Unsaturated flow in coarse porous media

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 interactionsthat take place in both coarse- and fine-grained unsaturated soils. Experimental observations of water flow throughcoarse porous media are presented to gain greater understanding of the processes and mechanisms that contribute to themovement and retention of water in coarse-grained unsaturated soils. The use of pendular ring theory to describe howwater is held within a porous material with relatively low volumetric water contents is explored. Experimental measure-ments of seepage velocity and volumetric water content were obtained for columns of 12 mm glass beads using digitalvideography to capture the movement of a dye tracer front at several infiltration rates. An estimated curve for hydraulicconductivity versus matric suction is shown and compared to a theoretical curve. The method is shown to provide areasonable 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 sol-eau qui se produisent dans les sols non saturés à gros grains de même qu’à grains fins. On présente des observationsexpérimentales de l’écoulement d’eau à travers des milieux poreux grossiers pour acquérir une plus grande compréhen-sion des processus et mécanismes qui contribuent au mouvement et à la rétention de l’eau dans les sols non saturésà gros grains. On explore l’utilisation de la théorie du pendule annulaire pour décrire comment l’eau est retenue àl’intérieur d’un matériau poreux avec des teneurs en eau volumétriques relativement faibles. Des mesures expérimenta-les de la vitesse d’infiltration et de la teneur en eau volumétrique ont été obtenues pour des colonnes de billes de verrede 12 mm utilisant la vidéographie numérique pour capter le mouvement du front d’un traceur coloré à plusieurs vites-ses d’infiltration. Finalement, on montre une courbe estimée de la conductivité hydraulique en fonction de la succionmatricielle et on la compare à la courbe théorique. On montre que la méthode fournit un outil de prédiction raison-nable.

Mots clés : courbe caractéristique sol-eau, courbe de conductivité hydraulique, fonction de perméabilité à l’eau, écrancapillaire, succion matricielle.

[Traduit par la Rédaction] Reinson et al. 262

Introduction

In recent times, designers of effective cover and liner sys-tems have taken advantage of the capillary barrier phenom-ena to reduce infiltration (Nicholson et al. 1989). Capillary

barriers in an unsaturated soil profile are formed usingcoarse-grained materials that drain to low degrees of satura-tion, and hence low unsaturated hydraulic conductivities,under small matric suctions. The layer of coarse-grained ma-terial has a lower unsaturated hydraulic conductivity thanthe overlying finer grained material and acts to limit infiltra-tion. Barbour and Yanful (1994) and Stormont and Morris(1998) show that reliable predictions of water retention andflow in unsaturated coarse-grained soils are critical for accu-rately modeling drainage through layered soil systems. TheUnified Soil Classification System (USCS) defines coarse-grained soils as comprising two parts: sand ranging from0.075 to 4.75 mm and gravel ranging from 4.75 to 75 mm.

Prediction and measurement of soil water retention andflow in coarse-grained materials is challenging. In this study,a coarse-grained porous medium composed of spheres isconsidered as a step towards a better general understandingof coarse-grained materials. The soil-water characteristiccurve (SWCC) of coarse-grained materials is explored using

Can. Geotech. J. 42: 252–262 (2005) doi: 10.1139/T04-070 © 2005 NRC Canada

252

Received 16 October 2002. Accepted 16 June 2004.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 19 February 2005.

J.R. Reinson. Diavik Diamond Mines Inc., PO Box 2498,Station Main, 5007-50th Avenue, Yellowknife, NT X1A 2P8,Canada.D.G. Fredlund. Department of Civil Engineering, Universityof Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9,Canada.G.W. Wilson.1 Department of Mining Engineering, TheUniversity of British Columbia, 6350 Stores Road, 5th Floor,Vancouver, BC V6T 1Z4, Canada.

