Groundwater flow and Coupled analysis

Duc Bui

Groundwater flow and Coupled analysis

Abstract
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The research focuses on enhancing the PLAXIS software for effective simulation of groundwater flow and coupled flow-deformation analysis in both saturated and partially saturated soils. It presents a comprehensive approach by modifying the existing undrained analysis and incorporates the Barcelona Basic Model to address the mechanical behavior of unsaturated soils. The study includes detailed formulations of governing equations, verification of numerical methods, and emphasizes the need for a simultaneous solution of displacement and pore pressure in practical applications.

Figures (410)

Fig. 1.1: A visualisation of soil mechanics showing the role of the surface flux boundary conditions (Fredlund, 1996) (groundwater flow boundary conditions). In the areas where the upward flux (i.e. evaporation and evapotranspiration) exists, suction above phreatic level increases (and degree of saturation decreases) and the water level is lowered with time while in the case of downward flux (i.e. precipitation) suction decreases (and degree of saturation increases) and the water level rises with time. In the case of zero net surface flux, the pore water pressure profile become in equilibrium at a hydrostatic condition (Figure 1.1).

Fig. 5.1: Effect of the parameter g, on the retention curve, (g, = 2.0 and g, =-1.0)

Fig. 5.3: Effect of the parameter g, on the retention curve, (g, = 1.0 and g, = 2.0) The effective saturation is defined as:

Fig. 5.4: Mualem - Van Genuchten: head - degree of saturation

Fig. 5.7: Linearized V an Genuchten: head - relative permeability

Fig. A 1.4: Active pore pressure in time vs height in PLAXIS 3D

Fig. A1.3: Degree of saturation in time vs height in PLAXIS 2D

Fig. A1.7: Degree of saturation in time vs height (existing PlaxFlow)

Fig. A3.2: Active pore pressure in time vs height in PLAXIS 2D

Fig. A3.4: Active pore pressure in time vs height in PLAXIS 3D

Fig. A3.3: Degree of saturation in time vs height in PLAXIS 2D

Fig. A3.5: Degree of saturation in time vs height in PLAXIS 3L

Fig. A3.6: Active pore pressure in time vs height (existing PlaxFlow) Fig. A3.7: Degree of saturation in time vs height (existing PlaxFlow)

Fig. B1.4: Active pore pressure in time vs height in PLAXIS 3D

Fig. B1.2: Active pore pressure in time vs height in PLAXIS 2D

Fig. 1.7: Degree ot saturation in time vs height (existing Plaxrlow)

Fig. B3.2: Active pore pressure in time vs height in PLAXIS 2D

Fig. B3.5: Degree of saturation in time vs height in PLAXIS 3D

Fig. B3.4: Active pore pressure in time vs height in PLAXIS 3D

Fig. B3.5: Degree of saturation in time vs height (existing PlaxFlow)

Fig. B4.3: Active pore pressure in time vs height in PLAXIS 3D

Fig. B4.4: Active pore pressure in time vs height (existing PlaxFlow)

Fig. C 1.3: Degree of saturation in time vs height in PLAXIS 2D

Fig. C 1.4: Relative permeability in time vs height in PLAXIS 21

Fig. C 1.6: Degree of saturation in time vs height in PLAXIS 3D

Fig. C 1.7: Relative permeability in time vs height in PLAXIS 3D

Fig. C 1.8: Active pore pressure in time vs height (existing PlaxFlow)

Fig. C 1.9: Degree of saturation in time vs height (existing PlaxFlow)

Fig. C 2.3: Degree of saturation in time vs height in PLAXIS 2D

Fig. C 2.4: Relative permeability in time vs height in PLAXIS 2D

Fig. C 2.5: Active pore pressure in time vs height in PLAXIS 3D Fig. C 2.6: Degree of saturation in time vs height in PLAXIS 3D

Fig. C 2.7: Relative permeability in time vs height in PLAXIS 3D

Fig. U 2.y: Degree of saturation in ume vs height (exisung Plaxtlow)

