Coulomb Collapse Theory

A theory of rollover formation based on Coulomb failure, or breakup (Suppe 1985) of the hanging wall onto the footwall block, has been advanced by Xiao and Suppe (1988, 1992), Groshong (1989), and others. According to this theory, rocks fail along Coulomb fracture surfaces oriented at about 20 deg to 30 deg to the maximum principal stress (σ₁) (Billings 1972), which in the case of extensional deformation is subvertical. This theory describes many of the features observed on seismic sections, and the model commonly mimics observed rollover geometries subject to depth correction (Fig. 11-1). An understanding of this theory of genesis of rollovers can help you better interpret rollover geometries in poor data areas and thereby generate better prospects (Tearpock et al. 1994).

The elements of the Coulomb theory are shown in Figure 11-2a through e. Figure 11-2 represents the initial prefaulted or pregrowth state. This simple example shows an incipient major fault that has, for purposes of demonstration, a single concave upward bend. In nature, major listric normal faults are normally curved, and this more realistic case can be duplicated by introducing several concave upward bends into a model.

As the hanging wall block moves over the footwall block, a small void opens between the hanging wall and the footwall blocks, as described by Hamblin (1965) (Fig. 11-2b). Gravitational forces cause the hanging wall block to instantaneously collapse into the hole (created by the sliding) along the Coulomb failure surfaces. In Figure 11-2b the Coulomb collapse angle is 70 deg with respect to the horizontal. Beds in the hanging wall shear parallel to the Coulomb failure surfaces, and the material fills the hole, causing the beds to extend (Fig. 11-2c).

Observe in Figure 11-2b and c that material experiences rotation as it passes through the Coulomb shear surface that is fixed to the concave upward bend in the major fault. This is a locus of deformation that extends upward through the strata and, for that reason, this shear surface is called an active axial surface (Fig. 11-2c). Deformation occurs only along active axial surfaces, which are affixed to the bends in listric faults.

Initially, the material that passes through the bend in the normal fault is deformed at the active axial surface and is translated basinward. This material lies adjacent to the inactive axial surface, basinward of which the strata are undeformed (Fig. 11-2c). The inactive axial surface rides passively along the straight portion of the fault surface as movement progresses. As this surface is not affixed to the bend in the fault surface that creates the void, the inactive surface is passive and does not cause the hanging wall beds to dip toward the fault surface. Notice that the slip on the fault surface is the distance between the active and inactive fault surfaces, parallel to the fault surface.

This process is more readily understood if the fault model is subject to another increment of sliding (Fig. 11-2d). Of course, in nature these increments are infinitesimal. The sliding opens another void between the hanging wall and the footwall blocks, and the active axial surface that formed in Figure 11-2c translates basinward (Fig. 11-2d). However, the void instantaneously closes by gravitational shear failure, pinning the active axial surface to the bend. The resultant structure shown in Figure 11-2e contains a graben-like feature adjacent to the steepest portion of the major fault and is a monoclinally shaped rollover structure. Sedimentary compaction within the basin creates basinward dip and closes the structure in the direction to the right in the figure.

You can better understand these processes by redrawing Figure 11-2e and cutting the hanging wall block from the footwall block with scissors. Then subject the major fault to another increment of motion and collapse the suspended material onto the footwall parallel to the Coulomb failure surfaces.

Growth Sedimentation

In areas like the Gulf of Mexico, most major normal faults are active growth faults. This means that sedimentation occurs simultaneously with fault slip, and thus the stratigraphic intervals are subject to vertical expansion across the fault surface (i.e., the intervals thicken on the downthrown side of the growth fault). How does this syndepositional sedimentation affect the structure, and can an analysis of this growth aid us in finding hydrocarbons?

Referring back to Figure 11-2c, let us assume that a layer of sediment is deposited across the hanging wall and footwall portions of our structure (Fig. 11-3a). The graben-like feature contains more accommodation space than the top of the rollover, so the sediments will be thickest over the graben and thinner over the footwall and the crestal portions of the monoclinal rollover. We saw in Figure 11-2b through e that the active axial surface is fixed to the bend in the major fault and that it represents the locus of deformation of the hanging wall beds. The inactive axial surface is passive and migrates basinward. Notice in Figure 11-3a that the inactive surface does not extend upward into the recently deposited growth sediments. The active axial surface, however, is a locus of active deformation that affects the entire body of growth sediments. When deformation is viewed as an incremental process, a growth axial surface must connect the active axial surface to the inactive axial surface, as shown in Figure 11-3a. You can convince yourself of this statement by visualizing a thin layer of additional growth sediments deposited above layer 1. This layer will be horizontal. Another small increment of sliding along the major fault causes Coulomb collapse that deforms this recently deposited layer in the region where the active and the growth axial surfaces converge (Fig. 11-3a). An additional increment of sliding, combined with growth sedimentation, produces the geometry observed in Figure 11-3b. The growth wedge expands as the growth sediments move through the active axial surface.

For larger rollover structures, sands are more likely to be deposited in the graben, whereas suspended sediments are more likely to be dominant across the top of the rollover. Therefore, facies changes and stratigraphic traps are predicted to occur in the vicinity of the growth axial surface. A more realistic example of growth faulting is presented in the following section on projecting large growth faults to depth.

The growth axial surface can be located on a seismic section by using dip domain analysis (Tearpock et al. 1994). This surface is located between the steeper-dipping beds at the front of the structure and the flatter-dipping beds located at the crest. After you locate the growth axial surface, you can determine the stratigraphic interval that was deposited as the structure grew. This is important, as no structure exists to trap hydrocarbons prior to the growth phase. To determine the growth phase of the structure, follow the growth axial surface downward. The growth axial surface subparallels the main fault during the time of growth, thus delimiting the history of the structural growth. The growth axial surface terminates at the inactive axial surface, and this point indicates the top of the pregrowth strata (Fig. 11-3a and b). No structure existed during the pregrowth phase of sedimentation to trap hydrocarbons (Fig. 11-2a).

Example of a Rollover Structure.

Some listric faults are observed to dip more gently with depth and then, at greater depth, to increase in dip (Fig. 11-12). This creates a concave downward bend in the major fault. Above this bend, the Coulomb collapse theory predicts that the deformation takes place along basinward-dipping conjugate shear surfaces. Shear can occur along two conjugate surfaces (Billings 1972) and, in our example, this shear occurs in the clockwise direction (Fig. 11-4a).

The Brazos Ridge, in the northern Gulf of Mexico, can be used to demonstrate the Coulomb breakup theory for a more realistic case. The Corsair fault is the major listric normal fault in the Brazos Ridge area. In the most general sense, the Corsair fault dips at about 45 deg, flattens to about 10 deg, and then steepens to approximately 20 deg, as represented in a computer model in Figure 11-4c. The Brazos region is subject to synchronous deformation and sedimentation, and thus the sediments deposited in the hanging wall block are growth sediments. If the simple Corsair fault model described in Figure 11-4c were subject to gravitational (basinward) sliding, two voids would be created, one above the concave-up portion and the other adjacent to the concave-down portion of the master fault. Clockwise collapse above the deeper concave downward bend (Fig. 11-4a) and counterclockwise collapse above the concave upward bend (Fig. 11-4b) would produce the geometry observed in Figure 11-4c. Notice that there is a relationship between the shape of the fault and the shape of the rollover in the hanging wall.

If deposition rates are high, as in the Brazos Ridge area, the sediments will fill the low area adjacent to the shallowest portions of the major fault, thin over the top of the rollover structure, and then thicken basinward (Fig. 11-4c). The vertical expansion of stratigraphic intervals, as observed in this example, demonstrates why the sediments are referred to as growth sediments. Also notice the growth axial surface in Figure 11-4c. It subparallels the growth fault, and it terminates with depth at an inactive axial surface that dips toward the fault (compare to Fig. 11-3b). The model in Figure 11-4c contains rollover amplitudes that are higher than observed along the Brazos Ridge (Fig. 11-1a). However, compaction combined with synthetic and antithetic crestal faulting reduces the amplitudes of rollovers.

The thickness of the crestal growth section can vary with the shape of the fault. In the case of the Brazos Ridge, the large expansion fault that creates the structure dips at about a 20-deg angle at depth. Thus, as the hanging wall moves over the footwall, the hanging wall slides down a 20-deg dipping fault surface. This slope causes the hanging wall to drop in elevation relative to the footwall, and the downward motion creates ample space for the accumulation of thick growth section over the crest of the structure (Figs. 11-3a and 11-12). Later we show that this fault geometry creates productive four-way closures (Fig. 11-21).

However, if a large expansion fault flattens at depth and is subhorizontal, then as the hanging wall moves over the footwall, the crest of the rollover structure does not change elevation with respect to the footwall (Fig. 11-5). This fault geometry creates the classic half-graben structure, with accommodation space developing between the fault and the flank of the half-graben structure but not over the crest of the structure. In this case the growth sediments are more likely to thin onto the flank of structure, creating stratigraphic traps. However, on actively growing structures, this thinning results from an interaction of stratigraphic and structural processes, classified under the heading of tectonostratigraphy (Chapter 13). Figure 11-5 shows a seismic profile of a half-graben structure from the Central Sumatra Basin, Indonesia. As the structure grows, growth sediments deposited in the half-graben structural low will be subject to axial surface deformation, as shown in Figure 11-5. The growth sedimentary section thins dramatically across the growth axial surfaces and onto the flank of structure (Suppe et al. 1992; Shaw et al. 1997). As growth sedimentation is typically episodic, alternating between high and low sedimentation rates, the local subsidence rate resulting from fault motion may temporarily exceed the sedimentation rate. In this case, coarser sediments may fill the accommodation space in the half-graben (Xiao and Suppe 1992). As the sedimentation versus subsidence rates fluctuate, the tectonostratigraphic process creates facies changes and unconformities along strike and down-dip of the growth axial surfaces.

Provided that information exists on the shape of a fault that has created a rollover, the Coulomb breakup theory can be used to predict the rollover angle (θ) or bed dips. Based on Coulomb deformation, the rollover angle (θ) is related to the change in fault dip (f) through a complicated set of trigonometric formulae, which can be represented by graphs (Fig. 11-6a and b). The assumptions used in the graphs are that the material in the hanging wall is subject to Coulomb collapse and that the structure has not experienced sedimentary compaction, as described later in this chapter in the section Compaction Effects along Growth Normal Faults. Thus, these diagrams strictly apply to the nongrowth phase of the sedimentation and will approximate the rollover angle within growth sediments. Let us assume that a seismic section images the shallow and deeper portions of a listric fault but does not image the bed dips. If the above conditions are met and if the initial fault dip (β) and the fault dip at a deeper level (α₁) can be observed on a depth-converted seismic section, then the rollover dip can be estimated from Figure 11-6a or b. Figure 11-6a assumes a Coulomb breakup angle of 60 deg, whereas Figure 11-6b assumes a 70-deg breakup angle. For example, the Brazos Ridge generally has a major fault that initially dips at about 45 deg and flattens abruptly to about 15 deg. Therefore, the change in fault dip f= 30 deg. From Figure 11-6a and b, the predicted bed dips at the front of the rollover structures are estimated to be 18 deg to 22 deg. Even though the Brazos Ridge growth sediments have been subject to compaction, these results compare favorably to observed bed dips, which average about 20 deg at the bend in the fault (see Fig. 11-1a and b). For presentation purposes, Figure 11-6a and b can also be used to generate generic or idealized models of real rollover structures. The generic structure shows how the rollover structure formed.

Rollover Geometry Features

A major advancement in the understanding of structures occurred when Hamblin (1965) recognized that rollover anticlines form as hanging wall beds slip along listric normal faults. As the hanging wall slides along the fault surface, the hanging wall conforms to the shape of the fault surface by collapsing onto the fault surface. Hamblin indicated that the process could result from vertical or Coulomb collapse (Fig. 11-7). Numerous authors tested the two mechanisms and favor Coulomb collapse over the vertical collapse mechanism (White et al. 1986; Worrall and Snelson 1989; Groshong 1989; Bischke and Suppe 1990b; Dula 1991; Nunns 1991; Withjack et al. 1995). Xiao and Suppe (1992) show, from empirical observations using forward models, that many rollovers form as a result of Coulomb collapse at angles of 65 deg to 70 deg with respect to the horizontal. Industrial software programs are available that incorporate the vertical and inclined collapse algorithms to model rollover structures (Rowan and Kligfield 1989).