1Corresponding author (e-mail: [email protected]).

classic capillary theory to determine the air-entry value andexperimental observation and pendular ring theory (Gvirtz-man and Roberts 1991) to estimate the point of residual sat-uration. Pendular rings describe the water retained at thecontact point of a soil grain with another grain. Theoreticallyderived values are compared with experimental measure-ments of soil–water interactions in coarse-grained unsatu-rated porous media obtained using direct visual observationfrom a series of controlled experiments. The laboratorystudy used a porous medium (i.e., glass beads) that was uni-form in size and shape and considered to be “ideal” for thisstudy (Reinson 2001). Water movement and retention are re-corded using high-speed videography. The research programpresented in this paper involved (i) investigation of capillarytheory to estimate the SWCC for uniform glass beads with adiameter of 12 mm, (ii) experimental observations of waterretention in the glass bead medium, (iii) development of anSWCC based on theory and experiment, (iv) development ofa methodology to measure seepage velocities through uni-form glass beads, (v) measurement of the seepage velocities,(vi) estimation of the hydraulic conductivity versus matricsuction curve (i.e., k versus (ua – uw), where ua is the pres-sure in the air phase and uw is the pressure in the waterphase), and (vii) comparison of the estimated k versus (ua –uw) curve to a theoretical curve predicted by the method pro-posed by Brooks and Corey (1964).

Background

There is precedent for the study of water movement andretention in porous media using ideal soil systems, both forexperimentation and for the development of theory. In 1856,Henri Darcy published his results for the flow of waterthrough sand filters used in connection with public fountains(Darcy 1856). Darcy determined that for laminar flow, a lin-ear relationship existed between the gradient producing flowand the resulting flow rate. Whitney (1889) used an idealsoil 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 mo-tion of groundwater and related coefficients of permeabilityof soil to pore shapes in systems of equally packed spheres.

The retention of water in an unsaturated soil system wasinvestigated by Haines (1925), who developed several trigo-nometric formulae describing the cohesive forces within thewater phase of a given soil volume. A circular air–waterinterface was assumed and an equation derived for the inter-particle force at the contact point in terms of the angularradius of the wetted area. Fisher (1926) reviewed the theo-retical approach Haines adopted and revealed that the omis-sion of the tension in the air–water interface introduced anerroneous factor into the formulae. Several corrections werethen made to the derivation provided by Haines, and calcula-tions were developed for the half-volume of water at thecontact point (i.e., the volume of water revolved around thecontact point of two spheres), the water tension at the con-tact point, and the interparticle force (Fisher 1926).

Waldron et al. (1961) tested the validity of the theory de-veloped by Fisher (1926) by extracting water from saturatedpacks of uniformly sized glass beads. The study revealed theamount of water retained by the glass beads greatly ex-

ceeded the quantity predicted using the expression given byFisher. Waldron et al. stated that an additional mechanismdue “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 agree-ment may also be due to the fact that Fisher’s expression didnot include a term for the adsorptive mechanism between thewater and the solid.

Horton and Hawkins (1965) investigated the infiltrationpath of water through a vertically layered system composedof glass spheres of three different sizes. The authors demon-strated that the capillary potential of the small spheres wasgreater than that of the larger spheres. The smallest spheressupported water by surface tension throughout the length ofthe column after the column was saturated and drained. Thewater was found to be supported at progressively lower lev-els in the medium as the sphere size increased.

Gvirtzman and Roberts (1991) developed a conceptualmodel that describes the spatial distribution of two immisci-ble fluids in the pore space of packed spheres. The modelquantitatively analyzed the interfacial area between wettingand nonwetting fluids and the fluids and the solid spheres asa function of degree of saturation. A potential application ofthe model is to quantify the air–water interface in the unsat-urated zone. General equations were developed for the vol-ume of water stored at grain to grain contacts and for thesurface area of various packing arrangements.

Theoretical estimation of the soil-watercharacteristic curve

Essential to understanding unsaturated soil-water systemsis the soil-water characteristic curve (SWCC). The SWCC isthe relationship between soil suction (matric suction) andwater content (gravimetric or volumetric) or degree of satu-ration. The curve represents the storage capability of the soiland defines the amount of water retained in the pores undervarious matric suction values. Stated in another way, theSWCC represents the relationship between the amount ofwater within a soil volume and the energy in the water phase(matric suction). Barbour (1998) illustrated the applicationof the SWCC as a conceptual, interpretive, and predictivemodel for understanding unsaturated soil behaviour.

