Introduction Saturated-Unsaturated Consolidation This example looks at the volume change behavior of a saturated-unsaturated column under loading, wetting and drying conditions. Feature Highlights GeoStudio feature highlights include: Using the fully coupled stress-pore-pressure formulation to analyze a soil column with both a saturated and unsaturated zone Using a step-boundary function to apply a surface load only for the first time step Applying hydraulic boundary conditions to simulate wetting and drying conditions Computing swelling and shrinking responses due to pore-pressure changes without any other applied loads The use of higher-order 8-noded elements 3 Configuration and setup The configuration and setup is shown in Figure. It is basically a column for a D analysis. The sides are prevented from moving by specifying zero x-displacement boundary conditions on the sides. The bottom is fixed and the top is allowed to move freely in the vertical direction. Elevation - m..9.8.7.6.5.4.3....9.8.7.6.5.4.3... -. -.....3.4.5 Figure One-dimensional column configuration The initial pore-pressure conditions are established by defining a water table at mid-height of the column. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page of 9
4 Material properties The Material Category required for a coupled analysis is the Effective Parameters w/ PWP Changes. The material is assigned the Linear-Elastic constitutive model with E = kpa and ν =.334 (/3). The volumetric water-content function used is presented in Figure, and the hydraulic conductivity function is presented in Figure 3. The saturated K is x -5 m/sec. The slope of the volumetric water content function for positive pore-pressures is defined by m v which in this case has been specified as./kpa. This is equivalent to /E = / =.. The value of m v specified is actually not important in this case, since for a coupled analysis SIGMA/W computes m v from the specified E modulus..5 sat-unsat Vol. Water Content (m³/m³).4.3... Matric Suction (kpa) Figure The volumetric water content function SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page of 9
sat-unsat.e-5 X-Conductivity (m/sec).e-6.e-7.e-8.e-9.e-. Matric Suction (kpa) Figure 3 The hydraulic conductivity function 5 Analysis: Surface Load In this analysis we want to apply a surface load and then allow the resulting excess pore-pressure to dissipate. Since SIGMA/W is based on an incremental formulation, we cannot apply a surface load and keep it there during the dissipation stage. The surface load must be applied only during the first time step, and then it must be numerically removed for the remaining time steps. This can be accomplished with a step-function such as shown in Figure 4. Step function Normal Boundary Stress (kpa) 8 6 4 4 6 8 Time (sec) Figure 4 Surface load boundary function The time step duration is sec. SIGMA/W obtains the function values at time n and (n-). In this case they are and zero. The difference is kpa, which is the surface load applied for the first integration time step. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 3 of 9
For all other time steps the applied load increment is zero. For example, the function value at t = is and at t = is also, and therefore the applied load increment is zero as intended. Figure 5 shows the pore-pressure from applying the surface pressure of kpa (the unit of water has been set to kn/m 3 for discussion convenience). Where the soil is saturated, the pore-pressure increase is exactly kpa; that is, the entire load has gone into the pore-pressure. Where the soil is unsaturated, there is no pore-pressure increase, also as expected. In between, there is a smooth transition, which reflects the shape of the volumetric water content function. Excess pwp - Pore-Water Pressure (kpa) Figure 5 Pore-pressure from the applied load Figure 6 shows the dissipation of the excess pore-pressure. Below the initial water table position, the pore-pressure dissipates while, above this level, the pore-pressure increases slightly. Ultimately, the porepressure distribution returns to a hydrostatic straight line that is slightly shifted to the right. This causes the zero-pressure line (watertable) to rise slightly, which is due to the volume decrease of the saturated zone. In Figure 6 at sec, the final position of the pore-pressure distribution has not totally reached the ultimate long term position. However, the migration towards this final position is evident. This porepressure dissipation response is completely logical and intuitively correct. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 4 of 9