Fig. C 3.3: Degree of saturation in time vs height in PLAXIS 2D Fig. C 3.4: Relative permeability in time vs height in PLAXIS 2D

Fig. C 3.6: Degree of saturation in time vs height in PLAXIS 3D

Fig. C 3.7: Relative permeability in time vs height in PLAXIS 3D

Fig. U 3.9: Degree ot saturation in time vs height (existing Plaxtlow)

Fig. C 4.3: Degree of saturation in time vs height in PLAXIS 2D

Fig. C 4.2: Active pore pressure in time vs height in PLAXIS 2D

Fig. C 4.6: Degree of saturation in time vs height in PLAXIS 3D

Fig. C 4.4: Relative permeability in time vs height in PLAX 1S 2D

Fig. C 4.7: Relative permeability in time vs height in PLAXIS 3D

Fig. C 4.9: Degree of saturation in time vs height (existing PlaxFlow)

‘ig. C4.8: Active pore pressure in time vs height (existing PlaxFlow)

Fig. D1.5: Active pore pressure in time vs height in PLAXIS 3D

Fig. D1.3: Degree of saturation in time vs height in PLAXIS 2D

Fig. D1.4: Pore pressure at the Gauss point at the top of the column in time in PLAXIS 2D

Fig. D1.6: Degree of saturation in time vs height in PLAXIS 3D

Fig. D1.7: Pore pressure at the Gauss point at the top of the column in time in PLAXIS 3D

Fig. D1.8: Active pore pressure in time vs height (existing PlaxFlow)

.D1.9: Degree of saturation in time vs height (existing PlaxFlow)

Fig. D1.10: Pore pressure at the Gauss point at the top of the column in time (existing PlaxFlow)

Fig. D2.4: Pore pressure the top of the column in time in PLAXIS 2D

Fig. D2.5: Active pore pressure in time vs height in PLAXIS 31

Fig. D2.7: Pore pressure the top of the column in time in PLAXIS 3D

Fig. D3.4: Degree of saturation in time vs height in PLAXIS 2D

Fig. D3.7: Degree ot saturation in time vs height in PLAX 1S 3D

Fig. D3.5: Pore pressure at node and Gauss point at the top of the column in time in PLAXIS 2D

Fig. D3.8: Pore pressure at node and Gauss point at the top of the column in time in PLAXIS 3D

‘ig. D3.9: Active pore pressure in time vs height (existing PlaxFlow)

Fig. D3.10: Degree of saturation in time vs height (existing PlaxFlow)

Fig. D3.11: Pore pressure at the Gauss point at the top of the column in time (existing PlaxFlow)

Fig. D4.5: Pore pressure at node and Gauss point at the top of the column in time in PLAXIS 2D

Fig. D4.3: Active pore pressure in time vs height in PLAXIS 2D Fig. D4.4: Degree of saturation in time vs height in PLAXIS 2D

Fig. D4.8: Pore pressure at node and Gauss point at the top ot the column in time in PLAXIS 3D

Fig. D4.9: ig. D4.9: Active pore pressure in time vs height (existing PlaxFlow)

Fig. D4.10: Degree of saturation in time vs height (existing PlaxFlow)

Fig. D4.8: Pore pressure at the Gauss point at the top of the column in time (existing PlaxFlow)

Tab. G1.1: Input data for loam layer (B9)

This lesson demonstrates the applicability of PLAXIS to agricultural problems. The potato field lesson involves a loam layer on top of a sandy base. Regional conditions prescribe a water level at the position of the material interface. The water level in the ditches remains unchanged. The precipitation may vary on a daily basis due to weather conditions. The calculation aims to predict the variation of the water content in the loam layer in time as a result of time- dependent boundary conditions. Fig. G 1.1: Potato field geometry

Tab. G1.3: Prescribed flux (Qpin =0 and @nax= 1m The precipitation fluxes are given in Table G1.3. The threshold values for ponding and precipitation (evapotranspiration) are chosen as 1 m and 0 at the top of the boundary, respectively.

