liquid saturation assuming vapor pressure lowering according to Kelvin's equation (T = 20øC, a - 1 x /Pa,/3-1.6, Ss = 1, Sr = 0.045). 3 P=P [ b sat -'i... RH= t... ' "?' -- RH= "...?-.../.... _:...:.. [ / RH= /' _ S=.999 In (10) the capillary pressure, Pc, is given by the van Genuchten function [Van Genuchten, 1980] (equation(11)) where a and/3 are curve-fitting parameters (a is related to the air-entry pressure), S is the liquid saturation, and the subscripts r and s 10-9, 10 ø denote residual and full saturation, respectively. Figure 4 shows a plot of the ratio of the surrounding vapor pressure to the saturated vapor pressure (i.e., relative humidity ) for some typical van Genuchten parameters of the host rock at Yucca Figure 5. Evaporation rate of a water droplet with and without consideration of curvature effects on the vapor pressure at Mountain (Table 1). The relative humidity decreases with dethe surface of the droplet, Pb (RH, relative humidity of surcreasing liquid saturation in the rock due to increased capilroundings; S, liquid saturation of host rock). larity. The evaporation rate (equation(8)) can now be expressed in conjunction with (9), (10), and (11). Recall that the driving where force for evaporation is mass transfer from the surface of the droplet, where the vapor pressure, Pb, is greater than the P saturated vapor pressure according to (9). The surrounding RH = Psat(T) (13) vapor pressure, P, is less than or equal to the saturated vapor pressure and can b_ expressed by the relative hum!dity, RH, of Figure 5 plots the evaporation rate as a function of droplet radius for three different relative humidities in the host rock the surroundings as shown in Figure 4. Therefore, for an isothermal system the evaporation rate is assuming an ambient temperature and pressure of 20øC and 90 a function of the relative humidity, RH, of the surroundings kpa (see Table 1 for other input parameters). Two conditions and the droplet radius as follows: for the vapor pressure at the surface of the water droplet are also shown. The solid line represents a condition where the 4 rrdrb(pb - Ri-iPsat) vapor pressure at the surface of the droplet, Pb, is equal to the the = RT (12) saturated vapor pressure, Psat (i.e., no curvature effects). The dashed line represents the condition where the vapor pressure at the surface of the droplet is greater than the saturated vapor pressure because of curvature effects as given by (9). The kgplt. RH vs. S results indicate that the evaporation rate increases with in }-- creasing droplet radius and decreasing relative humidity. The increasing droplet radius increases the available area for diffusion, and the decreasing relative humidity increases the driving potential for mass transfer to the surroundings. Figure 5 illustrates that the influence of curvature on the vapor pressure of the water droplet is small for droplets with radii greater than 100/xm. As the radius increases, the droplet is more like a flat surface, and the evaporation rate approaches that of a free surface of liquid. Evaporation from a free surface of liquid may also have important applications to film flow along fracture surfaces [Tokunag and Wan, 1997]. In addition, the effect of the curvature is less pronounced as the relative humidity of the surroundings decreases. At low relative humidities, the difference in vapor pressures is dominated by the surrounding vapor pressure, Pa. For relative humidities below , the evaporation rate is unaffected by curvature effects for the entire range of droplet sizes considered. It is interesting to note that for a matrix liquid saturation of 0.8, the corresponding "surrounding" relative humidity of the hos.t rock is using the typical capillary pressure parameters given in Figure 4. These results indicate that the evaporation of water droplets

