Drainability of Granular Bases for Highway Pavements

Transcription

1 TRANSPORTATION RESEARCH RECORD Drainability of Granular Bases for Highway Paveents BRUCE M. McENROE The best easure of the drainability of a granular base is the iniu degree of saturation that can be achieved through gravity drainage in the field. The aount of water that can drain fro a base course depends not only on the physical properties of the aterial, but also on the cross-sectional geoetry of the paveent syste. A finegrained base ay reain fully saturated under the largest suction that can be developed through gravity drainage. A forula for the iniu degree of saturation in the granular base is developed fro Brooks and Corey's forula for water retention in unsaturated porous edia. The relationship for drainable porosity in the FHWA subdrainage design anual tends to overestiate the aount of drainage fro fine-grained bases and greatly underestiate the aount of drainage fro coarse-grained bases. If the iniu degree of saturation for a granular base is sufficiently low, it will drain fairly quickly. The recoended ethod for the estiation of drainage ties is a onediensional analysis of the saturated flow below the phreatic surface. This analysis accounts for the nonunifor spatial distribution of drainable porosity. Casagrande and Shannon's procedure, which is recoended by FHWA, tends to underestiate drainage ties, particularly for base courses that are relatively thin. The recoended procedures for subdrainage analysis have been ipleented in the SUBDRAIN coputer progra of the Kansas Departent of Transportation. The relationship is k =Ky µ in which K is the intrinsic pereability of the granular aterial and 'Y and µ are the specific weight and viscosity of the fluid, respectively. The coefficient of pereability, k, has diensions of LIT (length/tie). The intrinsic pereability, K, has diensions of L 2 Another easure of drainability is the lowest degree of saturation that can be achieved through gravity drainage in the field. The degree of saturation, s, is defined as the ratio /n, in which is the voluetric water content and n is the porosity (the voluetric water content at coplete saturation). The lowest degree of saturation that can be achieved in the field through gravity drainage is denoted sin The difference between the water content at saturation and the lowest water content that can be achieved in the field through gravity drainage is tered the drainable porosity, nd. The porosity; the drainable porosity, and Sin are related as follows: (1) Paveents with inadequate subsurface drainage deteriorate uch faster than well-drained paveents. If the base course of the paveent is saturated or nearly saturated, wheel loads can cause water and base aterial to be puped out through joints and cracks and at paveent edges, which eventually underines the paveent. Because it is virtually ipossible to keep water fro entering - paveents through joints and cracks over the long run, good drainage is essential for paveent longevity. The AASHTO procedure for paveent design (1) incorporates a drainage coefficient as a key input. The value of this coefficient depends on the quality of drainage of the paveent syste and the percentage of tie that the road bed is exposed to oisture levels near saturation. The AASHTO design guide relates the quality of drainage to the tie required for the reoval of water fro the base course, but it does not specify what degree of drainage or level of saturation constitutes ''reoval.'' In FHWA's coputer progra (2), the drainability of the base is easured by tie required for a saturated base to drain to water content equal to 85 percent of the water content at saturation. One easure of the drainability of a base course is its coefficient of pereability (Darcy pereability), k. The coefficient of pereability depends upon the intrinsic pereability of the granular aterial and the specific weight and viscosity of the fluid. Departent of Civil Engineering, 26 Learned Hall, University of Kansas, Lawrence, Kans In current practice, the drainable porosity of the base aterial is usually estiated fro the coefficient of pereability by eans of a relationship that appears in graphical for in FHWA's report Highway Subdrainage Design (3). The algebraic for of this relationship is nd =.355k 235 (3) for k in eters per day. The corresponding forula for the iniu degree of saturation is s. = k.235 n (4) for k in eters per day. Equation 3, which is strictly epirical, was fitted to easured values of the coefficient of pereability and the drainable porosity for soils of varied gradations and densities. The report states that it "should be used with caution, particularly at the extreities where data were lacking or were quite scattered." Despite this warning, FHWA's DAMP progra (2) obtains the drainable porosity of the granular base fro Equation 3 exclusively. It does not allow the user to enter another value for the drainable porosity. Drainage ties are norally estiated, directly or indirectly, fro forulas published by Casagrande and Shannon (4) in (2)

