Computers & Geosciences 70 (2014) 15–25
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Computers & Geosciences journal homepage: www.elsevier.com/locate/cageo
A new algorithm for evaluating 3D curvature and curvature gradient for improved fracture detection Haibin Di, Dengliang Gao n West Virginia University, Department of Geology and Geography, Morgantown, WV, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 19 February 2014 Received in revised form 3 May 2014 Accepted 5 May 2014 Available online 14 May 2014
In 3D seismic interpretation, both curvature and curvature gradient are useful seismic attributes for structure characterization and fault detection in the subsurface. However, the existing algorithms are computationally intensive and limited by the lateral resolution for steeply-dipping formations. This study presents new and robust volume-based algorithms that evaluate both curvature and curvature gradient attributes more accurately and effectively. The algorithms first instantaneously fit a local surface to seismic data and then compute attributes using the spatial derivatives of the built surface. Specifically, the curvature algorithm constructs a quadratic surface by using a rectangle 9-node grid cell, whereas the curvature gradient algorithm builds a cubic surface by using a diamond 13-node grid cell. A dip-steering approach based on 3D complex seismic trace analysis is implemented to enhance the accuracy of surface construction and to reduce computational time. Applications to two 3D seismic surveys demonstrate the accuracy and efficiency of the new curvature and curvature gradient algorithms for characterizing faults and fractures in fractured reservoirs. Published by Elsevier Ltd.
Keywords: Curvature Curvature gradient Seismic attribute Fracture detection Algorithm
1. Introduction Discontinuity attributes such as seismic coherence have been widely used to visualize and highlight major faults that are discernable from seismic data (Bahorich and Farmer, 1995; Luo et al., 1996; Marfurt et al., 1998; Gersztenkorn and Marfurt, 1999; Marfurt et al., 1999; Cohen and Coifman, 2002); however, discontinuity attributes are basically qualitative and cannot be used to detect faults and fractures that fall below the seismic resolution. With the application of of Gauss curvature in 3D seismic interpretation (Lisle, 1994), curvature attribute has been popular for characterizing fractures from a more physical perspective and in a more quantitative manner (Roberts, 2001; Sigismondi and Soldo, 2003; Al-Dossary and Marfurt, 2006; Sullivan et al., 2006; Blumentritt et al., 2006; Klein et al., 2008; Chopra and Marfurt, 2007a, 2007b, 2010, 2011). Roberts (2001) discussed the applications of different curvature attributes and presented a workflow for measuring curvature based on 3D interpreted horizons. However, horizon-based curvature estimates are very sensitive to the quality of seismic data. Any noise in seismic data adds to the difficulty for an interpreter to accurately and efficiently pick seismic horizons, which increases
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Corresponding author. Tel.: +1 304 293 3310. E-mail address:
[email protected] (D. Gao).
http://dx.doi.org/10.1016/j.cageo.2014.05.003 0098-3004/Published by Elsevier Ltd.
the risk of introducing interpreter bias into curvature analysis. With the development of computer-aided dip-steering algorithms (Marfurt et al., 1998; Marfurt and Kirlin, 2000; Marfurt, 2006; Barnes, 2007), Al-Dossary and Marfurt (2006) improved the process by calculating volumetric curvature. In particular, they applied a running window semblance-based method to calculate the first derivatives of a seismic reflector, and a fractional-order derivative method to compute the second derivatives of the reflector. For horizontal or gently-dipping horizons, the algorithm is close to accurate curvature estimates; however, for steeply-dipping horizons, the algorithm will “undesirably mix geology of different formations” (Al-Dossary and Marfurt, 2006). This limitation results from the fact that the fractional approach calculates the reflector dip change on time slices as an approximation of the desired reflector second derivatives. Curvature gradient (Gao, 2013), defined as a new derivative of curvature, is a different indicator of seismic reflector geometry and compliments curvature attribute for improved fracture characterization. Following a description of the concept, Gao (2013) presented an equation of computing curvature gradient in 2D space. For 3D curvature gradient, he implemented an approximation algorithm, in which two gradients of a curvature cube along inline and crossline directions are combined to evaluate the curvature gradient with reduced computational time. However, this method assumes local linear nature of curvature gradient, which is not accurate in most cases.