In a listric normal fault system, as the hanging wall beds move through bends in the fault surface, small voids open between the hanging wall and the footwall (Fig. 11-7). Large gravitational forces instantaneously close these voids, forcing the hanging wall to conform to the shape of the footwall. This collapse mechanism causes the hanging wall beds to dip, or diverge, toward the fault surface (Fig. 11-8). Furthermore, the strata steepen in dip as they move across bends in the fault, so the more listric the fault surface, the steeper the dip of the beds (Fig. 11-8).

Motion of the hanging wall down the fault surface creates accommodation space for growth sediments. The recently deposited sediments, having existed for only a short period of time, have not moved through many fault bends and therefore exhibit gentle bed dips (Fig. 11-8). The deeper and older sediments have moved through numerous fault bends (Xiao and Suppe 1992). Thus, older growth sediments dip at higher angles than shallow growth sediments (Fig. 11-8).

Several observations can be made about rollover fold shapes. First, the more listric the fault, the steeper the dip of the growth section. Second, there is a relationship between fault shape and rollover shape. If fault shape is known, then rollover geometry can be predicted with a high degree of certainty (Xiao and Suppe 1992; Shaw et al. 1997). Conversely, if rollover shape is known, then fault shape can be predicted, but with less certainty. Predicting fault shape from rollover shape is an inverse technique and calculation errors accumulate. These errors originate from uncertainties resulting from depth correction, geological and stratigraphic variations, and simple measurement errors. However, both techniques are important tools in extensional structural interpretation.

Projecting Large Normal Faults to Depth

We present a graphical method for projecting large normal faults to depth using hanging wall beds that have been subject to noticeable rollover. We also present methods for estimating fault shape and position from the shape of a well-constrained rollover structure.

In order to apply this method, two features must be known: (1) a marker bed that exhibits noticeable rollover and (2) the true dip of the fault at the hanging wall cutoff level of the marker beds. A profile is constructed that must be a dip profile, so that it is perpendicular (as close as possible) to strike of the fault and strike of the beds. A fault surface map can help you determine the dip direction of the profile. The structural information can come from well log data, depth-corrected seismic profiles, or cross sections. If possible, start the projection process by locating a marker bed above the known level of the main expansion fault. Then use the known portions of the fault to calibrate and test the projection procedure. If satisfied with the results, apply the method to the unknown levels of the fault.

Procedures for Projecting Large Normal Faults to Depth

1. If using a seismic profile, depth-correct the seismic section over the interval of concern. Depth-correct the section on the workstation using local checkshot or other more modern depth conversion methods such as employing formation tops to calibrate models derived from time surfaces and processing velocities.

2. If secondary faulting offsets the marker bed, restore the bed to its unfaulted position before proceeding further. This is readily done by scanning or digitizing the marker bed into a graphics program and restoring the upthrown and downthrown cutoffs of the marker bed (Bischke and Tearpock 1999).

3. In the graphics program, locate a deep marker bed on a level that intersects the main fault. The bed should exhibit obvious rollover and intersect the imaged portion of the main fault (Fig. 11-9a).

   

4. Fit tangents to the marker bed in order to segment the rollover into dip domains (see Cross-Section Construction and Kink Method Approximation sections in Chapter 10). Each dip domain contains a region of roughly uniform bed dips (Shaw et al. 1997). Curved rollovers will possess several dip domains (Figs. 11-8 and 11-9a). Our generic example has four important dip domains: dip domains 1, 2, 3, and 4 (Fig. 11-9b). Where the tangent lines intersect, construct five dip domain boundary lines (S₁, S₂, S₃, S₄, and S₅) to each of the dip domain panels (Fig. 11-9b). The dip domain boundaries are drawn at an angle of either 68 deg or 112 deg. In Figure 11-9b, dip is measured clockwise, with 0 deg to the right and 180 deg to the left. If the marker bed dips toward the main fault, then these domain boundaries dip at about 112 deg with respect to the horizontal, and they dip in the antithetic direction toward the main fault. If the marker bed dips away from the main fault, then these domain boundaries dip at about 68 deg in the synthetic direction, or in the same direction as the main fault. As an alternative to the above assumed angles, you can derive the collapse angle from the geometry of rollover structures, as described later in the chapter (Bischke and Tearpock 1999).

5. Project the first dip domain boundary S₁ downward to where it intersects the straight-line projection of the main fault surface (Fig. 11-9c). Where the two surfaces intersect at point (1), the fault dip decreases and the fault becomes more listric.

6. As the first step to determine the fault position beyond the first bend in the main fault, construct horizontal lines between the right and the left dip domain boundaries, or between boundaries S₁ and S₂, S₂ and S₃, and so on (Fig. 11-9b and c). Then, parallel to the dip domain boundaries and between the marker bed and the horizontal line, measure the distances D₁, D₂, and D₃. In other words, measure distances D₁ and D₂ along the 112-deg inclined dip domain boundaries. Similarly, measure distance D₃ along the 68-deg inclined dip domain boundary S₅ (Fig. 11-9c).

7. Next, project the known portion of the main fault forward, from the left domain boundary at point (1) to the projection of the right dip domain boundary at point (2) (Fig. 11-9c). Where the straight-line projection of the main fault intersects the right domain boundary S₂, measure the distance D₁ from point (2) upward and in line with the right domain boundary S₂. This procedure determines the position of point (3). The position of the main fault is now known at point (1) and at its projected position at point (3), defined by distance D₁. Next, construct the projected portion of the fault surface between points (1) and (3), establishing the position of the fault beneath dip domain 1 (Fig. 11-9c).

8. Project the main fault forward from point (3) to point (4) (Fig. 11-9c) and repeat step 7 for the next fault bend (Fig. 11-9d). For this fault bend, measure distance D₂ from point (4) at the right domain boundary S₃ (Fig. 11-9d).

9. If the rollover has a horizontal crest, this means that the fault does not change dip beneath the flat crest of the rollover. Thus, the main fault dips at the same angle beneath the crest of the rollover (Fig. 11-9d). In other words, the fault is straight beneath dip domain 3 in Figure 11-9b. Project the fault beneath the flat crest to dip domain boundary S₄ at point (5). At point (5) the fault turns down and becomes convex where the beds turn down, but what is the dip of the fault beyond dip domain boundary S₄?

10. Repeat step 7 again, but in this case, the fault turns down at the axial surface where the marker bed turns down. So subtract distance D₃ from point (6). Lastly, we construct the convex projection of the fault surface beneath dip domain 4 (Fig. 11-9d).

This technique is very precise on ideal or generic rollovers, but it suffers from measurement errors and geologic variations on real rollover structures, where these measurement errors add through each fault bend. Therefore, the projection technique may deteriorate after projecting the fault surface through three or four fault bends.

Field Example.

The method has application to all extensional regimes. A field example of a rollover structure from the Gulf of Mexico basin illustrates the procedures used to employ the fault projection method.

The example is from a 3D depth-corrected profile from the Burgentine Lake Field, onshore Texas (Fig. 11-10) (Bischke and Tearpock 1999). The horizons and fault surface were tied to well log data and mapped. The profile was depth-corrected using checkshot data. A graphical technique for determining the Coulomb collapse angle (described in the next section) derives a collapse angle ψ of 63 deg for the structure (Fig. 11-10a). This angle is slightly less than the 67-deg to 68-deg collapse angles observed on some Gulf of Mexico rollovers (Xiao and Suppe 1992). Employing the steps described in the preceding section on procedures, and using a ψ of 63 deg, we generate the cross section shown in Figure 11-10b. The projected portion of the fault surface lies over the depth-corrected portion of the fault surface. In this example, the agreement between theory and observation is very good. Therefore, we have shown that the method has application in working with real rollover structures.

Determining the Coulomb Collapse Angles from Rollover Structures

We conclude this section by describing a graphical technique for determining the Coulomb collapse angle. In Bischke and Tearpock (1999), we present the mathematical basis for these graphical methods. Here we describe the graphical method.

Using a dip-oriented, depth-corrected seismic profile, begin by locating a deep dip domain that lies adjacent to a large expansion fault, as outlined by the dashed lines in the hanging wall block in Figure 11-11a. The fault is represented by the lines L and S. A tangent line ℓ is fit to the steepest dipping beds adjacent to the fault surface, and another tangent line is fit to the flat crest of the rollover. The technique requires the determination of two displacement parameters: (1) the throw T on the fault and (2) the vertical distance V between the marker bed’s footwall position and the position of the marker bed at the crest of the rollover (Fig. 11-11b).

At the position of the marker bed’s hanging wall cutoff, construct the vertical throw vector line T on the depth-corrected profile (Fig. 11-11b). Determine the vertical distance V. Next, extend the throw line T by the distance T-V (Fig. 11-11b). Determine length () by first locating the point of intersection between tangent line ℓ, at the front, and the tangent line at the crest of the rollover. Length is the horizontal distance between this intersection point at the crest and the vertical line T (Fig. 11-11b). Measure length at the same structural level as the tangent line drawn at the flat crest of the rollover structure. Next, position a line of length () adjacent to the marker bed’s footwall cutoff position (Fig. 11-11b). The final step in the process is to determine the value of the Coulomb collapse angle ψ (Fig. 11-11b). The angle ψ is equal to the inclination of the dip domain boundary (Fig. 11-11a). From the right-hand termination of line (), positioned adjacent to the marker bed’s footwall cut off, construct a dashed line downward to the lower extension of the T-V line (Fig. 11-11b). Lastly, measure the Coulomb collapse angle ψ from the horizontal, using a protractor. At Burgentine Lake, the Coulomb collapse angle ψ is 63 deg if measured counterclockwise, as dip (Fig. 11-10b).

In conclusion, Hamblin’s (1965) inclined collapse mechanism of rollover formation combined with dip domain analysis provides a graphical method for projecting normal faults into poor data regions. The fault projection method, when compared to depth-corrected profiles of fault shape, yields reasonable results. The technique is sensitive to measurement errors and may deteriorate after projecting a fault through more than a few fault bends. The method is also dependent on the collapse angle (Bischke and Tearpock 1999), so we presented a graphical method for calculating the Coulomb collapse angle for rollover structures.

Backsliding Process

One can envision these large listric growth faults as being within a large, slowly moving landslide (Xiao and Suppe 1992). As the hanging wall fault block slips, a void opens between the hanging wall and the footwall, and the overlying material instantaneously collapses along Coulomb shear surfaces, producing the rollover (Figs. 11-2 and 11-4). We learned from Figure 11-6a and b that the greater the concave bend (ϕ) in the major normal fault, the greater the rollover angle (θ). If the dip along the major fault decreases with depth, then eventually the rollover angle may increase to an angle such that the bed dip (θ) exceeds the dip on the major fault (Fig. 11-13). Frictional failure could then occur along the bedding surfaces that form the front limb of the rollover structure, and not just along the Coulomb shear surfaces. The higher the dip of the rollover structure, the closer the sedimentary beds come into parallelism with the antithetic Coulomb failure direction. At some angle of inclination, perhaps less than the Coulomb shear angle, the bedding planes are likely to present less frictional resistance than a sequence of sedimentary layers. Thus, frictional failure along the bedding surfaces would be favored over the Coulomb shear surface (of 60 deg to 70 deg), which cuts across the layering at a high angle (Fig. 11-2b). These potential bedding plane slip surfaces are likely to occur in weak layers, such as overpressured shale zones, which would present the least possible frictional resistance.

We therefore conclude that backsliding along bedding surfaces is a mechanism that can account for the apparent termination of the slip along the downward dying synthetic faults and antithetic faults. Using the examples of the Brazos Ridge, the backsliding appears to initiate where the rollover angle exceeds about 10 deg. The backsliding is a second-order effect relative to the total amount of slip along the major normal fault, about one part of bedding plane slip to about every seven parts of the slip along the Corsair fault. Although the bedding plane slip within the frontal limb of the rollover structure is small compared to the total slip, we shall see that it can have a marked effect on the amplitude of the rollover structure.

Backsliding Model.

Exactly how does this backsliding mechanism operate? We present two examples: the first outlines the backsliding process, and the second, a more realistic model, assumes that the backsliding consumes about 1 part in 10 of the amount of slip that occurs along the major detachment fault. Depending on geological conditions, however, backsliding may at times be the dominant process.