The key elements of an SWCC are illustrated in Fig. 1.The volumetric water content decreases slowly from satura-tion at zero suction to lower values of water content withincreasing matric suction. The water content change is dueto consolidation of the soil matrix for a compressible soilmatrix. Pore spaces remained saturated until the air-entryvalue is reached, defined as occurring at “the matric suctionthat must be exceeded before air recedes into the soil pores”(Fredlund and Rahardjo 1993). The air-entry value is there-fore the matric suction value at which the largest pores beginto drain. The volumetric water content now decreases morerapidly due to the emptying of soil pores. The residual watercontent (Fig. 1) occurs when an increase in matric suctiondoes not produce significant change in the volumetric watercontent (Fredlund and Rahardjo 1993). At this point, air hasentered all the pore spaces within the material and the re-maining water is held primarily at grain to grain contacts.This research begins with an examination of classic capillary

© 2005 NRC Canada

Reinson et al. 253

theory for the estimation of air entry of an arrangement ofspheres and the work of Gvirtzman and Roberts (1991) toestimate the point of residual saturation. The theory is devel-oped by addressing the physics of matric suction, air-entryvalue, and residual saturation in turn.

Matric suctionMatric suction represents the balance of the forces acting

across the air–water interface in a porous medium. It is de-fined as the difference between the air (ua) and water (uw)pressures, or (ua – uw). A concave curvature of the air–waterinterface towards the higher pressure forms because the co-hesive forces between the water molecules at the air–waterinterface are not equal in all directions (Fig. 2). This is simi-lar to a bubble where the pressure inside is greater than thepressure outside. The radius of this curvature can be used torelate the pressure difference (∆u) across the curved surfaceto the surface tension.

Balancing the forces in the vertical direction (Fig. 2) re-sults in

[1] 2 2T uRs ssin sinβ β= ∆

where 2Rs sin β is the length of the curved surface projectedonto the horizontal plane; and Rs and Ts represent the radiusand surface tension (0.07275 N/m at 20 °C), respectively.

Equation [1] can be simplified to the following form forthe case of a pressure difference across a two-dimensionalcurved surface:

[2] ∆u T R= s s/

Air-entry valueThe point of air entry occurs when the largest pores with-

in the porous media begin to desaturate. The air-entry valueis a function of the size of the largest pores, which are largerfor coarse-grained soils and smaller for fine-grained soils.Coarse-grained soils may start to drain immediately follow-ing the application of a small matric suction.

The largest pores in a well-packed system of uniformspheres will occur within a cubic packing arrangement, asshown in Fig. 3a. Figure 3b shows the geometry of a simple

© 2005 NRC Canada

254 Can. Geotech. J. Vol. 42, 2005

Fig. 1. Typical soil-water characteristic curve plotted from 0.1 to100 000 kPa for a drying cycle.

Fig. 2. Surface tension phenomena; forces acting on a two-dimensional curved surface (after Fredlund and Rahardjo 1993).

Fig. 3. (a) Cubic packing arrangement (after Graton and Fraser1935). (b) Geometry of a simple cubic layer.

cubic layer. Classic capillary theory can be used to calculatethe matric suction corresponding to the air-entry value asfollows: the geometry of the right-angled triangle is formedby taking a line from the contact point of two of the spheresto the centre of the inner capillary circle, as shown inFig. 3b. This provides a relationship between the radius ofthe capillary circle (r) and the radius of the sphere (R):

[3] rR

R= −cos 45

Fredlund and Rahardjo (1993) show that when the contactangle between the water and the glass sphere is assumed tobe zero, eq. [4] applies:

[4] ( ) /u u T Ra w s s− = 2

where ua is the pressure in the air phase; uw is the pressurein the water phase; and (ua – uw) is the matric suction, or thepressure difference across a three-dimensional surface.

The matric suction, (ua – uw), at the air-entry point can becalculated by assuming that the contact angle between thewater and the glass spheres is zero and substituting eq. [3]into eq. [4] (i.e., r equals Rs):

[5] ( )/(cos )

u uT

R Ra w

s− =−

245

Point of residual saturationGvirtzman and Roberts (1991) modeled the static condi-

tion of a wetting fluid, composed of water present at sphereto sphere contacts, and showed that it was applicable at lowdegrees of saturation. The meniscus of water between twouniform spherical particles is illustrated in Fig. 4a and is de-scribed as a pendular ring. The pendular ring has centres ofcurvature in opposite directions. Further consideration of thegeometry of the single pendular ring shown in Fig. 4b re-veals that two trigonometric relations can be written for thependular radii (r1 and r2), sphere radius (R), and β* as givenin eqs. [6] and [7]:

[6] r R1 = −⎛

⎝⎜

⎞

⎠⎟1 cos

cosβ

β

∗

∗

[7] r R21= + −⎛

⎝⎜

⎞

⎠⎟sin cos

cos

* *

*

β ββ

where r1 is the radius of the arc between the tangency pointswith the spheres, r2 is the radius of rotation of the ring, andβ* is the angle between the straight line through the centre ofthe spheres and a line through the centre of one sphere at thepoint of tangency with the pendular ring. The geometry con-sidered here is for uniform spheres with equal radii (R).