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In this paper, we develop new algorithms to compute 3D volumetric curvature and curvature gradient attributes that are analytically accurate and computationally efficient. Our algorithms first construct local surfaces to represent the geometry of 3D seismic reflectors at each sample within a seismic volume. As a second-derivative-related geometric attribute, curvature is measured using a quadratic surface defined by a rectangle 9-node grid cell. As a third-derivative-related geometric attribute, 3D curvature gradient is measured using a cubic surface defined by a diamond 13-node grid cell. Then curvature and curvature gradient are computed using newly-developed equations. Following a description of the new algorithms, they are applied to two 3D seismic surveys of fractured reservoirs from the Stratton field in Texas and from Teapot Dome in Wyoming. Both new algorithms generate attribute cubes that better define potential faults and fractures in fractured reservoirs. Appendices A and B provide a detailed derivation of the analytical equations for computing 3D curvature and 3D curvature gradient attributes along the dip direction, respectively.
2. A new algorithm for 3D curvature Curvature k is evaluated as the ratio of dip angle change with respect to the arc length at a given point on a curve. In 3D space, a seismic volume is often described using an x–y–z coordinate system, in which x-, y- and z-coordinates denote the inline, crossline and depth/time directions, respectively. Using this system, a seismic reflector can be locally fit by a three-dimensional surface, which is often denoted as z ¼ zðx; yÞ. On the surface, numerous types of seismic curvature can be evaluated at each sample. Roberts (2001) presented 12 different curvatures, among which dip curvature is one of the most effective to represent the structural geometry and to detect geological features. Here, we present a detailed mathematical derivation (see Appendix A), which results in Eq. (1) for computing 3D seismic curvature along the dip direction. k3D ¼
1
1
½1 þ A21 þ A22 3=2 ðA21 þ A22 Þ
½A21 B1 þ A22 B2 þ 2A1 A2 B3
Fig. 1. Flowchart of the 3D curvature algorithm. The input is a regular amplitude volume. At each sample, the algorithm calculates the curvature along the dip direction to create a curvature volume.
ð1Þ
dz dz and A2 ¼ dy represent the reflector first derivatives, where A1 ¼ dx or apparent dips, along inline (x-) and crossline (y-) directions, 2 2 2 d z d z d z respectively; B1 ¼ dx 2 , B2 ¼ dy2 and B3 ¼ dxdy represent the reflector second derivatives, or derivatives of vector dip. Based on Eq. (1), we develop a new algorithm which computes 3D volumetric curvature accurately and efficiently. The curvature algorithm consists of three steps (Fig. 1). First, at a given sample in a seismic volume, a local quadratic 9-node surface is constructed to represent the 3D geometry of the seismic reflector at that point, based on volumetric estimates of reflector apparent dips. Then, the new algorithm calculates the derivative terms required in Eq. 1, including both the first and second derivatives of the quadratic surface. Finally, 3D curvature at the target sample is estimated by substituting Eq. (1) with these derivatives. The above steps are repeatedly executed from one sample to another. Consequently, a regular amplitude volume is transformed into a volume of seismic curvature. In our new curvature algorithm, automatic construction of a quadratic surface is the key to accurate volumetric curvature estimates. Curvature is a second order geometric attribute, and thus the built surface should be quadratic to calculate the reflector second derivatives. As shown in Fig. 2, to build such a quadratic surface that represents the reflector geometry centered at sample A, the algorithm uses a rectangle 9-node grid cell (Roberts, 2001). Fig. 3 illustrates the workflow for automatically building a quadratic surface within the 9-node grid cell shown in Fig. 2.
Fig. 2. The rectangle 9-node grid cell for constructing a quadratic surface to represent the 3D geometry of the local seismic reflector centered at sample A.
Specifically, construction of the quadratic surface is achieved by extending the corresponding reflector from the target trace to its neighboring traces. At the given sample A, backward and forward apparent dips are considered for locating the reflector at its neighboring traces. Fox example, to find sample B that is on the
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17
Fig. 3. Flowcharf of constructing a quadratic surface using a rectangle 9-point grid cell (shown in Fig. 2). A total of 12 estimates of reflector apparent dips are required.