4 ß _ 2668 HO: EVAPORATION OF PENDANT WATER DROPLETS 10_6... i. s=l.0 ' - :, ] ø Evaporation Versus Infiltration in a Single Fracture The formulated rate of evaporation (equation (12)) can be used to estimate the infiltration required to balance the rate of evaporation in a single fracture, assuming that the size of the droplet can be correlated to the size of the fracture in which it originated. This method provides a threshold limit of the infiltration above which water will propagate through the fracture. For liquid flow below this threshold infiltration, the evaporation exceeds the infiltration and water will not propagate through the fracture. This section details the formulation and assumptions associated with estimating the "evapoinfiltration threshold," or EIT, for a single fracture. Consider a single fracture that is approximated as a vertical tube in the rock with constant cross-sectional area, AT. The radius of the cross-sectional area is assumed to be equal to the radius of a water droplet, rb, that exists in the fracture. The mass flow rate, ta T, through this single fracture can be written as follows: Figure 6. Evaporation rate of a water droplet considering curvature effects on the vapor pressure at the surface of the droplet, Pb, over a large range of droplet radii (RH, relative humidity of surroundings; S, liquid saturation of host rock). (14) where qf is the liquid Darcy velocity in the fracture. Equating the evaporation rate (equation(12)) and the mass flow rate (equation(14)) and solving for the Darcy velocity, qf, yields in nearly saturated environments is possible. However, the the following expression: augmentation of evaporation because of liquid-vapor interface 4D(Pb- RI-IPsat) curvature of the droplets is negligible if the droplet radius is q/ = prtrb greater than 100 /am or if the relative humidity of the sur- (15) roundings is less than For very small droplets (-1/am or less) in nearly saturated environments (RH - 100%), the curvature may significantly enhance the evaporation rate over the evaporation rate from a flat surface of water. Figure 6 shows a plot of the evaporation rate (using(9) for Pb) as a function of droplet radius over a large range of droplet radii. Inclusion of very small droplet radii (<1 /am) reveals a competition between the dependence of water droplet vapor Equation (15) gives the Darcy velocity that specifies the EIT for a single fracture. If the Darcy velocity in a single fracture is less than that given by (15) for a given water droplet radius (and hence fracture radius), then evaporation will exceed the water flow rate and no propagation will occur. On the other hand, if the Darcy velocity is greater than that given by (15), then the water flow rate will exceed the evaporation rate and liquid propagation through the fracture will occur. pressure and water droplet surface area on droplet radius. As Figure 7 shows a plot of the EIT for flow in a single fracture the radius decreases below 10-2/am the vapor pressure of the as a function of droplet radius and relative humidity of the water droplet becomes significantly larger than the saturated vapor pressure. The evaporation rate therefore increases as the droplet radius decreases, even though the surface area avail- kgplt.slngle frac EIT able for evaporation decreases. It is important to note that the lower limit of r/, (10-4 /am) in Figure 6 approaches that of a 106 I... I... k... -m-i= i... single water molecule. While investigators have supported the use of Kelvin's equation for liquid droplets as small as 10-3/am [Fisher and Israelachvili, 1979], the plots in Figure 6 are used... RH=.9932 ' :i' purely to illustrate the behavior of (12) at small values of % and are not meant to accurately depict physical evaporation at 102 ' -""RH:.9998 i scales less than 10-3 tam. For radii greater than 1 tam the water droplet vapor pressure 1.. S=.999 '"'"" ' '"'"'"' i., is not significantly greater than the saturated vapor pressure, 0 RH-! and the evaporation rate increases as the surface area in : S=I creases with increasing droplet radius. It is also interesting to note that for a surrounding relative humidity of 1.0, the evap i... Lh ' oration rate appears to be constant for droplet radii greater than 1 tam. The increase in surface area, which increases the 10 '4... I i evaporation rate for relative humidities less than 1, is balanced 101! by the decrease in water droplet vapor pressure according to (9). Thus the rate of evaporation appears to remains constant over the range of droplet radii shown in Figure 6, but in truth, Figure 7. Evapoinfiltration thresholds (EITs) as a function the evaporation decreases asymptotically zero as the droplet of droplet radii and surrounding relative humidities for flow size increases. and evaporation in a single fracture.

