From intuition to statistics in building subsurface structural models
Combining forward trishear models with stochastic global optimization algorithms allows a quantitative assessment of the uncertainty associated with a given structural model.
J. P. BRANDENBURG, FARUK O. ALPAK, STEVE NARUK and JOHN SOLUM, Shell
Elements of fault propagation folds (such as toe-thrusts and detachment folds) are often incompletely imaged in seismic surveys. Steeply dipping to overturned bedding and the areas adjacent to faults are particularly problematic, and are often associated with data wipeout zones. In this article, forward models of fault propagation folds using the trishear algorithm are used to generate and statistically analyze such families of results. The methodology is applied to examples of incompletely imaged structures related to deepwater hydrocarbon reservoirs, and results are compared to prior manual palinspastic restorations and borehole data. In addition to extrapolating existing data into seismic data gaps, this methodology is useful for extending structural interpretations into other areas of limited resolution, such as subsalt. This technique can be used for rapid reservoir appraisal, and may have other applications for seismic processing, well planning and borehole stability analysis. BALANCED CROSS-SECTIONS Creation of balanced geological cross-sections is required to validate subsurface models in structurally complex areas.1 This time-consuming process depends heavily on the intuition of the interpreter. The result is typically a single balanced section, which in reality is one of a family of similar results. Numerical models of growth of folds and faults, combined with history-matching algorithms from the reservoir engineering and geophysical communities, can be used to build statistically constrained families of balanced sections, thereby reducing uncertainty in structural models. For complex folded and faulted oil and gas fields, a correct structural interpretation is important to understand trap geometry, reservoir distribution and compartmentalization. This process is difficult, even with high-quality data. Borehole data, such as core and logs, provide very accurate localized information, while seismic data provide spatial coverage but lack sufficient spatial detail. A key step in validating structural subsurface models is to restore the reservoir layers to their undeformed state. A cross-section that can be restored without leaving gaps or overlaps is said to be balanced. If the model cannot be restored as a balanced cross-section, it contains fundamental inconsistencies that can lead to suboptimal development decisions. The first steps in this process depend on the modeler making some educated guesses.
A challenge often faced during this process is the interpolation of picked horizons through areas of ambiguous or missing seismic data. In a number of situations, robust seismic reflectors can become completely obscured due to complex geological structure. This is a common effect in the forelimb of fault propagation folds, where steeply inclined bedding and offset across faults scatter acoustic energy, leading to data wipeout zones, Fig. 1. As steep dips and offsets produce similar effects, the interpreter is left to make a judgement call about how much of the wipeout to attribute to faulting and how much to folding. Stratigraphy and bedding inclination inferred from borehole logs can be used to validate and refine the interpretation, but this still requires the use of a balanced structural model to connect the highly localized borehole information with the larger-scale but incomplete seismic data.2 Finally, although a final section may be balanced, it is, in reality, one member of a family of similar balanced sections that can account for the same structure. RELATING FAULTS TO FOLDS A number of methods are used to connect discontinuous data in the process of balancing cross-sections. Although these methods vary in complexity and sophistication, they all relate the final structure to an inferred process by which it was formed. Classic techniques invoke geometric rules for the present-day structure, such as conservation of line lengths across multiple faults. More contemporary calculations involve a forward model that explicitly represents the process by which the structure evolved. One such forward-modeling technique in frequent use is the trishear method.3,4 Trishear is a kinematic representation for the incremental folding of strata ahead of an advancing fault tip, behind which the fault is a freely slipping plane. This is a parameterization of the process by which anticlines associated with large thrusts (fault propagation folds) develop. Although the physical processes taking place near fault tips are significantly more complicated, the larger-scale mechanical folding process has been shown to be well approximated by trishear.5
A limitation of trishear is that it was originally formulated for a planar fault; the present implementation uses radial coordinate remapping to connect a curved fault to a horizontal detachment (Fig. 2), allowing for modeling of more geologically realistic geometries. This remapping is similar to reconstruction methods that invoke a radius of curvature for the curved portion of the fault,6 and is an improvement over other methods that require an abrupt change in fault orientation.7 Such highly curved fault geometries are more appropriate for offshore fold-and-thrust-belt hydrocarbon plays such as in the Gulf of Mexico, deepwater northwest Borneo and deepwater Nigeria. CHOOSING MODEL PARAMETERS As with most structural tools, trishear contains several tuning parameters. In addition to a variety of parameters defining the geometry of the fault (such as fault dip, depth of detachment and, in this case, radius of curvature), two model-specific parameters control the geometry of the related folding. These are the propagation-to-slip ratio (P/S: how far the tip of the fault advances per increment of fault slip), and the trishear angle (how broadly the folding is distributed adjacent to the fault).8 These are typically chosen by trial and error during cross-section balancing. Since the trishear algorithm is not computationally intensive, it is possible to systematically generate and compare large numbers (on the order of 10,000–100,000) of models. Because of this, inverse methods have been successfully used to solve for best-fit tuning parameters via grid search,4,9,10 gradient optimization methods11 and stochastic global optimization methods.12 Quality of fit is usually expressed as the magnitude of an error calculated as a cumulative offset of the final trishear model from the observed stratigraphic horizons, although angular information such as structural dips can also be incorporated.13 The goal of any of these approaches is to find a set of values for the trishear parameters