10 Steady-State Fluid Flow with a Free Surface

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1 Steady-State Fluid Flow with a Free Surface Steady-State Fluid Flow with a Free Surface 10.1 roblem Statement This numerical simulation analyzes the steady-state seepage flow through a homogeneous embankment with vertical slopes exposed to different water levels and resting on an impermeable base. The total discharge, Q, and the length of seepage face, s, are compared to the exact solutions. The problem tests FLAC s formulation for unconfined groundwater flow. Figure 10.1 shows the geometry and boundary conditions of the problem. h s h impermeable base L Figure 10.1 roblem geometry and boundary conditions The dimension and elevations are: L =9m h 1 =6m h 2 = 1.2 m

2 10-2 Verification roblems The following material properties are used: 10.2 Analytic Solution permeability (k) (m/sec)/(a/m) porosity (n) 0.3 water density (ρ w ) 1000 kg/m 3 water bulk modulus (K w ) 1000 a soil dry density (ρ) 2000 kg/m 3 gravity (g) 10 m/sec 2 Dupuit s formula (see, for example, Davis and DeWiest (1966), p. 186) gives the exact solution for the total flow rate (per unit model thickness) as Q = kρ w g h 2 1 h2 2 2L (10.1) For this particular case, the value for Q is m 3 /s. The length, s, of the seepage face as a function of the characteristic dimensions of the section was obtained by olubarinova-kochina, and is given in Figure 10.2 (see, e.g., Harr (1991), pp ). For this particular problem: h 2 /h 1 = 0.2, L/h 1 = 1.5, and the value of s/h 1 is evaluated from the graphs at 0.1, which gives an elevation s = 0.6 m FLAC Model Two cases have been studied, corresponding to two different initial conditions: CASE 1: The embankment is initially dry (saturation = 0), and the sides are suddenly exposed to the water. CASE 2: The water level at both sides is initially the same (h 1 = h 2 = 6 m), followed by a sudden drawdown at the right side (h 2 = 1.2 m). The grid and boundary conditions are the same for both cases (see Figure 10.3). The only difference is the initial pore pressure distribution. In Case 1, saturation and pore pressure are zero inside the mesh; in Case 2, saturation is 1 for all gridpoints, and the pore pressure inside the mesh follows a gravitational gradient. The material properties are assigned as described above. Since only the steady-state fluid-flow solution is of interest in this problem, the mechanical behavior of the soil and its interaction with the groundwater flow are not addressed (SET mech off). The absolute values of soil density, homogeneous permeability and porosity are not relevant to the final solution, and the water bulk modulus is given a small value, compatible with free surface numerical stability, to speed up the

3 Steady-State Fluid Flow with a Free Surface 10-3 calculation to steady state. (The criterion used for numerical stability is K w > 0.3 ρ w gl x, where L x is the maximum horizontal zone dimension in the vicinity of the free surface, as discussed in Section in Fluid-Mechanical Interaction.) h/h Figure 10.2 Seepage face solution after olubarinova-kochina

4 10-4 Verification roblems JOB TITLE : May-08 9:32 step 500 Flow Time E E-01 <x< 9.500E E+00 <y< 8.000E+00 Grid plot 0 2E 0 Fixed Gridpoints ore-pressure Figure 10.3 FLAC grid and fixed pore pressure locations 10.4 Results and Discussion As a preliminary to the numerical simulations, the time needed to reach steady state may be estimated using the definition of characteristic time, t c = L c 2 c (10.2) where L c is the characteristic length of the problem, and c is the diffusivity. For a flow-only problem, c = kk w n (10.3) Using L = L c and the above property values, we obtain t c = s. Division of t c by the explicit fluid flow timestep of s (type RINT limits for the timestep) gives a prediction of approximately 4500 steps to reach steady state. The flow simulation for the two cases is carried out using the SOLVE command. The evolution towards steady state is monitored using the FISH function flow, which calculates inflow and

5 Steady-State Fluid Flow with a Free Surface 10-5 outflow at the left model boundary and right model boundary, respectively. Inflow is positive if water flows into the model, while outflow is positive if water flows out; at steady state, the free surface is a streamline, and inflow is equal to outflow. The evolutions of the inflow and outflow rates are plotted and compared to the analytical steady-state solution (shown as a solid line) in Figures 10.4 and It may be observed that steady state is reached at the estimate of 5000 steps obtained earlier. The difference in the flow patterns leading to steady-flow is seen by comparing Figures 10.6 and 10.7, which show the flow vectors after 500 steps for Case 1 and 2, respectively. In both cases, the final flow pattern is similar (see Figures 10.8 and 10.9). The calculated length of seepage extends in those figures from the tail water elevation up to the point on the downstream embankment slope where the magnitude of the flow vector becomes zero. As may be seen, the numerical seepage length compares well with the sketched analytical solution References Davis, S. N., and R. J. DeWiest. Hydrogeology. J. Wiley, Harr, M. E. Groundwater and Seepage. Dover, 1991.

