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Prediction of the Soil-Water Characteristic Curve from Grain-Size Distribution and Volume-Mass Properties Murray D. Fredlund D.G. Fredlund G.W. Wilson Department of Civil Engineering University of Saskatchewan Saskatoon, Sask. S7N 5A9 Fax: (306) 966-5427 Email: mdf128@engr.usask.ca ABSTRACT This paper presents a method of estimating the soil-water characteristic curve from the grain-size distribution curve and volume-mass properties. The grain-size distribution is divided into small groups of uniformly-sized particles. A packing porosity and soil-water characteristic curve is assumed for each group of particles. The incremental soil-water characteristic curves are then summed to produce a final soil-water characteristic curve. Prediction of the soil-water characteristic curve from grain-size distribution allows for a inexpensive description of the behavior of unsaturated soils. The soil-water characteristic curve forms the basis for computer modelling of processes in unsaturated soils. water characteristic curves and grain-size INTRODUCTION distribution curves for a mixture of sand, silt, This paper presents a model for the and clay were obtained from SoilVision prediction of the soil-water characteristic (Fredlund, 1996), which contains over 6000 curve, (SWCC), based on the particle-size soils. The soil-water characteristic curves were distribution, dry density, void ratio, and specific then fitted with the Fredlund & Xing (1994) gravity of a soil. The model first fits a equation. This provided an approximation for modification of the Fredlund & Xing (1994) the curve fitting parameters in the Fredlund & equation to the grain-size distribution curve Xing (1994) equation classified according to (Figure 1). The grain-size distribution curve is dominant particle size. Parameters used in the then analyzed as an incremental series of Fredlund & Xing (1994) equation for soils particle sizes from the smallest to the largest in composed entirely of sand or entirely of clay order to build an overall soil-water are easy to obtain. Uniform soils containing characteristic curve. Small increments of only mid-range particle sizes are more difficult uniform-sized particles are transposed to obtain to obtain and as a result some estimation is a SWCC representing the average particle size. required. Once the entire grain-size distribution curve is incrementally analyzed, the individual soil- During development of the algorithm to water characteristic curves are superimposed to predict the SWCC, it was decided that give the SWCC for the entire soil. provision must be made for the storage of grain-size information. If grain-size information In order to build the general SWCC, it must was to be stored, a method of mathematically be assumed that the SWCC for each uniform representing each grain-size curve should be particle size is relatively unique. Typical soil- found. The benefits of a mathematical fit would be two-fold. A grain-size curve fit with a 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 mathematical equation would then allow fit with an equation. This idea was then further computations to be performed on the implemented in the form of a least-squares curve. It was reasoned that a prediction of the curve-fitting algorithm which allowed for fitting soil-water characteristic curve would be of the grain-size distribution data. possible if the grain-size distribution could be 100% 90% Fit curve 80% Experimental Percent Passing 70% 60% 50% 40% 30% 20% 10% 0% 0.0001 0.001 0.01 0.1 1 10 100 Particle Size (mm) Figure 1 Fit of grain-size curve using a modified Fredlund & Xing (1994) equation (# 10741) The second benefit of mathematically research by Wagner presented several representing each grain-size curve was that it lognormal distributions capable of fitting the would provide coefficients of indices by which grain-size curve. Providing a meaningful grain-size curves could be classified. This representation of the grain-size data in the would allow the ability to search the database extremes proved difficult for a lognormal for soils with grain-size curves within a distribution. specified band. This technique has proven invaluable in performing sensitivity analyses on Due to similarity between the shape of the soil parameters. grain-size distribution and the shape of the soil- water characteristic curve, a different approach was taken. The Fredlund & Xing (1994) THEORY FOR MATHEMATICALLY equation, which had previously been used to fit REPRESENTING THE GRAIN-SIZE SWCC data, provided a flexible and continuous DISTRIBUTION CURVE equation that could be fit by the nonlinear regression using three parameters. The Previous research work to fit the grain-size equation was modified to permit the fitting of curves was reviewed (Wagner, 1994). The grain-size curves. The modified equation, [0.1], 2 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 allowed for a continuous fit and proper definition of the extremes of the curve. 