Unsaturated flow in coarse porous media

Transcription

1 252 NOTE / NOTE Unsaturated flow in coarse porous media Jeff R. Reinson, Delwyn G. Fredlund, and G. Ward Wilson Abstract: Design of effective capillary barrier systems requires a thorough understanding of the soil water interactions that take place in both coarse- and fine-grained unsaturated soils. Experimental observations of water flow through coarse porous media are presented to gain greater understanding of the processes and mechanisms that contribute to the movement and retention of water in coarse-grained unsaturated soils. The use of pendular ring theory to describe how water is held within a porous material with relatively low volumetric water contents is explored. Experimental measurements of seepage velocity and volumetric water content were obtained for columns of 12 mm glass beads using digital videography to capture the movement of a dye tracer front at several infiltration rates. An estimated curve for hydraulic conductivity versus matric suction is shown and compared to a theoretical curve. The method is shown to provide a reasonable predictive tool. Key words: soil-water characteristic curve, hydraulic conductivity curve, water permeability function, capillary barrier, matric suction. Résumé : La conception de systèmes d écrans capillaires exige une compréhension approfondie des interactions soleau qui se produisent dans les sols non saturés à gros grains de même qu à grains fins. On présente des observations expérimentales de l écoulement d eau à travers des milieux poreux grossiers pour acquérir une plus grande compréhension des processus et mécanismes qui contribuent au mouvement et à la rétention de l eau dans les sols non saturés à gros grains. On explore l utilisation de la théorie du pendule annulaire pour décrire comment l eau est retenue à l intérieur d un matériau poreux avec des teneurs en eau volumétriques relativement faibles. Des mesures expérimentales de la vitesse d infiltration et de la teneur en eau volumétrique ont été obtenues pour des colonnes de billes de verre de 12 mm utilisant la vidéographie numérique pour capter le mouvement du front d un traceur coloré à plusieurs vitesses d infiltration. Finalement, on montre une courbe estimée de la conductivité hydraulique en fonction de la succion matricielle et on la compare à la courbe théorique. On montre que la méthode fournit un outil de prédiction raisonnable. Mots clés : courbe caractéristique sol-eau, courbe de conductivité hydraulique, fonction de perméabilité à l eau, écran capillaire, succion matricielle. [Traduit par la Rédaction] Reinson et al. 262 Introduction In recent times, designers of effective cover and liner systems have taken advantage of the capillary barrier phenomena to reduce infiltration (Nicholson et al. 1989). Capillary Received 16 October Accepted 16 June Published on the NRC Research Press Web site at on 19 February J.R. Reinson. Diavik Diamond Mines Inc., PO Box 2498, Station Main, th Avenue, Yellowknife, NT X1A 2P8, Canada. D.G. Fredlund. Department of Civil Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada. G.W. Wilson. 1 Department of Mining Engineering, The University of British Columbia, 6350 Stores Road, 5th Floor, Vancouver, BC V6T 1Z4, Canada. 1 Corresponding author ( gww@mining.ubc.ca). barriers in an unsaturated soil profile are formed using coarse-grained materials that drain to low degrees of saturation, and hence low unsaturated hydraulic conductivities, under small matric suctions. The layer of coarse-grained material has a lower unsaturated hydraulic conductivity than the overlying finer grained material and acts to limit infiltration. Barbour and Yanful (1994) and Stormont and Morris (1998) show that reliable predictions of water retention and flow in unsaturated coarse-grained soils are critical for accurately modeling drainage through layered soil systems. The Unified Soil Classification System (USCS) defines coarsegrained soils as comprising two parts: sand ranging from to 4.75 mm and gravel ranging from 4.75 to 75 mm. Prediction and measurement of soil water retention and flow in coarse-grained materials is challenging. In this study, a coarse-grained porous medium composed of spheres is considered as a step towards a better general understanding of coarse-grained materials. The soil-water characteristic curve (SWCC) of coarse-grained materials is explored using Can. Geotech. J. 42: (2005) doi: /T04-070