that minimizes this error. However, both the manual estimate and computed choices for parameters suffer from issues concerning the non-uniqueness of the solution. For any acceptable trishear model, there is a family of similar results that satisfy the incomplete seismic and wellbore constraints with a similar quality of fit.13 Moreover, trishear posed as an inverse problem contains a number of local minima that affect the efficiency of any gradient-based optimization.11 Although many of the local minima may be discarded as physically meaningless, this still indicates the inherent non-uniqueness of the approach. Rather than a hindrance, this non-uniqueness can be exploited to quantify the actual uncertainty of the solution12 and, therefore, of the trap itself. Because of this presence of local minima, the present implementation approaches the problem using stochastic global optimization techniques.12,14 Stochastic global optimization combines systematic decomposition of the parameter space with random walks, such that the algorithm will visit many local minima in its search for one optimal global minimum and, therefore, can avoid the trapping problem inherent in gradient methods. Such methods are used in geophysical signal processing14 and for tuning geological parameters to match production data in reservoir models.15 The present implementation applies the “neighborhood algorithm” stochastic global optimization method16 to find optimal values for trishear parameters. Taken at face value, the neighborhood algorithm produces one best-fit solution that is just as non-unique as that from other methods. However, by sampling all of the local minima visited in the search for a global minimum, a family of acceptable solutions can be systematically characterized and ranked.12
FAULT PROPAGATION FOLDS WITH OPTIMIZED TRISHEAR To test this application, stochastic global optimization was used to analyze two incompletely characterized fault propagation fold structures. The first is a synthetic example reminiscent of some salt-related structures in the deepwater Gulf of Mexico. In this case, nucleation and growth of a thrust fault have been controlled by the presence of a salt roller left by the withdrawal of autochthonous salt, Fig. 3a. In addition to the forelimb data wipeout zone, seismic imaging is further complicated by the presence of an allochthonous salt canopy overhanging the structure, Fig. 3b. The thinning of strata across the fold is interpreted to be pre-kinematic and controlled by the salt roller. In order to reconstruct this structure with a forward model, the starting condition must include this thinning. Previous trishear formulations have mainly assumed flat-lying, parallel strata as the starting condition, and have only considered syn-tectonic growth.17 Here, the pre-kinematic effect of the salt roller is added as an additional model constraint, Fig. 3c. This adds three unknown parameters to be solved for, making a total of nine, which is well within the capability of the neighborhood algorithm. The optimized solution (global minimum) determined by this method is shown in Fig. 3d. The error in this case is calculated as the normalized distance of all control points (green curves in Fig. 3b) to the nearest point in the final trishear model. As expected, the algorithm also identified a number of parameter combinations leading to solutions that were only marginally less accurate. Figure 4 shows this family of solutions ranked by the amount of residual offset relative to the global minimum, or best, result. (For example, models with a 1% deviation are no more than 1% less accurate than the best model.) In this example, the inverse problem is well-constrained and the solutions are clustered tightly around the global minimum. The second example is an application of this method to the offshore Venezuela fold example shown in Fig. 1b.18 This structure is not salt related, but is additionally complicated by pre-kinematic changes in thickness. To account for this effect, the salt-roller approximation is extended to a small pre-kinematic bump near the center of the structure and to a much larger thinning toward the footwall side of the fault. This extension increases the number of parameters in the inversion, which in turn broadens the family of acceptable solutions. The ranked results of the analysis are shown in Fig. 5. As in the previous Gulf of Mexico example, each horizon is now constrained to a zone of probability in the forelimb data wipeout zone. Additionally, the original seismic data in this problem reveal very little about the location of the detachment, which has largely been inferred. The detachment from the original interpretation18 lies within the shallow end of the range of solutions. However, the forward model results do preclude the existence of the red fault in the original interpretation, potentially changing a fault-dependent trap into a completely fold-dependent one.
DISCUSSION Combining forward trishear models with stochastic global optimization algorithms allows a quantitative assessment of the uncertainty associated with a structural model. For example, reservoir volumes in toe-thrust settings can be largely affected by the position of the thrust fault where it is obscured by the forelimb data wipeout zone.2 By applying this methodology, the possible positions of the thrust in this zone can be ranked statistically. This same approach can be applied to assess possible footwall geometries in thrust settings, where steeply dipping bedding both forms potential hydrocarbon traps and completely obscures the seismic image. Likewise, the prediction of steep, seismically unimaged bedding can be useful for well planning and geosteering. ACKNOWLEDGMENTS
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THE AUTHORS |
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J. P. BRANDENBURG joined the Clastic Reservoir Research Team at Shell as a geoscientist in 2008 with degrees in geology from the University of Michigan (PhD and MSc) and Michigan State University (BSc). |
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| FARUK O. ALPAK is a research scientist with Shell International E&P. He holds a PhD degree in petroleum engineering. Before joining Shell, Dr. Alpak worked at the Schlumberger-Doll Research Center as a visiting scientist on mathematical modeling and inversion projects. | |
| STEVE NARUK is currently Shell’s Principal Technical Expert in structural geology. He also serves as Team Leader of both the Clastic Reservoir Research Team and the Reservoir Modeling Foundations Team within Shell’s Expertise and Deployment organization. | |
| JOHN SOLUM joined Shell as a structural geologist in 2007 after serving on the faculty of Sam Houston State University and with the US Geological Survey. He holds degrees from the University of Michigan (PhD and MSc) and from Utah State University (BSc). | |