5 HO: EVAPORATION OF PENDANT WATER DROPLETS 2669 surrounding environment. Equation (9) is used to calculate the droplet vapor pressure in (15). Figure 7 shows that as the relative humidity increases, the driving force for evaporation decreases and the amount of flow through the fracture that can be evaporated decreases. Also, as the fracture radius increases, the EIT decreases because the area available for liquid flow increases and less velocity is required to obtain a sufficient flow rate that exceeds the evaporation rate. Continuum Infiltration, q (mrn/year) 4. Evaporation Versus Infiltration in a Fracture Network The rate of evaporation (equation(12)) can also be used to estimate the net infiltration required to balance the rate of evaporation in a fracture network assuming that flow in the matrix is negligible. The method used before to determine the EIT for a single fracture is now extended to formulate the EIT for a fracture network with a uniformly applied infiltration. Consider a three-dimensional fractured rock domain with cross-sectional area, Ac, and height, H, as shown in Figure 8. The fractures are conceptualized as continuous uniform vertical tubes in the rock with constant cross-sectional area, zlf. Assuming that infiltration is steady and distributed equally among all of the fractures, the mass flow of water through a network of fractures can be expressed in the following mass balance: pqac = rhfn (16) where q is the Darcy infiltration (m/s) over the continuum cross-sectional area, A c; rhf is the mass flow rate of water through a single fracture (kg/s); and N is the number of fractures in the domain. The number of fractures, N, is equal to the number of fractures per unit area, f, multiplied by the continuum cross-sectional area, A c. In addition, the number of fractures per unit area, f, is defined as the fracture porosity, tkf, divided by the fracture cross-sectional area, 4f. Using these relations in (16) yields the following relation for the mass flow of water through a single fracture given a specified continuum infiltration, q' Cross- I Sectional Area, Af 2r b Cro ss- Sectional Area, A c Water Droplet Figure 8. Sketch of the fractured domain used to develop a balance between infiltration and evaporation in a network of fractures. yields the following final form of the mass flow of water through a single fracture: pq rrr rhf = 8kf, b (19) The mass flow of water through a single fracture (equation (19)) is set equal to the mass flow of evaporation from a single water droplet (equation(12)). The resulting equation is solved for the continuum infiltration rate, q: 32Dk,b(P - RHPsat) q = prtr} (20) In the above equation the infiltration rate, q, is a threshold infiltration (m/s). Infiltration rates above this threshold exceed pq pqaf the rate of evaporation in a fracture and water will propagate through the fracture. Infiltration rates below this threshold are less than the rate of evaporation, and propagation of water The radius of each fracture tube is assumed to be equal to through the fracture cannot occur. The above formulation the radius of the water droplet (see Figure 1) so that Af = assumes that evaporation occurs only from a single droplet The fracture porosity can also be expressed as a function of the formed by either asperities or cavities within each fracture water droplet radius using the following definition of bulk (Figure 1). fracture permeability, kf, t,' Figure 9 shows the evapoinfiltration threshold (EIT) given by (20) as a function of droplet radius (and hence fracture = = 8) radius) and relative humidity of the surroundings. The vapor pressure of the water droplet, Pt,, is calculated using (9) (inwhere kf is the permeability of a single fracture. Poiseuille flow cludes curvature effects). As the droplet radius increases, the in a circular tube has been assumed to derive the above ex- EIT decreases because more infiltrating water is being concenpression for the intrinsic fracture permeability, kf. The bulk trated into fewer fractures. As the relative humidity of the fracture permeability, kf, 6, has been estimated from air- surroundings decrease, the EIT increases because more evappermeability tests in the field to be on the order of m 2 oration occurs in the "drier" environments (refer to Figure 5) (Y. Tsang, personal communication, Earth Sciences Division, Lawrence Berkeley National Laboratory, 