The curved membrane of a pendular ring can be visual-ized in three dimensions as a “wheel rim” or “saddleshaped” for a partial ring. Equation [8] expresses the matricsuction at the interface by using the Laplace equation basedon pressures acting across the membrane:

[8] ( )u u Tr r

a w s− = − +⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 1

1 2

where ( )u ua w− is the pressure difference across the mem-brane or the matric suction, Ts is the surface tension, and r1and r2 are as previously defined. The similarity betweeneqs. [2] and [8] can be noted.

In an unsaturated porous medium, r1 of the pendular ringis concave towards the air and therefore creates a negativepressure. Similarly, r2 is concave toward the water and there-fore creates a positive pressure. Figure 4b defines both radiias positive values, however, and therefore the negative signappears in front of r1 in eq. [8].

An expression for matric suction as a function of spheresize and pendular ring size, which is described by the angleto the point of tangency (β*), is found by substitutingeqs. [6] and [7] into eq. [8]:

© 2005 NRC Canada

Reinson et al. 255

Fig. 4. (a) Meniscus of water, as a pendular ring. (b) Geometryof a pendular ring.

[9] ( )u ua w− =

T

R R

s −−⎛

⎝⎜

⎞

⎠⎟

++ −⎛

⎝⎜

⎞

⎠⎟

1

1

1

1coscos

sin coscos

* *

*

ββ

β ββ

∗

∗

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

A theoretical value for matric suction can be calculated byassuming the radius of the spheres (R) and varying the angleto the point of tangency (β*).

Expressions to calculate the volume of pendular rings (Vp)have been previously derived by several authors (Haines1925; Rose 1958; Orr et al. 1975). The equation presentedby Rose (1958) is as follows:

[10] V Rp = − − −⎡

⎣⎢⎢

⎧⎨⎪

⎩⎪2 2 2 23π β β β βcos tan sin tan* * * *

+ −⎛⎝⎜

⎞⎠⎟

−⎛

⎝⎜

⎞

⎠⎟

⎤

⎦

⎥⎥

⎫⎬⎪

⎭⎪

π β ββ2

12

**

*

coscos

The soil structure should be considered for the applicationof these formulae in the context of an unsaturated soil.Graton and Fraser (1935) investigated numerous packingarrangements of spheres. The extreme porosities for idealpackings are the cubic and rhombic arrangements shown inFig. 5, with porosities of 47.64% and 25.92%, respectively.The number of equivalent pendular rings within a given vol-ume can be determined by considering the geometry of cu-bic and rhombic unit cells in Fig. 5.

The cubic unit cell clearly comprises 12 quarter pendularrings and therefore contains three complete pendular rings.The number of pendular rings in the rhombic unit cell is notas clear. Upon inspection, it can be seen that the rhombicunit cell can be divided into one octahedron and two tetrahe-drons, as shown in Fig. 6. The octahedron contains 12 partsof a ring that make up 109.47° of a full pendular ring (i.e.,the geometry shown in Fig. 4b rotated 109.47° around thevertical y axis, as compared with rotation of 360° to producea full pendular ring). One tetrahedron contains six parts of aring that make up 70.53° of a full pendular ring (times twotetrahedrons), thus a rhombic cell contains six completependular rings.

The relationship between the matric suction and volumet-ric water content of pendular rings can be derived for agiven radius of spheres using eq. [9] to calculate matric suc-tion. Equation [10] is used to calculate the volume of a sin-gle pendular ring. Volumetric water content is obtained bymultiplying the volume of an individual pendular ring by thenumber of pendular rings for a given packing arrangementand dividing by the volume of the respective unit cell. Equa-tions [9] and [10] are linked by the angle to the point oftangency (β*). The relationship of matric suction and volu-metric water content can then be plotted. A major assump-tion for this calculation is that the angle to the point oftangency (β*) is constant within the unit cell.