2 Reinson et al. 253 classic capillary theory to determine the air-entry value and experimental observation and pendular ring theory (Gvirtzman and Roberts 1991) to estimate the point of residual saturation. Pendular rings describe the water retained at the contact point of a soil grain with another grain. Theoretically derived values are compared with experimental measurements of soil water interactions in coarse-grained unsaturated porous media obtained using direct visual observation from a series of controlled experiments. The laboratory study used a porous medium (i.e., glass beads) that was uniform in size and shape and considered to be ideal for this study (Reinson 2001). Water movement and retention are recorded using high-speed videography. The research program presented in this paper involved (i) investigation of capillary theory to estimate the SWCC for uniform glass beads with a diameter of 12 mm, (ii) experimental observations of water retention in the glass bead medium, (iii) development of an SWCC based on theory and experiment, (iv) development of a methodology to measure seepage velocities through uniform glass beads, (v) measurement of the seepage velocities, (vi) estimation of the hydraulic conductivity versus matric suction curve (i.e., k versus (u a u w ), where u a is the pressure in the air phase and u w is the pressure in the water phase), and (vii) comparison of the estimated k versus (u a u w ) curve to a theoretical curve predicted by the method proposed by Brooks and Corey (1964). Background There is precedent for the study of water movement and retention in porous media using ideal soil systems, both for experimentation and for the development of theory. In 1856, Henri Darcy published his results for the flow of water through sand filters used in connection with public fountains (Darcy 1856). Darcy determined that for laminar flow, a linear relationship existed between the gradient producing flow and the resulting flow rate. Whitney (1889) used an ideal soil system to qualitatively study soil-water retention. Briggs (1897) published results on the mechanics of soil moisture, and Slichter (1898) undertook a theoretical study for the motion of groundwater and related coefficients of permeability of soil to pore shapes in systems of equally packed spheres. The retention of water in an unsaturated soil system was investigated by Haines (1925), who developed several trigonometric formulae describing the cohesive forces within the water phase of a given soil volume. A circular air water interface was assumed and an equation derived for the interparticle force at the contact point in terms of the angular radius of the wetted area. Fisher (1926) reviewed the theoretical approach Haines adopted and revealed that the omission of the tension in the air water interface introduced an erroneous factor into the formulae. Several corrections were then made to the derivation provided by Haines, and calculations were developed for the half-volume of water at the contact point (i.e., the volume of water revolved around the contact point of two spheres), the water tension at the contact point, and the interparticle force (Fisher 1926). Waldron et al. (1961) tested the validity of the theory developed by Fisher (1926) by extracting water from saturated packs of uniformly sized glass beads. The study revealed the amount of water retained by the glass beads greatly exceeded the quantity predicted using the expression given by Fisher. Waldron et al. stated that an additional mechanism due to the pressure drop across the air water interface, which is involved in the retention of water in coarse-grained, non-interacting materials was present. The lack of agreement may also be due to the fact that Fisher s expression did not include a term for the adsorptive mechanism between the water and the solid. Horton and Hawkins (1965) investigated the infiltration path of water through a vertically layered system composed of glass spheres of three different sizes. The authors demonstrated that the capillary potential of the small spheres was greater than that of the larger spheres. The smallest spheres supported water by surface tension throughout the length of the column after the column was saturated and drained. The water was found to be supported at progressively lower levels in the medium as the sphere size increased. Gvirtzman and Roberts (1991) developed a conceptual model that describes the spatial distribution of two immiscible fluids in the pore space of packed spheres. The model quantitatively analyzed the interfacial area between wetting and nonwetting fluids and the fluids and the solid spheres as a function of degree of saturation. A potential application of the model is to quantify the air water interface in the unsaturated zone. General equations were developed for the volume of water stored at grain to grain contacts and for the surface area of various packing arrangements. Theoretical estimation of the soil-water characteristic curve Essential to understanding unsaturated soil-water systems is the soil-water characteristic curve (SWCC). The SWCC is the relationship between soil suction (matric suction) and water content (gravimetric or volumetric) or degree of saturation. The curve represents the storage capability of the soil and defines the amount of