1997) and is assumed to be a constant in this study. Equation (18) then yields an expression for the fracture porosity as a function of droplet radius assuming that the water droplet radius is equal to the fracture radius. Expressing the fracture porosity and fracture cross-sectional area in terms of water droplet radius in (17) and greater infiltration rates can be accommodated. Because isotopic evidence suggests that flow through fractures at great depths has occurred recently at Yucca Mountain [Wolfsberg et al., 1996], one can postulate that the average infiltration (for the past 50 years) must have been greater than the threshold infiltration (EIT) given in Figure 9 for a specified fracture (droplet) size and relative humidity. The fracture size H

6 2670 HO: EVAPORATION OF PENDANT WATER DROPLETS factor through the formulation would result in a threshold infiltration that was reduced by a factor of 10. In other words, if only 10% of the available fractures contained water, then the mass flow in each of those fractures would be increased by a factor of 10 and the total infiltration, q, would have to be reduced by a factor of 10. If evaporation within the fractures were more pervasive than just from a single droplet (e.g., evaporation along the surface of fracture film flow), then the amount of infiltration that could be applied would be greater than that given by (20). Similarly, if imbibition of water from the fractures into the rock matrix occurred at a steady rate, then the total amount of infiltration that could be applied would also be increased. Another assumption is that the threshold infiltration shown in Figure 9 pertains only to steady conditions. If transient events occur, then large pulses of infiltration from individual events might be sufficiento propagate water through the frac- Figure 9. Evapoinfiltration thresholds (EITs) as a function tures to great depths [Altman et al., 1996; Nitao and Buscheck, 1991]. However, if steady state models are used, Figures 7 and 9 provide a novel method of estimating a lower bound to of droplet radii and surrounding relative humidities for flow steady infiltration based on water flow (and evaporation) in and evaporation in a network of fractures (k,, = 10-3 m2). fractures. Finally, although this paper presents a theoretical framework to describ evaporation of water droplets in fractures, the can be estimated using (18) and the following relations that accuracy of the models presented herein remains to be tested. define fracture porosity, 4, and fracture spacing, d: The author recommends that to the extent possible, experimental studies be performed to assess the accuracy of the ck = fa (21) droplet evaporation and EIT models presented in this paper. d = f-l/2 (22) Until further verification can be performed, the models should be used with caution, especially when estimating EITs in frac- Plugging (21) and (22) into (18) yields an expression relating ture networks due to the highly idealized assumption of flow the fracture size to the fracture spacing: occurring steadily through a set of uniform cylindrical tubes. 5. Conclusions Using an estimated fracture spacing of 0.74 m [Wilson et al., The evaporation of water droplets in fractures has been 1994] in the host rock at Yucca Mountain, (23) yields a fracture investigated with consideration of liquid-vapor interface curradius of 730/xm. Assuming that the average liquid saturation vature. The following conclusions have been reached based on of the rock matrix is 0.8 (yielding a relative humidity of ), the results of this study: the EIT is approximately 10-3 mm/yr. Thus the average infil- 1. Evaporation of pendant water droplets that are formed tration applied uniformly to all fractures must have exceeded at asperities or cavities in fractures is possible, even in nearly 10-3 mm/yr in recent times for water flow to propagate saturated environments, because of the convex liquid-vapor through the fractures without being evaporated. However, it is interface curvature that increases the water droplet vapor presimportant to recall the major assumptions that have led to this sure above the saturated vapor pressure of a flat or concave conclusion for infiltration through a network of fractures: liquid surface found within the pores of the matrix. 