Figure 11-14 illustrates the backsliding mechanism. Initially, slip along the master fault must occur to an extent such that a critical rollover failure angle is exceeded (Fig. 11-14a). This critical failure angle depends on local geological conditions, such as the pore pressures, and is likely to vary from area to area. As the pore pressure is unlikely to be large in pregrowth sediments, backsliding is most likely to occur within the growth sedimentary package. At some time, a weak rock, such as an overpressured shale, moves through the concave upward bend in the normal fault surface. When this occurs, an increment of backsliding along the bedding surfaces that comprise the frontal portions of the rollover structure fills the lower part of the void created by the basinward sliding (Fig. 11-14b). Initially, the plane of detachment propagates along the bedding plane surface and, in the case of the Brazos Ridge, up to the master synthetic fault. At this location, the synthetic shear associated with the concave downward bend in the major fault turns the strata downward in the basinward direction and forms the upper portions of the rollover (Figs. 11-4c and 11-12).

At the position of the master synthetic fault, the shear within the weak horizon can follow one of four possible paths. First, the propagating bedding plane slip (overpressured) could also turn downward and follow the bedding surfaces across the top of the rollover structure. This possibility seems unlikely, as large amounts of energy would be required. Second, the propagating slip could cut across the more gently dipping bedding surfaces located near the top of the rollover to a pre-existing antithetic fault (plane of weakness), and then follow the older antithetic fault to the surface. This would create small, triangular-shaped structures. Third, at the position of the master synthetic, the growing bedding plane fault could follow a Coulomb failure surface up to the sea floor, creating a new antithetic fault (Fig. 11-14b). This deformation path appears to be the mechanism of least resistance and the path most favored by many compensating faults. This mechanism would also result in antithetic faults that appear to terminate along the master synthetic, which are normally observed on our Brazos Ridge seismic section example. Fourth, the hanging wall block above the bedding plane detachment and the major fault and master synthetic fault could detach and slump toward the major fault (not shown in Fig. 11-14). This mechanism would produce a surface of detachment along the master synthetic. This mechanism could be a dominant process in certain areas. We favor the third, and to a lesser extent, the second mechanism to explain the observed antithetic faulting that we have studied in examples such as the Brazos Ridge.

The third mechanism would cause another void to open along the newly formed antithetic fault (Fig. 11-14b). This hole can be filled only by collapse along the basinward-dipping Coulomb failure surfaces (Fig. 11-14c), forming a keystone structure. According to the balanced deformation model just described, another increment of backsliding will result in the formation of another antithetic, which forms above the master synthetic and the previously formed antithetic.

Another consequence of this model is that the bedding plane slip is likely to occur along shale horizons that could be related to fluctuations in sea level (Payton 1977). As a result, the intervening horizons are more likely to contain sands, or reservoir rock. Notice in Figure 11-12 that the antithetic faults die downward into a unit of semicoherent reflections between 2.1 sec and 2.4 sec. This unit is an overpressured shale, and we have observed normal faults that die into overpressured shale intervals in other structures.

Example from Corsair Trend.

A more realistic, generic example is illustrated in Figure 11-15. Here the major fault initially dips at 45 deg, decreases to 10 deg on the flat, and then steepens basinward to 20 deg. This fault configuration is similar to the shape of the Corsair fault in Figure 11-12. Also, the Coulomb failure angle is assumed to be 70 deg and the growth phase of the sedimentation to begin at Horizon 1 (Fig. 11-15a). In Chapter 13, we describe how to distinguish the growth interval from the nongrowth interval using growth plots. Since the sediments deposited beneath Horizon 1 are pregrowth sediments, and since the backsliding mechanism is likely to initiate in an overpressured horizon, the backsliding process is not likely to begin until a critical dip angle is reached within the growth sediments. At this stage, the first increment of backsliding occurs and a keystone structure forms (Fig. 11-15b) at Horizon 1. Notice in this example that the distance between the concave upward bend and the concave downward bend in the fault is greater than in the example in Figure 11-14.

Additional sliding along the major fault produces the geometry present in Figure 11-15c, and another increment of backsliding results in Figure 11-15d. The backsliding has the effect of creating keystone blocks that become younger upward as the blocks grow larger. The basinward sliding along the major fault has deactivated the antithetic fault that formed in Figure 11-15b, and this antithetic fault stopped growing after Horizon 2 was deposited (Fig. 11-15c) and was buried by the more recent growth sediments. The backsliding also has the effect of reducing the rollover amplitude. After the deposition of Horizon 4, the geometry appears as shown in Figure 11-15e. At this stage, the lower two antithetic faults, which formed during the interval of time that Horizons l to 3 were deposited, have moved through the locus of deformation (active axial surface) that is associated with the concave downward bend in the major fault. The clockwise shear associated with this deformation has the effect of slightly rotating the antithetic faults clockwise (Fig. 11-15e). During the interval of time that Horizon 5 is deposited (Fig. 11-15f), slip along the major fault has advanced to the stage that the newly forming antithetics initiate to the right of the clockwise shear-active axial surface. Thus, the antithetics that form after the deposition of Horizon 4 will not be rotated clockwise by the concave downward bend in the major fault (Fig. 11-15g). Additional increments of backsliding are shown in Figure 11-15h and i.

Figure 11-15i is a generalization of the deformation that occurred during the deposition of Horizons 1 through 7. Dashed lines are drawn at the base of the antithetic faults where the slip entered the bedding planes and at the top of the antithetic faults where they ceased to grow and became inactive (Fig. 11-15i). These lines can be considered axial surfaces that are associated with the growth phase of the antithetic crestal faulting and the formation of the keystones (Tearpock et al. 1994).

Again, notice that the more recent antithetic faulting and keystones form to the left of the older rollovers. If this model is correct, then it can be tested by data. Figures 11-16 and 11-17 are two seismic lines from the northern Gulf of Mexico and Brunei (Southeast Asia) respectively. On both of these lines, the antithetic faults have a tendency to be younger upward and to be older in the deeper sediments. The pattern of downward dying faults and keystone blocks is similar to our example in Figure 11-15.

Let us briefly review the consequences of the above-described deformation.

1. Growth antithetic faults form basinward and above synthetic faults located near the crests of rollovers.

2. The antithetics become younger upward with the older antithetics being positioned basinward and terminating at a deeper level.

3. Slip along the master synthetic may die with depth, and slip along the antithetic faults appears to terminate at the position of the master synthetic.

4. The deformation mechanism forms a keystone structure, or a graben, which with the antithetic faults has the effect of reducing the amplitude of the rollover structure.

5. The deformation and sedimentary mechanisms form faults that die in both the upward and downward directions.

6. The process may be controlled by the sedimentary cycle.

Prospect Example

A good application of this method arises when analyzing a prospect. Examine Figure 11-31, showing Well No. 1, which penetrates a large growth fault, and Well No. 2, which has a discovery at the D Sand level on a rollover structure. Seismic data from this area are not of the best quality, but they image gentle south dips. The proposed well, to be drilled up-dip of the discovery well, looks like a lead-pipe cinch! Would you participate in the proposed well? If not, why not?

Let’s do a quick examination of the cross section in Figure 11-31. We first observe that the normal fault is planar in shape, dipping at an angle of about 55 deg. The discovery and proposed wells appear to be in the hanging wall rollover anticline formed by the fault, which is penetrated by Well No. 1.

Several questions come to mind. Why is the fault in Figure 11-31 not listric? Why do the beds dip or roll toward a fault surface that is a planar surface? Geoscientists know that rollover structures are associated with listric fault surfaces; that is, rollover structures form above fault surfaces that are not planar. In the section on the origin of rollover in this chapter, we demonstrate that only above listric fault surfaces do hanging wall strata form a rollover anticline.

Let’s take a look at the fault surface shown in Figure 11-31 and refer back to Figure 11-29. Figure 11-29 allows us to estimate fault dips from lithology. The lithology in the footwall of the fault in Well No. 1 allows us to predict the dip and shape of the fault. Well No. 1, which penetrates the fault near −4000 ft, contains about 50% sand in the footwall block between the −4000-ft and −8000-ft levels (−1200 m to −2400 m). The average depth of the sandy section is −6000 ft (−1800 m). On Figure 11-29, a fault at a depth of −6000 ft (−1800 m) in a 50% sand section will dip at an angle that lies halfway between the fault shape in pure sandstone and the fault shape in pure shale. We can fit a protractor to this figure to obtain a fault dip of about 55 deg. The fault shown on Figure 11-31 dips about 55 deg between −4000 ft and −8000 ft, exhibiting good agreement between observation and theory.

Notice that Well No. 1 penetrates a predominantly shale section below −8000 ft (−2400 m). Thus, the footwall section below −8000 ft should be subject to significant compaction. Faults in compacting shales will dip at gentler angles than faults traversing sand-rich sections. Let’s take another look at Figure 11-29, but this time we assume a pure shale section between −8000 ft and −20,000 ft (−2400 m to −6100 m). The average depth of this pure shale section is about −14,000 ft (−4250 m). Fit a protractor to the fault shape on the pure shale curve (Fig. 11-29) at an average depth of −14,000 ft. Figure 11-29 predicts that the fault will dip at an average of 40 deg below −8000 ft. If the fault dips at an average of 40 deg below −8000 ft, then the fault should follow the path shown in Figure 11-32. The listric bend in the fault causes the strata to roll toward the fault and to form a rollover anticline in the hanging wall block (see section on origin of rollover). Furthermore, the listric shape of the fault causes the strata to dip at a higher angle than those shown in Figure 11-31 and causes the fault to be farther to the south than previously interpreted. The result of our quick analysis is to predict the possibility that the productive D Sand will be faulted out in the proposed well (Fig. 11-32). What appeared at first to be a sure thing may not be. An alternative to the quick-look method is to use a spreadsheet and the procedures outlined in this section to write a program using Eqs. (11-6), (11-7), (11-8), (11-9), and (11-10). The program can be used to predict the fault dip through a varying sandstone/shale section.

The importance of this example is to stress that there is often a geometric relationship of fault shape to fold shape, and it can be used when generating geologic interpretations. The cross section shown in Figure 11-31 does not conform to certain basic geologic principles. Therefore, a good possibility exists that the interpretation is in error. The error may be small or large. The quick-look technique indicates that the error, in this case, may be large enough to result in a dry hole. This quick analysis should direct the prospect generator to additional work to either substantiate the interpretation, as shown in Figure 11-32, or modify the interpretation based on the principles shown in this section on the effects of compaction on the shape of growth normal faults.

The compaction work presented by Xiao and Suppe (1989) can result in accurate predictions of fault shape where compaction is the predominant process. The method is stable and tends to average the well log data. As the method predicts the average fault dip based on the average lithology over an interval, the method averages through intervals that do not obey the idealized compaction formulae. Thus, the method tends to average out local and aberrant secondary cementation intervals that may affect the compaction curves. Deformation of the fault surface, either by salt or by another fault, may affect the results of the method.

Inverting Fault Dips to Determine Sand/Shale Ratios or Percent Sand

Inverting fault dips is a technique used to estimate the percent sand in a prospective interval, which is valuable information for a prospect. When well log data are not available on the level of interest, and provided that a normal fault is present, it may be possible to estimate the amount of sand off of a seismic line. The seismic line must be depth-corrected. With an estimate of the percent sand over the prospect area, a certain amount of risk can be mitigated, better volumetrics can be estimated, and overall economic evaluation can be improved.

Normal faults can be listric in shape and dip at lower angles with increasing depth, or the faults can be antilistric and dip at higher angles with increasing depth (Fig. 11-29). Observation by industry indicated that growth normal faults commonly dip at shallower angles in shale sections and at higher angles in sand sections (Gow 1962; Roux 1978). Industry uses this relationship between fault shape and lithology (Roux 1978) to predict sand intervals from depth-corrected seismic profiles. If on a depth-corrected, dip-oriented seismic section a growth fault steepens at depth, then the fault may be entering a sand-rich interval (Bischke and Tearpock 1993). In this section, we develop a technique based on the Xiao-Suppe equation [Eq. (11-6)] to estimate percent sand from the shape of faults on depth-corrected seismic sections.

In the Brazos Ridge area in the northern Gulf of Mexico, many of the growth normal faults at shallow depths, and within the crests of rollover structures, dip at angles of 40 deg to 45 deg, and with greater depths dip at higher angles (50 deg to 60 deg). These antilistric normal faults steepen into a known section of higher sandstone/shale ratios (approximately 1.0 to 1.5). These normal faults are the basis for a method of using fault dips to estimate sandstone/shale ratios (Bischke and Suppe l990a). The basic concept for accomplishing this task has been presented in the previous section and is implicit in Eq. (11-6).