water retained in the pores under various matric suction values. Stated in another way, the SWCC represents the relationship between the amount of water within a soil volume and the energy in the water phase (matric suction). Barbour (1998) illustrated the application of the SWCC as a conceptual, interpretive, and predictive model for understanding unsaturated soil behaviour. The key elements of an SWCC are illustrated in Fig. 1. The volumetric water content decreases slowly from saturation at zero suction to lower values of water content with increasing matric suction. The water content change is due to consolidation of the soil matrix for a compressible soil matrix. Pore spaces remained saturated until the air-entry value is reached, defined as occurring at the matric suction that must be exceeded before air recedes into the soil pores (Fredlund and Rahardjo 1993). The air-entry value is therefore the matric suction value at which the largest pores begin to drain. The volumetric water content now decreases more rapidly due to the emptying of soil pores. The residual water content (Fig. 1) occurs when an increase in matric suction does not produce significant change in the volumetric water content (Fredlund and Rahardjo 1993). At this point, air has entered all the pore spaces within the material and the remaining water is held primarily at grain to grain contacts. This research begins with an examination of classic capillary

3 254 Can. Geotech. J. Vol. 42, 2005 Fig. 1. Typical soil-water characteristic curve plotted from 0.1 to kpa for a drying cycle. Fig. 2. Surface tension phenomena; forces acting on a twodimensional curved surface (after Fredlund and Rahardjo 1993). theory for the estimation of air entry of an arrangement of spheres and the work of Gvirtzman and Roberts (1991) to estimate the point of residual saturation. The theory is developed by addressing the physics of matric suction, air-entry value, and residual saturation in turn. Fig. 3. (a) Cubic packing arrangement (after Graton and Fraser 1935). (b) Geometry of a simple cubic layer. Matric suction Matric suction represents the balance of the forces acting across the air water interface in a porous medium. It is defined as the difference between the air (u a ) and water (u w ) pressures, or (u a u w ). A concave curvature of the air water interface towards the higher pressure forms because the cohesive forces between the water molecules at the air water interface are not equal in all directions (Fig. 2). This is similar to a bubble where the pressure inside is greater than the pressure outside. The radius of this curvature can be used to relate the pressure difference ( u) across the curved surface to the surface tension. Balancing the forces in the vertical direction (Fig. 2) results in [1] 2Ts sinβ = 2 urs sinβ where 2R s sin β is the length of the curved surface projected onto the horizontal plane; and R s and T s represent the radius and surface tension ( N/m at 20 C), respectively. Equation [1] can be simplified to the following form for the case of a pressure difference across a two-dimensional curved surface: [2] u = T s / R s Air-entry value The point of air entry occurs when the largest pores within the porous media begin to desaturate. The air-entry value is a function of the size of the largest pores, which are larger for coarse-grained soils and smaller for fine-grained soils. Coarse-grained soils may start to drain immediately following the application of a small matric suction. The largest pores in a well-packed system of uniform spheres will occur within a cubic packing arrangement, as shown in Fig. 3a. Figure 3b shows the geometry of a simple

4 Reinson et al. 255 cubic layer. Classic capillary theory can be used to calculate the matric suction corresponding to the air-entry value as follows: the geometry of the right-angled triangle is formed by taking a line from the contact point of two of the spheres to the centre of the inner capillary circle, as shown in Fig. 3b. This provides a relationship between the radius of the capillary circle (r) and the radius of the sphere (R): R [3] r = R cos 45 Fredlund and Rahardjo (1993) show that when the contact angle between the water and the glass sphere is assumed to be zero, eq. [4] applies: [4] ( ua uw) = 2Ts/ Rs where u a is the pressure in the air phase; u w is the pressure in the water phase; and (u a u w ) is the matric suction, or the pressure difference across a three-dimensional surface. The matric suction, (u a u w ), at the air-entry point can be calculated by assuming that the contact angle between the water and the glass spheres is zero and substituting eq. [3] into eq. [4] (i.e., r equals R s ): 2T [5] ( ua uw) = s R/(cos 45) R Fig. 4. (a) Meniscus of water, as a pendular ring. (b) Geometry of a pendular ring. Point of residual saturation Gvirtzman and Roberts (1991) modeled the static condition