7 dr ln 1 + [0.1] 1 d Pp (d ) = 1 − ln 1 + d r gm ga gn ln exp(1) + d dm where: Pp(d) = percent passing a particular grain-size, d ga = fitting parameter corresponding to the initial break in the grain-size curve, gn = fitting parameter corresponding to the maximum slope of grain-size curve, gm = fitting parameter corresponding to the curvature of the grain-size curve, d = particle diameter (mm), dr = residual particle diameter (mm), dm = minimum particle diameter (mm) Aberg, 1996) It appeared that a theoretical THEORY OF PREDICTING THE approach to the problem would hopefully SOIL-WATER CHARACTERISTIC provide superior predictions. CURVE FROM THE GRAIN-SIZE The second approach was theoretical and DISTRIBUTION involved converting the grain-size distribution The mathematical fit of the grain-size to a pore-size distribution which was then distribution led to the development of an developed into a SWCC (Arya, 1981). This algorithm capable of predicting the soil-water research was duplicated and compared to characteristic curve. A review of current experimental data. Difficulty was encountered research showed that one of two approaches in generating a reasonable SWCC along the have typically been taken in the prediction of entire range. Predicted soil-water characteristic the soil-water characteristic curve from grain- curves typically showed abnormal “humps” and size. The first approach entails a statistical fell to zero volumetric water content long estimation of properties describing the SWCC before the experimental data was completely from grain-size and volume-mass properties desaturated (Figure 2). (Gupta, 1979; Ahuja, 1985; Ghosh; 1980; 3 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 0.35 Actual Volumetric Water Content 0.3 0.25 Estimation from pore-size 0.2 0.15 0.1 0.05 0 0.01 1 100 10000 1000000 Suction (kPa) Figure 2 Illustration of abnormalities associated with prediction of SWCC from pore-size distribution A new approach is proposed for predicting the soil-water characteristic curve from the This resulted in the production of two plots, grain-size distribution curve. It was assumed one for the ‘n’ parameter, and one for the ‘m’ that a soil composed entirely of a uniform, parameter. These plots described the variation homogeneous particle size would have a unique in the ‘n’ and ‘m’ parameters with grain-size. soil-water characteristic curve. The shape of This allowed n and m parameters to be the SWCC for pure sands, pure silts and pure estimated for any soil composed of uniform clays was known. Using a best-fit analysis with diameter particles. the Fredlund & Xing (1994) equation, three parameters were computed for each soil type. The grain-size distribution curve can be It was then assumed that these parameters divided up into small divisions of uniform soil could be associated with a dominant particle particles. Starting at the smallest diameter size, size on the grain-size plot. The uniqueness of a packing porosity was estimated (Harr, 1977) the soil parameters was confirmed by querying for each division and a soil-water characteristic the SoilVision database for plots of the ‘n’ and curve estimated as shown in Figure 3. The ‘m’ parameters versus the percent sand, silt, divisional soil-water characteristic curves can and clay of a soil. It was hypothesized that as a then be summed starting with the smallest soil tended towards uniformity, the ‘n’ and ‘m’ particle size and continuing until the volume of parameters would show a trend towards a pore space is equal to that of the entire particular value. The particle sizes falling heterogeneous soil. The result is a theoretically between pure clays, pure silts and pure sands predicted soil-water characteristic curve. were then approximated. 4 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 100% 90% Fit curve 80% Experimental Percent Passing 70% 60% 50% 40% 30% 20% 10% 0% 0.0001 0.01 1 100 Particle Size (mm) Figure 3 Small divisions of particle size used to build complete SWCC fit the experimental data as well as IMPLEMENTATION OF THE SWCC experimental data points and generated data PREDICTION INTO SOILVISION points on the best fit curve. Figure 5 shows the Information relevant to describing the grain- second page of the grain-size distribution form. size distribution is organized in a single form in the SoilVision knowledge-based system. Figure The header on the form allows for a number 4 shows the grain-size form for the knowledge- of helpful functions and algorithms. If soil data based system. Two pages are required to consists of % Coarse, % Sand, % Silt, %Clay present the information. The first page contains or D10, D20, D30, D50, or D60 data on page parameters controlling the fit of grain-size, the two of the main soil form, pressing a button smallest particle diameter, the error between will convert this data into experimental points the fit data and experimental data, the error along the grain-size distribution graph. Once between predicted SWCC and experimental experimental data is obtained, pressing Fit data predicted and experimental data, and Curve! will initiate the linear regression algorithm that will best-fit the equation to counters which Access® uses to identify experimental data. The results of the fit can be individual records. Page one also contains the viewed by pressing the Graph! button and a packing porosity field which controls the soil-water characteristic curve can be predicted prediction of the soil-water characteristic by pressing the Predict SWCC... button. curve. Page two displays the equation used to 5 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 