For example, if the initial fault dip (θ₀) and porosities (ϕ₀) are known and if the average porosities (ϕ) are calculated at all depth levels using Eqs. (11-7) to (11-10), then the fault dip (θ) at the level of interest can be calculated from Eq. (11-6). This is the forward modeling procedure described previously. Forward modeling is more accurate than inverse modeling, and ratios such as sand/shale ratio are sensitive.

If, however, the fault dip (θ) is constrained by good depth-corrected seismic sections, then the average porosities (ϕ) for sand and for shale can be calculated from Eqs. (11-7 to 11-9) at any depth level. This is called inverse modeling, which is sensitive to measurement errors. As porosities are dependent on lithology [see Eqs. (11-7) to (11-10)], we can estimate the (sand or shale/depth curve in any area, provided that we have information on local porosity depth curves. This information can be derived from the nearest wells, which in frontier areas could be many miles from the area of interest.

First, we must initiate a sandstone/shale ratio formula. This can be accomplished using Figure 11-33. In practice, the observed height of the figure is the top and the bottom of a portion of an interpreted growth normal fault on a depth-corrected seismic section. This distance represents a column of interbedded sandstone and shale. Using a deck-of-cards analogy, the sandstone layers are removed from the deck and placed at its bottom.

We now calculate from Figure 11-33:

where

Using the law of sines,

and substituting and rearranging yields

or

where

  Sandstone/shale ratios can be converted to percent sand by the following equation:

As θsh and θss can be obtained from Eq. (11-6) and the porosity from Eqs. (11-7) to (11-10), we can calculate the average porosity, or sand/shale, curve with increasing depth.

Before proceeding to a test case, we outline a number of factors that affect the accuracy of our methods.

1. Inversion methods are sensitive, and errors occur when measurements are not accurate. The method is most accurate when penetrating thick sand sections interbedded with thick shale formations, such as exist in the Brazos Ridge and Segno Field examples presented in the next section (Figs. 11-34, 11-36, and 11-37).

   

   

   

   

2. Accordingly, accurate measurements are necessary.

3. The method applies only to growth normal faults and thus to areas that are experiencing both active sedimentation and extension.

4. Although the method can be applied wherever velocity information is available, the velocity function is critical to obtaining reliable results because it is used to determine depth of the fault. Velocity determinations are discussed in Chapter 5.

5. The method in its present stage of development involves only compaction, and thus deformation of normal faults by other processes, such as salt flow, may invalidate the method.

6. The method is restricted to the depth to which the seismic reflections can be resolved.

7. The method applies only to sandstone/shale lithologies, and not to carbonates. However, faults also steepen into carbonate reservoirs.

Example from Corsair Fault Offshore Texas

In the following test of the method from data located on the Corsair fault, Texas, we compare a depth-corrected growth normal faults to well log data (Bischke and Suppe 1900a). In order to compare observed well log results, which resolves horizons to within feet to seismic data, in which the resolution is a function of depth, the well log data must be converted to longer wavelengths. This is accomplished by first determining the sand and shale horizons from SP logs, as discussed earlier (Fig. 11-28), and then measuring the wavelength of coherent reflectors from the seismic section. The frequency content of the seismic section is thus determined. As the frequency content decreases with depth, the wavelength of the reflections increases with depth. This wavelength is then depth-corrected and passed over the SP data as a moving average, with the length of the operator increasing with depth. The result of this averaging process from an example well is the sandstone/shale depth curve shown in Figure 11-34. The processed data are near several antilithic faults located on a rollover above the Corsair fault, Texas (Bischke and Suppe 1990a, their Figs. 8 and 9). The data are plotted at a frequency of two seismic wavelengths. This long-wavelength well log data can be directly compared to normal fault calculations based on seismic data.

Typical results, calculated using Eq. (11-13) and based on a growth fault located near Well No. 9, are shown in Figure 9 of Bischke and Suppe (1990a). This fault is the closest fault to Well No. 9. The results approximate the well log data. The seismic data was processed utilizing stacking velocities. The normal fault, which terminates at a depth of −7000 ft, has been used to predict the higher sandstone/shale ratios present in the −4800-ft to −6300-ft level. Other examples, located closer to the main fault, encounter a sedimentary dumb or more precisely a prograding wedge, which has higher sandstone/shale ratios ranging from 2 to 4. The prograding wedge thins and dies out near the crest of structure. A typical calculated example is shown in Fig. 11-35, in which high sandstone/shale ratios occur near 1 between a depth of 5500 ft to 6000 ft.

Elements of the sandstone-shale theory were known to industry for many years and were applied worldwide. As a final example, we examine a field in Texas where an early form of the method was applied in the 1960s.

Example from Segno Field, Polk County, Texas.

The Segno Field rollover structure formed downthrown to an east-west trending normal fault (Fig. 11-36a). Examine the footwall portion of cross section A-A′ constructed from dense well control on the eastern flank of the field (Fig. 11-36b). This section from Gow (1962) through the Tertiary Jackson, Cook Mountain, Sparta, and Weches shale sections shows that the normal fault dips at angles of 30 deg to 40 deg. Gow wrote on the profile, “Note that fault plane dip steepens in sand formations” (Gow 1962). In the Yegua and Wilcox sand sections the fault surface steepens to about 50 deg to 55 deg (Fig. 11-36b).

A change in fault dip angle from 30–40 deg to 50–55 deg is typically apparent on depth-corrected seismic profiles. Indeed, Roux (1978) presents an example of a fault that changes dip through a thick sand and shale section on a vintage seismic time profile. This profile is in Lowell’s book on structural styles (Lowell 1985). Industry recognized, and at times used, this “change in fault dip technique” to locate sand sections during the 1960s and 1970s. Subsequent bright spot analysis apparently replaced the technique, but geoscientists are again using the technique where direct hydrocarbon indicators are not applicable or not in support of amplitude versus offset (AVO) analysis.

We obtained a seismic line that crosses the crest of the structure. The line was digitized and depth-corrected using interval velocities. Interval velocities approximate true velocities and are always available to the geoscientist. On the profile (Fig. 11-36b), the fault dips at high angles through the Yegua and Wilcox sands and at less steep angles through the Jackson, Cook Mountain, and Sparta shale sections (Bischke and Tearpock 1993). After picking the fault on the vintage seismic line, the inflection points on the fault surface were depth-corrected for input into a sand percentage prediction program. This program uses formulas presented in the previous section. Program results shown in Figure 11-37 are comparable to Well No. 8 in the footwall block (Fig. 11-36a), located adjacent to the seismic line (Bischke and Tearpock 1993). In Figure 11-37, the diamond-shaped bar on the left shows the resulting computations from the Atwater-Miller sand compaction formula [Eq. (11-10)], whereas the open box bar on the right defines the Xiao-Suppe compaction curve results [Eqs. (11-8) and (11-9)]. The Atwater-Miller sand compaction formula is more conservative than the Xiao-Suppe formula. The closed box bar (middle) is the percent sand based on the Well No. 8 log.

When subject to computation, the interpreted fault trace predicts a sand response on the level of the Yegua sand between the depths of about −5000 ft and −7000 ft. The Atwater-Miller and Xiao-Suppe bars bracket the sand percentage in Well No. 8 and predict a sand percentage of about 60% (Fig. 11-37). Between the −5000-ft and −6700-ft depths in Well No. 8, the sand percentage is about 60% (Bischke and Tearpock 1993). An examination of Figure 11-37 indicates that a shale section should exist at about the −7000-ft to −8800-ft level and that sand should exist below −8800 ft. Well No. 8 contains shale between −6700 ft and −8200 ft before encountering the Wilcox sand (Fig. 11-36b). These estimates, which are a good match to the well data, are based on imprecise interval velocities, and better velocity estimates and seismic data should improve the depth percent sand and sandstone/shale predictions.

The method works best on seismic sections where you can pick the fault trace accurately and where thick sand and shale sections are present. Fault surfaces also respond to interbedded sand/shale sections, and the section need not be predominantly sand (Bischke and Tearpock 1993; Tearpock and Bischke 1991). Why drill a well to a depth of 20,000 ft if there is no direct evidence for sand at that level, without first using fault dip analysis to help reduce your sand risk?

This method has application to petroleum exploration, as was recognized by industry years ago (Gow 1962; Roux 1978). Normally, when a well is drilled, one of the least known parameters is the sand/shale ratio of the reservoir. As the section in the Segno Field example contains shale from −7000 ft to −8800 ft (Fig. 11-37), the method also produces estimates for seal thicknesses. Such information can be used by a geoscientist or engineer to better determine the target depths for wells and to improve the estimates of potential reserves prior to drilling. The method can be applied to the third dimension, so 3D percent sand maps can be constructed to better select well locations.

We have seen examples of exploration well locations being moved by as much as 1000 ft on the basis of this analysis. If a prospect has good structural closure over a large lateral area, the specific choice of the actual well location may be based on where the prospective intervals have the highest percent sand. This technique can be used, under the circumstances discussed, to make such determinations. All that is required to apply the method is an array of small antilistric, growth normal faults. The usefulness of the method increases as well control becomes limited and the distance between wells increases.

Topics

During a field study of rollover structures, Hamblin (1965) realized that rollovers form when the hanging wall beds move through bends in the causative listric fault. Typically, at shallow depths, the beds dip at shallow angles and bed dips steepen where the master normal fault increases its curvature (ϕ) (Fig. 11-38). This means that the greater the curvature of the large master fault, the more the beds diverge or dip toward the master fault. As growth faults typically increase curvature with increasing depth, the structural relief (R) also increases with increasing depth (Fig. 11-38). A comparison of the orange, yellow, and blue horizons shows how R and ϕ increase with depth (see Table 11-1 for a glossary of terms).

The mechanics of the rollover formation process is well understood and has been discussed by many geoscientists (e.g., Dula 1991; Nunns 1991; Xiao and Suppe 1992; Withjack et al. 1995; Tearpock and Bischke (2003) but has been subject to few studies in recent years. This section reviews only the most basic aspects of rollover formation. Xiao and Suppe (1992) showed, using theoretical analysis and empirical data, that if three primary parameters are known, then the geometry of rollover structures could be accurately modeled. These primary parameters are the Coulomb collapse angle (ψ), the shape of the listric normal fault, or the change in normal fault dip angle (ϕ), and the growth or expansion index (Ei) (see Fig. 11-39a to c). Ei = Gn/Gf is calculated as the ratio between the corresponding hanging wall (Gf) and footwall horizon thicknesses (Gn). Thorson (1963) was first to define the concept of growth, which does not involve time (consult Chapter 13 for more about growth structures).

For the Gulf of Mexico and Indonesia, the Coulomb collapse angles are about 67 deg to 70 deg (Xiao and Suppe 1992; Shaw et al. 1997; Bischke and Tearpock 2003) (see Fig.11-11). The change in fault angle (ϕ) can be obtained from depth-corrected seismic (Tearpock and Bischke 2003) and the expansion index from depth-corrected seismic or well log data. Thus, the basic theory and procedures exist to accurately predict hanging wall rollover geometries from using structural balancing software packages. Nomograms for predicting the shape of the bed dips in the hanging wall structure are presented in Tearpock and Bischke (1991, 2003). Prospectors typically know that the greater the change of the listric fault angle (ϕ), the steeper the bed dips on the flank of the rollover and the greater the structural relief (R) to the crest of the rollover structure (Fig. 11-38).

This section addresses both pregrowth and synsedimentary or growth structures. On pregrowth structures there is a general mathematical relationship between the structural relief (R), the throw (T) on the fault, and the change in fault dip (ϕ) [see Eq. (11-18) in the section Pregrowth Structure and Calculations], or

where K is a different function for pregrowth or for growth faulted structures.

In pregrowth modeled structures, the closed form equation derivations show that the K function is primarily influenced by the change in fault dip angle (ϕ) (see Pregrowth Structure and Calculations). However, other parameters also influence the shape of pregrowth rollover structures, and in general, K is a function of

[Fig. 11-39b; also see Eq. (11-18) in Pregrowth Structure and Calculations], where (β) = the initial or steepest dipping portion of the master fault and (α) = the dip of the deep portion of the fault beneath the crest of the rollover (Fig. 11-38). Except for the Coulomb collapse angle (ψ), the remaining parameters, such as the structural relief (R)/throw (T), can be measured from depth-corrected seismic sections. All calculations apply to the top of the horizons. However, the base of Horizon H1 is the top of Horizon H2.