same reflector as A, the apparent dip at sample A toward the previous trace along the inline (x-) direction, here denoted as the inline backward apparent dip θxB , is measured; similarly, the inline forward apparent dip θxF is used to locate sample C (Fig. 4). The use of backward and forward apparent dips guarantees that the curve linking three samples A, B and C represents the target reflector well. Consequently, in 3D space, the eight neighboring samples, B, C, D, E, G, H, J and K, can be located, all of which lie on the same reflector as sample A, and linking all the nine samples leads to the desired 3D quadratic surface that well represents local reflector geometry, which can be illustrated by the following equation (modified from Roberts (2001)). z¼
1 1 B1 x2 þ B2 y2 þB3 xy þ A1 x þ A2 y 2 2
ð2Þ
The workflow illustrated in Fig. 3 indicates that the efficiency of our algorithm greatly depends on volumetric dip estimates. There are several approaches for measuring reflector dips, including discrete scanning (Finn, 1986; Marfurt et al., 1998; Marfurt and Kirlin, 2000; Marfurt, 2006), plane-wave destructor (Fomel, 2002), and 3D complex seismic trace analysis (Taner et al., 1979; Scheuer and Oldenburg, 1988; Barnes, 2007). Among these, 3D complex seismic trace analysis is most computationally efficient and convenient for implementation. Generally, this method uses a 3D generalization of instantaneous frequency, and reflector dip is rapidly evaluated as the negative ratio of spatial frequency and instantaneous frequency (Barnes, 2007). tan θ ¼
fx fz
ð3Þ
where f x denotes the spatial frequency and f z denotes the instantaneous frequency. The new curvature algorithm implements the method of 3D complex seismic trace analysis into measuring the required 12 apparent dips (shown in Fig. 3). Additionally, to avoid numeric unstability, we average a set of spatial frequencies and instantaneous frequencies within a vertical window. For example, the
Fig. 4. Schematic diagram of forward and backward apparent dips along the inline (x-) direction at a sample location: θxB is the backward apparent dip from sample A to sample B, and θxF is the forward apparent dip from sample A to sample C.
inline backward and forward apparent dips θxB and θxF at sample A(x,y,z) (shown in Fig. 4) are evaluated as tan θxB ¼
f xB fz
ð4aÞ
tan θxF ¼
f xF fz
ð4bÞ
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where fz ¼
1 N 1 Zðx; y; z þiΔzÞ Zðx; y; zÞ þ arg arg ∑ N i ¼ 0 4π iΔz Z ðx; y; zÞ Zðx; y; z iΔzÞ N 1 Zðx þ Δx; y; z iΔzÞ Zðx þ Δx; y; zÞ þ arg ∑ arg Zðx; y; zÞ 12π N Δx i ¼ 0 Zðx; y; z iΔzÞ Zðx þ Δx; y; z þ iΔzÞ þ arg Zðx; y; z þ iΔzÞ
f xF ¼
N 1 Zðx Δx; y; z iΔzÞ ∑ arg 12π N Δx i ¼ 0 Zðx; y; z iΔzÞ Zðx Δx; y; zÞ Zðx Δx; y; z þ iΔzÞ þ arg þ arg Zðx; y; zÞ Zðx; y; z þ iΔzÞ
f xB ¼
in which Zðx; y; zÞ denotes the complex or analytic traces at sample A; argðÞ denotes the argument of a complex number; Δz denotes sampling interval; Δx denotes the spatial interval along inline direction; and 2N þ 1 denotes the vertical analysis window size.Next, our algorithm computes the quadratic coefficients of the surface equation (Eq. 2) as the desired reflector second derivatives used in Eq. (1). Using the rectangle 9-node grid cell shown in Fig. 2, three sets of the reflector first and second derivatives can be calculated, and averaging these three sets by Eqs. (5a and 5b) and (6a–6c) leads to a stable estimate of all the five derivative terms used in Eq. 1. Our test comparison indicates that not only the results are more accurate and noise resistant, but also the process is computationally more efficient. z z z z z z dz G B K H J C A1 ¼ ¼ ave ; ; ð5aÞ dx 2 Δx 2 Δx 2 Δx dz zE zD zK zJ zH zG ¼ ave ; ; ð5bÞ A2 ¼ dy 2 Δy 2 Δy 2 Δy B1 ¼
2 d z zC þ zB 2zA zK þ zH 2zE zJ þ zG 2zD ¼ ave ; ; 2 2 2 2 dx Δx Δx Δx
ð6aÞ
B2 ¼
2 d z zE þ zD 2zA zK þ zJ 2zC zH þ zG 2zB ¼ ave ; ; dy2 Δy2 Δy2 Δy2
ð6bÞ
B3 ¼
zK zH zJ þ zG d z ¼ dxdy 4ΔxΔy
2
ð6cÞ
where zi ði ¼ A–KÞ refers to the values at 9 nodes in the grid cell. Δx and Δy are the spatial intervals along inline (x-) and crossline (y-) directions, respectively. aveðÞ denotes an averaging operator. Roberts (2001) presented similar equations-based on 3D picked horizons.
