FINITE ELEMENT STOCHASTIC ANALYSIS

Transcription

1 FINITE ELEMENT STOCHASTIC ANALYSIS Murray Fredlund, P.D., P.Eng., SoilVision Systems Ltd., Saskatoon, SK ABSTRACT Numerical models can be valuable tools in te prediction of seepage. Te results can often be misleading if proper variational analysis is not performed. Measurement of te soil-water caracteristic curve (SWCC) or saturated and unsaturated ydraulic conductivity information is often neglected or minimized. Te modeller is terefore forced to develop a seepage model based on estimated soil properties. Te development of suc models based on a best estimate of soil parameters is very often of little value. A stocastic analysis of te results of finite element models is often required in order to properly interpret wat information a numerical model is providing. RÉSUMÉ 1. INTRODUCTION Te use of numerical models in te regulatory process as increased dramatically in te past few years. Te use of suc finite element or finite difference models as become accepted practice. Tere is great benefit in te use of tese models as tey provide to us a snapsot of te teoretical beavior of a system given very specific constants. Te danger of tese models, owever, is tat tey can easily provide us wit a false sense of security. In many cases te models do not accommodate te material eterogeneity of a pysical system. Consideration is often not provided for experimental error. It is commonplace, for example, for measurements of saturated ydraulic conductivity to vary between ½ to 2 orders of magnitude wen measurements are taken on te same soil sample. As results are often sensitive to ydraulic conductivity, tis presents a significant modeling problem. It may be stated tat a single run of a numerical model given static parameters tells us very little wit regards to te beavior of a pysical system. Tis paper examines metods of accommodating stocastic variation in seepage numerical models suc tat realistic laboratory and field sampling variables may be accommodated. 2. SEEPAGE THEORY In virtually all studies of flow in te unsaturated zone, te fluid motion is assumed to obey te classical Ricards equation (Hillel, 198;Bear, 1972). Tis equation may be written in several forms. Te tree forms of te unsaturated flow equation are identified as te -based form, te θ-based form, and te mixed form. A transient form of te H-based formulation is presented below. x Were: x y ( ψ ) ky( ψ ) = m y 2 w γ w t = total ead, kx = ydraulic conductivity of te soil in te x ky = ydraulic conductivity of te soil in te y gw = te unit weigt of water (9.81 kn/m3), m2w = te slope of te soil-water caracteristic curve. Te partial differential equation essentially equates flow into and out of a unit volume to te resulting cange in storage. Te equation appears simple but is plagued by a number of difficulties in obtaining solutions using te finite element or finite difference metod. Te storage curve and te permeability are bot igly non-linear for an unsaturated soil. Tis causes numerical instability tat may be reduced troug te application of automatic mes refinement. Te form of te Ricards equation presented above is also susceptible to water-balance errors. Tere are several advantages to te θ-based form. One advantage is tat it can be formulated to be perfectly mass-conservative. It is not commonly used, owever, because tis form of te Ricards equation degenerates in fully saturated media and because material discontinuities produce discontinuous θ profiles. Te -based form of te Ricards equation is te most commonly implemented form. Its primary drawback is tat it can suffer from poor mass-balance in transient problems. Tis problem is exacerbated by problems wit a igly non-linear soil-water caracteristic curve.