General Eq. (11-14) may be used to determine the deep rollover structure and stratigraphy of pregrowth horizons, where the fault surface and seismic reflections image but where the seismic resolution may be of poor quality and/or the sequence stratigraphic relationships are uncertain. Thus, the seismic does not need to be perfectly coherent.

On the more common growth or synsedimentary faults, K is a function of

[see Eq. (11-18) in Pregrowth Structure and Calculations], where (ψ) = the Coulomb collapse angle, (ρ) = the growth axial surface (Xiao and Suppe 1992), and (τ) = angle that the rollover limb dips toward the master fault, defined by inflection points located near the crest and base of the limb (Fig. 11-39b or 11-41).

On pregrowth and growth structures, the method and its equation may also be used to check interpretations for consistency and structural validity where structure is complex or where the correlations are uncertain. Again, the seismic section must be depth-corrected. Following Xiao and Suppe (1992), we introduce simple, modeled derived structures to better understand the properties of real models based on seismic data. They classified models into pregrowth structures and growth structures. The pregrowth, or no-growth, structures are simpler than growth structures and are studied first. The application of these models to existing Gulf of Mexico rollover structures follows. At first, the models are simple, approximating curvilinear surfaces as straight lines, and become increasingly complex.

Lastly, real structures are analyzed. Xiao and Suppe (1992) realized that real structures could be modeled with smaller straight-line segments. In fact, real listric fault surfaces are often curved and have multiple active axial surfaces emanating from small bends in the fault surface (Xiao and Suppe 1992).

Linear Growth

Linear or near-linear growth is not always present but can occur when the R/T picks of seismic horizons, when processed in a spreadsheet, generate linear or near-linear regression coefficients in an x/y equation. The PF (see Table 11-1 for definitions) regressions may have high R-square values. More specifically, the variable R and T depth changes at increasing depths, and the depths of the R and T values adjacent to the master fault are modeled by a linear regression equation with high correlation coefficients. These high correlation coefficients of the measured and calculated depths result in a near linear (T/R) versus depth graph (e.g., Fig. 11-40, and see Chapter 13). Linear or near-linear growth is interesting in that linear functions with high R-square calculations are able to make predictions. For example, if the position of sands or a high %TOC horizon exists in the footwall, and these identical beds have not been drilled in the downthrown block, then the approximate depth of the downthrown beds maybe estimated utilizing the near-linear formula.

The correlations were conducted using well log parasequences and on high seismic acoustic impedance contrast data. Of course, the parasequence data had more correlations than the seismic data. The zero crossings on the seismic were not included in order to simplify the interpretation of the seismic profile. It is generally accepted that if random error is not included, then fewer independent variable correlations are required to justify linearity, even as low as 2 to 3 (Warner 2013). Later in the discussion of syntectonic models, we visually explain, using seismic data, how to identify near-linear growth using growth axial surfaces (Fig. 11-39c).

The cause of this linear or near-linear growth is not well studied but may result from uniform sedimentary loading, causing the master fault to generate linear growth. Linear growth was first recognized in paleontological data (Shaw 1964), then in contractional (compressional) setting using seismic, well logs and outcrop data (Suppe et al. 1992; Suppe et al. 1997; and Shaw et al. 2005). The latter case is interesting in that in order to get linear growth, the sediments must flood over the uplifted structure at a steady state. Linear growth was also recognized in extensional well log data, and segments of linear growth exist between some sequence boundaries (Bischke 1994b; Sanchez et al.1997; Chatellier and Porras 2001, 2004) and was also recognized in field studies (Bischke et al. 1999). Linear growth is also confirmed in well log data collected along strike-slip faults (Shaw et al. 1994). When using well logs, the parasequence correlations can be typically determined within less than a meter (several feet), so the growth data graphs have a high degree of accuracy (Bischke 1994b).

Furthermore, linear or near-linear growth is not uncommon and exists in all environments (Tearpock and Bischke 2003) (for examples, see Figures 13-8, 13-9, 13-11, 13-12, 13-14, 13-17, and 13-21). For example, seven parasequences on depth-corrected seismic had an R-square = 0.982. Knowing beforehand from the seven parasequence examples presented above that only one of the examples was linear suggests that linear growth may be more common than realized. Linear growth just exists within sedimentary sequences. We do not suggest that linear growth is always present, as published examples in the preceding references also contain nonlinear sedimentary growth. Despite the evidence of linear or near-linear growth along normal faults, no studies have been published on how linear growth occurs or on its extent. On the other hand, linear growth associated with contractional structures has been subject to numerous studies (see preceding references).

Unlike contractional structures in which the growth sediments must flood over the uplifted structures, as long as the normal fault is slipping, extensional normal faults create accommodation space to trap growth sediments. It is important to understand that growth measurements involve thickness changes and are not directly related to time (Suppe et al.1992). Unlike growth sedimentation, empirical determinations of sedimentation rates are known to scale—or obey fractal mathematical relationships, which plot as a negative sloping power law on a log/log scale (Sadler 1999). In other words, mean sedimentation rates are high over short time periods, and sedimentation rates are smaller over long time periods. In effect, longer time periods allow for more and/or larger hiatus, which include nondeposition or erosion (Sadler 1999). The important point is that sedimentation rates are strongly dependent on the time frame of reference.

Industry acknowledges that thin growth sections on normal faults (let us assume a growth section of 100 feet, or 33 m) were deposited over shorter time frames than growth sections measuring several miles thick (assume 2 miles, or 3.2 km) (for extensional examples, see Tearpock and Bischke 2003; for contractional examples, see Echavarria et al. 2003). Thus, in analogy with sedimentary rate data, thin growth sections are unlikely to contain many large or numerous hiatus (sequence boundaries, unconformities, or erosion), while thick growth sections are likely to contain more and larger hiatus. We are not familiar with any study that addresses these issues for normally faulted sections.

Calculations and Examples

This section shows how to calculate and describe equations for pregrowth and growth structures. Pregrowth structures are discussed first. The following pregrowth and growth models contain both measured and calculated values (see definitions described in the next section, Measured and Calculated Values on Graphs). Only the most salient aspects of rollover formation are reviewed, and the details of rollover formation are covered by Xiao and Suppe (1992) and in the first five sections of Chapter 11 in Tearpock and Bischke (2003).

Pregrowth Structure and Calculations.

Rolled-over beds (hanging wall anticlines) are caused by a void that develops between the hanging wall and foot wall of a normal fault, as the hanging wall beds slip (S) along the footwall fault surface (Hamblin 1965) (Fig. 11-39a). The weight of the hanging wall instantaneously closes the void caused by Coulomb shear collapse (ψ). The shear, which parallels the collapse angle, produces the rolled-over bed geometry (Hamblin 1965; Xiao and Suppe 1992) (Fig.11-39b).

In Figure 11-39b, the amount of the collapse required to fill the void is shown by the vector D, which parallels the collapse angle (ψ). In other words, the magnitude of vector D represents the amount of displacement required to cause the hanging wall to collapse onto the footwall (Hamblin 1965), closing the void. This collapse process forms the flank of the rollover, which dips toward the fault between the active and inactive axial surfaces and forms the structural relief (R) (Fig. 11-39b). The vector D is the vector Ax in Xiao and Suppe (1992). The structural relief is defined by the inflection points on the flank of the structure. The vector D vector is determined by the amount of slip (S) or throw (T), the Coulomb collapse angle (ψ), the change in dip (ϕ), and the dip of the fault (α) at depth as the hanging wall moves over the footwall (Fig. 11-39b). (Consult Table 11-1 for the definition of symbols.)

A graphical or inverse model solution to the structure consists of several steps. First, a copy of the modeled hanging wall is constructed from the fault surface, and the hanging wall slips over the footwall. Next, parallel to the dip of the shallow portion of the fault surface (β), project the amount of slip (S) below the bend in the major fault surface. Then, parallel to the inactive axial surface, add the D vector to the tip of the slip vector (S) (Fig. 11-39b). The D vector determines the position of the deep fault surface (α) and also determines the flank dip angle (τ) of the rollover.

Using the law of sines, a mathematical solution from Figure 11-39b is as follows:

where

where

Substituting Eqs. (11-16) and (11-17) into Eq. (11-15) and employing trigonometric identities eliminates π from Eqs. (11-15) and (11-16), and all of the angles are first-quadrant angles. This formulation yields

and rearranging yields

or

If ψ = β, then Eq. (11-18) simplifies to

where

Equation (11-22) yields the Rn values if the Tn values are measured. Equation (11-20) is plausible, as the major fault dip (β) could initiate or fracture at a 67-deg to 70-deg angle, or for dips approximating the Coulomb collapse angle ψ [see Fig. 11-39c to calculate R/T for Eq. (11-20)]. A study of the shallow portions of large expansion faults using depth-corrected data had a range of values between 60 and 45 deg for β. Figure 11-40 was created using Eq. (11-18) for Figures 11-39b and 11-41, assuming 50-deg and 60-deg angles for β, α = 20 deg, and ϕ = 40 deg. Even with four sine equations, the results are nearly linear. The accuracy of Figure 11-40 can be checked against Figures 11-39b and 11-41. In addition, from depth-corrected data and knowing the footwall stratigraphy and fault shapes, from well control or seismic data, Eq. (11-18) can be used to predict the throw or structural relief across large pregrowth faults and thus the downthrown stratigraphy (Fig. 11-41).

A Pregrowth Structure

A pregrowth structure (Fig. 11-41) is distinguished from a growth structure in that the rolled-over beds form a band or dip domain of uniform horizontal width that typically converges toward a large bend in the listric fault. However, if the growth is a syntectonic or growth fault and the growth increases with depth, then the growth-faulted rolled-over beds grow wider with increasing depth (Xiao and Suppe 1992) (Figs. 11-38 and 11-44). In Figure 11-39b, notice that the bed is rolled over at the bend in the fault surface (ϕ). The active axial surface emanates from this bend and dips at angle of about 67 deg. The (ψ) surface is active as shear along the (ψ) surface propagates into the hanging wall and defines where the beds roll over (Xiao and Suppe 1992).

In the hanging wall, the vertical distance between the top, or crest, of the structure and the depth at which the dipping beds are near horizontal defines the structural relief (Rn). Let us assume in Figure 11-41 that (1) sands are present in the footwall at the top of horizon 5, (2) in the hanging wall and in the vicinity of horizon 5, the data are semicoherent and cannot be correlated from the hanging wall into the footwall using reflection character, and (3) all of the wells in the hanging wall penetrate only as deep as horizon 4. Which is the correct solution to the depth of horizon 5? To determine the correct horizon, geoscientists can measure the Rn/Tn ratios for horizons 5a, 5b, or 5c (Fig. 11-42).

A convenient way of illustrating the correct horizon position—5a, 5b, or 5c—is by projection and then displaying the Rn/Tn results on an Rn/Tn versus depth(Zn) plot (Fig. 11-42). Depth(Zn) is the total depth to the base of the Rn/Tn or Hn values, typically the subsea depths to the horizons or to a datum below the surface. These plots have been called Δd/d or throw (T) versus depth (Z) plots in the literature (Bischke 1994b; Pochat et al. 2004). Again, depth(Zn) is the total depths to the base of the throw(Tn) and structural relief(Rn) vectors. When applied to well log data, these plots have an accuracy of better than 5 ft/10,000 ft, or a 0.05% error, at a depth of 10,000 ft (3049 m) (Bischke 1994b). Seismic resolution accuracy at this depth is substantially lower than well log data (Sheriff 1980), and thus the error is higher. Geoscientists can determine seismic resolution from vertical Fresnel zone nomograms that are a function of frequency, that decreases with depth, velocity, and wavelength (Sheriff 1980).

Figure 11-42 is a Δd/d or structural relief(Rn)/throw(Tn) versus depth(Zn) graph for the measured data in Figure 11-41. Also, note that the base of Horizon H1 is the top of Horizon H2. Projecting the base of horizons H1 to H4 and the R2/T2 to R5/T5 data and to depth leads to the obvious conclusion that the Horizon H5b data point on the graph is the correct answer. Notice that the H5b horizon R5/T5 data matches Rn/Tn data for the top of horizons H2, H3, and H4. Also, notice that the R5b/T5b data point has a unique depth(Zn) position on the graph of 9.05 units (Fig. 11-42). This follows, as the Rn/Tn measurements for the top of horizons H2 to 5b all have the same values of Rn/Tn = 0.80 (Figs. 11-41 and 11-42). The Rn-square or FP correlation values (see Table 11-1 for definition) for these horizons is 1.0. This makes sense, as the pregrowth sequence was deposited prior to the faulting and the faulting is post-depositional (the Tn values in Fig. 11-41 all have the same value). Thus, all of the horizons must have the same throw(Tn) values, in this case equal to 3.55 units. A plot of the throw(Tn) values versus depth(Zn) would confirm these conclusions. The calculated Rn/Tn value is 0.80, which is calculated from Eq. (11-18).