3. A new algorithm for 3D curvature gradient ' Curvature gradient k (Gao, 2013), is defined as the spatial derivative of curvature with respect to the arc length. Here, we develop what we feel to be the first accurate and applicable equation for computing 3D curvature gradient along the dip direction (See Appendix B). k3D' ¼ 3 þ
1
1
ð1 þ A1 2 þ A2 2 Þ3 ðA1 2 þ A2 2 Þ3=2 1
1
ð1þ A1 2 þ A2 2 Þ2 ðA1 2 þ A2 2 Þ3=2
ðA1 2 B1 þ A2 2 B2 þ 2A1 A2 B3 Þ2
Fig. 5. Flowchart of the 3D curvature gradient algorithm. The input is a regular amplitude volume. At each sample, the algorithm calculates the curvature gradient along the dip direction to create a curvature gradient volume.
curvature gradient algorithm consists of three steps (Fig. 5). First, at a given sample in a seismic volume, the algorithm constructs a cubic surface to represent the local geometry of seismic reflectors. Then, it computes not only the first and second derivatives but also the third derivatives of the cubic surface. Finally, 3D curvature gradient is evaluated by substituting Eq. (7) with these derivatives. The above steps are repeatedly executed from one sample to another. Consequently, a seismic amplitude volume is transformed into a volume of seismic curvature gradient. By comparing the curvature-gradient equation (Eq. 7) to the curvature equation (Eq. 1), we notice that the algorithm for 3D curvature gradient would be more complicated than that for 3D curvature. Curvature gradient is related to not only the first and second derivatives, but also to the third derivatives of a seismic reflector. Thus, the computation of curvature gradient needs a cubic surface, instead of a quadratic one used in the curvature algorithm, because curvature gradient is related to the reflector third derivatives while a quadratic surface is only accurate enough to evaluated the second derivatives. Consequently, a more advanced grid cell than the rectangle one (shown in Fig. 2) should be used for constructing this cubic surface. Fig. 6 demonstrates a diamond grid cell with 13 nodes. Fig. 7 illustrates the workflow of building the cubic surface (Eq. 8) within this diamond grid cell, which requires a total of 16 estimates of apparent dips. 1 1 1 1 1 1 z ¼ C 1 x3 þ C 2 y3 þ C 3 x2 y þ C 4 xy2 þ B1 x2 þ B2 y2 þ B3 xy þ A1 x þ A2 y 6 6 2 2 2 2
½A1 3 C 1 þA2 3 C 2 þ 3A1 2 A2 C 3 þ3A1 A2 2 C 4
ð7Þ
ð8Þ
where d3 z d3 z d3 z d3 z C 1 ¼ dx 3 , C 2 ¼ dy3 , C 3 ¼ dx2 dy, and C 4 ¼ dxdy2 represent the reflector third derivatives. Based on Eq. (7), we develop a new algorithm to compute 3D volumetric curvature gradient accurately and efficiently. The
To facilitate the construction of a cubic surface, the curvature gradient algorithm also implements 3D complex seismic trace analysis for volumetric dip estimates. Next, using the diamond grid cell of 13 nodes (Fig. 6), the algorithm computes two linear coefficients of the cubic surface
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equation as the first derivatives (Eqs. 9a and 9b), three quadratic coefficients as the second derivatives (Eqs. 10a–10c), and four cubic coefficients as the third derivatives (Eqs. 11a–11d) z z z z z z z z dz G N B K H J M C ; ; ; A1 ¼ ¼ ave dx 2Δx 2Δx 2Δx 4Δx dz zE zD zK zJ zH zG zQ zP A2 ¼ ¼ ave ; ; ; dy 2y 2y 2y 4y
B1 ¼
2
B3 ¼
zK zH zJ þ zG d z ¼ dxdy 4 Δx Δy
C1 ¼
d z zN zM 2zC þ 2zB ¼ 2Δx3 dx3
C2 ¼
d z zQ þ zP 2zE þ2zD ¼ dy3 2Δy3
ð11bÞ
C3 ¼
3 zK zJ þ zH zG 2zE þ 2zD d z ¼ 2 Δx 2 Δy dx dy
ð11cÞ
C4 ¼
zK þzJ zH zG 2zC þ 2zB d z ¼ dxdy2 ΔxΔy2
ð10cÞ
3
ð9aÞ
ð11aÞ
3
ð9bÞ
2 d z zC þ zB 2zA zK þ zH 2zE zJ þ zG 2zD zN þ zM 2zA ¼ ave ; ; ; 2 2 2 2 2 dx Δx Δx Δx 4Δx
ð10aÞ
B2 ¼
19
2 d z zE þ zD 2zA zK þ zJ 2zC zH þ zG 2zB zQ þ zP 2zA ¼ ave ; ; ; Δy2 Δy2 Δy 2 4Δy2 dy2
ð10bÞ
Fig. 6. The diamond 13-node grid cell for constructing a cubic surface to represent the 3D geometry of the local seismic reflector centered at sample A.