of a wetting fluid, composed of water present at sphere to sphere contacts, and showed that it was applicable at low degrees of saturation. The meniscus of water between two uniform spherical particles is illustrated in Fig. 4a and is described as a pendular ring. The pendular ring has centres of curvature in opposite directions. Further consideration of the geometry of the single pendular ring shown in Fig. 4b reveals that two trigonometric relations can be written for the pendular radii (r 1 and r 2 ), sphere radius (R), and β * as given in eqs. [6] and [7]: 1 cosβ [6] r1 = R cos β sinβ [7] r2 = R + cos β 1 * cos β * * where r 1 is the radius of the arc between the tangency points with the spheres, r 2 is the radius of rotation of the ring, and β * is the angle between the straight line through the centre of the spheres and a line through the centre of one sphere at the point of tangency with the pendular ring. The geometry considered here is for uniform spheres with equal radii (R). The curved membrane of a pendular ring can be visualized in three dimensions as a wheel rim or saddle shaped for a partial ring. Equation [8] expresses the matric suction at the interface by using the Laplace equation based on pressures acting across the membrane: [8] ( ua uw) = T 1 1 s + r r 1 2 where ( ua uw) is the pressure difference across the membrane or the matric suction, T s is the surface tension, and r 1 and r 2 are as previously defined. The similarity between eqs. [2] and [8] can be noted. In an unsaturated porous medium, r 1 of the pendular ring is concave towards the air and therefore creates a negative pressure. Similarly, r 2 is concave toward the water and therefore creates a positive pressure. Figure 4b defines both radii as positive values, however, and therefore the negative sign appears in front of r 1 in eq. [8]. An expression for matric suction as a function of sphere size and pendular ring size, which is described by the angle to the point of tangency (β * ), is found by substituting eqs. [6] and [7] into eq. [8]:

5 256 Can. Geotech. J. Vol. 42, 2005 [9] ( u u ) = a T s w 1 1 cos β R cos β + sinβ R 1 + cos β 1 * cos β * * Fig. 5. Unit cells of the (a) cubic and (b) rhombic packing arrangements (after Graton and Fraser 1935). A theoretical value for matric suction can be calculated by assuming the radius of the spheres (R) and varying the angle to the point of tangency (β * ). Expressions to calculate the volume of pendular rings (V p ) have been previously derived by several authors (Haines 1925; Rose 1958; Orr et al. 1975). The equation presented by Rose (1958) is as follows: 3 [10] Vp = 2πR 2 2cos β tanβ 2sinβ tanβ * * * * * π * cos β + β 2 * cos β 1 2 The soil structure should be considered for the application of these formulae in the context of an unsaturated soil. Graton and Fraser (1935) investigated numerous packing arrangements of spheres. The extreme porosities for ideal packings are the cubic and rhombic arrangements shown in Fig. 5, with porosities of 47.64% and 25.92%, respectively. The number of equivalent pendular rings within a given volume can be determined by considering the geometry of cubic and rhombic unit cells in Fig. 5. The cubic unit cell clearly comprises 12 quarter pendular rings and therefore contains three complete pendular rings. The number of pendular rings in the rhombic unit cell is not as clear. Upon inspection, it can be seen that the rhombic unit cell can be divided into one octahedron and two tetrahedrons, as shown in Fig. 6. The octahedron contains 12 parts of a ring that make up of a full pendular ring (i.e., the geometry shown in Fig. 4b rotated around the vertical y axis, as compared with rotation of 360 to produce a full pendular ring). One tetrahedron contains six parts of a ring that make up of a full pendular ring (times two tetrahedrons), thus a rhombic cell contains six complete pendular rings. The relationship between the matric suction and volumetric water content of pendular rings can be derived for a given radius of spheres using eq. [9] to calculate matric suction. Equation [10] is used to calculate the volume of a single pendular ring. Volumetric water content is obtained by multiplying the volume of an individual pendular ring by the number of pendular rings for a given packing arrangement and dividing by the volume of the respective unit cell. Equations [9] and [10] are linked by the angle to the point of tangency (β * ). The relationship of matric suction and volumetric water content can then be plotted. A major assumption for this calculation is that the angle to the point of tangency (β * ) is constant within the unit cell. Fig. 6. The rhombic unit cell divided into two tetrahedrons and one octahedron (after Graton and Fraser 1935). Fig. 7. Theoretical soil-water characteristic curves for cubic and rhombic packing arrangements of 12 mm glass beads, based on pendular