2 24 TRANSPORTATION RESEARCH RECORD 1434 The ethods for estiation of drainage ties in FHWA's subdrainage design anual (3) and the DAMP progra (2) are based on these forulas. The basic fors of Casagrande and Shannon's relationships were derived through a siplified one-diensional analysis in which the phreatic surface (water table) was considered planar at all ties. To copensate for the error introduced by this approxiation, they incorporated an undeterined coefficient in their analysis as a correction factor and used experiental data to deterine its values for various conditions. This paper presents a new analysis of the drainage of a saturated granular base. Starting fro basic principles of water retention and flow in porous edia, this analysis leads to soe new ethods for the estiation of iniu degrees of saturation, drainable porosities, and drainage ties. It also provides a basis for the evaluation of the current ethods. 1\vo exaple probles illustrate the practical application of the recoended procedures. MINIMUM DEGREE OF SATURATION Theory Any granular aterial has a characteristic drainage curve that relates the degree of saturation to the pore-water suction head (negative pressure head), \fl. The drainage curve is best deterined fro easureents of the water content at equilibriu for successively larger suction heads in the laboratory. An approxiate drainage curve can be coputed fro grain-size distribution and bulk density data (5). The drainage curves of ost granular aterials can be approxiated closely by the forula of Brooks and Corey (6), (5) on the cross-sectional geoetry of the paveent syste. The geoetry of the paveent section deterines the axiu porewater suction at any point in the base. Figure 1 shows a paveent section with a granular base and edge drains. The subgrade is considered ipervious. If the granular base is saturated and then allowed to drain under the force of gravity with no further inflow and no evaporation, drainage will eventually cease. In this state of static equilibriu, the suction head at any point is equal to its height above the water table in the edge drain (\fl = w + z) and, fro Equation 5, the corresponding iniu degree of saturation at this level, sin(z), is 1 Z:::;; lfla - W { S, + - S, W + Z Sin (z) = (1 ) ( lfla )>. z > lfla - W At elevations z :::;; lfla - w, the granular base will not drain at all. The drainable porosity at any level is the difference between the porosity and the iniu water content at that level: n,az) = n [1 - Sin (z)] (8) Equation 8 follows fro Equation 2. The average iniu saturation at a distance x fro the edge drain, sin(x), is the average of sin(z) over the thickness of the granular base: x+d -Sin (x) = d 1 J Sin (z) dz (9) The evaluation of the right-hand side of Equation 9 leads to an algebraic forula for sin(x): 1 xs~ (7) The ters s,, lfl and A. in Equation 5 are constants for a particular aterial. The values of these constants are deterined by fitting Equation 5 to the data that ake up the drainage curve. The constant s, is tered the residual saturation. It is the degree of saturation that is approached asyptotically at very large suction heads. The constant lfla is tered the air-entry head. It is the suction head below which the aterial reains fully saturated. The diensionless constant A. is tered the pore-size distribution index. The ore unifor the aterial, the larger the value of A.. Laliberte et al. (7) showed that the values of s,, lfl and A. are related to the porosity and intrinsic pereability of the granular aterial and the specific weight, viscosity, and surface tension of the fluid according to the forula Sin (x) = s, + (1 - s,) { t\la - : - x + t\i~ [(w + d + x)1->d(l - A.) _,1,aI->.1} < 'I' X1 < X - X2 s, + (1 - s,) ( \fl~ ) [(w + d + x) 1 ->d 1 - A. z (1) (1 - s,) n cr 2 A. -"------'-----= 5 K \fl: -y 2 A. + 2 (6) in which K is the intrinsic pereability of the granular aterial and -y, µ, and er are the specific weight, viscosity, and surface tension of the fluid, 'respectively. The for of this relationship has a theoretical basis. The value of the constant on the right-hand side was deterined experientally. The drainable porosity of the base course of a paveent depends not only on the physical properties of the aterial, but also x::.o Subgrade FIGURE 1 Cross section of paveent with granular base and edge drain.