3
ð11dÞ
where zi ði ¼ A–Q Þ refers to values at 13 nodes in the grid cell. 4. Results To verify the value of the new curvature and curvature gradient algorithms, we calculate both attributes for two 3D seismic datasets, one being time data from the Stratton field in Texas and the other being depth data from Teapot Dome in Wyoming. In the Stratton data, the reservoir structure is dominated by a north-trending listric fault and associated rollover anticlines as well as fracture systems subparallel to the fold hinge. As a baseline, the structure contour map of the horizon at approximately 1850 ms is shown in Fig. 8a, indicating that the formation gradually dips from the eastern portion towards the west. In order to highlight the potential faults and fractures resulting from the roll-over bending, the seismic volume was processed using the proposed curvature and curvature gradient algorithms. For the convenience of result visualization and comparison, both attributes are displayed on the same seismic reflector (Fig. 8). In the attribute maps, four major north-trending faults are clearly depicted from the steeply-dipping horizon (denoted by dotted lines), verifying the accuracy of our algorithms on highlighting faults and fractures. Specifically, faults are delimited by the juxtaposition of positive curvature and negative curvature (Fig. 8b), which are directly delineated by curvature gradient (Fig. 8c). Perspective chair displays of curvature and curvature gradient images, along with a seismic line, help better illustrate
Fig. 7. Flowchart of constructing a cubic surface using a diamond 13-point grid cell (shown in Fig. 6). A total of 16 estimates of reflector apparent dips are required.
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Fig. 8. Application of the new algorithms to the 3D seismic volume over the Stratton field in Texas. (a) Structure contour map of the horizon at approximately 1850 ms, gradually dipping from the eastern area toward the west. (b) The corresponding curvature image generated from the proposed curvature algorithm. (c) The correpsonding curvature gradient image generated from the proposed curvature gradient algorithm. Four north-trending faults (denoted by dotted lines) are highlighted to demonstrate their spatial relationship to curvature and curvature gradient.
the expressions of faults and fractures by the two attributes (Fig. 9). Positive and negative curvature highlight the upthrown and downthrown fault blocks, respectively (Fig. 9b), whereas the fault planes are directly highlighted by curvature gradient (Fig. 9c). Here, integrating curvature with curvature gradient helps differentiate structural features of reservoir formations, which is instrumental in fractured reservoir characterization. In the second example, we use Kirchhoff prestack depthmigrated data in Teapot Dome computed by Aktepe (2006). Fig. 10a displays the structure contour map of the horizon at approximately 4600 ft, which depicts the northwest-trending anticline and associated northeast-trending cross-regional transfer faults. Fig. 10b and c displays the corresponding curvature and curvature gradient images, respectively. Compared to curvature, curvature gradient helps characterize the subtle fractures that are not easily discernable from curvature attribute (Fig. 10c). At a deeper horizon at of approximately 6000 ft in the Tensleep reservoir interval, curvature gradient better defines two sets of lineaments (Fig. 11c). One set trends to the northwest and is subparallel to the regional folds that have been well documented in previous studies (Cooper et al., 2006). The other set trends obliquely to the hinge of the fold, which have been also confirmed by outcrop studies and image log analysis (Sterns and Friedman, 1972; Cooper et al., 2006; Schwartz, 2006).
Fig. 9. Perspective chair display of (a) structure contour, (b) curvature attribute, and (c) curvature gradient attribute along with a seismic line. White lines indicate three major north-trending faults interpreted in Fig. 8.
