HM_004 Uniaxial Sedimentation using Layer Deposition
Contents
This example extends the uniaxial sedimentation example HM_003 by considering overpressure build up and dissipation during the sedimentation of a 1000m column represented by layer sedimentation. The layer sedimentation algorithm is more general than the Eulerian boundary algorithm and allows general intersection of the new sediment layer with existing sediment. The disadvantage, however, is that the layers are deposited as layers of discrete thickness rather than the continuous process used with the Eulerian boundary.
Specific issues considered are:
1Deposition using layer sedimentation for coupled problems.
The example documentation assumes that the user has undertaken following examples beforehand:
1Mech_002 Uniaxial Burial of a 2000m of Sediment. This describes the layer sedimentation algorithm.
2HM_001 Introduction to Hydro-Mechanical Analysis - Uniaxial Consolidation.
3HM_002 Uniaxial Sedimentation using Pre-Existing Sediment.
4HM_003 Uniaxial Sedimentation using Eulerian Deposition.
The objective is to simulate the sedimentation and consolidation of1000m of sediment over a period of 1.0 Ma. Only 20m of pre-existing sediment is represented and sedimentation is represented by the layer sedimentation algorithm. 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 for Case 2 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_004\Data\hm_004_Case1.dat. The basic data includes: 1 Stratigraphy_definition and Stratigraphy_horizon data for the pre-existing layer. 2Sedimentation data for the new layers 3A 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. 4Material 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. 5Fluid properties (Fluid_data) for water. 6Time 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). 7Global damping (Damping_global_data) for the geomechanical field with 2% percentage damping (see Mech_002). 8One global load case corresponding to zero pore pressure on the surface of the model (line 3). 9Support 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. 10Point 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. 11Mesh control (Mesh_control) and Structured mesh generation data (Unstructured_mesh_data) defining a uniform mesh with 40 elements. 12 Couple control (Couple_control_data) and control data (Control_data). This is identical to example see HM_002.
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The stratigraphy associated with the initial sediment must be defined when the layer sedimentation procedure is used. This requires 1Defining the existing stratigraphy layer depositional order and the group associated with each layer (via Stratigraphy_definition). 2Defining the topology of the top surface horizon for each stratigraphy layer (via Stratigraphy_horizon).
Stratigraphy_definition The stratigraphy order is specified in order starting from the deepest sediment layer. Each stratigraphy layer must comprise a single group (see Group_data). It is recommended that, for each layer, the stratigraphy unit name, stratigraphy horizon name and group name are identical. This greatly simplifies data definition and result interrogation, especially when the number of stratigraphy layers becomes large. If this convention is adopted, ParaGeo will internally identify the required associations between unit, horizons and groups, so that only the unit order is specified on the Stratigraphy_definition data structure.
Stratigraphy_horizon The stratigraphy horizon defines the top surface of each pre-existing stratigraphy unit. It is defined by a set of geometry lines in 2-D or a set of geometry surfaces in 3-D. A stratigraphy_horizon data structure must be defined for each stratigraphy unit.
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Layer sedimentation is defined using three data structures: 1Sedimentation_data which defines the control data for deposition of a single layer; e.g. unit name, material, sedimentation type, etc. 2Sedimentation_parameters which defines default settings for the deposition of subsequent layers. 3Sedimentation_horizon which defines the target topology for the stratigraphy horizon of the new layer.
Note that a stratigraphy_horizon is an existing horizon defined by geometry entities (lines 2-D, surfaces 3-D) that form part of the model, whereas a Sedimentation_horizon is a faceted surface that is not connected with the model. During layer sedimentation new geometry entities are created to define the new layer, and the stratigraphy_horizon for the new layer is formed using a combination of these new geometry entities and pre-existing geometry (if required).