Fig. G1.4.2D: Active pore pressure after steady state calculation (PLAXIS 2D)

Fig. G1.5.2D: Degree of saturation after steady state calculation (PLAXIS 2D)

Fig. G1.7.2D: Degree of saturation after 4.5 days (PLAXIS 2D)

Fig. G1.9.2D: Degree of saturation after 9 days (PLAXIS 2D)

Fig. G 1.5.3D: Degree of saturation after steady state calculation (PLAXIS 3D) Fig. G1.4.3D: Active pore pressure after steady state calculation (PLAXIS 3D)

Fig. G 2.1: Triangular earth dam with a constant head on the left side and seepage on the right side

Fig. G2.3.2D: Groundwater head (PLAXIS 2D)

Fig. G2.6.3D: Total discharge at cross section B-B: Q=0.4982 m3/day/m (PLAXIS 3D)

Fig. G3.2.2D: Finite element mesh (PLA XIS 2D) Fig. G3.1: Geometry of the problem

Figure G3.1 shows the geometry and boundary conditions of the problem. As indicated in the figure there is a 10 m wide impermeable dam founded on a soil layer of 10m thick. The bottom of the soil layer is considered to be impermeable. A 5.0 m wall is placed under the dam. At the left side of the dam (as shown in figure) the water level is 15.0 m while at the right side the water level is 13.0 m. The wall is simulated by means of an impermeable interface. The element mesh is locally refined around the wall particularly at the tip of the wall. 6 noded and 15 noded elements are alternately used. The problem is also simulated with PlaxFlow to compare outputs. Permeability of soil is 1.0 m/day.

Fig. G3.4.2D: Groundwater head (PLAXIS 2D)

Fig. G3.4.3D: Groundwater head (PLAXIS 3D)

Fig. G4.3.2D: Groundwater head (PLAXIS 2D)

Fig. G4.5.3D: Total discharge Q=0.1531 m°’/day/m (PLAXIS 3D - 10 noded tetrahedral elements)

Fig. G4.4.3D: Flow field (PLAXIS 3D - 10 noded tetrahedral elements) Fig. G4.3.3D: Groundwater head (PLAXIS 3D)

Fig. G5.3.2D: Active pore pressure (PLAX IS 2D)

Fig. G5.4.3D: Flow field (PLAXIS 3D) Fig. G5.3.3D: Active pore pressure (PLAXIS 3D)

Fig. G6.3.2D: Finite element mesh (PLAXIS 2D - 15 noded elements Fig. G6.2: Monograph for Muskat problem (Kang- Kun Lee and Darrell I. Leap, 1997)

Fig. G6.4.3D: Flow field and phreatic surface (PLAXIS 3D)

Fig. G7.1.2D: Geometry and FE mesh (PLAXIS 2D)

In this example a column of soil with a well inside is simulated. Geometry of the problem and the finite element mesh is shown in Figure G7.1. Phreatic line is at the top boundary to generate fully saturated soil. The side and the bottom boundaries are closed for flow, therefore inflow or outflow can only occur through the top boundary. The total discharge of well is consider 1 m°/day. This example is calculated in PLAXIS 2D (plane strain and axi-symmetric) and in PLAXIS 3D.

Fig. G7.2.2D.2: Flow field (PLAXIS 2D - axi-symmetric)

Fig. G7.3.2D.1: Total discharge leaving the domain; Q=1.0 m°/day/m (PLAX! 2D - plane strain)

Fig. G7.3.2D.2: Total discharge leaving the domain; Q=1.0 m°/day/rad (PLAXIS 2D - axi-symmetric)

Fig. G7.3.3D: Total discharge leaving the domain; Q=1.0 m°/day (PLAXIS 3D) Qummary:

In this example a block of soil with a drain inside is simulated with PLAXIS 2D and PLAXIS 3D. Geometry of the problem and the finite element mesh is shown in Figure G8.1. The block is 50 m long, 20 m high (and 10 m wide in case of 3D) and the drain is exactly in the middle. The initial water level is at 18 m. Permeability of soil is 1 m/day and coarse material (from standard data set) is used for retention curve. Steady state type of calculation is used. The minimum head of drain is 10 m.