Then we test and compare the results and the computational efficiency of our algorithms with the traditional algorithms, using the 3D Stratton data which contains 100 inlines, 200 crosslines, and 1500 samples per trace with an sampling interval of 2 ms. As the first test, two different methods for dip estimates, 3D complex seismic trace analysis and discrete scanning, are implemented to construct the quadratic surface used in the new curvature algorithm and the cubic surface used in the new curvature gradient algorithm. Both results for curvature and curvature gradient are shown in Figs. 12 and 13, respectively. As demonstrated, the results are very similar, but the computational time of discrete scanning is 10 times more than that of 3D complex seismic trace analysis (Table 1). Next, the curvature cube from our algorithm is compared to that from the traditional curvature algorithm by Al-Dossary and Marfurt (2006). As shown in Fig. 14, curvature estimates from both algorithms are similar for the eastern area where the reflector is horizontal, whereas our algorithm provides a better delineation in the west where the reflector dips steeply. Finally, Fig. 15 displays the comparison of curvature gradient from our new algorithm and the approximation algorithm by Gao (2013), and more accurate estimates of curvature gradient are produced by the new algorithm as denoted by arrows.
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Fig. 10. Application of the new algorithms to the 3D seismic volume over Teapot Dome in Wyoming. (a) Structure contour map of a deformed horizon at approximately 4600 ft, demonstrating a northwest-trending anticline and associated faults perpendicular to the fold hinge. (b) The corresponding curvature image generated from the proposed curvature algorithm. (c) The correpsonding curvature gradient image generated from the proposed curvature gradient algorithm. More structural details are revealed by curvature gradient.
5. Discussion The success of our algorithms depends mainly on the accuracy of surface construction. At each sample in a seismic volume, building a 9-node quadratic and 13-node cubic surface requires a total of 12 and 16 dip estimates, respectively. Rapid and accurate volume-based evaluation of reflector dip plays a critical role in enhancing both the accuracy and computational efficiency of the new curvature and curvature gradient algorithms. Compared to the discrete scanning dip estimate (Finn, 1986; Marfurt et al., 1998; Marfurt and Kirlin, 2000; Marfurt, 2006) and plane-wave destruction filter (Fomel, 2002), 3D complex seismic trace analysis is computationally more efficient and convenient for implementation (Taner et al., 1979; Scheuer and Oldenburg, 1988; Barnes, 2007). However, numeric unstability is a major concern and should be addressed when applying this approach. More work is expected for the developing efficient dip-steering method, based on which the accuracy of automatic surface construction can be further improved and thereby our algorithms can produce even better estimates of seismic curvature and curvature gradient attributes. Curvature and curvature gradient attributes are both dependent on the measuring direction on a surface in 3D space. The focus of our algorithm description lies on dip curvature and dip curvature gradient, which represent the attributes measured along the true dip direction, one of the important direction for subsurface structural interpretation. Nevertheless, the new algorithms can be easily extended to other important directions, such as strike direction and two principle directions, by developing the corresponding applicable equations. In the new algorithms, the rectangle 9-node grid cell and the diamond 13-node grid cell are the simplest but eligible cell for
constructing a quadratic surface and a cubic surface, respectively. A cell with less than 9 nodes cannot provide the required information about the reflector second derivatives to compute curvature attribute, and a cell with less than 13 nodes cannot provide the required information about the reflector third derivatives to compute curvature gradient attribute. The algorithms would become more robust if a larger and more complicated grid cell, for example a rectangle 25-node cell, is used. However, surface construction at every sample using the 25-node grid cell needs a total of 40 estimates of reflector apparent dip, which would lead to a fourfold increase in computation time.
6. Conclusion We have developed new and efficient algorithms for robust estimate of 3D volumetric curvature and curvature gradient attributes. Both algorithms consist of two steps: (1) to automatically construct a local surface representing the geometry of 3D seismic reflectors, and (2) to compute attributes using the coefficients of the built surface equation. The new 3D curvature algorithm constructs a quadratic surface using a 9-node grid cell, whereas the new curvature gradient algorithm constructs a cubic surface using a more complicated 13-node grid cell. Dip estimate is accomplished based on the method of 3D complex seismic trace analysis, yet averaging instantaneous and spatial frequency within a vertical analysis window is applied to keep computation stability. The major advantages of our algorithms over the existing ones are the enhanced accuracy and efficiency of delineating faults and fractures in 3D space. Applications of the new algorithms to both
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Fig. 11. (a) Structure contour map of the Tensleep Formation in the Teapot Dome survey. (b) The corresponding curvature image generated from the proposed curvature algorithm. (c) The corresponding curvature gradient image generated from the proposed curvature gradient algorithm. Two sets of lineaments are recognizable by curvature gradient.