Figure 11-40 is a Rn/Tn versus depth(Zn) graph, which has properties similar to those of Δd/d plots in that outlying points constitute an anomaly. The cause of an anomalous point must be investigated, and this technique is discussed in detail in Chapter 13. The possibility that the horizon points H5a and H5c are the result of variable geology are unlikely. This conclusion follows, as the horizons were continuous when deposited and were deposited in a near horizontal position prior to the faulting. Excluding the possibility of a channel, stratigraphy does not change across a nonexistent fault. The possibility of a channel can be checked on adjoining lines. A possible but less likely explanation is that a major unconformity terminates by coincidence at the fault surface. Again, this explanation is unlikely, as the nonexistent fault was not moving and could not represent a buttress to onlapping sediments. Also, adjacent lines may resolve this possibility. Another option is to create a figure similar to Figure 11-42 using different depth(Z) values, such as points H5a and H5c, which do not match shallower Rn/Tn = 0.8. If the problem is an improper depth correction and a velocity error, then the velocities can be checked and the graph will help correct the incorrect velocities to a proper value. The correct depth will correspond to the depth in which Rn/Tn = 0.8. Lastly, another option exists when confronted with an anomalous situation in which it is not possible to correlate from the footwall into the hanging wall using seismic reflection character. In pregrowth structures, as the hanging wall along the fault surface must fit back into the footwall, structural balancing or, more specifically, retrodeformation may help (see Chapter 10, Fig. 10-10).

The next plot (Fig. 11-43) is based on Figure 11-41 and contains a deep single bend (ϕ) below Horizon H5. As the beds have not moved through the bend in the fault, the data points shown in Figure 11-43 are impossible in uniform thickness pregrowth strata, as the downthrown and upthrown beds must have the same thicknesses and thus the same Rn/Tn values. This interpretation of the data is possible as a result of poor seismic data or poor stratigraphic correlations. The Rn/Tn versus depth(Zn) graph (Fig. 11-43) looks reasonable except for the FP calculation of 0.357. In addition to the low FP value, the Rn/Tn plot in Figure 11-43 looks more random than the data in Figure 11-42. Assuming that the best linear fit is the most probable answer, possible problems exist in horizon correlations H4 and H5. These horizons appear more anomalous than correlations H2 and H3, which could be better. If there are any questions as to data anomalies, the following ratios can be graphed. Ratios, such as Rn/Tn ratios, are more sensitive than direct depth measurements such as the throw (T), which in this case, because the faulting is post-depositional, has a uniform value of 3.55 units (Fig. 11-41).

Accuracy of Methods

One way to address accuracy or correctness is to calculate the FP and R-squares of the data. Another is to calculate the % differences of the data. Percentage differences have the property that the order of the numbers do not matter. The % difference takes the averaged sum of the largest and smallest horizon thicknesses (Gn) (Fig. 11-44), then divides this number into the horizon thickness differences between the largest value minus the smallest value multiplied by 100. As there are no horizon thickness differences in the correct values measured on Figure 11-42, the % difference = 0.0, which means that the data have zero variations. The following discussion on growth and real structures contains % differences that are much larger. Large % differences can affect the accuracy of any projection method, but methods exist to recognize, address, and mitigate large % differences in real structures. (See Fig. 11-45 for the general equation).

The calculated values for graphs use Eqs. (11-18) and (11-28) to (11-30), and the measured values often used higher-imaged scales (200% to 400%), a magnifier, and typically a 30 scale (1/30 of an inch). Thus, the measured values have errors of up to 0.0167 in. = (1/(30 × 2) = (0.042 cm) (the viewing scale is shown as a percentage above the measured scale at 1/60 inches).

Growth Structure and Calculations

Figure 11-44 shows a modeled growth structure labeled to calculate growth on a simple fault (0 is on the left, and π is on the right). Measurements are taken from the horizontal, and ρ can exceed 90 deg, but in the formula π − ρ is less than 90 deg. As described in the section Details for Constructing Triangles, the object of the following calculations is to create a triangle often, but not always, bounded by an axial surface dip ψ. In this case, the triangle is formed by the vector Mn, flank dip vector Nn, and ρn (see Fig. 11-44).

Down to the Right Faults

The following formula is for a down to the right fault, or one that flattens to the right. Thus, if by convention, 0 is next to the shallow portion of the master fault, and π is above the deeper portion, then down to the right faults satisfy Eq. (11-28) (see Table 11-1 for definitions of terms). In this case, for (ρn > ψ) and from the law of sines and Figure 11-44,

where

where, as before, R1 is the structural relief for Horizon H1, but (T1 + Gf) is the throw plus the sum of the growth section from the base of the horizon being calculated to some datum located at or below the sea floor (Fig. 11-62a). Similarly, Gh1 is the total sedimentary or growth section for the horizon being studied to a datum, which in this case is from the base to the top of Horizon H1 (Fig. 11-44).

Substituting Eqs. (11-24) and (11-25) into Eq. (11-23) yields

However, in general and as the horizon deepens and the flank dip often increases, the total Depth(Zn) of the base of Horizon Hn increases and thus, in general, from Figure 11-44,

where

1. n = 2, 3, 4, . . . m for top of Horizon Hn

2. ψ = Coulomb shear collapse angle, typically 67 to 70 deg

3. ρn = dip of the growth axial surface in Figure 11-44, which in this case is the angle π–ρ

4. τn = frontal dip of the rollover structure defined from inflection points, located near the crest and base of structure for the top of Horizon Hn

5. Ghn = total thickness of growth sediments for the base of Horizon Hn (which has a thickness Gn), which is equal to the total depth(Zn) of the horizon being studied. The Ghn depth is taken relative to a datum depth that can exist below the surface or sea floor. The chosen datum can affect the accuracy of Eq. (11-28).

One exception to Eq. (11-28) is that in multibend or nonlinear structures, if the right side of the triangle is not ρn and is replaced by ρm, where m = a, b, c, . . .z, then, Eq. (11-28) becomes

(See sections The Growth Axial Surface Increases Dip and The Growth Axial Surface Decreases.)

Formula for Down to the Left Faults.

This equation follows the same formulation as Eq. (11-28), and its derivation is also

but with 0 on the right.

Measured and Calculated Values on Graphs.

For measured values, the figures were made larger at a scale of 200% to 400%, and the Rn/Tn values are measured directly with a scale ruler and a magnifying glass, resulting in a measurement error of about 1/60. For syntectonic structures, defined by Eqs. (11-28) to (11-30), the calculated and measured values are compared to each other on the graphs. The posted calculated values on all of the graphs use angular measurements defined from the right-hand side of Eqs. (11-28) to (11-30) (calculated values on Fig. 11-45). Notice that from the left-hand side of Eqs. (11-28) to (11-30) is the ratio

This ratio is measurable. Thus, the measured values for Eq. (11-31) are first obtained by measuring the lengths of Rn and Ghn. The left-hand side of Eqs. (11-28) to (11-30) is then determined from Eq. (11-31) by dividing Rn by the measured value of the total growth (Ghn), or depth(Zn), from a chosen datum to the horizons being studied. The Ghn = depth(Zn) values are much larger than the Rn values.

Syndepositional Growth Models

In nature, syndepositional growth structure is more common than pregrowth structure and is structurally more complex. For example, geoscientists have recorded growth on normal faults using expansion indices (Thorsen 1963) or throw (Tn) versus depth(Zn) diagrams (Bischke 1994b; Sanchez et al. 1997; Tearpock and Bischke 2003). Expansion indices (Ei) typically compare downthrown horizon or sequence package thicknesses (Gn) divided by the equivalent upthrown sequence thicknesses, here defined as Gn/Gfn (see Fig. 11-46 for the definition of depth(Zn) and Gn/Gf). An alternative is to correlate thickness on the crest of the rollover structure to upthrown thicknesses, here defined as Gc/Gf (Fig. 11-46, right-hand side). If the growth is roughly uniform, then Eq. (11-18) in the Pregrowth Structure and Calculations section can be used to confirm the correlations. If the growth axial surface changes rapidly, then Eq. (11-28) is modified to correct for the change in growth (Fig. 11-52).

This discussion starts with the simplest models first, by developing models that have uniform or a small amount of growth. For this low-growth analysis, the late and rapidly growing sediments infilling the most recently deposited sediments are not included in any calculation. For example, if the thickness of Gf is about 0.0 in the models, then the expansion index calculations Gn/Gf would be near infinite, making the % difference evaluations useless. Thus, Horizon H1 (Fig. 11-46) will be included only when investigating large-growth sedimentation models. The initial models will concentrate on simple uniform growth. These models have a straight-line growth axial surface (ρ) (Xiao and Suppe 1992) and where no variations in sedimentary thicknesses exist. This assumption permits for a simple understanding of the methods, which are then made more realistic. Typically, lower R-square values indicate a more complex model, and thus R-square values are reported in the text. Understand that the R-square values reflect the correlation calculations!

Uniform Growth

In the simple uniform sedimentation model (Fig. 11-46), the measurements and calculations start at the top of Horizon H2. In this model, the downthrown stratigraphic horizon thicknesses do not change, and the growth and corresponding Rn/Tn measurements are linear. The geometry of syndepositional structures expand, and the horizontal width of the structure grows wider with depth. The growth axial surface (ρ) defines the top or crest of structure (Xiao and Suppe 1992) and is a constant in this model (Fig. 11-46). If (ρ) appears to be linear, as in Figure 11-46, then the linear regression results will exhibit high R-square values, and the growth will be linear. This occurs in real or modeled data. If the growth axial surface does not change dip, and the horizons (Gn) do not change thickness, then the expansion index Gn/Gf does not change and the growth is uniform. Again, the vertical distance between the top, or crest, of the structure and its flank, which is the depth at which the beds no longer dip, define the structural relief (Rn). Thus, R1 is the vertical distance between a horizontal line drawn at the crest of structure (that terminates at the position of the growth axial surface) and a point on the flank of the structure where the horizon is roughly horizontal. Again, we assume that the position of the top of Horizon H6 is not clearly known (Fig. 11-46).

Figures 11-47a and b are an Rn/Tn and Rn/Ghn versus depth(Zn) plot. The mathematical formulation of expansion faults is covered in the Pregrowth Structure and Calculations section. Notice that the Rn/Ghn plot (Fig. 11-47b) does not require footwall data, as Ghn exists only in the hanging wall. The measured structural relief(Rn)/throw(Tn) and calculated Rn/Ghn values are near linear with an R-square very close to 1. Measurement errors are the cause of the imperfections. The misinterpreted horizons 6a and 6c clearly deviate from the more linear trend on Figure 11-47a. Ratios, such as Rn/Tn versus depth(Zn) graphs, are more sensitive than direct measurements of Rn and Tn, both of which can contain minor errors. Studies of the models generally show that the structural relief(Rn) versus depth(Zn) graphs tend to have the highest R-square results from the linear regression. Figure 11-47b contains the same data plotted in Figure 11-46, but the data are calculated using Eq. (11-16) and angles. Growth measurements (Ghn) is always measured from the datum for horizon Gn to the base of the horizon being studied [Fig.11-50 and Eq. (11-13)]. In this data set, the direct measurements result in an R-squared of 0.9919 (Fig. 11-47a). The calculated data from Eq. (11-18) result in an R-squared = 1.0 (Fig. 11-47b). The result is constant, as the angles do not change between horizons. The measured values result in an R-square of 0.9, which results from direct measurement errors (consult Measured and Calculated Values on Graphs for definitions of measured versus calculated values). Data with a positive slope on these Rn/Tn diagrams means that the structure is growing larger with depth(Zn). As a result of this growth, the sedimentary wedge grows thicker. On Figure 11-47a, the total depth to the top of horizon 6b is 10.3 units. The position of the R5/T5/base depth(Z5) (which defines the top of horizon 6) value has a unique position on the linear, calculated uniform growth graphs (see the Gh1 = 2.45 and Ghn = 4.45 calculated depths on Figure 11-46). Thus, by projection, the top of Horizon H6b is the correct horizon top for uniform growth, and not horizons H6a or H6c.The data set need not be perfectly linear to conduct this projection exercise. An R-square near or at 1.0 means a high correlation is involved in the projection process.