3 McEnroe 25 in which X1 = [ :. - w - d L x 2 = [ :. - W. L \j/ - W - d < O 1 - w - d s; "' < L w \jla - w - d > L _a "' - w < s; \jla - w < L "' - w _a >L (11) (12) The average drainable porosity at a distance x fro the edge drain, n.t(x), can be deterined fro the porosity and the average iniu water content at this location as follows: nax) = n [l - Sin(x)] (13) The spatially averaged iniu saturation for the entire base course, Si"' is the average of sin(x) fro x = to x = L: Sin = -1 [-Sin (x) dx L o (14) The evaluation of the right-hand side of Equation 14 leads to an algebraic forula for sin: S. = + (l _ ) {~ + (\j/ - w) (x2 - X1) S, S, L dl (x~ - xi) 2dL d.l(l - A.) (2 - A.) - (w + d + x 1 ) 2 ->- - (w + l)'-' + (w + x,r' J _ \jla (x2 - Xi)} d.l(l - A.) 1 e;.i::: c 8 ~ :::J -ro 6 (/) Cl> 4 ~ O> Cl> "'C 2 c ~ ~-~~~-~ Eq k, /day FIGURE 2 Miniu degree of saturation versus coefficient of pereability for Exaple 1. (15) The average drainable porosity for the entire base course, Nd, can be deterined fro the porosity and the spatially averaged iniu saturation as follows: (16) For the base to drain at all, the air-entry head of the granular aterial ust be less than the elevation difference between the top of the base at the crown and the water table in the edge drain (\jla < W + d + L). The following exaple illustrates the relationship between the coefficient of pereability of the base aterial and the iniu degree of saturation. It also provides a coparison of Equations 15 and 16 and Equation 3. Exaple 1: Miniu Degree of Saturation for Typical Paveent Section Proble: The base course of a paveent is to have a slope of.2 /, a thickness of.1 (3.9 in.), and a half-width (L in Figure 1) of 7. (23. ft). The botto of the drainpipe is to be.1 (3.9 in.) below the botto of the base (w =.1 ). The base is to be constructed of a well-sorted granular aterial. This type of aterial would have a porosity of about.4, a residual saturation on the order of.1, and a pore-size distribution index on the order of 4. The objective is to deterine the (spatially averaged) iniu degree of saturation of the base for aterials with coefficients of pereability fro 1 to 1 /day (33 to 3,3 ft/day). Solution: According to Equation 15, the iniu degree of saturation of the base is deterined by four geoetric variables (d, w, L, and ) and three properties of the aterial (s,, \j/, and A.). Equation 6 provides an estiate of the air-entry head based on other properties of the aterial and the fluid. The solid curve in Figure 2 shows Sin fro Equation 15 ford=.1 (3.9 in.), w =.1 (3.9 in.), =.2 /, L = 7. (23. ft), s, =.1, and \j/ = 1.333/C 112 for \j/ in eters and k in eters per day. The forula for \j/, fro Equation 6, is based on n =.4, 'Y = 981 N/ 3, µ =.131 N s/ 2, and a =.742 Ni (water at 1 C). The foregoing exaple deonstrates that a granular base ust be fairly coarse to drain well. In this exaple, aterials with coefficients of pereability less than 15 /day do not drain at all because their air-entry heads are too large (\j/ > w + d + L). Materials with coefficients of pereability below 33 /day (11 ft/day) will reain ore than 85 percent saturated. A iniu saturation of 5 percent requires a coefficient of pereability of 64 /day (21 ft/day). On the other hand, nearly all of the pore water will drain by gravity if the aterial is very coarse. Materials with c~efficients of pereability above 15 /day ( 49 ft/day) will drain to below 2 percent of saturation. Figure 2 also shows Equation 3, the relationship incorporated in FHWA's DAMP progra (2). This forula appears to overestiate the aount of drainage fro fine-grained aterial and to greatly underestiate the aount of drainage fro coarse-grained aterials. DYNAMICS OF DRAINAGE One-Diensional Analysis with Spatially Varied Drainable Porosity Figure 1 shows the drainage of a granular base with no inflow or outflow through the paveent or subgrade. Drainage starts fro