Sedimentation_Data Sedimentation_data is the primary data structure for definition of sedimentation of a new stratigraphy unit. One data structure is required for each new layer and the sedimentation process is associated with a single analysis stage (step). Consequently in the current problem, where ten additional layers are deposited on the pre-existing layer, the analysis consists of 11 steps. Sedimentation_parameters is used to specify data that is common to all sedimentation steps. Consequently the only data specified to add a new layer is the unit name.
Sedimentation_Parameters Sedimentation_parameters set default values of the sedimentation parameters that will be used for each sedimentation step by default. The value of any parameter can be replaced for a particular sedimentation step by specifying the same parameter with a different value on the is the Sedimentation_data data structure for that step. The principal data includes: 1The sedimentation horizon name (Sediment_horizon_name); i.e. the Sedimentation_horizon to be used in defining the top surface of the new layer. 2The name of the material properties to be used for the new material (Material_name). 3The duration (Duration) of the initialisation of gravity loading for the new layer. This is generally set to less than the time for the stage so that additional dynamic relaxation can occur prior to deposition of the next layer. Note that by default, the Sedimentation_type is "Absolute" where the Sedimentation_horizon defines the top surface of the new layer. Two alternative sedimentation types are available 1"Relative" - where a sedimentation horizon defines the topology of the new top surface but its location is defined via a layer thickness (Reference_thickness) and reference location (Reference_location). 2"Drape" - Where a uniform thickness of material is added across the complete model.
Sedimentation_Horizon The Sedimentation_horizon data structure defines the topology of a top surface horizon for sedimentation of a new stratigraphy layer. The horizon is defined as a faceted surface; i.e. 2-noded facets in 2-D, 3-noded facets in 3-D. The node numbering and coordinates defining the faceted surface are defined in the data structure and do not form part of the nodes in the mesh. The horizon may be defined as either stationary or moving; e.g. increasing in elevation or prograding, via a displacement component magnitudes and a time curve. A sedimentation horizon may be used to define more that one layer; e.g. a prograding horizon is often used to define sedimentation of all layers. Alternatively multiple sedimentation horizons may be specified and used for sedimentation of individual layers.
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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_004\results and the high definition history files can be displayed in excel file hm_004.xlsx. Two cases are considered: •Case 1 - where the permeability (k) = 1.0E-18 m2 •Case 2 - where the permeability (k) = 1.0E-19 m2. These cases correspond to the conditions used for Case 1 and Case 2 in example HM_003 Uniaxial Sedimentation using an Eulerian Boundary.
Case 1 Permeability (k) = 1.0E-18 m2 The evolution of pore pressure as a function of column height is shown in the figure below.
The evolution of pore pressure as a function of time exhibits a marked change at time 1.0 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
Comparison of the predicted pore pressure with solution obtained by continuous sedimentation using an Eulerian boundary (example HM_003) shows that the discrete algorithm provides an almost identical solution. The discontinuous nature of the layer sedimentation algorithm does result in more high frequency noise in the solution and requires smaller time steps in both the geomechanical and pore pressure fields. For example HM_003_Case1 uses 100 steps in the porous flow field whereas 100 steps/layer deposition (total of 1000 steps) is used in HM_004_Case_1.
Comparison of Pore Pressure Evolution at the Base of the Column as a Function of Time for HM_003_Case1 and HM_004_Case1
Case 2 Permeability (k) = 1.0E-19 m2 As expected the evolution of pore pressure as a function of time exhibits significantly higher pore pressure than Case 1 where (k) = 1.0E-18 m2.
Pore Pressure and Stress Evolution at the Base of the Column as a Function of Time
Comparison of the predicted pore pressure with solution obtained by continuous sedimentation using an Eulerian boundary (example HM_003) shows that the discrete algorithm provides an almost identical solution. As with case 1, the discontinuous nature of the layer sedimentation algorithm results in more high frequency noise in the solution and requires smaller time steps in both the geomechanical and pore pressure fields.
Comparison of Pore Pressure Evolution at the Base of the Column as a Function of Time for HM_003_Case2 and HM_004_Case2 |
