Curvature. Definition of curvature in 2D and 3D

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1 Curvature Deformation associated with folding and faulting alters the location and shape of geological surfaces. In Move, the strain caused by deformation can be quantified using restoration algorithms and strain tracking available in the 3D Kinematic Modelling module. An alternative approach involves analysing the shape of the deformed geometry, which may be used as a proxy for strain in cases where computation of the full strain tensor is not required. The Surface Geometry tool in Move allows for quick analysis of curvature on mesh surfaces and grids. The quantification of a surface geometry as a proxy for strain is well documented in geological and materials science literature. One of the measures commonly used is curvature, which has a real physical meaning and can be used for objective comparison between objects. In a geological setting, curvature is often used for firstorder estimations of fracture orientations (e.g. Fischer and Wilkerson 2000). The basic assumption is that rock layers deform as elastic plates, so that layer-parallel strain is directly related to the curvature of the surface. Pollard and Fletcher (2005) provide a summary and description of how curvature can be analysed and interpreted in geological applications. Definition of curvature in 2D and 3D Figure 1 illustrates 2D curvature on a line at points A-B-C and D-E-F. Normals (black dashed lines) are drawn perpendicular to the connector lines (red lines) between points A-B and B-C, and D-E and E-F (blue dots). The intersection of these normals corresponds to the centres of circles, so-called osculating or kissing circles (in green), that pass through the points A-B-C and D-E-F. Figure 1 Illustration to explain curvature in 2D (after Martel 2015).

2 The curvature kk is the curvature of the osculating circle and is defined to be the reciprocal of the radius. It is calculated by the equation: kk = 1/rr Where rr is the radius of the osculating circle (R and Q in Figure 1). Due to the inverse relationship with radius, larger k values correspond to smaller circles and tighter curves, indicating a greater change in the slope of a curved line over a given length. The definition of curvature in 3D is analogous to that in 2D, although in this case, two separate values of curvature - the principal curvatures Kmax and Kmin - are needed to fully describe 3D curvature. By definition, Kmax and Kmin are orthogonal and represent the largest and smallest circles respectively, which can be fitted through neighbouring vertices of a mesh or grid. At any given location, the two orthogonal 2D planes that contain the Kmax and Kmin circles represent the two curvature directions. Pre-processing surfaces for curvature analysis Before analysing the curvature of a surface, it is important to ensure that analysed surface undulations represent geological features rather than artefacts from model building. It is therefore recommended that surfaces are pre-processed using the Smoothing tool to remove artefacts such as spikes created by auto tracking of seismic reflections or undulations due to surface creation algorithms, because noise introduced by model building may conceal the geologically induced curvature. Figure 2 shows the Smoothing tool in Move. Figure 2 The Smoothing tool user interface.

3 Performing curvature analysis in Move The Surface Geometry tool in Move allows analysis of surface properties such as dip, dip azimuth, area, thickness, cylindricity and curvature. The tool is accessed from the Data and Analysis panel and provides a number of different options depending on the method chosen. For curvature analysis, surfaces are added to the Object Collection box, prompting Move to calculate best-fit default grid parameters for the collected objects. Changing the size/cells of the grid will alter the resolution of structures in the resulting analysis. Figure 3 Surface Geometry (Curvature) tool user interface.

4 With the appropriate surface(s) collected, there are seven curvature options. These can be seen below the collection box in Figure 3. The seven options are summarised below: Simple Measure of the rate of change of dip across the surface. For each face, it measures the divergence from the average dip of the chosen triangle and its surrounding 3 triangles. Gaussian (Angle Deficit) Calculates on the edge vertices of triangles. This is the measure of the overlap when the faces surrounding the point are flattened. This will give a result similar to the Ktotal option using the grid but provides only a value and no resultant curvature directions. Kmax The direction of maximum curvature. The colour map shown on the surface shows the curvature value along the direction of the red vector. Kmin The direction of minimum curvature. The colour map shown on the surface shows the curvature value along the direction of the blue vector. Ktotal Defined as Kmax x Kmin, Ktotal is known as Gaussian Curvature but differs from the Gaussian (Angle Deficit) method of the Surface Analysis tool. It provides both curvature intensities as well as Kmax and Kmin orientations. Ktotal describes how developable a surface is, a mathematical term for whether a surface can be flattened into a plane without overlap or separation of faces on the surface. A Ktotal of zero describes a surface that is either flat or cylindrically folded (i.e. curved along a single direction at that point so that Kmin = 0). If Ktotal is not zero, deformation was either polyphase or noncylindrical, in which case Ktotal may be a useful proxy for the non-cylindrical component of strain (Lisle, 1994). Kmean This is defined as 0.5 x (Kmax+Kmin) and is also referred to as mean curvature. Kmean is a more appropriate proxy for strain than Ktotal in areas of cylindrical or weakly non-cylindrical folding as it is less sensitive to small Kmin values. Kgeological This method assigns a set of predefined numbers to the surface that correspond to different fold shapes as defined in (Pollard and Fletcher 2005). The numbers are outlined below: Basin (3) Synform (2) Synformal Saddle (0.75) Plane (0.20) Antiformal Saddle (-0.5) Antiform (-1.5) Dome (-2.5) Saddle (-4) This method can be used to objectively identify and partition geological structures.