4 26 TRANSPORTATION RESEARCH RECORD 1434 an initial condition of coplete satura!ion at tie t =. The objective of the analysis is to deterine the degree of saturation of the base at ties t >. The priary direction of flow is downslope parallel to top of the subgrade. The vertical distribution of porewater pressure is essentially hydrostatic everywhere except over the edge drain and very near x =, where the vertical curvature of the strealines is significant. Most of the fl.ow occurs in the zone of positive pore-water pressures below the phreatic surface (the surface of atospheric pressure). The drainage of the base course is analyzed as a proble of one-diensional unconfined saturated flow in the zone of positive pressures below the phreatic surface, with a spatially varied drainable porosity. The degree of saturation above the phreatic surface at a distance x fro the edge drain is assued to be sin(x). The corresponding water content above the phreatic surface is n - nix). The continuity equation for the flow in the zone below the phreatic surface is One-Diensional Analysis with Constant Drainable Porosity A ore approxiate analysis with a constant drainable porosity leads to a siple algebraic forula for th_e tie to 5 percent drainage. In this analysis, the spatially averaged drainable porosity, Nd, is substituted for the local drainable porosity, nix), in the governing differential equation. With this siplification, the proble can be stated in ters of the diensionless variables Kt T = LNd x X= L h H= L (21) (22) (23) (17) D d L (24) in which h(x,t) is the elevation of the phreatic surface and q(x,t) is the discharge (per unit width) in the -x direction. The equation of otion is the Dupuit discharge forula for unconfined seepage over a sloping bed: The goverµing equation is ah(x,1) - ~ {[H(XT) - X] ah(x,t)} at ax ' ax (25) q(x,t) = K [h(x,t) - x] ah(x,t) ax (18) The substitution of the right-hand side of Equation 18 for q(x,t) in Equation 17 yields the governing equation with h(x,t) as the dependent variable: _ ah(x,t) a { ah(x,t)} ntt(x) -- - K - [h(x,t) - x] -- = dt dx dx (19) The initial condition is h(x,o) = x + d, which represents coplete saturation with no excess pressure. The lower boundary, x =, is the brink of the edge drain. The appropriate boundary condition at this location is a hydraulic gradient of unity (8). The upper boundary, x = L, is the crown of the road. Syetry requires that no fl.ow cross this boundary. This requireent is satisfied by a horizontal phreatic surface (hydraulic gradient of zero) until h becoes zero. The average degree of saturation at any tie can be deterined fro the phreatic-surface profile: S(t) = L 1 J.L {- _ h(x,t) - x} Sin (x) + [1 - Sin(x)] d dx (2) This atheatical odel of the drainage process has been ipleented in the SUBDRAIN coputer progra of the Kansas Departent of Transportation. This progra solves Equation 19 for the stated initial and boundary conditions by a nonlinear iplicit finite-difference schee. The progra returns the average degree of saturation at the end of each tie step. It also returns the tie to 85 percent saturation (S =.85) and the tie to 5 percent drainage (S = Sin +.5Nin). and the initial condition is H = D + X. The boundary conditions can also be stated in ters of these diensionless variables. H(X,1) is deterined entirely by D, the diensionless thickness of the base. The degree of drainage at any diensionless tie, U, is the fraction of drainable pore space that has been drained: 1 - s U= 1 - Sin (26) Its value at any tie can be calculated fro the diensionless phreatic-surface profile: U(1) = f ~(X,1) - X] dx (27) Because D deterines H(X,T), it also deterines U(1). Figure 3 shows the relationship between the diensionless tie to 5 percent drainage, Tso, and the diensionless thickness of the base, D, as deterined fro nuerical solutions of the governing equation for any values of D. These nuerical results are fitted closely by the siple epirical forula Tso =.63 D In diensional for, this forula is (28) LNd d Tso =.63 K L (29) ( ) -! Forulas for ties to other degrees of drainage could be developed in the sae way.

