Geomechanical Modelling Geomechanical restoration is a quantitative method of modelling strain during geological deformation. Geomechanical methods incorporate the elastic properties of the rock and therefore in some restoration or forward modelling objectives more realistically model the response of rock masses to deformation. In Move, the Geomechanical Modelling module allows you to model the evolution of structures through time, quickly and easily (Figure 1a). The strain resulting from a geomechanical restoration can be converted to attributes and viewed in Move using the Strain Capture tool (Figure 1b). These attributes can then form the basis for fracture network prediction. Strain can be modelled in both a forward and reverse modelling sense at any restoration step, thus providing a method to predict strain at any point through geological time. Critically, this makes it possible to predict fracture networks or stress systems at important geological steps, such as the time of hydrocarbon maturation and migration or mineral deposition. Here we describe the theory behind geomechanical restoration, as well as outlining a workflow used to model and capture strain from folding and faulting. Time step 1 Time step 2 Time step 3 Figure 1: a) Restoration steps from a model of the present day geometry (time step 3) to the initial starting geometry (time step 1); b) Inversion of restoration steps showing present day model colour mapped for the magnitude of e1 (minimum shortening direction). Warm colours (reds and oranges) represent high shortening with cold colours (greens and blues) representing low shortening.
Theory In contrast to kinematic restorations, geomechanical restorations consider the mechanics of the rock and honour physical laws, namely the conservation of mass, momentum, and energy. Geomechanical Modelling uses mass-spring systems (c.f. Terzopoulos et al., 1987; Provot, 1995; Baraff & Witkin, 1998; Bourguignon & Cani, 2000), which approximate surfaces or volumes as a series of triangular elements comprised of nodes, connected by damped springs that dissipate deformational energy during the restoration (Figure 2a). The mass-spring system mimics the response of the rock mass to natural forces and is computationally faster than Finite Element methods. More advanced restorations can be conducted using a modified damped spring system (Figure 2b), within which damped spring orientations do not correspond to the triangle edges but rather, are user-defined and can be used to incorporate anisotropy in mechanical properties (e.g. Bourguignon & Cani 2000). Figure 2: a) All mass spring implementations use a surface comprising a series of nodes (black) connected by damped springs (green), red dashed lines indicate restoration to target surface and the movement of springs (blue arrows); b) More complex implementations use a modified mass spring system where springs (red and green) are aligned in a user defined orientation and cross link triangle edges (grey) separating triangle nodes (black).
Defining and imposing anisotropy in mechanical properties allows a model to account for variations in the mechanical response of the rock mass depending on orientation, or geological response to deformation. For example, for geomechanical restoration of a symmetrical fold, damped spring orientations can be defined that are fold axis parallel and fold axis perpendicular. Defining the stiffness and damping of these springs (for example, stiff, fold-axis parallel damped springs and more flexible fold-axis perpendicular damped springs), will ensure that unfolding will extend the surface perpendicular to the fold-axis (parallel to the inferred shortening direction), as expected from basic geological principles. Conversely, using randomised spring directions for a symmetrical fold may produce artificial fold-axis parallel strains. The deformation processes defined in a restoration, such as unfolding by flattening a surface to a datum, or removing fault displacement by closing fault gaps, introduce forces in the mass-spring system. It is the nature of the mass-spring system to minimise strain by dissipating the deformational energy (forces) within the rock mass. Changes to the length of each damped spring will convert (store) a component of the energy associated with the applied force to elastic potential energy within the spring (Figure 3). Additionally, each damped spring will transmit a component of the applied force to the adjacent damped springs. The component of the force that is converted into elastic potential energy or transmitted, is dependent upon the elastic properties of the rock mass, the Young s Modulus and the Poisson s ratio, which are defined in the Stratigraphy & Rock Properties table. It is the progressive absorption and transmission of the deformational energy through adjacent damped springs that dissipates deformational forces in the rock mass and minimises energy within the mass-spring system. Figure 3: Deformational energy or force is accommodated (dissipated) within a mass spring system by changes to the length of the damped springs; a) Force applied to the surface. The component of the deformational force that is converted into elastic potential energy by changes to the length of the damped springs and b) The component of the deformational force transmitted to adjacent springs, the value varies depending on the elastic properties of the rock mass.
In practice, geomechanical restorations are conducted incrementally over finite time-steps. At every timestep, the force acting on each damped spring within the mesh is calculated based on the distortion of each triangular element resulting from the previous restoration time step. For simple mass-spring systems, the distortion of each triangle corresponds exactly to the distortion of each damped spring, which is situated along the triangle edges. Within a modified mass-spring system, the triangle is distorted based upon the axial and shear forces associated with the change in length of the perpendicular damped springs within the triangle element (Figure 4). Once all forces are balanced the restoration is complete. Axial Forces c) Angular (shear) Forces V U U, V represent spring directions Figure 4: Distortion of the damped springs within a modified mass spring system during restoration (a) produce axial (b) and shear (c) forces on the triangle edges. Method In this section, we outline how strain produced from folding or faulting can be captured incrementally during a restoration. A three-dimensional model of the fault-cored St Corneli Anticline in the southern Spanish Pyrenees (Shackleton et al., 2011) is used as an example (Figure 5).