Figure 4 Page one of the grain-size distribution form Figure 5 Page two of the grain-size distribution form error. A good curve fit of the grain-size curve CONCLUSIONS is essential for the prediction of a reasonable The readapted Fredlund & Xing (1994) soil-water characteristic curve. The minimum equation produces a satisfactory fit of the particle size was also found to have an grain-size distribution. Figure 3 shows that the influence on the prediction of the soil-water experimental data can be fit with a minimal characteristic curve prediction. If the minimum 6 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 particle size variable was too low, the the accuracy of the prediction algorithm overabundance of clay size particles woul appears to be reasonable. Results tended to be dominate the prediction. If the minimum sensitive to the packing porosity and more particle size was too high, an absence of research is required in this regard. Soils with smaller particles would result in the soil drying experimental data for both the grain-size curve out prematurely. and the soil-water characteristic curve were extracted from the database. The results of The prediction of soil-water characteristic comparisons between experimental and curve from the grain-size distribution was predicted data can be seen in Figure 6, Figure found to be particularly accurate for sands, and 7, Figure 8, Figure 9, Figure 10, Figure 11, reasonably accurate for silts. Clays, tills and Figure 12, and Figure 13. loams were more difficult to predict although 7 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 100% 90% Fit curve 80% Experimental Percent Passing 70% 60% 50% 40% 30% 20% 10% 0% 0.0001 0.001 0.01 0.1 1 10 100 Particle Size (m m ) Figure 6 Grain-size distribution fit for a Sand (# 10720) 0.40 0.35 Volumetric Water Content Predicted from Grain-size 0.30 Experimental 0.25 0.20 0.15 0.10 0.05 0.00 0.1 1 10 100 1000 10000 100000 1000000 Soil Suction (kPa) Figure 7 Comparison between experimental and predicted curves for Sand (# 10720) 8 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 100% 90% Fit Curve 80% Experimental Percent Passing 70% 60% 50% 40% 30% 20% 10% 0% 0.0001 0.001 0.01 0.1 1 10 100 Particle Size (mm) Figure 8 Grain-size distribution fit for a Loamy Sand (# 10741) 0.40 Volumetric Water Content 0.35 Predicted from Grain-size 0.30 Experimental 0.25 0.20 0.15 0.10 0.05 0.00 0.1 1 10 100 1000 10000 100000 1000000 Soil Suction (kPa) Figure 9 Comparison between experimental and predicted curves for a Loamy Sand (# 10702) 9 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 100% 90% Fit curve 80% Experimental Percent Passing 70% 60% 50% 40% 30% 20% 10% 0% 0.0001 0.001 0.01 0.1 1 10 100 Particle Size (mm) Figure 10 Grain-size distribution for a Sand (# 350) 0.40 Volumetric Water Content 0.35 Predicted from Grain-size 0.30 Experimental 0.25 0.20 0.15 0.10 0.05 0.00 0.1 1 10 100 1000 10000 100000 1000000 Soil Suction (kPa) Figure 11 Comparison between experimental and predicted data for a Sand (# 350) 10 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 100% 90% Fit Curve 80% Experimental Percent Passing 70% 60% 50% 40% 30% 20% 10% 0% 0.0001 0.001 0.01 0.1 1 10 100 Particle Size (mm) Figure 12 Grain-size distribution for a Silt Loam (# 10861) 0.40 Predicted from Grain-size Volumetric Water Content 0.35 Experimental 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.1 1 10 100 1000 10000 100000 1000000 Soil Suction (kPa) Figure 13 Comparison between experimental and predicted data for a Silt Loam (# 10861) 11 3rd Brazilian Symposium on Unsaturated Soils, Rio de Janeiro, Brazil, April 22-25, 1997 Fredlund, M.D., Sillers, W.S., Fredlund, D.G., ACKNOWLEDGEMENTS Wilson, G.W., 1996, Design of a I wish to acknowledge the help of G.W. knowledge-based system for unsaturated Wilson in forming some of the ideas for this soil properties, Third Canadian Conference prediction. Also of note was the help I recieved on Computing in Civil and Building from Sai Vanapalli locating previous research Engineering, pp. 659-677 done in this field. Ghosh, R.K., 1980, Estimation of soil- moisture characteristics from mechanical REFERENCES properties of soils, Soil Science Journal, Vol. 130, No. 2, pp. 60-63 Aberg, B., 1996, Void sizes in granular soils, Journal of Geotechnical Engineering, Vol. Gupta, S.C., and Larson, W.E., 1979, 122, No. 3, pp. 236-239 Estimating soil-water retention characteristics from particle size distribution, Ahuja, L.R., Naney, J.W., and Williams, R.D., organic matter percent, and bulk density, 1985, Estimating soil-water characteristics Water Resources Research Journal, Vol. 15, from simpler properties or limited data, Soil No. 6, pp. 1633-1635 Sci. Soc. Am. Journal., Vol. 49, pp. 1100- 1105. Harr, M.E., 1977, Mechanics of particulate media, McGraw - Hill International Book Arya, L.M., and Paris J.F., 1981, A Company, New York, 27-33 physicoempirical model to predict the soil moisture characteristic from particle-size Wagner, L.E., and Ding, D., 1994, distribution and bulk density data, Soil Representing aggregate size distributions as Science Society of America Journal, Vol. modified lognormal distributions, American 45, pp. 1023-1030. Society of Agricultural Engineers, Vol. 37, No. 3, pp. 815-821 12

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