Excess pwp sec sec 3 sec 7 sec 5 sec 3 sec 63 sec 6 sec 5 sec 5 sec sec - Pore-Water Pressure (kpa) Figure 6 Pore-pressure dissipation Figure 7 shows the settlement profile as the excess pore-pressure dissipates. Above the initial water table there is some immediate compression, but there is virtually no compression in the saturated zone. This is because water is nearly incompressible and, therefore, there is no volume change until some of the water is squeezed out of the saturated zone. Once the excess pore-pressure begins to dissipate, there is some compression in the initial saturated zone, but above the initial watertable there is essentially no further compression. Eventually, the settlement profile would be a straight line with depth reflective of the soil-grain structure stiffness represented by E. The end result will be the same as if the load had been applied to a fully unsaturated column. There is, however, a time delay due to the flow of the water. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 5 of 9
Settlement Y-Displacement (m) Figure 7 Settlement due to consolidation 6 Analysis: Wetting and swelling In this analysis we start with the same hydraulic conditions as previously. The initial pore-pressure distribution is hydrostatic with negative pore-pressures above the initial watertable and positive below the initial watertable. Next, we apply a surface boundary condition of H = m. This means the pore-pressure is zero. Physically, this is equivalent to saying it is raining on the surface just enough to keep the pore-pressure at zero, but not enough to cause surface ponding. In this case specifying a Head boundary condition means water will flow into the sample. The infiltration will result in a decrease in effective stress and therefore the soil will swell. Figure 8 shows the pore-pressure profiles at various times. Initially, the pore-pressure at the ground surface is -kpa. When it starts to rain, the pore-pressure immediately becomes zero as per the specified boundary condition. The negative pore-pressure then begins to diminish with time and if it were to rain long enough, all the voids would fill with water and the pore-pressure would return to being hydrostatic with zero at the surface and kpa at the bottom, exactly as one would expect. As shown in Figure 9, the sample begins to swell near the top and then continues to swell as the porepressures increase (become less negative in the top half). Ultimately the volume change profile is a straight line, the same as in the previous loading analysis. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 6 of 9
Excess pwp - Pore-Water Pressure (kpa) Figure 8 Pore-pressures changes with infiltration Settlement Y-Displacement (m) Figure 9 Swelling due to increase in pore-pressure 7 Analysis: Drying and shrinking This analysis repeats the previous swelling analysis, but forces water to be extracted (drying) by specifying a boundary condition on the surface of H = zero. Since the elevation is m, this means the specified pore-pressure at the surface is - kpa. Initial the pore-pressure was -, but then went to - SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 7 of 9
due to some drying. The drying will cause the effective stress to increase, which in turn will lead to shrinkage, as shown in Figure. Excess pwp - Pore-Water Pressure (kpa) Figure pore-pressure profiles during drying Settlement Y-Displacement (m) Figure Shrinking during drying Now we have the exact reverse of the swelling. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 8 of 9
Of significance is the time required to reach the final long-term condition. The time to reach the final shrinking condition is much longer than to reach the final swelling condition. The reason for this is the hydraulic conductivity variations with negative pore-pressures. In the drying case, there is a larger zone of unsaturated soil where the hydraulic conductivity is much lower than where the soil is saturated. 8 SIGMA/W coupled behavior In the SIGMA/W coupled formulation, the stiffness of the soil grain structure E is related to the porepressure stiffness m v. Generally, where the pore-pressure is positive m v is equal to /E. This represents fully saturate conditions. So at each Gauss integration point in an element, SIGMA/W looks at the pore-pressure, and if the porepressure is greater than or equal to zero, m v is computed from the specified E value. Where the pore-pressure is negative, SIGMA/W computes an m v from E and an m v from the slope of the volumetric water content function. If the m v from the function is smaller than the m v from E, m v is set to the value computed from E (this is the fully saturated case). If the m v from the function is higher than the m v computed from E, the value from the function is used in the coupled analysis (this is the unsaturated case). Referring back to the loading analysis above, the m v in the upper unsaturated zone is higher than /E and, consequently, the soil will behave in an unsaturated manner and there is essentially no response to the surface loading. In summary, the slope of the volumetric water content function relative to the specified E value becomes very important in a saturated-unsaturated coupled analysis. 9 Conclusion This one-dimensional fully coupled analysis of a case with both satuated and unsaturated zones demonstrates that the SIGMA/W formulation gives solutions in accordance with what is intuitively correct. The results are explainable and all have the correct trend. SIGMA/W Example File: Sat-Unsat column tests.doc (pdf) (gsz) Page 9 of 9