Figure 7 shows the relationship between matric suctionand volumetric water content calculated using eqs. [9] and[10] for both rhombic and cubic ideal packing arrangementsof 12 mm diameter spheres. The pendular ring calculationsbecome invalid once the water content increases to the pointwhere the rings coalesce. Examination of Fig. 4 indicatesthat coalescence occurs at an angle of 45° for cubic packedspheres, with the assumption that the contact angle is zero.Consideration of three spheres in a rhombic packingarrangement indicates that the rings coalesce at 30°. Thecurves are therefore only valid to the right of the point of45° for the cubic packing curve and 30° for the rhombicpacking curve. It can be seen that the curves to the leftof these points are inaccurate due to the coalescence ofpendular rings. The calculated water content at saturationranges from 42% (rhombic packing) to 15% (cubic packing).These calculations do not match the porosity of the rhombicand cubic unit cells of 25.92% and 47.64%, respectively.

© 2005 NRC Canada


Fig. 5. Unit cells of the (a) cubic and (b) rhombic packing ar-rangements (after Graton and Fraser 1935).

Fig. 6. The rhombic unit cell divided into two tetrahedrons andone octahedron (after Graton and Fraser 1935).

Fig. 7. Theoretical soil-water characteristic curves for cubic andrhombic packing arrangements of 12 mm glass beads, based onpendular theory.

Experimental methods

A set of column experiments was conducted to determinethe unsaturated hydraulic conductivity function and SWCCof unsaturated glass beads. Digital videography was used toobserve the formation and characteristics of pendular ringsand record the infiltration of dye tracers through the porespaces of a column of 12 mm glass beads. The experimentalapparatus, including the column and a seepage reservoir, areshown in Fig. 8. The apparatus consisted of a clear lucitecolumn 1100 mm high with an i.d. of 100 mm and an o.d. of110 mm. The column was filled to a height of approximately900 mm with 12 mm transparent glass beads. The dry den-sity (1.23 g/cm3) and specific gravity (3.1 g/cm3) of the po-rous beads were measured, and the porosity of the packedglass beads was calculated to be 61%.

The seepage reservoir consisted of a lucite tube with avolume of approximately 1500 mL, solid lucite cover, andsealed base plate. Approximately 150 pieces of small-diameter (approximately 1 mm i.d.) rubber tubing werethreaded through holes in the base plate and sealed with sili-cone. The opposite ends of the tubing were threaded througha lucite plate that was grooved to fit at the top of the col-umn. The seepage reservoir was partially filled with waterand then connected to a rheostatic peristaltic pump that al-lowed the infiltration rate to be accurately controlled.

For the dye tracer experiments, steady-state flow condi-tions were established in the glass bead column through theapplication of a constant infiltration rate for a period of 2 h.Videography data were collected using three infiltrationrates: 0.630, 0.150, and 0.024 mm/s. Infiltration rates are ex-pressed as volume per total cross-sectional area of columnper unit time, which is equivalent to the units used to de-

scribe rainfall rates. A coloured dye tracer was then appliedto a single point on the surface of the glass beads within thecolumn. The tracer dye was chosen based on a review ofseveral dyes that revealed standard commercial “blue foodcolouring” to be the most visible for digital imaging.

The camera was placed adjacent to the column and fo-cused on the centre of the transparent column. Video imageswere captured at three distinct vertical positions for each in-filtration rate, at vertical flow lengths of 30, 50, and 170 mmdepth from the top surface of the glass beads within thecolumn. Images were recorded at a single depth from thesurface of the column for a given dye application. Thesteady-state infiltration rate was maintained until the dyewas removed from the column, and the dye tracer applica-tion was repeated. Several sets of images were recorded foreach vertical position and for each infiltration rate.

The video imagery was examined to determine the timingof the passage of the dye front past the camera location. Thedigital images were enhanced to clarify the position of thedye within the glass beads. Image enhancement involvedchoosing an image in which the dye was easily seen andmodifying the colour of the dye in the image. A range ofblue shades was found to define the colour of the dye; thedominant shade was replaced with blue, and the lighter anddarker shades were replaced with green and purple, respec-tively. Once the enhancement was complete for one image,the same enhancement was repeated for all images in eachseries. The elapsed time between images was calculated byrecording the frame number from the recorded video imageand the video frame rate of 29.97 frames/s. Once the digitalvideo images were reviewed, the velocities of the dye frontthrough the glass beads was calculated by measuring thevertical flow length and dividing by the elapsed time.

