HM_003 Uniaxial Sedimentation using Eulerian Deposition
Contents
This example extends the uniaxial consolidation example HM_001 by considering overpressure build up and dissipation during the sedimentation of a 1000m column. Deposition is represented by a moving Eulerian boundary defined via a part-geometry and boundary surface data. In this case the height of the column grows gradually throughout the sedimentation time.
Specific issues considered are:
1Problem set up and simulation of large-scale coupled problems.
2Deposition using an Eulerian boundary defined via a part-geometry and boundary surface data
The example documentation assumes that the user is familiar with mesh generation and single field geomechanical modelling functionality and the following examples should undertaken beforehand:
1Mech_001 Mechanical Analysis Introduction
2HM_001 Introduction to Hydro-Mechanical Analysis - Uniaxial Consolidation
3HM_002 Uniaxial Sedimentation using Pre-Existing Sediment
The Part_geometry_set data structure defines a geometry set; e.g. a group of lines or surfaces, that does not form part of the simulation domain. Part_geometry_set's are used to define boundary conditions e.g. Prescribed_boundary_data. The geometry is defined in terms of lines (Part_line) in 2-D and surfaces (Part_surface) in 3-D. It is important that standard geometry lines (Geometry_line) and surfaces (Geometry_surface) are not used to define Part_geometry_set's. Part geometry is faceted so that Part_line is defined as faceted polyline and Part_surface is a tessellated surface. Part geometry is unaffected by remeshing operations.
Part geometry is defined as a single or series of Part_geometry_set's, with each set requiring: 1A list of part-geometry entities in the set. 2The definition of each geometry entity i.e. Part_line's or Part_surface's. 3The nodal coordinates of the nodes defining the geometry entities in the set Part_nodal_data ISET. Optionally the part-geometry may be assigned a deformation by defining a list of updated coordinates, via Part_nodal_update together with a Time_curve.
Part-Geometry Defined as a Single Part_line with coordinates updated using Part_nodal_update
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The Prescribed_boundary_data structure defines a special prescribed displacement boundary condition defined using either a Lagrangian or Eulerian description.
Lagrangian Description The geometry of the model is prescribed to be located on a rigid boundary surface; e.g. the basement geometry. The advantage of this load type is that the rigid boundary surface may be assigned a prescribed movement corresponding the evolution of the sediment outside the model; e.g. differential uplift, fault offset, etc. This type of differential displacement would be difficult to apply directly to the model due to the continuous remeshing that is used to accommodate the large deformations.
Eulerian Description With the Eulerian prescribed boundary the mesh is free to move relative to the boundary surface and material is continuously added or removed to the model so the model domain is consistent with the boundary surface geometry. This option is used for example to define sedimentation or erosion processes in a geomechanical analysis.
By default a Lagrangian boundary description is used and the primary data required is 1The geometry entities in the model to be assigned to the boundary. These are geometry lines in 2-D and geometry surfaces in 3-D. 2The part-geometry set to be used 3The time curve associated with the update of the part-geometry is the prescribed boundary is moving during the simulation.
The Prescribed Boundary is Defined by Assigning Part-Geometry Set 1 to Lines L1 and L2
Sandbox Simulation with Extension and Vertical Offset of a Boundary Surface
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The objective is to simulate the sedimentation and consolidation of 1000m of sediment over a period of 1.0 Ma. Only 20m of pre-existing sediment is represented and continuous sedimentation is represented by an Eulerian boundary defined using the Part_geometry and Prescribed_boundary_data keywords.
The model is analysed in uniaxial strain conditions; i.e. the vertical sides of the model are constrained in the horizontal direction, and the base of the model is constrained in the vertical direction. Gravity is the only load type and the top surface of the model is prescribed zero pore pressure (pf); i.e. free drainage, resulting in a gradual dissipation of pore pressure in the model. The gravity load for the pre-existing sediment is applied over 0.1 Ma and then sedimentation of the column takes place between 0.1 and 1.1 Ma. Unstructured triangle (TPM3V) is used with a target element size of 20m. Adaptive remeshing is also required due to the sedimentation process. Remeshing is triggered at a distortion of 15% with the objective of maintaining the element size at 20m.
Material Properties The material corresponds to "HM_002_elastic" in the "training.mdb" material database, The material parameters is chosen to match the parameters assumed the simulations presented by Wangen (2010, p.403). The primary mechanical properties for this material are:
The additional material parameters related to the porous flow field are:
Notes 1The simulation uses isotropic permeability. In general, however, soils exhibit transverse isotropic flow with the horizontal permeability often between 5 to 100 times larger than the vertical permeability. In this case orthotropic permeability may be defined via Permeability_x, Permeability_y and Permeability_z. Furthermore, permeability is dependent on the current porosity, so that in applications where significant compaction is used permeability is defined as a function of porosity. This may be either isotropic via Permeability_vs_porosity or orthotropic via Permeability_x_vs_porosity, Permeability_y_vs_porosity and Permeability_z_vs_porosity. 2The Biot constant (Biot_constant) defines the contribution of the pore pressure to the total stress via the effective stress relationship
The Biot constant may either be user defined (as in this case) or may be computed automatically via the relationship
3The material is specified as fully saturated ( Fluid_saturation = 1.0). 4The material is saturated with fluid number 1.