5 McEnroe 27 Siplified One-Diensional Analysis by Casagrande and Shannon Casagrande and Shannon's odel of the drainage process (4) can be expressed concisely in ters of the diensionless variables T, U, and Das Tso T = C(D) f(d, U) (3).1 in which f(d, U) = D + 2U 2U - D In D 2D - 2UD + 1 l + In (2-2U) (D + 1) U$.5 (31) D FIGURE 4 Tie to 85 percent of saturation versus coefficient of pereability for Exaple 2. D + 1 -Dln-- D u >.5 Exaple 2: Drainage Ties for Typical Granular Base and C(D) = D- 113 (32) The function f(d, U) is the solution for T fro their siplified one-diensional analysis with a planar phreatic surface. The function C(D) is a correction factor that was introduced to better fit the results of soe laboratory and field experients. A forula for the tie to 5 percent drainage can be obtained by the substitution of.5 for U in Equation 3. This forula is ( D + 1) Tso = ( D- 113 ) 1 - D In -- (33) Equation 33 is plotted in Figure 3. For large values of D, Equation 33 closely approxiates the nuerical results fro the coplete one-diensional analysis with a constant drainable porosity. For sall values of D, Equation 33 appears to underestiate T 5 considerably. The following exaple illustrates the relationship between the coefficient of pereability of the base aterial and two easures of the drainage tie for a typical paveent section. It also provides a coparison of three different ethods for estiating these drainage ties. (!! 24.c c:e ~ 18 :::s cu ~ 12 'cf. LO co 6.s ' I \ \ \\\ \ \\ Method #2 """:,_ ' 1 Method #1 Cl> E i= 1 1 k, /day FIGURE 3 Coparison of two approxiate ethods for tie to 5 percent drainage. 1 Proble: As in the previous exaple, the base course of a paveent is to have a slope of.2 /, a thickness of.1 (3.9 in.), and a half-width (L in Figure 1) of 7. (23. ft). The botto of the drainpipe is to be.1 (3.9 in.) below the botto of the base (w =.1 ). The base is to be constructed of a wellsorted granular aterial. This type of aterial would have a porosity of about.4, a residual saturation on the order of.1, and a pore-size distribution index on the order of 4. The objective is to deterine the ties to 85 percent saturation (S =.85) and the ties to 5 percent drainage (U =.5) for base aterials with coefficients of pereability fro 1 to 1 /day (33 to 3,3 ft/day). Solution: The drainage ties are estiated by three different ethods: 1. Coplete one-diensional analysis with spatially varied drainable porosity fro Equations 1-13 (SUBDRAIN progra), 2. Coplete one-diensional analysis with a constant, spatially ayeraged drainable porosity fro Equations (odified SUBDRAIN progra), and 3. Siplified one-diensional analysis by Casagrande and Shannon ( 4) with constant drainable porosity fro Equation 3 (DAMP progra). Figure 4 copares the results for the tie to 85 percent of saturation. These results for Methods 1 and 3 do not differ greatly except fork< 4 rn/day (13 ft/day). A coparison of the results for Methods 1 and 2 shows that the ore approxiate ethod yields considerably shorter estiates of drainage ties for aterials with pereabilities less than about 1 /day (33 ft/day). The drainable porosity is actually larger near the centerline of the road than near the sides because of the difference in elevation. Most of the water that drains fro the base ust travel a distance greater than L/2 to reach the edge drain. This is why Method 1, which uses spatially varied drainable porosities, yields longer drainage ties than Method 2, which uses a spatially averaged drainable porosity. The two ethods yield nearly identical drainage ties for very coarse aterials because the spatial variability of drainable porosity is very sall for these aterials in this syste. Figure 5 copares the results for the tie to 5 percent drainage. The drainage ties for Method 3 are based on different drain-