© 2005 NRC Canada

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).

Video and still images were also recorded during periodsof wetting, during drain down, and at steady-state condi-tions, both with and without infiltration, to allow examina-tion of water distribution at the pore scale and facilitate theobservation of pendular rings within the pore space. Thesaturated hydraulic conductivity of the glass beads was de-termined using constant-head infiltration testing followingAmerican Society for Testing and Materials (ASTM) stan-dards D2434-68 and D5856-95.

Results and discussion

Figure 9 shows a detailed image of the glass bead columnobserved at the 30 mm vertical flow length during static con-ditions. The pore space of a simple rhombic layer within theglass bead column is visible. Figure 9 demonstrates the pres-ence of pendular ring structures within the glass bead col-umns. An approximate angle to the point of tangency isindicated in Fig. 9. Observations of the pendular ringsformed in the pore spaces indicated that the pendular ringscoalesced at angles of tangency smaller than those predictedby the ideal geometry shown in Fig. 4. Angles of tangencywhen the pendular rings “jumped” to a coalesced arrange-ment 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 con-tinuous water phase can be used to estimate the point ofresidual saturation. These observations indicate that thependular rings do not strictly obey eqs. [9] and [10], and thatresidual saturation occurs at a matric suction greater than thematric suctions that correspond to angles of tangency of 45°and 30° indicated in Fig. 7. Attempts to experimentally mea-sure the suction in the glass bead columns using tensiom-eters were unsuccessful. An approximation for the matricsuction at the point of residual saturation can be made usingeqs. [9] and [10] and the observed point of tangency angle of20°. The calculated matric suction is 0.148 kPa.

Figures 10 and 11 demonstrate typical images collectedduring the dye infiltration studies. Figure 10 shows a series

of images that were extracted from the video clip of tracerdye moving through the column with an applied infiltrationrate of 0.024 mm/s. The images on the right side demon-strate the results of the digital enhancement procedure andbetter illustrate the movement of the dye front. Figure 11shows a set of colour-enhanced close-up images of a 50 mmsection of glass beads as the dye front moves through thebeads, starting at image 080 and ending with image 135.

The infiltration rate and the measured dye front velocitywere used to determine the flux velocity (vflux) and seepagevelocity (vseep) for each dye tracer experiment. The flux ve-locity is defined as the volumetric discharge (Q) over the to-tal cross-sectional area (A) perpendicular to the direction ofdischarge and through which the discharge flows. Freeze andCherry (1979) refer to flux velocity as the “specific dis-charge” or as the Darcy velocity for saturated materials. Theterm flux velocity is preferred in this study because it is notassociated with saturated conditions. In the videography ex-periments, the flux velocity is therefore equal to the infiltra-tion rate.

Seepage velocity, for saturated conditions, is defined asthe flux velocity (vflux) divided by the porosity (n). Seepagevelocity for unsaturated conditions is the flux velocity di-vided by the volumetric water content (θw). The measureddye velocities are considered to be equivalent to seepage ve-locities.

Table 1 summarizes the results of the seepage velocitymeasurements based on the digital videography. Table 1 alsoshows the corresponding volumetric water contents and de-grees of saturation calculated on the basis of the measuredseepage velocities. The volumetric water content was calcu-lated as the flux velocity divided by the seepage velocity.Three measured points relating seepage velocity to measuredwater content are therefore established for the columns ofglass beads. The measurements of the saturated hydraulicconductivity of the glass beads indicated ksat was equal to0.055 m/s.

The experimental observations can now be combined withthe previously developed theory to construct an SWCC and

© 2005 NRC Canada


Fig. 9. Pore space of a simple rhombic layer formed by 12 mm glass beads within the column of glass beads.

an unsaturated hydraulic conductivity curve for the glassbead media. Figure 12 presents an SWCC for the glass beadcolumns. Also presented for comparison is the SWCC mea-sured experimentally by Stormont and Anderson (1999) fora pea gravel.