theory. Figure 7 shows the relationship between matric suction and volumetric water content calculated using eqs. [9] and [10] for both rhombic and cubic ideal packing arrangements of 12 mm diameter spheres. The pendular ring calculations become invalid once the water content increases to the point where the rings coalesce. Examination of Fig. 4 indicates that coalescence occurs at an angle of 45 for cubic packed spheres, with the assumption that the contact angle is zero. Consideration of three spheres in a rhombic packing arrangement indicates that the rings coalesce at 30. The curves are therefore only valid to the right of the point of 45 for the cubic packing curve and 30 for the rhombic packing curve. It can be seen that the curves to the left of these points are inaccurate due to the coalescence of pendular rings. The calculated water content at saturation ranges from 42% (rhombic packing) to 15% (cubic packing). These calculations do not match the porosity of the rhombic and cubic unit cells of 25.92% and 47.64%, respectively.

6 Reinson et al. 257 Fig. 8. Lucite column through which the digital video images were taken (left) and seepage reservoir at the top of the lucite column (right). Experimental methods A set of column experiments was conducted to determine the unsaturated hydraulic conductivity function and SWCC of unsaturated glass beads. Digital videography was used to observe the formation and characteristics of pendular rings and record the infiltration of dye tracers through the pore spaces of a column of 12 mm glass beads. The experimental apparatus, including the column and a seepage reservoir, are shown in Fig. 8. The apparatus consisted of a clear lucite column 1100 mm high with an i.d. of 100 mm and an o.d. of 110 mm. The column was filled to a height of approximately 900 mm with 12 mm transparent glass beads. The dry density (1.23 g/cm 3 ) and specific gravity (3.1 g/cm 3 ) of the porous beads were measured, and the porosity of the packed glass beads was calculated to be 61%. The seepage reservoir consisted of a lucite tube with a volume of approximately 1500 ml, solid lucite cover, and sealed base plate. Approximately 150 pieces of smalldiameter (approximately 1 mm i.d.) rubber tubing were threaded through holes in the base plate and sealed with silicone. The opposite ends of the tubing were threaded through a lucite plate that was grooved to fit at the top of the column. The seepage reservoir was partially filled with water and then connected to a rheostatic peristaltic pump that allowed the infiltration rate to be accurately controlled. For the dye tracer experiments, steady-state flow conditions were established in the glass bead column through the application of a constant infiltration rate for a period of 2 h. Videography data were collected using three infiltration rates: 0.630, 0.150, and mm/s. Infiltration rates are expressed as volume per total cross-sectional area of column per unit time, which is equivalent to the units used to describe rainfall rates. A coloured dye tracer was then applied to a single point on the surface of the glass beads within the column. The tracer dye was chosen based on a review of several dyes that revealed standard commercial blue food colouring to be the most visible for digital imaging. The camera was placed adjacent to the column and focused on the centre of the transparent column. Video images were captured at three distinct vertical positions for each infiltration rate, at vertical flow lengths of 30, 50, and 170 mm depth from the top surface of the glass beads within the column. Images were recorded at a single depth from the surface of the column for a given dye application. The steady-state infiltration rate was maintained until the dye was removed from the column, and the dye tracer application was repeated. Several sets of images were recorded for each vertical position and for each infiltration rate. The video imagery was examined to determine the timing of the passage of the dye front past the camera location. The digital images were enhanced to clarify the position of the dye within the glass beads. Image enhancement involved choosing an image in which the dye was easily seen and modifying the colour of the dye in the image. A range of blue shades was found to define the colour of the dye; the dominant shade was replaced with blue, and the lighter and darker shades were replaced with green and purple, respectively. Once the enhancement was complete for one image, the same enhancement was repeated for all images in each series. The elapsed time between images was calculated by recording the frame number from the recorded video image and the video frame rate of frames/s. Once the digital video images were reviewed, the velocities of the dye front through the glass beads was calculated by measuring the vertical flow length and dividing by the elapsed time.