Fluid Properties The properties of the pore fluid must be specified. The principal parameters are:
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The initial data file for the project is: HM_003\Data\hm_003_Case1.dat. The basic data includes: 1 Part_geometry and Prescribed_boundary_data to define the sedimentation surface (see later) 2A single group which is assigned the "hm_002_elastic" properties defined using Group_control_data and Group_data data structures. The group is active in all fields (see hm_002). The Porous_flow_type = 4 i.e. a coupled geomechanical/porous flow. A structured mesh of 4-noded elements is used. 3Material properties (Material_data) for "hm_002_elastic" exported from material database training.mdb using the "Set data for geomechanical only" option. The material data is stored in hm_002_elastic.mat in the HM_002\Data directory. 4Fluid properties (Fluid_data) for water. 5Time scaling data (Time_scaling_data) with target time step 1E-5 Ma; i.e. ca. 100,000 steps for the sedimentation time (see Mech_002). 6Global damping (Damping_global_data) for the geomechanical field with 2% percentage damping (see Mech_002). 7One global load case corresponding to zero pore pressure on the surface of the model (line 3). 8Support data (Support_data) defining: (a) Geomechanical field: line 1 fixed vertically and lines 2 and 4 fixed horizontally. (b) Porous flow field: Line 3 is defined as prescribed. 9Point history (History_point) data: (a) Set 1: Output of stress and pore pressure at the base of the column (b) Set 2: Output of pore pressure at 21 points evenly distributed throughout the depth of the column. 10Mesh control (Mesh_control) and Structured mesh generation data (Unstructured_mesh_data) defining a uniform mesh with 40 elements. 11 Couple control (Couple_control_data) and control data (Control_data). This is identical to example see HM_002.
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The Part_geometry_set data structure defines a geometry set; e.g. a group of lines or surfaces, that does not form part of the simulation domain. For the current case, a single part-geometry set (Set=1) is defined that corresponds to the sedimentation surface. This is defined by a single Part_line with one facets and two nodes.
Geometry Description for Part_line 1
The coordinates in the initial configuration, defined via Part_nodal_data (Set=1), are:
Initial Nodal Coordinates for Part Geometry Set 1
The location of these nodes is updated by specifying Part_nodal_upate (Set=1) to coordinates:
Update Nodal Coordinates for Part Geometry Set 1
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Analytical equations for the development of overpressure in in a basin at constant sedimentation rate were developed by Gibson (1958) and further discussed in Gibson (1967) and Wangen (2010) amongst others. The solution assumes a shallow basin where the porosity reduction is negligible; i.e. the geometrical impact of compaction may be neglected. The key assumption of the model is that Porosity (φ) is a linear function of the vertical effective stress (σ'v); i.e.
Substitution in the mass conservation equation and solution for the special case of constant sedimentation rate and zero surface pore pressure then gives
Note that rearrangement of the expression for porosity gives
Wangen (2010, p.403) provides overpressure predictions for a particular set of material parameters comprising;
A solution, obtained by very approximate integration using a multi-point trapezoidal rule, is provided in Excel worksheet hm_003\Results\hm_003_Case1.xlsm\Analytical.
Overpressure at 1 Ma for three Different Permeability Values
Overpressure at the Base of the Column vs. Time for three Different Permeability Values References Gibson, R. E. 1958. The progress of consolidation in a clay layer increasing in thickness with time. Géotechnique, 8, 171–182. Gibson, R. E., England, G. L. & Hussey, M. J. L. 1967. The theory of one-dimensional consolidation of saturated clays. Géotechnique, 17, 261-273. Wangen, M. 2010. Physical Principles of Sedimentary Basin Analysis, Cambridge University Press.
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The result files for the project are in directory: HM_003\results and the high definition history files can be displayed in excel file hm_003_Case1.xlsx. Two cases are considered: •Case 1 - where the time curve for the sedimentation surface is linear •Case 1A - where the time curve for the sedimentation surface is an S-Curve; i.e. giving a smoother transition at the start and end of the sedimentation process.
Case 1 Linear Time Curve The evolution of pore pressure as a function of column height is shown in the figure below.
Contours of Pore Pressure Between Time t=0.1 and t=1
The evolution of pore pressure as a function of time exhibits a marked change at time 1.1 corresponding to the end of sedimentation. The overpressure is almost completely dissipated by the end of the simulation at t = 2Ma. The vertical and horizontal effective mean stresses increase during this period to compensate for the reduction in pore pressure.
Pore Pressure and Stress Evolution at the Base of the Column as a Function of Time
The graphs of pore pressure, vertical effective stress and horizontal effective stress as a function of depth are created by exporting data for "graph on a line" from ParaView at time t = 1.0 Ma and t = 2.0 Ma.
Effective Vertical/Horizontal Stress and Pore Pressure as a Function of Depth
Case 1A S-Curve Time Curve The case with an S-Curve sedimentation profile results in a smooth transition from the phase of overpressure build up during sedimentation to overpressure reduction post-sedimentation.
Pore Pressure and Stress Evolution at the Base of the Column as a Function of Time
Comparison of the pore pressure vs. time response for Case 1 and Case 1A shows that the minor oscillations observed at the abrupt end of sedimentation in Case 1 are almost completely eliminated in Case 1A.
Pore Pressure as a Function of Time for Case 1 and Case 1A
The predicted overpressure profile as a function of depth at t = 1.0Ma is very close to the analytical solution. An exact match is not expected due to the slightly different loading regimes in the analytical model and the simulation.
Comparison of the Predicted Pore Pressure and the Analytical Solution at time t = 1.0 Ma
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