5 Simple and Gaussian (Angle Deficit) options run directly with a mesh surface whereas the other five options (Kmax, Kmin, Ktotal, Kmean and Kgeological) appended by Using grid in their title require the user to define a grid before running. The five Using grid options are interrelated and when Display Principal Curvature Directions is toggled on in the tool, the Kmax and Kmin directions are visualized as red and blue vectors respectively (Figure 4). Figure 4- Diagram illustrating the relationships between folding and joints (after Ramsay and Huber 1987). Both conjugate fractures and those orthogonal to bedding/curvature are shown. Kmax direction shown in red and Kmin in blue. Curvature as a proxy for fractures The Fracture Modelling module in Move uses strain to model fractures. As curvature can be viewed as a proxy for strain intensity, curvature data may be used as an input for fracture modelling without the need to model the geological evolution in detail. The curvature parameters required for fracture modelling include curvature intensity as well as plane orientations with normals that contain the maximum (Kmax) and minimum (Kmin) curvature directions, which can be saved as mesh attributes by toggling on Create all Grid Curvature Attributes during attribute creation. From the dip and dip azimuth of the planes that contain Kmax and Kmin, fractures can be created and visualized using the Create Fracture Set window (Figure 5). This is accessed via the Create Fractures button in the Vertex Attribute Analyser, located in the Data and Analysis panel. Figure 6 shows an example of curvature-related fracture orientations displayed in Move. If calculated fracture orientations match observed fractures that are assumed to have formed due to folding, the approach of using curvature as a proxy for strain-based fractures is likely valid. In this case, a full curvature-based fracture model can be run to predict fractures in areas of higher uncertainty using Move s Fracture Modelling module.

6 Figure 5 Create Fracture Set window for creating fractures from curvature directions. The above calculations are based solely on the geometry of the analysed surface and not the strain history, so it is often used as a first-order evaluation to test if observed fracture orientations are related to fold geometries. The relationship between fold curvature directions and fracture orientations has been well documented in the work of Ramsay and Huber (1987; Figure 4), Stearns and Friedman (1972) and Cooper (1992). Figure 6 Surface of the St Corneli Anticline, Spain, colour mapped for maximum curvature. Fracture planes (red) have been created orthogonal to the maximum and minimum curvature directions. Modelled fracture orientations can be compared to observed fractures that are assumed to be related to regional folding to test if the use of curvature-based fractures is appropriate.

7 References Cooper, M., 1992, The analysis of fracture systems in subsurface thrust structures from the Foothills of the Canadian Rockies: Thrust tectonics, pp Fischer, M.P. and Wilkerson, M.S., 2000, Predicting the orientation of joints from fold shape: Results of pseudo three-dimensional modeling and curvature analysis. Geology, 28(1), pp Lisle, R.J., 1994, Detection of zones of abnormal strains in structures using Gaussian curvature analysis: AAPG Bulletin, 78(12), pp Martel, S., 2015, Structural Geology, lecture notes distributed in Department of Geology and Geophysics at The University of Hawaii at Manoa, August Pollard, D. D. and Fletcher, R.C., 2005, Fundamentals of structural geology, Cambridge University Press. Ramsay, J. G. and M. I. Huber, 1987, The Techniques of Modern Structural Geology. Volume 2: Folds and Fractures. New York, Academic Press. Stearns, D. W. and M. Friedman, 1972, Reservoirs in fractured rock: Geologic exploration methods in R. E. King, ed., Stratigraphic oil and gas fields: AAPG Memoir 16, p If you require any more information about calculating curvature in Move, then please contact us by or call: +44 (0)