In normally faulted structures, delta (Δd)/d (change in depth/depth) plots of stratigraphic sequences are also often near linear, indicating near-linear stratigraphic growth within sedimentary sequences (Bischke 1994b; Sanchez et al. 1997; and Bischke et al. 2006). Thus, near-linear growth is not uncommon (Tearpock and Bischke 2003) (Figs. 13-8, 13-9, 13-11, 13-12 13-14, 13-17, and 13-21).

The graph represented in Figure 11-48 is similar to the graph shown in Figure 11-46, except that the displacements are larger and horizons H7, H8, and H9 have moved through the active axial surface into an inactive zone. In this case, these horizons have moved through the bend in the fault bend defined by (ϕ), and these horizons dip onto the deep portion of the fault defined by α (Fig. 11-46). Notice that the calculations for throw(Tn) on Figure 11-48 for horizons H7 to H9 do not change and have the same values. The constant values mean that the region to the right of the active axial surface is inactive and lacks any deformation.

Low Variances in Growth Sedimentation

Similarly, if the sedimentation differences have relatively low slopes, then the growth plots are nearly linear (Fig. 11-49) and are similar to Figure 11-47a. As the growth approximates uniform growth, the calculated Rn/Tn values have an R-square = 0.99, and thus this model, like the previous model, is relatively simple. Hereafter both the growth and sedimentation variances are posted on Figure 11-49 and on all of the remaining higher-growth figures. The following discussion on growth contains % differences that are much larger than low-growth sedimentation differences.

The Rn or Tn % differences have no meaning on growth structures, as the Rn or Tn values often increase and have a larger value with increasing depth(Zn). However, the horizon thicknesses Gn or the Ghn/Gfn values can define differences in sedimentation thicknesses that affect growth and its linearity. Thus, Gn % differences will define sedimentation variations on growth structures. On Figure 11-49, the Gn % differences have a 27% variation.

High Variations in Growth

Thorsen (1963) showed from correlations of well log data that expansion index Ghn/Gfn defined displacements on listric growth faults. The growth typically increases with increasing depth, reaches a maximum, and then decreases (Fig. 13-3). Thus, growth on listric faults is typically not random. Nevertheless, we test a nonuniform, or random, sedimentation model on a relatively high horizon thickness Gn, hanging wall % difference data (Figs. 11-50 and 11-51a and b) scenario. As a result of the large % difference values of 113%, the Rn/Tn data are less reliable with a measured R-square of 0.9335 (Fig. 11-51a) that indicates lower correlations, but this R-square value could indicate a less certain interpretation.

However, the calculated Rn/Ghn plot (Fig. 11-51b) is near linear with a correlated R-square result = 0.988 even though the Gn % difference variance is 113%. When the % difference data exceed about 50% to 70%, the correlation of the data may or may not significantly decrease. Also, posted on Figure 11-50 are the rounded measured and calculated values for R2/Gh2 and R2/Gh3 and the Rn/Tn values. (See Measured and Calculated Values on Graphs for the definition of calculated and measured values).

Figure 11-51c tests the accuracy of the correlation, projection method assuming that the position of the top of Horizon H6 is unknown. This is accomplished by excluding Horizon H6 data from the data set and using only the H2 to H5 data on the plot. The recalculated Rn/Ghn values are shown in Figure 11-51c (notice that a comparison of Fig. 11-51b and c show slight differences in the y, Rn/Ghn, and R-square values). Thus, it is possible to predict the top Horizon H6 by evaluating the two y/x equations in Figure 11-51c. Simply insert the x = depth value for the top of Horizon H6, which is 9.25, into the two equations (Fig. 11-51c). The measured and calculated solutions are 0.49 and 0.476 respectively. Next, the relative % errors can be determined from the results of the two plots by (1) taking the measured and calculated values from Figure 11-51b, then (2) subtracting the values from the Figure 11-51c results, and (3) determining the % relative error by multiplying the difference by 100. The measured and calculated relative % errors are 1.8% and 0.085% respectively (Fig. 11-51c).

The Growth Axial Surface Increases Dip

In the modeled examples described, the growth axial surface remains constant. Figure 11-52 shows an example of the growth axial surface changing dip and dipping at a lower angle, thus increasing the growth-angle (ρ1) dips at 139 deg and then dips at ρ2 = 153 deg at greater depths. In Eq. (11-28), the ρ1 and ρ2 angles are subtracted from (ψ) = 70 deg. Thus, there are two sources of changing growth in Figure 11-52: the change in the horizon thicknesses and the change in the dip of the growth axial surface. A more gently dipping growth axial surface means that the growth decreases (Xiao and Suppe 1992). On seismic sections, this change in the growth axial surface is recognized by a shift in the position of the crest and flank of the rollover structure. In this case, the crest of the rollover shifts to the right. If, however, the growth increases, then the growth axial surface bends downward, and the crest of the structure narrows and shifts to the left (Fig. 11-54). Furthermore, this change in growth creates a discontinuity in the data (Fig. 11-53a), indicating that the problem is nonlinear. This nonlinearity must be adjusted for in the measurements and calculations. Typically, solutions to nonlinear problems are not trivial.

This topic is also briefly discussed in the Pregrowth Structure and Calculations section, where a detailed derivation of Eq. (11-18) is presented. The following is a description of how to calculate the right-hand side of Eq. (11-28) to adjust for nonlinearity. Notice on Figure 11-52 that the growth axial surface (ρ1) for the crest of horizons H1 and H2 does not change position. Thus, (ρ1) does not change in the calculations for the first two horizons. The solution for the top of Horizon H3 in Eq. (11-28) is to form a triangle that includes the angles (ρ1), ψ, and the value of the bed dip τ2 [use radians in Eq. (11-28) for a computer-generated solution].

At the crest of Horizon H4 (or the bottom of Horizon G3), the growth axial surface abruptly changes dip from ρ1 = 139 to ρ2 = 153 deg, and this change in the growth axial surface shifts the position of the crests of horizons H4 and H5 to the right. In order to adjust for this nonlinear behavior, the triangle formed does not include the growth axial surface (ρ2) but rather the angle ρa = 142.5 deg. Thus, in Eq. (11-28), the ρm replaces the angle ρn, and Eq. (11-29) contains the term ρm − ψ in the numerator (consult Fig. 11-52 for variable ρ values). The lengths M3 and N3 and the angle ρa = 142.5 deg are shown in bold on Figure 11-52. Also, notice that for the base of Horizon H4, the value of the growth axial surface changes to ρb = 145 deg, both N4 and M4 lengthen, and τ4 increases. This change must also be incorporated into Eq. (11-28) [see Eq. (11-29)], and the values can change with every horizon. However, this procedure corrects for the nonlinearity of the growth.

Figure 11-53a and b shows the measured and calculated results from Figure 11-52 for an Rn/Tn and Rn/Ghn versus depth(Zn) graphs. Figure 11-53 has a relatively large Gn % difference of 61% and a growth axial surface that changes from 139 deg to 153 deg, which has the result of decreasing sedimentary growth. These large variability changes have the effect of creating a distinct discontinuity in the tops of horizons H3 and H5 on the Rn/Tn graph (compare the tops of H2 to H4 and H4 to H5 on Fig. 11-53a). In spite of the discontinuity on Figure 11-53a, the Rn/Ghn R-square valves are near unity.

Increasing or decreasing variable growth is present on many rollovers that exhibit a divergent bed dip pattern and is not uncommon (Fig. 11-38). However, real structures can have a high degree of correlation similar to the modeled values. The number of different-sized triangles can vary for each sedimentary sequence on some rollover profiles. To solve for nonlinear growth, one simply recognizes where the growth axial surface is located on the seismic section. The growth can increase or decrease several times, and this changing growth can be calculated by adjusting the sides and angles of the triangle. Also recognize that even if the structure is nonlinear in its formation, precise information on model-derived Rn/Tn data are available in the nonlinear portion of the curve between imaged horizons. For example, on Figure 11-53a, an equation exists for horizons H4 to H5. This follows as two points determine a straight line!

The Growth Axial Surface Decreases Dip

Figure 11-54 is an example of the effects of increasing horizon growth on linearity of the regression analysis. This increasing growth axial surface model was created by decreasing the dip of the growth axial surface in Figure 11-54. In this new model, the growth axial surface ρ1, which is 139 deg, is the same as in Figure 11-52. However, ρ2 is decreased in dip to 130 deg from 153 deg in Figure 11-52. This change in the dip of the growth axial surface also makes the problem nonlinear like Figure 11-52, and the problem is solved as described in the prior section, The Growth Axial Surface Increases Dip. The equation shown on Figure 11-52 determines the values of the plot in Figure 11-55b. Also, this change in the dip of the growth axial surface causes the crest of the structure to narrow, and the crest of structure shifts to the left.

Relative to Figure 11-52, Figure 11-54 has a lower Gn % difference = 46% and has a relatively high calculated linear regression R-square of 0.9516 (Fig. 11-55a). Some seismic lines image a decreasing growth axial surface. When the % difference values approach about 50%, the R-squares tend to be lower, as seen in Figure 11-55a. Nevertheless, one could argue that a 5% error is better than guessing. In Figure 11-55b, the calculated and measured linear regression R-square values are 0.9975 and 0.9821 respectively; these values are not significantly different. The similar R square-linear regression results are characteristic of the Rn/Ghn graphs.

Figure 11-52 shows how to correct for nonlinear data measurements. In the model, the growth axial surface changes from ρ1 = 139 deg degrees above Horizon H3 to ρ2 = 153 deg degrees at depth. The change in the dip of the growth axial surface causes the limb of the structure to widen with increasing depth. A triangle must be formed in order to employ Eq. (11-29) and solve for one of the data points. This triangle is shown in bold for the top of Horizon H4 and is defined by the growth axial surface ρa = 142.5 deg, N3, M3, and Gh3 in Figure 11-52.

Large Growth, Multibend Structure

This multibend model contains large growth variations as the frontal limb dip angles change from 12 to 32 deg (Fig. 11-56). In this model, there are two fault bends and two active axial surfaces emanating from the bends. Also, for each horizon, there are two dips that define the structural relief and a growth axial surface equal to 150 deg. Consequently, the structure has two distinct dip domains. Therefore, each domain produces two different structural reliefs (Rnm) and (Tnm) for each throw. The two structural reliefs are defined as Rna for the shallow dip domain and Rnb for the domain near the fault that is deeper. This means that Rna/Ghna depths are shallower than the Rnb/Ghnb depths (consult Fig. 11-56 for the posted, measured, and calculated values for R1a/Gh1a and R1b/Gh1b). Notice that the Rna dips nearer to the crest of structure are smaller than the Rnb dips. Thus, there are two groups of problems to be solved in Figure 11-56. This geometry is similar to some large growth faults (Fig. 11-16).

Domain 1.

The first domain group of solutions involves the structural relief of domain R1a. This involves the growth axial surface ρ 1a = 150 deg, N1a, and the active surface (ψ) to form the first triangle. The second problem also involves the growth axial surface related to ρ 1a dipping at 150 deg, the active axial surface, and N2a. The third set of solutions involves the triangle formed by ρ 1a and ρ 3b dipping at 110 deg and N3a. and not the angle (ψ) (Fig. 11-56). As the left-hand side of the triangle is not the angle (ψ) but ρ3b = 110 deg for this triangle, (ψ) is replaced by ρ3b = 110 deg when formulating Eq. (11-29). Also, for the measured values of Rna/Ghna, the Ghna values are measured at 1.5, 3.5, and 5.95 units.

Domain 2.

The second domain Rnb/Ghnb are solved next. The three triangles are formed by ρ 1b = 158 deg, N1b, and the shallowest active axial surface (ψ) at the first bend in the fault. Next, ρ 2b = 131, Nb2, and the active axial surface form a triangle. The last problem involves ρ b3 dipping at 110 deg, Nb3, and the active axial surface located at the second bend in the fault (Fig. 11-56). Horizon H4 is in the inactive zone of no deformation and has the same values as H3 (Fig. 11-56). In this case, the Ghnb values are larger than Ghna values in the first domain.