1. Infiltration is steady and distributed among uniformly 2. Evaporation rates of water droplets depend on the dropspaced fractures approximated as circular tubes. Imbibition let radius and relative humidity of the surroundings. As the into the matrix is neglected. droplet radius increases, the evaporation rate increases be- 2. Evaporation in a fracture occurs from a pendant water cause of the increased surface area available for diffusion. As droplet with an available evaporative surface area that can be the relative humidity of the surroundings decreases, the driving defined by a mean radius of curvature equal to that of a sphere potential for mass transfer increases and the evaporation rate the si e of the fracture aperture. Evaporation in each fracture increases. only occurs from a single droplet (assumed to be formed at 3. The effect of liquid-vapor interface curvature on enasperities or cavities (Figure 1)). hancing evaporation of droplets is only significant for small 3. The fracture porosity is a function of the bulk fracture droplets (<1 /am) and nearly saturated environments (RH permeability (assumed to be constant at 1 x 10-3 m 2) and 100%). For droplets greater than 100/am, the droplet behaves water droplet radius, which is defined by the size of the frac- similarly to a flat surface of liquid, and for surrounding relative ture aperture. humidities less than 99%, the driving force for evaporation is Given the above assumptions, it is useful to examine the dominated by the surrounding vapor pressure and not the impact on the conclusions if deviations from these assumptions droplet vapor pressure. occur. If the infiltration were not evenly distributed among all 4. Evapoinfiltration thresholds (EITs) have been develavailable fractures, the number of available fractures for the oped by balancing the amount of evaporation from a water analysis, N, would be less, say by a factor of 10. Carrying this droplet in a fracture with the infiltrating mass flow through the

7 HO: EVAPORATION OF PENDANT WATER DROPLETS 2671 fracture. For infiltration rates above the EIT the liquid flow through the fractures exceeds the evaporation rate, and propagation through the fractures can occur. For infiltration rates below the EIT, infiltration is less than the evaporation in each fracture and propagation cannot occur. Acknowledgments. The author would like to thank Mike Wilson, Jack Gauthier, and Peter Davies for their insightful discussions and careful reviews of this paper. This work was performed under Work Agreement WA-335, Rev 0, WBS This work was supported by the Yucca Mountain Site Characterization Office as part of the Civilian Radioactive Waste Management Program, which is managed by the U.S. Department of Energy, Yucca Mountain Site Characterization Project. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL References Altman, S. J., B. W. Arnold, R. W. Barnard, G. E. Bart, C. K. Ho, S. A. McKenna, and R. R. Eaton, Flow Calculations for Yucca Mountain Groundwater Travel Time (GWTF-95), SAND , Sandia Natl. Lab., Albuquerque, N.M., Bear, J., Dynamics of Fluids in Porous Media, p. 488, Elsevier, New York, Carey, V. P., Liquid-Vapor Phase-Change Phenomena, pp , Hemisphere, Washington, D.C., Fisher, L. R., and J. N. Israelachvili, Direct experimental verification of the Kelvin equation for capillary condensation, Nature, 277(5697), , Nitao, J. J., and T. A. Buscheck, Infiltration of a liquid front in an unsaturated, fractured porous-medium, Water Resour. Res., 27(8), , Tokunaga, T. K., and J. M. Wan, Water film flow along fracture surfaces of porous rock, Water Resour. Res., 33(6), , Van Genuchten, M. T., A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, , Vargaftik, N. B., Tables on the Thermophysical Properties of Liquids and Gases in Normal and Dissociated States, 2nd ed., John Wiley, New York, Wilson, M. L., et al., Total-system performance assessment for Yucca Mountain--SNL second iteration (TSPA-1993), SAND , Sandia Natl. Lab., Albuquerque, N.M., Wolfsberg, A. V., B. A. Robinson, and J. T. Fabryka-Martin, Migration of solutes in unsaturated, fractured rock at Yucca Mountain: Measurements, mechanisms and models, in Symposium V.' Scientific Basis for Nuclear Waste Management XIX, edited by W. M. Murphy and D. A. Knecht, pp , Mat. Res. Soc., Warrendale, Pa., C. K. Ho, Sandia National Laboratories, P.O. Box 5800, MS 1324, Albuquerque, NM ( ckho@nwer.sandia.gov) (Received February 19, 1997; revised August 15, 1997; accepted September 2, 1997.)