2 Celia (199) proposed a mixed form of te Ricards equation tat was designed to improve te mass-balance of te -based formulation. x Were: x y ( ψ ) ky( ψ ) kx ky θ θ = y t = total ead, = ydraulic conductivity of te soil in te x = ydraulic conductivity of te soil in te y = volumetric water content. Translation of te governing equation to 3D is presented below. x were: x y y z ( ψ ) ky( ψ ) kz( ψ ) kx ky kz gw θ θ = z t = total ead, = ydraulic conductivity of te soil in te x = ydraulic conductivity of te soil in te y = ydraulic conductivity of te soil in te z = te unit weigt of water (9.81 kn/m3), = volumetric water content. Te issues regarding te solution of te 3D governing partial differential equation are similar to 2D. Wile te equations appear concise, tere are significant numerical pitfalls related to te solution of tese equations. Te results can vary dramatically dependant upon te soil properties provided. 3. NEED FOR STOCHASTIC ANALYSIS It was during a first-year laboratory assignment tat te importance of variation was impressed on me. We were presented wit our electrical system and a formula governing its beaviour. Our assignment was to run te system, collect input data, ten use te formula to calculate results. Our laboratory group did precisely as was asked and to our surprise, received a surprisingly low mark. Te reason for our low mark, it was discovered, was tat we ad neglected to incorporate possible limits of variation in our answer, given te possible variation errors in our input parameters. Sixteen years later I ave discovered tat te state of practice of geotecnical engineering does not apply te concepts presented in first-year engineering wen it comes to numerical modelling of soil processes. Numerical models are routinely set up and run in geotecnical practice wit little regard for possible variation in input parameters. A single run of a typical finite element model gives little information, and sould not be eavily weigted in te regulatory process. Variational analysis in some form sould always be performed to answer te question, Wat is te model truly telling us? 4. EXAMPLE 1: UNSATURATED FLOW IN CLAY DAM One example of a seepage analysis involves determining te amount of flow troug te unsaturated (vadose) zone in an eart dam. A clay core is implemented in tis example to dissipate te ead accumulated on te upstream side of te dam. Te pysical dimensions of te dam are presented in Figure 1. Te model was setup and run using te SVFlux (Fredlund, 22) seepage software package. Figure 1 Cross-section of example dam (Stianson, 24) Te amount of flow proceeding over te clay core and troug te unsaturated zone is largely controlled by te unsaturated ydraulic conductivity. Te unsaturated ydraulic conductivity is frequently estimated, as laboratory costs may be quite ig ($7, - $, CAD). In typical estimation metods te slope of te unsaturated ydraulic conductivity curve is related to te slope of te soil-water caracteristic curve (SWCC) troug a power function. In tis particular example te Modified Campbell (Fredlund, 1996,24) metod of estimating te unsaturated ydraulic conductivity was used in tis example. Te Modified Campbell metod uses te Fredlund and Xing (1994) equation to represent te soilwater caracteristic curve as te basis for it s estimation of unsaturated ydraulic conductivity. Te Modified Campbell equation is presented below. Te slope of te unsaturated portion of te curve is controlled principally by te p parameter. ψ ln 1 + r 1 k( ψ ) = ( ks kmin ) 1 + k 6 m f n f ln 1 + ln exp(1) r + ψ a f were: r = suction at residual water content, p min

3 Ψ = soil suction, k s = saturated ydraulic conductivity, k min = minimum ydraulic conductivity, a f = Fredlund and Xing a parameter, n f = Fredlund and Xing n parameter, m f = Fredlund and Xing m parameter, p = Modified Campbell p parameter. A Monte Carlo analysis was ten set up tat allowed te p parameter in te Modified Campbell metod to vary suc tat te mean value was 5 and te standard deviation was 2. 2 variations were generated and te resulting normal distribution is presented in Figure 2. Te conductivity of te core material was not modified illustrates te importance of a sensitivity analysis wen interpreting model results. It would be easy to run tis example using average model parameters. Suc model parameters would give te impression tat approximately 5% to 6% of flow goes over te core and troug te unsaturated zone (using average values of for te p parameter of te Modified Campbell equation. Suc an impression would be misleading wit regards to te results presented by te full stocastic analysis. It sould be noted tat te current model was set up using average soil properties. It would also be possible to get variation in results troug te variation of oter soil parameters suc as te air-entry value (AEV) of te soilwater caracteristic curve used for te primary dam material. Frequency More Figure 2 Variation of te "p" parameter in te Modified Campbell equation A series of 2 runs were used wit te Monte Carlo analysis. Te results are presented in Figure 3. Bin 5. EXAMPLE 2: FOOTING DEFORMATIONS Calculations involving stresses and deformations beneat a strip footing form an important part of te design process. Te deformations caused by te application of load are central to te design tolerances. In a way similar to oter finite element models tere are certain soil properties tat are sensitive and soil properties tat are not sensitive. Te determination of te sensitive soil properties is of paramount importance. Suc a determination can be made troug te use of stocastic analysis. Te geometry for te mentioned example problem is sown in Figure 4. A footing is placed at te corner of a 3m x 3m soil region. A load expression of 1 kpa is ten applied at te base of te footing and te resulting stresses and deformations are calculated. 9% 8% Percent Unsaturated Flow 7% 6% 5% 4% 3% 2% % % Modified Campbell P 12 Figure 3 Percent unsaturated flow as related to te slope of te unsaturated ydraulic conductivity curve Te results indicate tat te amount of unsaturated flow over te core is igly sensitive to te slope of te unsaturated ydraulic conductivity function. Overall, te amount of flow troug te dam as a percentage of total flow varies between 18% and 83%. Tis example