Fig. G8.3.2D: Degree of saturation (PLA XIS 2D)

Fig. G8.5.2D: Total discharge in soil; Q=4.473 m°/day/m (PLAXIS 2D) Fig. G8.4.2D: Flow field (PLAXIS 2D)

Fig. G8.5.3D: Total discharge in soil; Q=4.476 m°/day/m (PLAXIS 3D)

Fig. G9.2.3D: Geometry and FE mesh (PLAXIS 3D)

Fig. G9.3.2D6: Active pore pressures (6-noded elements)

Fig. G9.3.2D15: Active pore pressures (15-noded elements)

Fig. G10.1.3D: FE mesh (PLAXIS 3D 10 noded elements) Tab. G10.1: Van Genuchten parameters

Fig. G10.2: Comparison of phreatic level for non- homogeneous rectangular dam made by Bardet & Tobita (2002)

Fig. CA1.3.2D: Horizontal effective stresses (PLAXIS 2D)

Fig. CA1.2.3D: V ertical effective stresses (PLAXIS 3D)

Fig. CA1.4.2D: Vertical total stresses (PLAX IS 2D)

Fig. CA1.3.3D: Horizontal effective stresses (PLA XIS 3D)

Fig. CA1.4.2D: Horizontal total stresses (PLA XIS 2D)

Fig. CA1.4.3D: Horizontal total stresses (PLA XIS 3D)

The consolidation phenomenon will be practically finished when the argument of the exponential function is about 4 or 5 (Verruijt, 1993). This will be the case when T =2. Verification: The problem of one dimensional consolidation can be formulated by a differential equation (Terzaghi, 1923):

Fig. CA3.2: Active pore pressure for case 1a (PLAXIS 2D)

Fig. CA3.3: Vertical displacement for case 1a (PLAXIS 2D

Fig. CA3.5: Vertical displacement for case 1a (PLAXIS 3D Foundation)

Fig. CA3.9: Vertical displacement for case 1b (PLAXIS 2D)

Fig. CA3.8: Active pore pressure for case 1b (PLAXIS 2D)

Fig. CA4.1: Geometry boundary conditions for problem CA4

Fig. CA4.F: Increase of water head as a function of time in problem CA4

Fig. CA4.4; Active pore pressure for case 2a (PLAXIS 3D Foundation)

Fig. CA4.7: Vertical displacement for case 2a (PLAXIS 3D Foundation)

Fig. CA4.10: Active pore pressure for case 2b (PLAXIS 3D Foundation)

Tab. CA5.1: Input data for loam layer (B9)

The example of Potato filed which is calculated by transient calculation, is chosen for fully coupled flow-deformation analysis (example G1). This lesson demonstrates the applicability of PLAXIS to agricultural problems. The potato field lesson involves a loam layer on top of a sandy base. Regional conditions prescribe a water level at the position of the material interface. The water level in the ditches remains unchanged. The precipitation may vary on a daily basis due to weather conditions. The calculation aims to predict the variation of the water content in the loam layer in time as a result of time-dependent boundary conditions. Fig. CA5.1: Potato field geometry

Fig. CA5.3.2D: Finite element mesh (15 noded elements)

Fig. CA5.5.2D: Degree of saturation after steady state phase (imposing head)

Fig. CA5.7.2D: Active pore pressure and external water load after 5 days

Fig. CA5.8.2D: Degree of saturation after 5 days

Fig. CA5.9.2D: Deformed mesh after 5 days

Fig. CA5.11.2D: Active pore pressure after 9 days

Fig. CA5.3.3D: Finite element mesh (10 noded elements)

Fig. CA5.9.3D: Deformed mesh after 5 days Fig. CA5.10.3D: Flow field after 5 days

Fig. CA6.2.3D: Geometry and finite element mesh of the dam and sub-soil (PLAXIS 3D)

Fig. CA6.4.2D: Degree of saturation for high reservoir level (PLAXIS 2D)

Fig. CA6.3.3D: Steady-state pore pressure for high reservoir level (PLAXIS 3D)

Fig. CA6.4.3D: Degree of saturation for high reservoir level (PLAXIS 3D.

