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.
a) b)
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. www.mve.com
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).
b)
a) 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). www.mve.com
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.
a) b)
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. www.mve.com
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.
a) b)
c)
Axial Forces
Angular (shear) Forces
V
U, V represent spring directions
U
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).
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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.
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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.
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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 www.mve.com