4 sown in Figure 7. Model runs were organized in terms of increasing Poisson s Ratio values. Figure 4 Geometry of strip footing In tis particular analysis te Poisson s ratio was varied via te Monte Carlo metod wit a mean value of.4 and a standard deviation of.3. 2 runs of te problem were generated and solved. Te distribution of Poisson s Ratio used in tis example is sown in Figure 5. Figure 6 Vertical stress distribution beneat strip footing Frequency Vertical deformation (mm) Run # 1m 4m 7m m 13m Poisson's Ratio More Figure 5 Frequency of Poisson's Ratio used in example Te model results sow an insensitivity to stresses in te y direction but a sensitivity to vertical deformations. Deformations were summarized at 3m dept increments beginning at a dept of 1m. Te stress deformation beneat te footing is sown in Figure 6. Te resulting deformations as a function of model run number are Figure 7 Deformations plotted as a function of model run. Stocastic analysis gives us a metod of determining variability of output. In tis example we can include te possible variation of output deformations given te possible variation in input soil properties. Te resulting analysis increases our ability to compreend te value of te numerical results. 6. CONCLUSIONS Probabilistic metods ave not been as widely used in geotecnical engineering as migt be expected. Often it is te difficulty in application tat results in avoidance. Te

5 application of probabilistic metods in finite element analysis as typically not been performed because models lacked i) an ability to batc solve a group of differing problems. Tis limitation as largely been overcome given te application of automatic mes generation and automatic mes refinement. Batces of problems may be set up and run wit varying soil properties and te mes will automatically optimize for eac scenario. Geotecnical Engineering Geotecnical Board and Board on Energy and Environmental Systems Commission on Engineering and Tecnical Systems, National Researc Council, Wasington, D.C., 1995 Stianson, J., Tode, R., 24, SVFlux Tutorial Manual, SoilVision Systems Ltd., Saskatoon, SK, Canada A single run of a finite element model is often of little value to te practicing engineer. A group of runs based on a certain varying of soil properties provides muc improved information regarding te possible variance of model output. Te value of finite element analysis is dramatically improved by implementing te tecniques of stocastic analysis. As a result, consultants can provide clients wit a statistical basis for teir modeling results. Te result of te application of tis tecnology will result in increased clarity of regulation guidelines as well as improved defensibility of modeling results by geoconsultants. Probabilistic metods, wile not a substitute for traditional deterministic design metods, do offer a systematic and quantitative way of accounting for uncertainties encountered by geotecnical engineers, and tey are most effective wen used to organize and quantify tese uncertainties for engineering designs and decisions. (NRC, 1995) 7. REFERENCES Bear, J., Dynamics of Fluids in Porous Media, Dover Publications Inc., New York Celia, M.A., Bouloutas, E.T., 199. A General Mass- Conservative Numerical Solution for te Unsaturated Flow Equation, Water Resources Researc, Vol. 26, No. 7, pp , July Fredlund, M.D., 24, SoilVision Teory Manual, SoilVision Systems Ltd., Saskatoon, SK, Canada Fredlund M.D., 1996, Design of a Knowledge-Based System for Unsaturated Soil Properties, M.Sc. Tesis, University of Saskatcewan, Saskatoon, Saskatcewan, Canada Fredlund, M.D., J. Stianson & M. Rykaart, 22, Applications of Automatic Mes Refinement in te SVFlux and CemFlux Software Packages, 55t Canadian Geotecnical Conference and 3rd Joint IAH-CNC/CGS Conference, October 2-23, Niagara Falls, ON, Canada Fredlund, D.G., and Xing, A., 1994, Equations for te soilwater caracteristic curve, Canadian Geotecnical Journal, Vol. 31, No. 3, pp Hillel, D., 198. Fundamentals of Soil Pysics, Academic, San Diego, California Probabilistic Metods in Geotecnical Engineering, Committee on Reliability Metods for Risk Mitigation in