The saturated volumetric water content of the glass beadcolumn at 0.001 kPa is shown to be 61%, which matches themeasured porosity. The measured porosity is higher than thetheoretical 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 po-rosity may also be due to wall or container effects. The air-entry value is shown for a volumetric water content of 61%and a matric suction of 0.058 kPa. The glass beads are con-sidered incompressible, and therefore no change in watercontent is considered between saturation and the air-entryvalue. The air-entry value is determined from considerationof the largest pore space created within a cubic packing ar-rangement. Equation [5] was used to calculate the matricsuction (0.058 kPa) that corresponds to the pore geometry

© 2005 NRC Canada

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 withan infiltration rate of 0.024 mm/s: (a) image 400, (b) image 800, (c) image 2340.

shown in Fig. 3. The higher saturated porosity indicates thatpores greater than this size likely exist within the glass beadstructure, and therefore the value of 0.058 kPa represents anupper reasonable limit on the air-entry value.

The point of residual saturation is estimated from the ex-perimental observations of pendular ring behaviour notedearlier and is shown to occur at 0.148 kPa. The volumetricwater content at this point is estimated using the volume ofthe pendular rings indicated from eq. [10], adjusted for thedifferences in porosity. The final point on the estimated

SWCC shown in Fig. 12 corresponds to a matric suctionvalue of 1.0 × 106 kPa, at which the water content of allsoils is zero (Fredlund and Rahardjo 1993).

Also presented in Fig. 12 for comparison is the SWCCmeasured experimentally by Stormont and Anderson (1999),who conducted testing on columns of various soil layer con-figurations, including a coarse-grained soil referred to as peagravel, for the investigation of capillary barriers. The SWCCwas determined by the hanging-column method. The grain-size distribution of the pea gravel was uniform, with D90,

© 2005 NRC Canada


Fig. 11. Series of enhanced images showing the movement of the dye over a 5 cm vertical flow length with an infiltration rate of0.15 mm/s: (a) image 080, (b) image 100, (c) image 112, (d) image 113, (e) image 125, ( f ) image 135.

D50, and D10 of 15, 8, and 5 mm, respectively. The measuredwetting SWCC for the pea gravel had an air-entry value ofapproximately 0.02 kPa (i.e., 2 mm of suction head). Thepoint of residual water content occurred at a volumetric wa-ter content of approximately 3.5% and at a matric suction ofapproximately 0.2 kPa (Stormont and Anderson 1999).

An unsaturated hydraulic conductivity curve can be esti-mated by combining the measured relationship of watercontent to hydraulic conductivity with the theoretically de-termined SWCC presented in Fig. 12. The matric suctionvalues that correspond to the volumetric water contentsshown in Table 1 can be estimated using the SWCC for theglass beads shown in Fig. 12. The matric suction valueswere plotted against values of hydraulic conductivity (i.e.,infiltration rate) that correspond to the same volumetricwater content. This provides an estimate of the hydraulicconductivity versus matric suction relationship (Fig. 13).

Wilson (1990) showed that the Brooks and Corey (1964)method was suitable for estimating the k versus matric suc-tion (ua – uw) curve for cohesionless sand. The authors arenot aware of the application of the Brooks and Coreymethod to a coarse uniform gravel, such as a pea gravel, foruse as a capillary barrier. Figure 13 shows a comparison ofthe measured k versus matric suction (ua – uw) curve for the12 mm glass beads with the theoretical curve predicted byBrooks and Corey. It can be seen that the measured and pre-

dicted curves show reasonable agreement. These results sug-gest that the Brooks and Corey method for predicting the un-saturated hydraulic conductivity of granular soils may beapplied to coarse granular soils considered suitable for acapillary barrier.

Conclusions

Pendular ring and capillary theories can be used to predictthe air-entry and residual saturation points on the soil-watercharacteristic curve (SWCC) for uniform coarse soils. Themeasurement of flux and seepage velocities can be usedto estimate the relationships between unsaturated hydraulicconductivity and volumetric water contents in unsaturatedcoarse porous media. Measurement of seepage velocities us-ing digital videography to record the high-speed soil–waterinteractions is shown to be a useful tool. Although measure-ment is difficult, the results of the research program confirmthat the hydraulic conductivity versus matric suction curve isextremely steep for coarse-grained soils.

References

Barbour, S.L. 1998. Nineteenth Canadian Geotechnical Collo-quium: The soil-water characteristic curve: a historical perspec-tive. Canadian Geotechnical Journal, 35: 873–894.

Barbour, S.L., and Yanful, E.K. 1994. A column study of staticnonequilibrium fluid pressures in sand during prolonged drain-age. Canadian Geotechnical Journal, 31: 299–303.

Briggs, L.J. 1897. The mechanics of soil moisture. US Departmentof Agriculture, Soils Bulletin 10.