Fig. 12. Comparison of the new curvature algorithm using two different dipsteering methods for quadratic surface construction. (a) 3D complex seismic trace analysis. (b) Discrete scanning.
Fig. 13. Comparison of the new curvature gradient algorithm using two different dip-steering methods for cubic surface construction. (a) 3D complex seismic trace analysis. (b) Discrete scanning.
time and depth data from two 3D seismic surveys over the Stratton field in Texas and Teapot Dome in Wyoming indicate that both new algorithms help better define and detect faults and fractures in an analytically more accurate and computationally more efficient manner.
Disclaimer This project was funded by the Department of Energy, National Energy Technology Laboratory, an agency of the United States Government, through a support contract with URS Energy & Construction,
H. Di, D. Gao / Computers & Geosciences 70 (2014) 15–25
Table 1 Comparison of computational time between 3D complex seismic trace analysis and discrete scanning. Algorithm
3D Complex seismic trace analysis
Discrete scanning
Curvature Curvature gradient
5 min 47 s 8 min 17 s
53 min 35 s 79 min 26 s
23
assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
Acknowledgments This study has been funded by the US Department of Energy/ NETL under the Contract RES1000023/217U to Dengliang Gao and by URS 2013 Outstanding NETL-RUA Research Award to Dengliang Gao (400.OUTSTANDIRD). Thanks go to two anonymous peer reviewers for their positive comments and constructive suggestions that helped improve the quality of the paper. Thanks also go to K. Marfurt for his offer of newly processed prestack depthmigrated seismic data over Teapot Dome in Wyoming. The Kirchhoff prestack depth migration was computed by S. Aktepe at the University of Houston. This paper is a contribution to the West Virginia University Advanced Energy Initiative (AEI) program.
Appendix A. Derivation of equations for computing 3D curvature along the dip direction
Fig. 14. Comparison of (a) the new curvature algorithm and (b) the traditional curvature algorithm (Al-Dossary and Marfurt, 2006). Both algorithms provide similar results in the eastern area where the reflector is horizontal, whereas (a) shows a better expression for the west where the reflector dips steeply.
In 3D space, curvature along the dip direction k3D , is defined as the derivative of dip angle with respect to arc length along the dip direction on a surface (Roberts, 2001; Sigismondi and Soldo, 2003). k3D ¼
dθ ds
ðA:1Þ
where θ denotes the reflector dip and s denotes the arc length along the dip direction. In the x-y-z coordinate system, Eq. (A.1) becomes applicable using the chain rule of derivative k3D ¼
∂θ dx ∂θ dy þ ∂x ds ∂y ds
ðA:2Þ
Let a surface be describe using a function z ¼ zðx; yÞ. Then, as shown in Fig. A.1, dip angle θ is represented by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dz dz 1 θ ¼ tan þ ðA:3Þ dx dy dz dz and dy denote the apparent dips along x and y directions, where dx respectively (modified from Marfurt (2006) and Gao (2013)). By taking a derivative of Eq. (A.3) with respect to x and y, ∂∂xθ and ∂θ are evaluated as ∂y 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 ∂θ ∂4 dz dz 5 1 tan ¼ þ dx dy ∂x ∂x
Fig. 15. Comparison of (a) the new curvature gradient algorithm and (b) the traditonal curvature gradient algorithm (Gao, 2013). Both results are generally similar, but different in details (denoted by arrows).