6 28 TRANSPORTATION RESEARCH RECORD 1434.c a) O> «J c ~ ~ 6 b LO.9 Q) E i= k, /day FIGURE 5 Tie to 5 percent drainage versus coefficient of pereability for Exaple 2. The recoended ethod for estiation of drainage ties is a one-diensional analysis of the saturated flow below the phreatic surface, with the local drainable porosities deterined fro Equation 1. The equation that governs the drainage process is solved nuerically by a finite-difference ethod. If the spatial variability of the drainable porosity is neglected, drainage ties are underestiated. The forulas of Casagrande and Shannon ( 4) underestiate drainage ties, particularly for systes in which d/l «1. The recoended procedures for subdrainage analysis have been ipleented in the SUBDRAIN coputer progra of the Kansas Departent of Transportation. These procedures could easily be incorporated into FHWA's DAMP progra and other siilar progras. able porosities, and therefore different volues of drainage, than the drainage ties for Methods 1 and 2. Method 2 underestiates the tie to 5 percent drainage for aterials with pereabilities less than about 1 /day (33 ft/day) because the spatial variability of the drainable porosity is relatively large for these aterials in this syste. CONCLUSIONS The best easure of the drainability of a granular base is the iniu degree of saturation that can be achieved through gravity drainage in the field. The aount of water that can drain fro a base course depends not only on the physical properties of the aterial, but also on the cross-sectional geoetry of the paveent syste. The geoetry of the paveent section liits the aount of suction that gravity can exert on the pore water. A granular base ust be fairly coarse to drain adequately. A fine-grained base ay reain fully saturated under the largest suction that can be developed through gravity drainage. Equation 15 can provide a good estiate of the iniu degree of saturation for a granular base. It incorporates both the water-retention properties of the drainage aterial and the crosssectional geoetry of the paveent. Equation 3, which appears in FHWA's subdrainage design anual (3) and. the DAMP coputer progra (2), tends to overestiate the aount of drainage fro fine-grained bases and to greatly underestiate the aount of drainage fro coarse-grained bases. Equation 3 does not account for the cross-sectional geoetry of the paveent. ACKNOWLEDGMENT This project was supported by the Kansas Departent of Transportation (KDOT) through the K-TRAN Cooperative Transportation Research Progra. Andrew Gisi served as the project onitor for KDOT. REFERENCES 1. AASHTO Paveent Design Guide. Aerican Association of State Highway and Transportation Officials, Washington, D.C., Carpenter, S. H. Highway Subdrainage Design by Microcoputer: Drainage Analysis and Modelling Progras (DAMP). Report FHWA IP FHWA, U.S. Departent of Transportation, Moulton, L. K. Highway Subdrainage Design. Report FHWA-TS FHWA, U.S. Departent of Transportation, Casagrande, A., and W. L. Shannon. Base Course Drainage for Airport Paveents. Proceedings of the Aerican Society of Civil Engineers, Vol. 77, Separate No. 75, Arya, L. M., and J. F. Paris. A Physicoepirical Model to Predict the Soil Moisture Characteristic fro Particle-Size Distribution and Bulk Density Data. Soil Science Society of Aerica Journal, Vol. 45, 1981, pp Brooks, R. H., and A. T. Corey. Hydraulic Properties of Porous Media. Hydrology Paper 3. Colorado State University, Fort Collins, Laliberte, G. E., A. T. Corey, and R.H. Brooks. Properties of Unsaturated Porous Media. Hydrology Paper 17. Colorado State University, Fort Collins, McEnroe, B. M. Maxiu Saturated Depth Over a Landfill Liner. Journal of Environental Engineering, Vol. 119, No. 2, Mar./Apr. 1993, pp Publication of this paper sponsored by Coittee on Subsurface Drainage.