6 10-6 Verification roblems JOB TITLE : Vertical Embankment - Case 1 12-May-08 9:34 step 3881 Flow Time E+08 HISTORY LOT Y-axis : 2 flow (FISH) 3 inflow (FISH) 4 outflow (FISH) X-axis : Number of steps -05 (10 ) (10 ) Figure 10.4 Flow rate evolution (Case 1) JOB TITLE : Vertical Embankment - Case 2 12-May-08 9:39 step 3843 Flow Time E+08 HISTORY LOT Y-axis : 2 flow (FISH) 3 inflow (FISH) 4 outflow (FISH) X-axis : Number of steps -05 (10 ) (10 ) Figure 10.5 Flow rate evolution (Case 2)

7 Steady-State Fluid Flow with a Free Surface 10-7 JOB TITLE : Vertical Embankment - Case May-08 9:41 step 500 Flow Time E E-01 <x< 9.500E E+00 <y< 8.000E Boundary plot 0 2E 0 Flow vectors max vector = 7.287E E Figure 10.6 Flow vectors after 500 steps (Case 1) JOB TITLE : Vertical Embankment - Case May-08 9:40 step 500 Flow Time E E-01 <x< 9.500E E+00 <y< 8.000E Boundary plot 0 2E 0 Flow vectors max vector = 1.729E E Figure 10.7 Flow vectors after 500 steps (Case 2)

8 10-8 Verification roblems JOB TITLE : Vertical Embankment - Case May-08 9:37 step 3881 Flow Time E E+00 <x< 1.150E E+00 <y< 5.000E+00 Boundary plot 0 2E 0 Flow vectors max vector = 1.367E E -6 seepage face (analytical) (*10^1) Figure 10.8 Steady-state flow vectors and seepage face solution (Case 1) JOB TITLE : Vertical Embankment - Case May-08 9:38 step 3843 Flow Time E E+00 <x< 1.150E E+00 <y< 5.000E+00 Boundary plot 0 2E 0 Flow vectors max vector = 1.368E E -6 seepage face (analytical) (*10^1) Figure 10.9 Steady-state flow vectors and seepage face solution (Case 2)

9 Steady-State Fluid Flow with a Free Surface Data File FREESURFACE.DAT ;roject Record Tree export ;*** Branch: Initial Dry **** new ;... State: h2aa.sav... config gw g def ini h2 h1 = 6. h2 = 1.2 bl = 9. ck = 1e-10 rw = 1e3 gr = 10. qt = ck*rw*gr*(h1*h1 - h2*h2)/(2.0*bl) end ini h2 gen h1 bl h1 bl 0 mo el ; --- roperties --- prop por.3 perm=ck den 2000 water den=rw bulk 1e3 ; --- Initial conditions --- ini sat 0 ; --- Boundary conditions --- ini pp 6e4 var 0-6e4 i 1 ini pp 1.2e4 var 0-1.2e4 i 31 j 1 5 fix pp i 1 fix pp i 31 ini sat 1 i 1 ini sat 1 i 31 j 1 5 ; --- Settings --- set mech off set grav=gr ; --- Fish functions --- def flow inflow=0.0 outflow=0.0 loop j (1,jgp) inflow=inflow+gflow(1,j) outflow=outflow-gflow(31,j) end loop flow=qt

10 10-10 Verification roblems end ; --- Histories --- hist nstep 50 hist pp i 15 j 1 hist flow hist inflow hist outflow ; --- Step --- step 500 ; --- Generate plots --- save h2aa.sav ;... State: h2a.sav... ; --- Step to steady-state --- solve save h2a.sav ;*** Branch: Initial Saturated **** new ;... State: h2bb.sav... config gw g def ini h2 h1 = 6. h2 = 1.2 bl = 9. ck = 1e-10 rw = 1e3 gr = 10. qt = ck*rw*gr*(h1*h1 - h2*h2)/(2.0*bl) end ini h2 gen h1 bl h1 bl 0 mo el ; --- roperties --- prop por.3 perm=ck den 2000 water den=rw bulk 1e3 ; --- Initial conditions --- ini sat 1 ; --- Boundary conditions --- ini pp 6e4 var 0-6e4 ini pp 1.2e4 var 0-1.2e4 i 31 j 1 5 ini pp 0 i 31 j 6 21 fix pp i 1 fix pp i 31

11 Steady-State Fluid Flow with a Free Surface ; --- Settings --- set mech off set grav=gr ; --- Fish functions --- def flow inflow=0.0 outflow=0.0 loop j (1,jgp) inflow=inflow+gflow(1,j) outflow=outflow-gflow(31,j) end loop flow=qt end ; --- Histories --- hist nstep 50 hist pp i 15 j 1 hist flow hist inflow hist outflow ; --- Step --- step 500 ; --- Generate plots --- save h2bb.sav ;... State: h2b.sav... ; --- Step to steady-state --- solve save h2b.sav ;*** plot commands **** ;plot name: grid plot hold grid fix ;plot name: Flow rate evolution plot hold history 2 line 3 line 4 line ;plot name: Flow vectors plot hold bound flow ;plot name: Stady-state flow vectors and seepage face label plot 1 (9.25,1.5) seepage face label plot 2 (9.25,1.4) (analytical) label line 3 (9.0,1.2) (9.0,1.6) plot hold grid flow label 1 green label 2 green label 3 lmagenta

12 10-12 Verification roblems

1 Slope Stability for a Cohesive and Frictional Soil

Slope Stability for a Cohesive and Frictional Soil 1-1 1 Slope Stability for a Cohesive and Frictional Soil 1.1 Problem Statement A common problem encountered in engineering soil mechanics is the stability