Inc. Neither the United States Government nor any agency thereof, nor any of their employees, nor URS Energy & Construction, Inc., nor any of their employees, makes any warranty, expressed or implied, or
1
2
2
1 dz d z dz d z ffi ¼ þ dz 2 dz2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dz 2 dx dx2 dy dxdy dz 2 þ dy 1 þ dx þ dx dy 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 ∂θ ∂4 dz dz 5 1 tan ¼ þ dx dy ∂y ∂y
! ðA:4aÞ
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and in order to simplify the computation of partial derivatives, the equation of 3D curvature (Eq. A.7) can be written as k3D ¼
" # 2 2 2 1 1 d z d z d z 2 þ tan ð φ Þ þ 2 φ Þ tan ð dxdy dz 2 dz 2 3=2 dx2 dy2 1 þ tan 2 ðφÞ þ dy 1 þ dx
ðB:3Þ dz ; where φ ¼ atan2ðdy
A¼
¼
2
1
2
1 dz d z dz d z ffi þ dz 2 dz 2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dy dy2 dx dxdy
dz 2 dz 1 þ dx þ dy þ dx dy
! ðA:4bÞ
dx ¼ ds cos θ cos φ
ðA:5aÞ
dy ¼ ds cos θ sin φ
ðA:5bÞ
where θ denotes true dip and φ denotes dip azimuth. Rearranging Eqs. (A.5a and A.5b) leads to
dy dz ¼ ds dy
"
2 2 # 1=2 " 2 2 # 1=2 dz dz dz dz þ 1þ þ dx dy dx dy
ðB:4bÞ
2
2
d z d z d z tan ðφÞ þ tan 2 ðφÞ þ2 dxdy dx2 dy2
ðB:4cÞ
By the derivative theory, Eq. (B.2) is represented as " ! ! # ∂B 3=2 ∂B 3=2 0 3=2 ∂C dx 3=2 ∂C dy þ k3D ¼ A C þB C þB ∂x ds ∂y ds ∂x ∂y
ðB:5Þ
Substituting Eq. (B.5) with Eqs. (A.6a and A.6b) and Eqs. (B.4a– B.4c) leads to ∂B 3=2 dx A C ¼ 3B 3 ∂x ds
Geometric knowledge provides
" # 1=2 " 2 2 # 1=2 2 2 dx dz dz dz dz dz ¼ þ 1þ þ ds dx dx dy dx dy
ðB:4aÞ
2 2 dz dz þ dx dy
2
Fig. A.1. Schematic diagram of defining reflector dip: θx is apparent dip in the x-direction, θy is apparent dip in the y-direction, θ is reflector dip, and φ is dip azimuth (modified from Marfurt (2006)).
denotes dip azimuth. Then let
1 1 þ tan 2 ðφÞ
B ¼ 1þ
C¼
dz Þ dx
"
dz dx
2
þ
# 2 # 3=2 " 2 2 2 dz dz d z dz dz d z C þ 2 dy dx dx dx dy dxdy
ðB:6aÞ ∂B 3=2 dy A C ¼ 3B 3 ∂x ds
"
dz dx
2
þ
# 2 # 3=2 " 2 2 2 dz dz d z dz dz d z C þ 2 dy dy dy dx dy dxdy
ðB:6bÞ and ðA:6aÞ
AB 3=2
∂C dx ¼ B2 ∂x ds
"
"
ðA:6bÞ
dz dx
2
þ
2 # 3=2 dz dy
3 3 2 2 3 # 3 dz d z dz dz d z dz dz d z þ þ2 dx dx3 dx dy dx2 dy dx dy dxdy2 ðB:7aÞ
Substituting Eq. (A.2) with Eqs. (A.4a and A.4b) and Eqs. (A.6a and A.6b) leads to k3D ¼
1
1
½1 þ A21 þ A22 3=2
ðA21 þ A22 Þ
AB ½A21 B1 þ A22 B2 þ 2A1 A2 B3
ðA:7Þ
3=2 ∂C dy
∂y ds
¼B
2
"
2 2 # 3=2 dz dz þ dx dy
"
dz dy
3
# 2 3 2 3 3 d z dz dz d z dz dz d z þ 2 þ dx dy dxdy2 dx dy dx2 dy dy3 ðB:7bÞ
Appendix B. Derivation of equations for computing 3D curvature gradient along the dip direction
finally, 3D curvature gradient along the dip direction is evaluated as
In 3D space, curvature gradient along the dip direction k3D' , is defined as the derivative of curvature with respect to arc length along the dip direction on a surface (Gao, 2013). 0 k3D
dk3D ¼ ds
þ ðB:1Þ
where k3D denotes 3D curvature along the dip direction (Eq. A.7) and s denotes the arc length along the dip direction. In the x-y-z coordinate system, Eq. (B.1) becomes applicable using the chain rule of derivative 0
k3D ¼
∂k3D dx ∂k3D dy þ ∂x ds ∂y ds
ðB:2Þ
1
0
k3D ¼ 3
1
ð1 þ A1 2 þ A2 2 Þ3 ðA1 2 þ A2 2 Þ3=2 1
1
ð1 þ A1 2 þ A2 2 Þ2 ðA1 2 þA2 2 Þ3=2
ðA1 2 B1 þ A2 2 B2 þ 2A1 A2 B3 Þ2
½A1 3 C 1 þ A2 3 C 2 þ 3A1 2 A2 C 3
þ 3A1 A2 2 C 4
ðB:8Þ
Appendix C. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.cageo.2014.05.003.
H. Di, D. Gao / Computers & Geosciences 70 (2014) 15–25
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