In situations where surfaces have been deformed by both faulting and folding, the strains should be calculated separately due to variations in strain magnitudes and orientations between these two deformation mechanisms. Separate fracture networks should be generated using the folding and faulting strain maps. In the St. Corneli example, folding was progressively restored (State 1 to 4; Figure 5a) and the heave component of faulting was restored in a final restoration step (State 5). As a result, the folding and faulting strains can be calculated separately within the Strain Capture tool by performing strain calculations between the relevant restoration steps (Figure 5a). In this example, we distinguish between an initial deformed state (State 1; Figure 5a) and several geomechanically restored states (States 2-5, Figure 5a). The geomechanical restoration initially restored the deformation due to folding (States 1-4), and then the strain due to faulting in the final restoration step (State 5). Figure 5: a) Tracking the strain over different geometric states with the Strain Capture tool while geomechanically restoring the St Corneli Anticline; b) Strain captured in a forward sense from geometric state 4 to state 1, displayed on Orcau Vell surface at geometric state 1 and colour mapped for the strain tensor e1 (direction of maximum lengthening); c) Strain captured in a forward sense from geometric state 4 to state 3, displayed on Orcau Vell surface at geometric state 3, colour mapped for the strain tensor e1.
The Geomechanical Modelling module can be accessed from the Advanced Modules sub-section of the Modules panel. Depending on the complexity of the restoration, the checkbox Show workflow tree can be selected, which allows the user to turn on or off parts of the workflow as required. You can find more details on the full workflow in Move2015 tutorials 24 and 27. These tutorials specifically focus on capturing strain associated with folding and faulting separately. Restoring folding The Unfolding sheet allows the user to specify a Datum or a Target Surface to unfold the surface or volume to. In the Template Beds window of the Horizons sheet, the mesh surfaces, GeoCellular volumes, or TetraVolumes to be geomechanically restored can be selected. The mass spring system calculations carried out in the tool will use the Young s Modulus and Poisson s Ratio assigned to the selected horizons in the Stratigraphy & Rock Properties database. Any rock units that will be restored passively with the template bed can be collected into the Passive Beds box. Closing fault gaps To close fault gaps within mesh surfaces, the fault gaps can be defined as cut-off traces when working in the Faults sheet (this option is not available for volumes). Fault cut-offs can be treated as hanging wall or footwall cut-offs (Figure 6). This way the tool will close fault gaps in a geologically meaningful way when restoring the surface geomechanically. Digitization options for cut-offs include manual line picking across the upper and lower cut-off and semi-automatic digitization by loop selection of the fault gap (Figure 6a). The nature of fault closure can be defined by setting a numerical value between 1.0 and 0.0 that determines whether the upper (1.0) or lower (0.0) cut-off is treated as fixed during the restoration. The selection will determine how strain is distributed between the footwall and hanging wall when the fault gap is closed (Figure 6b). The surface can be restored either to a horizontal target, therefore assuming faulting is after folding or alternatively by not unfolding the surface, for example by defining the target as a duplicate copy of the surface to be un-faulted, therefore assuming faulting is pre-folding.
Figure 6: a) The Geomechanical Modelling toolbox (red box) showing the workflow tree and workflow sheets. Fault cut offs are digitized by loop selection and highlighted in green (footwall cut off) and red (hanging wall cut off); b) Closed state of fault after geomechanical restoration of the blue surface. References Baraff, D., Witkin, A., 1998. Large Steps in Cloth Simulation. Proceedings of ACM Siggraph, pp. 43-54. Bourguignon, D., Cani, M.P., 2000. Controlling Anisotropy in Mass-Spring Systems. Daniel Thalmann and Nadia Magnenat-Thalmann and B. Arnaldi. 11th Eurographics Workshop on Computer Animation and Simulation (EGCAS), Aug 2000, Interlaken, Switzerland. Springer-Verlag, pp.113-123 Provot, X., 1995. Deformation Constraints in a Mass-Spring Model to Describe Rigid-Cloth Behavior. Graphics Interface, pp 147-155. Shackleton, J.R., Cooke, M.L., Vergés, J. and Simó. T. 2011, Temporal constraints on fracturing associated with faultrelated folding at Sant Corneli Anticline, Spanish Pyrenees: Journal of Structural Geology, 33, pp 5-19 Terzopoulos, D., Platt, J., Barr, A., Fleischer, K., 1987. Elastically Deformable Models. Proceedings of ACM Siggraph, Computer Graphics Volume 21(4), pp 205-214