As there are two dip domains for each horizon, in order to determine the correct values of Rn/Ghn, the value of R1a/Gh1a is added to the value of R1b/Gh1b values. Figure 11-56 shows the measured and calculated values R1a/Gh1a and R1b/Gh1b along with the sum of the two values.

The Pregrowth Structure and Calculations section provides additional discussion on changing growth axial angles and how to model changing growth axial surfaces on real rollover structures. The large Gn variations in this modeled structure test the viability of the Rn/Ghn projection methods (Figs. 11-56 and 11-57). The Rn/Ghn value calculations using Eqs. (11-28) and (11-29) have an R-square of 0.9986 even though Gn = 82% (Fig. 10-57). This linearity at high Gn values is probably the result of a constant growth axial surface. In this case, the measured R-square is equal to 0.9999 and is a direct linear measurement, measured with a scale. Real structures (e.g., Figs. 11-61, 11-62b and c) also can have high R-squares, as do modeled structures. However, the growth model study, which includes other models not shown, suggests that the projection methods are often less reliable if the Gn % difference calculations exceed about 50% to 70%. The study of model structures predicts that the only structures that cannot be projected to greater depths are structures in which random horizon thicknesses (Gn) are present or in which the growth axial surface (ρn) changes significantly.

Thorsen (1963), and other work on extensional growth structures (Bischke et al. 2006), suggests that random structures are not common. An exception would be structures that rapidly increase or decrease in growth with increasing depth or contain large unconformities. However, many real structures, described in the following Real Structures section, have near-linear R-squares.

In summary, as many real rollovers have curved frontal limbs, the two-bend models look more realistic [Fig. 11-56 has two fault bends (ϕ) of 20 deg each]. Each bend requires a τn and ρn calculation or measurement. How to calculate this type of structure is also covered in the Pregrowth Structure and Calculations section, but the calculations are relatively simple when using a spreadsheet. Every change in dip requires an axial surface (ψ). Again, if the left-hand side of the triangle is not the angle (ψ), but the angle ρμ, then for this triangle, ρμ replaces (ψ) in Eq. (11-28); in this case, use Eq. (11-29). Notice that on Figure 11-56, the structure moved through the deepest active axial surface and entered the inactive zone.

Real Structures

Real structures can be observed in both seismic and well log data. Seismic data is described first. Published public domain 3D, depth-corrected seismic data are not common, probably for several reasons, including proprietary issues. First, vertically expanded data, such as time data, expand the vertical component of the time section, making correlations and sequence boundaries more obvious. Second, depth correction takes more time. In contrast, in compressional regimes, non-depth-corrected seismic data can lead to major interpretation problems (Chapter 10), although this distortion of geometry is less problematic in extensional regimes. Nevertheless, simply converting a depth-corrected seismic line with a local or regional velocity/depth function permits geoscientists to view the prospect on a scale that is closer to one to one and presents a less distorted view of the actual geometry. Chapter 9 discusses in detail the modern depth-correction process, mapping methods, and techniques involving several 3D shale basin data sets.

Seismic

Southern Louisiana Rollover.

Figure 11-58a is a measured Rn and throw(Tn) diagram from an older depth-corrected 2D line taken from Xiao and Suppe’s (1992) paper on the origin of rollover. This seismic line was one of the first balanced lines for extensional structures. Horizon H6 was not included in this calculation, as its base is a pregrowth horizon. Horizon 5, which is a multibend horizon, was divided into two-line segments with different dips and depths (Zn) (see Fig. 11-56 for an example). Four active axial surfaces, ψ = 67 deg, were constructed at the position where the base of divergence of horizons H2 to H5 intersect the fault surface. The throw(Tn) correlations of y to x is high, and thus the R-squares results are high at 0.973, indicating roughly linear slip with increasing depth (Fig. 11-58a).

The base of horizons H2 to H4 have a growth axial surface with ρ = 139 deg, although the base of Horizon H5 shifts to the right and the value for ρ2 = 164 deg. This makes the rollover slightly nonlinear. However, both the measured and calculated Rn/Ghn data have a high R-squared even though the horizon thickness Gn % difference = 67% (Fig. 11-58b). This high variance is caused by the large change in the growth axial surfaces (ρn) and the horizon thicknesses (Gn) that change with depth. The high R-squares result from a near-uniform growth axial surface (ρn) for the base of horizons H2 to H4. In Figure 11-58b, the calculated and measured linear regression values are 0.996 and 0.965 respectively. Remember that for the left-hand side of the shallow triangle on the base of Horizon H5, which bends from τ = 5 to τ = 12 deg,ψ must be replaced by ρa = 98 deg in Eq. (11-29). This rollover expands and the horizons diverge or thicken with depth (consult the Gn numbers in Fig. 11-58b).

Next, let us assume in the case of this near-linear structure that a sand or high TOC shale exists in a footwall well below the top of Horizon H4 at a depth of 3.18 km (see Xiao and Suppe 1992, their Fig. 20, at 3.18 km) but has not been penetrated in the hanging wall. As the measured data are near linear, and the base of horizons H2 to H5 were mapped in the hanging wall and footwall, it is possible to estimate the depth of this high TOC horizon in the hanging wall within some degree of accuracy. This accuracy can be estimated using the determined R-square values and then comparing these values to a perfect R-square value of 1.0.

First, the % difference of the footwall depths are calculated for Horizon H4 at 2.92 km and the horizon of interest by calculating the depth to the top of Horizon H4 and subtracting the depth to the horizon of interest. This value is then divided by the Horizon H4 thickness (as shown in Fig. 11-59), which equates to 0.704% for the footwall. Thus, in the hanging wall, the high TOC zone is located at about at a thickness of 70.4% below the top of Horizon H4. Based on near-linear growth, Figure 11-59 also shows the estimated depth to where the high TOC horizon is located in the hanging wall. This calculated depth near the fault is at about 4.2 km (Fig. 11-59).

There is a check on this work that involves the linear regression y-x formula. As the measured data are near linear, the best depth estimate of 4.2 km for the TOC horizon is inserted into the growth formula. Next, the Rn/Ghn value times its depth should plot as a point along the measured near-linear trend (estimated depth on Fig. 11-58c). The y-x formula values were calculated before the insertion of the depth of the TOC horizon at 4.2 km in the hanging wall. The estimated value for Rn/Ghn = 0.1473 should lie along the measured data trend line (Fig. 11-58c).

Corsair Trend/Brazos Ridge Offshore Texas

Figure 11-60 is a tracing of a depth-corrected seismic line drawing of horizons construction by Xiao and Suppe (1992) with the help of a major oil company. A good discussion of the major producing Corsair trend, offshore Texas, is found in Vogler and Robison (1987). The 2D line is imaged in Figure 11-58a, b, and c.

Figure 11-1a is from the large Corsair fault, which exhibits obvious bed dip divergence associated with large changes in the dip of the Corsair fault and large throws exceeding 10,000 ft (3 km) (Christiansen 1983). The beds dip toward the fault with maximum dip of 24 deg at a depth of 2.5 km. The depth-corrected line tracing was constrained by numerous deep wells that were drilled into the Big Hum, producing zone to depths exceeding 17,000 ft (5 km) (Vogler and Robison 1987). The line drawing generated by Xiao and Suppe is published in Tearpock and Bischke (2003) (Fig. 11-1). In Figure 11-60, the line drawing was slightly restored by removing crestal faults with small displacements (Fig. 11-12). The growth on this large rollover decreases with depth, as ρ1 dips at 135 deg at shallow depths, and ρ2 dips at 161 deg at deeper depths (the dashed lines on Fig. 11-60).

In Figure 11-61, measurements taken from this two-bend depth-corrected line is presented in the form of as Rn/Ghn graph. In spite of the changes in the growth axial surfaces and large thickness changes (the Gn % difference = 100%), the R-squares are near 1.0 (Fig. 11-60).

Haynesville Shale Play, Texas/Louisiana

The next example is a 3D depth-corrected line from the Haynesville shale play from Texas/Louisiana (Fig. 11-62a). This salt-related example images a fault that bottoms in a salt weld at about 15,500 ft (4.7 km). The first three horizons exhibit linear growth with the beds dipping from 4.5 to 7.5 deg. The measured and calculated Rn/Ghn values have R-squares at about 1.0, with horizon thicknesses of Gn % difference = 40% (Fig. 11-62b). The slope of the graph is positive, indicating increasing growth with depth (flat to dipping beds).

The three horizons above the salt weld also exhibit linear growth, but in this case with a negative slope indicative of decreasing growth with increasing depth (Fig. 11-62c). Above the salt weld, the bed dips range from 9.5 to 10.5 deg. The measured R-square values are also at about 1.0, with a Gn % difference = 58%. Horizon H4 is a probable unconformity, does not fit the two linear trends, and exhibits decreasing growth.

Starting at the base of Figure 11-62c and working toward the surface, an interpretation of the tectono-stratigraphy is as follows. At first, the sediments loaded the salt as the sediments thickened upward to Horizon H5 (increasing growth, growth is in dimensionless time; see Chapter 13). The thicken upward sequence was linear and ended at Horizon H5. There was a change in growth resulting in the unconformity at Horizon H4. Then the growth decreased linearly (Fig. 11-62b) until the sedimentary loading expelled all of the salt, creating the weld. This event happens at the flat horizon above Horizon H1 (see Fig. 11-62a). This sequence style was first described by Thorsen (1963) using the expansion index (Ei) for the Norco, Bonnet Carre faults and in particular on the LaPlace fault (Fig. 13-3).

Well Log Data

The last example is a near-linear example of growth using well log data. This example uses missing section data from the Rabbit Island Field, Louisiana (Fig. 11-63a, interpreted by Dan Tearpock). The wells cut a downward dying fault and its fault surface map along with a seismic line is published in Tearpock and Bischke (2003) (see Fig. 13-27). At shallow depths, the missing section values increase, and then decrease below a depth of about 5600 ft (1 ft = 0.3048 m). The seismic line exhibits coherent data at about 11,000 ft, and a well at this level contains a complete stratigraphic section with no faulting to trap hydrocarbons (Tearpock and Bischke 2003). The missing section data are not precisely linear (mostly the result of the deepest correlation point) but are readily fit to a second-order polynomial with an X-square coefficient of only 7xE-06. However, if the missing section data are analyzed from the 5700-to 9700-ft level, the data are nearly linear (Fig. 11-63b). This portion of the Rabbit fault slipped at a near constant distance.

Unconformity in Shale Basin

Figure 11-64a shows a shale basin 3D seismic line with a large unconformity between the H3 and H4 horizons, indicating that not all growth is near linear and that unconformities can cause nonlinear growth. Figure 11-64b is a plot of the growth discontinuity (Bischke 1994b) in which Horizon H3 is clearly offset from the averaged R-square general trend using all four data points.

Most of the generic and real examples shown exhibited near-linear coefficient correlations of the crestal data. The example in Figure 11-64a is more complex and exhibits a changing growth axial surface (red lines), and thus this example has high R-squares (Fig. 11-64b). A large unconformity offsets the structure along the near horizontal portion of the dashed red line, as the unconformity shifts the crest of structure to the left.

Conclusions

This study started by discussing linear growth in general and then concentrated on which factors affect linear growth on generic models, finally applying these results to several real data. Many real examples were subject to the linearity tests. Growth becomes less linear if the % difference calculations exceed about 50% to 70%. Data become less linear if the growth is large and variable, alternating between increasing then decreasing growth, or if the growth axial surface (ρ) changes dip. Many seismic sections exhibit nonlinear growth, particularly if the section contains a large unconformity (Fig. 11-64).

Public domain, depth-corrected seismic data are not common in the literature, and thus firm conclusions regarding near-linear growth on numerous real structures is unavailable. The section Real Structures and examples presented in Chapter 13 on growth structures show that near-linear growth exists in well log data and on some older, depth-corrected 2D seismic data. Our study of 3D seismic and depth-corrected seismic show that episodes of near-linear data, obtained from Chapter 9, exist in shale play basins.

Linear growth may have applicability for the following reasons. First assume that linear growth is present in both the footwall and the hanging wall. If an interesting horizon, for example, either a hard shale or a high TOC section, is present in the footwall but has not been penetrated in the hanging wall, then projecting the footwall interval to depth in the hanging wall is possible. Next, run a linear regression analysis of the depths in the known footwall intervals. This analysis results in a y-x equation. The depth of the interesting interval in the hanging wall will lie on a Ghn/depth (Z) graph of the linear equation.