7 258 Can. Geotech. J. Vol. 42, 2005 Fig. 9. Pore space of a simple rhombic layer formed by 12 mm glass beads within the column of glass beads. Video and still images were also recorded during periods of wetting, during drain down, and at steady-state conditions, both with and without infiltration, to allow examination of water distribution at the pore scale and facilitate the observation of pendular rings within the pore space. The saturated hydraulic conductivity of the glass beads was determined using constant-head infiltration testing following American Society for Testing and Materials (ASTM) standards D and D Results and discussion Figure 9 shows a detailed image of the glass bead column observed at the 30 mm vertical flow length during static conditions. The pore space of a simple rhombic layer within the glass bead column is visible. Figure 9 demonstrates the presence of pendular ring structures within the glass bead columns. An approximate angle to the point of tangency is indicated in Fig. 9. Observations of the pendular rings formed in the pore spaces indicated that the pendular rings coalesced at angles of tangency smaller than those predicted by the ideal geometry shown in Fig. 4. Angles of tangency when the pendular rings jumped to a coalesced arrangement were observed to be approximately 20. The observed jump is often referred to as the Haines jump, as Haines (1925) was the first to note the phenomenon. The observed coalescence of the pendular rings to a continuous water phase can be used to estimate the point of residual saturation. These observations indicate that the pendular rings do not strictly obey eqs. [9] and [10], and that residual saturation occurs at a matric suction greater than the matric suctions that correspond to angles of tangency of 45 and 30 indicated in Fig. 7. Attempts to experimentally measure the suction in the glass bead columns using tensiometers were unsuccessful. An approximation for the matric suction at the point of residual saturation can be made using eqs. [9] and [10] and the observed point of tangency angle of 20. The calculated matric suction is kpa. Figures 10 and 11 demonstrate typical images collected during the dye infiltration studies. Figure 10 shows a series of images that were extracted from the video clip of tracer dye moving through the column with an applied infiltration rate of mm/s. The images on the right side demonstrate the results of the digital enhancement procedure and better illustrate the movement of the dye front. Figure 11 shows a set of colour-enhanced close-up images of a 50 mm section of glass beads as the dye front moves through the beads, starting at image 080 and ending with image 135. The infiltration rate and the measured dye front velocity were used to determine the flux velocity (v flux ) and seepage velocity (v seep ) for each dye tracer experiment. The flux velocity is defined as the volumetric discharge (Q) over the total cross-sectional area (A) perpendicular to the direction of discharge and through which the discharge flows. Freeze and Cherry (1979) refer to flux velocity as the specific discharge or as the Darcy velocity for saturated materials. The term flux velocity is preferred in this study because it is not associated with saturated conditions. In the videography experiments, the flux velocity is therefore equal to the infiltration rate. Seepage velocity, for saturated conditions, is defined as the flux velocity (v flux ) divided by the porosity (n). Seepage velocity for unsaturated conditions is the flux velocity divided by the volumetric water content (θ w ). The measured dye velocities are considered to be equivalent to seepage velocities. Table 1 summarizes the results of the seepage velocity measurements based on the digital videography. Table 1 also shows the corresponding volumetric water contents and degrees of saturation calculated on the basis of the measured seepage velocities. The volumetric water content was calculated as the flux velocity divided by the seepage velocity. Three measured points relating seepage velocity to measured water content are therefore established for the columns of glass beads. The measurements of the saturated hydraulic conductivity of the glass beads indicated k sat was equal to m/s. The experimental observations can now be combined with the previously developed theory to construct an SWCC and