© 2005 NRC Canada

Reinson et al. 261

Flux velocity,vflux (mm/s)a

Range of measuredseepage velocity usingdigital videography,vseep (mm/s)

Estimatedvolumetric watercontent,θw = vflux/vseep (%)b

Estimated degreeof saturation,S = θw/n (%)

0.630 45–58 1.40–1.10 2.000.150 14–40 1.10–0.38 0.910.024 1.8–3.0 1.30–0.80 1.60

aInfiltration rate.bThe seepage velocity used to estimate θw was the median value of the range shown in the second

column.

Table 1. Seepage velocity measurements based on digital videography.

Fig. 12. Estimated SWCC for 12 mm glass beads using a combi-nation of capillary and pendular theory and a measured SWCCfor pea gravel (after Stormont and Anderson 1999).

Fig. 13. Comparison of the estimated hydraulic conductivity (k)versus matric suction ( )u ua w− curve with the theoretical curveas predicted by Brooks and Corey (1964).

Brooks, R.H., and Corey, A.T. 1964. Hydraulic properties of po-rous media. Colorado State University, Hydrology Paper 3.

Darcy, H. 1856. Les fontaines publiques de la Ville de Dijon. Vic-tor Dalmont, Paris.

Fisher, R.A. 1926. On the capillary forces of an,deal soil: correc-tion of formulae given by W.B. Haines. Journal of AgriculturalScience, 16: 492–505.

Fredlund, D.G., and Rahardjo, H. 1993. Soil mechanics for unsatu-rated soils. John Wiley & Sons Inc., New York.

Freeze, R.A., and Cherry, J.A. 1979. Groundwater. Prentice-HallInc., Englewood Cliffs, N.J.

Graton, L.C., and Fraser, H.J. 1935. Systematic packing of spheres,with particular relation to porosity and permeability. Journal ofGeology, 43(8): 785–848.

Gvirtzman, H., and Roberts, P.V. 1991. Pore scale spatial analysisof two immiscible fluids in porous media. Water Resources Re-search, 27(6): 1165–1176.

Haines, W.B. 1925. Studies in the physical properties of soils. II. Anote on the cohesion developed by capillary forces in an idealsoil. Journal of Agricultural Science, 15: 529–535.

Horton, J.H., and Hawkins, R.H. 1965. Flow path of rain from thesoil surface to the water table. Soil Science, 100(6): 377–383.

Nicholson, R.V., Gillham, R.W., Cherry, J.A., and Reardon, E.J.1989. Reduction of acid generation in mine tailings through theuse of moisture-retaining cover layers as oxygen barriers. Cana-dian Geotechnical Journal, 26: 1–8.

Orr, F.M., Scriven, L.E., and Rivas, A.P. 1975. Pendular rings be-tween solids: meniscus properties and capillary force. Journal ofFluid Mechanics, 67(4): 723–742.

Reinson, J.R. 2001. Soil–water interactions in coarse porous me-dia. M.Sc. thesis, Department of Civil Engineering, Universityof Saskatchewan, Saskatoon, Sask.

Rose, W. 1958. Volumes and surface areas of pendular rings. Jour-nal of Applied Physics, 29(4): 687–691.

Slichter, C.S. 1898. Theoretical investigations of the motion ofground waters. US Geological Survey, 19th Annual Report,Part 2, pp. 295–384.

Stormont, J.C., and Anderson, C.E. 1999. Capillary barrier effectfrom underlying coarser soil layer. Journal of Geotechnical andGeoenvironmental Engineering, ASCE, 125(8): 641–648.

Stormont, J.C., and Morris, C.E. 1998. Method to estimate waterstorage capacity of capillary barriers. Journal of Geotechnicaland Geoenvironmental Engineering, ASCE, 124(4): 297–302.

Waldron, L.J., McMurdie, J.L., and Vomocil, J.A. 1961. Water re-tention by capillary forces in an ideal soil. Soil Science Societyof America Proceedings, 25: 265–267.

Whitney, M. 1889. Soil moisture. Journal of Agricultural Science,3: 167–173.

Wilson, G.W. 1990. Soil evaporative fluxes for geotechnical engi-neering problems. Ph.D. dissertation, Department of Civil Engi-neering. University of Saskatchewan, Saskatoon, Sask.

© 2005 NRC Canada


Documents

Unsaturated flow in coarse porous media