Fig. CA6.7.2D: Degree of saturation after rapid drawdown (PLAXIS 2D)

Fig. CA6.6.2D: Active pore pressure after rapid drawdown (PLAXIS 2D)

Fig. CA6.6.3D: Active pore pressure after rapid drawdown (PLAXIS 3D)

Fig. CA6.8.2D: Flow field after rapid drawdown (PLAXIS 2D)

Fig. CA6.9.2D: Active pore pressure after slow drawdown (PLAX IS 2D)

Fig. CA6.10.2D: Degree of saturation after slow drawdown (PLAXIS 2D)

Fig. CA6.13.2D: Degree of saturation for low reservoir level (PLAX IS 2D) Fig. CA6.12.2D: Steady-state pore pressure for low reservoir level (PLAXIS 2D)

Fig. CA6.13.3D: Degree of saturation for low reservoir level (PLAX IS 3D)

Fig. CA6.14.3D: Flow field for low reservoir level (PLAXIS 3D)

Fig. CA6.15.2D: Safety factors for different situations (PLAXIS 2D)

Fig. CA6.15.3D: Safety factors for different situations (PLAXIS 3D)

Fig. CA6.16.PF: Steady-state pore pressure distribution for high reservoir level (PlaxFlow)

Fig. CA6.19.PF: Steady-state pore pressure distribution for low reservoir level (PlaxFlow)

Fig. CA6.17.PF: Pore pressure distribution after rapid draw down (PlaxFlow)

Fig. USM 1.2: Shear stress versus axial strain for different suction (Plaxis)

Fig. USM1.5: Specific volume versus mean effective stress p‘ (Plaxis)

Fig. USM 1.4: Stress path in p’-q space (Plaxis)

Fig. USM 1.8: Stress path in p’-q space (Gonzalez & Gens, 2008)

Fig. USM1.9: Specific volume versus mean effective stress p’ (Gonzalez & Gens, 2008)

‘ig. USM1.11: a) Suction versus mean effective stress p’; b) suction versus deviatoric stress q; c) stress path in p’-q space; d) specific volume versus mean effective stress p’ (Sheng et al., 2003)

The material data of the soil is given in Table 2. For initialisation 100 kPa is assumed for preoverburden pressure (POP). Fig. USM2.1:; Geometry and FE mesh used for footing problem

Fig. 1: Skempton’s B-parameter versus degree of saturation

Tab. 1: Bulk modulus of water in the classical mode

Fig. 2: Geometry and finite element model of the problem

Excess pore pressures P., 94, (Pressure = negative) Maximum Value = -9.802 kN/m2 (Element 352 at Node 966) Minimum Value = -9.802 kN/m? (Element 267 at Node 903)

First of all, results of each case are provided and then the results are compared with each other. Figure 6 shows the stresses and active water pore pressures al the end of the initial phase (gravity loading). As Bishop stress is used, to explain how the effective stresses are calculated, degree of saturation is needed (Eq. 2) which is plotted in Figure 7.

Fig. 7: Results after the initial phase for coarse material: Degree of saturation

Fig. 9: Results after the initial phase for fine material: Degree of saturation

Fig. 10: Results after the phase 1 for coarse material: left) vertical total stress; middle) vertical effective stress; right) active pore pressure

Fig. 12: Results after phase 1 for coarse material: Degree of saturation

Fig. 14: Excess water pore pressure in fine material at the end of phase 1.

Fig. 15: Results after phase 1 for fine material: Degree of saturation

Here it is shown how to convert old PlaxFlow material data set to the new one. In the new material data set, Ca, is removed because it can simply be derived from the equivalent bulk modulus of water K,,/n and water weight \;:

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