8 Reinson et al. 259 Fig. 10. Series of non-enhanced (left) and enhanced (right) images showing dye front moving through a 17.5 cm column section with an infiltration rate of mm/s: (a) image 400, (b) image 800, (c) image an unsaturated hydraulic conductivity curve for the glass bead media. Figure 12 presents an SWCC for the glass bead columns. Also presented for comparison is the SWCC measured experimentally by Stormont and Anderson (1999) for a pea gravel. The saturated volumetric water content of the glass bead column at kpa is shown to be 61%, which matches the measured porosity. The measured porosity is higher than the theoretical porosity of a cubic packing arrangement (47%), which indicates that the packing of spheres in the column was not optimal and extra pore space was created. Extra porosity may also be due to wall or container effects. The airentry value is shown for a volumetric water content of 61% and a matric suction of kpa. The glass beads are considered incompressible, and therefore no change in water content is considered between saturation and the air-entry value. The air-entry value is determined from consideration of the largest pore space created within a cubic packing arrangement. Equation [5] was used to calculate the matric suction (0.058 kpa) that corresponds to the pore geometry

9 260 Can. Geotech. J. Vol. 42, 2005 Fig. 11. Series of enhanced images showing the movement of the dye over a5cmvertical flow length with an infiltration rate of 0.15 mm/s: (a) image 080, (b) image 100, (c) image 112, (d) image 113, (e) image 125, ( f ) image 135. shown in Fig. 3. The higher saturated porosity indicates that pores greater than this size likely exist within the glass bead structure, and therefore the value of kpa represents an upper reasonable limit on the air-entry value. The point of residual saturation is estimated from the experimental observations of pendular ring behaviour noted earlier and is shown to occur at kpa. The volumetric water content at this point is estimated using the volume of the pendular rings indicated from eq. [10], adjusted for the differences in porosity. The final point on the estimated SWCC shown in Fig. 12 corresponds to a matric suction value of kpa, at which the water content of all soils is zero (Fredlund and Rahardjo 1993). Also presented in Fig. 12 for comparison is the SWCC measured experimentally by Stormont and Anderson (1999), who conducted testing on columns of various soil layer configurations, including a coarse-grained soil referred to as pea gravel, for the investigation of capillary barriers. The SWCC was determined by the hanging-column method. The grainsize distribution of the pea gravel was uniform, with D 90,

10 Reinson et al. 261 Table 1. Seepage velocity measurements based on digital videography. Flux velocity, v flux (mm/s) a Range of measured seepage velocity using digital videography, v seep (mm/s) Estimated volumetric water content, θ w = v flux /v seep (%) b Estimated degree of saturation, S = θ w /n (%) a Infiltration rate. b The seepage velocity used to estimate θ w was the median value of the range shown in the second column. Fig. 12. Estimated SWCC for 12 mm glass beads using a combination of capillary and pendular theory and a measured SWCC for pea gravel (after Stormont and Anderson 1999). Fig. 13. Comparison of the estimated hydraulic conductivity (k) versus matric suction ( ua uw) curve with the theoretical curve as predicted by Brooks and Corey (1964). D 50, and D 10 of 15, 8, and 5 mm, respectively. The measured wetting SWCC for the pea gravel had an air-entry value of approximately 0.02 kpa (i.e., 2 mm of suction head). The point of residual water content occurred at a volumetric water content of approximately 3.5% and at a matric suction of approximately 0.2 kpa (Stormont and Anderson 1999). An unsaturated hydraulic conductivity curve can be estimated by combining the measured relationship of water content to hydraulic conductivity with the theoretically determined SWCC presented in Fig. 12. The matric suction values that correspond to the volumetric water contents shown in Table 1 can be estimated using the SWCC for the glass beads shown in Fig. 12. The matric suction values were plotted against values of hydraulic conductivity (i.e., infiltration rate) that correspond to the same volumetric water content. This provides an estimate of the hydraulic conductivity versus matric suction relationship (Fig. 13). Wilson (1990) showed that the Brooks and Corey (1964) method was suitable for estimating the k versus matric suction (u a u w ) curve for cohesionless sand. The authors are not aware of the application of the Brooks and Corey method to a coarse uniform gravel, such as a pea gravel, for use as a capillary barrier. Figure 13 shows a comparison of the measured k versus matric suction (u a u w ) curve for the 12 mm glass beads with the theoretical curve predicted by Brooks and Corey. It can be seen that the measured and predicted curves show reasonable agreement. These results suggest that the Brooks and Corey method for predicting the unsaturated hydraulic conductivity of granular soils may be applied to coarse granular soils considered suitable for a capillary barrier. Conclusions Pendular ring and capillary theories can be used to predict the air-entry and residual saturation points on the soil-water characteristic curve (SWCC) for uniform coarse soils. The measurement of flux and seepage velocities can be used to estimate the relationships between unsaturated hydraulic conductivity and volumetric water contents in unsaturated coarse porous media. Measurement of seepage velocities using digital videography to record the high-speed soil water interactions is shown to be a useful tool. Although measurement is difficult, the results of the research program confirm that the hydraulic conductivity versus matric suction curve is extremely steep for coarse-grained soils. References Barbour, S.L Nineteenth Canadian Geotechnical Colloquium: The soil-water characteristic curve: a historical perspective. Canadian Geotechnical Journal, 35: Barbour, S.L., and Yanful, E.K A column study of static nonequilibrium fluid pressures in sand during prolonged drainage. Canadian Geotechnical Journal, 31: Briggs, L.J The mechanics of soil moisture. US Department of Agriculture, Soils Bulletin 10.

11 262 Can. Geotech. J. Vol. 42, 2005 Brooks, R.H., and Corey, A.T Hydraulic properties of porous media. Colorado State University, Hydrology Paper 3. Darcy, H Les fontaines publiques de la Ville de Dijon. Victor Dalmont, Paris. Fisher, R.A On the capillary forces of an,deal soil: correction of formulae given by W.B. Haines. Journal of Agricultural Science, 16: Fredlund, D.G., and Rahardjo, H Soil mechanics for unsaturated soils. John Wiley & Sons Inc., New York. Freeze, R.A., and Cherry, J.A Groundwater. Prentice-Hall Inc., Englewood Cliffs, N.J. Graton, L.C., and Fraser, H.J Systematic packing of spheres, with particular relation to porosity and permeability. Journal of Geology, 43(8): Gvirtzman, H., and Roberts, P.V Pore scale spatial analysis of two immiscible fluids in porous media. Water Resources Research, 27(6): Haines, W.B Studies in the physical properties of soils. II. A note on the cohesion developed by capillary forces in an ideal soil. Journal of Agricultural Science, 15: Horton, J.H., and Hawkins, R.H Flow path of rain from the soil surface to the water table. Soil Science, 100(6): Nicholson, R.V., Gillham, R.W., Cherry, J.A., and Reardon, E.J Reduction of acid generation in mine tailings through the use of moisture-retaining cover layers as oxygen barriers. Canadian Geotechnical Journal, 26: 1 8. Orr, F.M., Scriven, L.E., and Rivas, A.P Pendular rings between solids: meniscus properties and capillary force. Journal of Fluid Mechanics, 67(4): Reinson, J.R Soil water interactions in coarse porous media. M.Sc. thesis, Department of Civil Engineering, University of Saskatchewan, Saskatoon, Sask. Rose, W Volumes and surface areas of pendular rings. Journal of Applied Physics, 29(4): Slichter, C.S Theoretical investigations of the motion of ground waters. US Geological Survey, 19th Annual Report, Part 2, pp Stormont, J.C., and Anderson, C.E Capillary barrier effect from underlying coarser soil layer. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 125(8): Stormont, J.C., and Morris, C.E Method to estimate water storage capacity of capillary barriers. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 124(4): Waldron, L.J., McMurdie, J.L., and Vomocil, J.A Water retention by capillary forces in an ideal soil. Soil Science Society of America Proceedings, 25: Whitney, M Soil moisture. Journal of Agricultural Science, 3: Wilson, G.W Soil evaporative fluxes for geotechnical engineering problems. Ph.D. dissertation, Department of Civil Engineering. University of Saskatchewan, Saskatoon, Sask.