# An Uncertainty Model for Fault Shape and Location

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## Abstract

Fault models are often based on interpretations of seismic data that are constrained by observations of faults and associated strata in wells. Because of uncertainties in depth migration, seismic interpretations and well data, there often is significant uncertainty in the geometry and position of the faults. Fault uncertainty impacts determinations of reservoir volume, flow properties and well planning. Stochastic simulation of the faults is important for quantifying the uncertainties and minimizing the impacts. In this paper, a framework for representing and modeling uncertainty in fault location and geometry is presented. This framework can be used for prediction and stochastic simulation of fault surfaces, visualization of fault location uncertainty, and assessments of the sensitivity of fault location on reservoir performance. The uncertainty in fault location is represented by a fault uncertainty envelope and a marginal probability distribution. To be able to use standard geostatistical methods, quantile mapping is employed to construct a transformation from the fault surface domain to a transformed domain. Well conditioning is undertaken in the transformed domain using kriging or conditional simulations. The final fault surface is obtained by transforming back to the fault surface domain. Fault location uncertainty can be visualized by transforming the surfaces associated with a given quantile back to the fault surface domain.

## Keywords

Structural uncertainty Uncertainty envelope Fault geometry realizations Well conditioning## 1 Introduction

In petroleum reservoirs, faults are generally modeled using seismic data, well data and a knowledge of the local geology. Thore et al. (2002) list several sources of uncertainty when modeling faults based on seismic data, but conclude that the main sources are uncertainty in the seismic interpretation combined with vertical and lateral uncertainty arising from the time-depth migration of the seismic data. The interpretation uncertainty is often a consequence of the poor quality of seismic data near faults, together with the fact that faults are usually represented as surfaces, even though they are three-dimensional zones of deformation. The interpretation uncertainty encompasses both the existence of faults, and the location and local shape of the faults. The depth migration uncertainty is due to uncertainties in the velocity model and the actual seismic signal path due to non-horizontal velocity contrasts. The interpretation error is assumed to be independent for each fault, whereas the error introduced by the time-depth migration is correlated between nearby faults. Although the fault-sealing properties and the existence of additional faults are the main sources of uncertainty in reservoir performance, the fault geometry and position also have significant effects, especially for wells located near major faults (Irving et al. 2010; Rivenæs et al. 2005). An uncertainty model for fault position and geometry enables this uncertainty to be updated based on well production data (Cherpeau et al. 2012; Irving and Robert 2010; Seiler et al. 2010; Suzuki et al. 2008).

Previous implementations of uncertainty modeling for fault geometries have been based on a range of fault parameterizations; Lecour et al. (2001) describe an uncertainty model for faults modeled as triangular surfaces. Local variability is examined with a P-field simulation, which can also include a global trend such as a shift, a change in dip or a more general random function. Caumon et al. (2007) use this modeling method to update the uncertainty of a reservoir simulation grid. Hollund et al. (2002) and Holden et al. (2003) also define a stochastic model for fault geometry. They parameterize the fault as a set of fault pillars defining a series of fault segments. However, their Markov chain Monte Carlo-based algorithm has severe performance problems and does not produce geologically realistic realizations (Røe et al. 2010). The fault model can also be parameterized by defining the fault surfaces based on a three-dimensional potential field. Mallet and Tertois (2010) generate stochastic realizations of faults and horizons by perturbing the UVT transform associated with the Geochron model (Mallet 2004). This model enables the conditioning of well data. Wellmann et al. (2010) have a more data-driven approach, where the fault model is updated by perturbing the input data set. This approach is well suited for capturing the uncertainty in the input data set, but underestimates the uncertainty in areas with less data.

The algorithm presented in this paper uses a parameterization where the faults are represented as tilted, regularly gridded surfaces. This makes it possible to use standard surface modeling techniques similar to those used for modeling stratigraphic surfaces as described by Abrahamsen et al. (1991). These techniques include the application of Gaussian random fields for generating surface realizations and the use of kriging to condition the surfaces to the well data. In the uncertainty model presented in this paper, the faults are simulated independently of each other. There is no correlation between different faults, except in the lengthening or shortening effect a fault has on other faults that it might truncate. Similar to Lecour et al. (2001), the algorithm in this paper uses a p-field based simulation technique where simulations are performed in a standard normally distributed domain. The well observations are translated into this domain, and the final realizations are translated back into the real domain using quantile mapping. The transformation is defined by the fault uncertainty envelope, constrained by a pair of surfaces that restrict the location of the simulated fault surface realizations. The smoothness of the simulated fault surface is controlled by the range of the variogram used. However, where Lecour et al. (2001) use a triangulated representation of the fault surfaces, the algorithm in this paper is based on a fault representation where each surface is represented as a function on a regular grid in a rotated coordinate system. Although this representation is not as flexible with respect to modeling complex fault surfaces, it does provide a simple parameterization of the fault surface uncertainty. The use of implicit fault truncations also allows the truncation rules to be updated based on different fault surface realizations. A consistent framework for conditioning the fault surface realizations with well picks and well paths is also introduced. The fault location can optionally be given an uncertainty at the well pick locations. The well paths conditioning points represent positions known to be on a particular side of the fault surface.

This framework provides a flexible and intuitive way to represent fault uncertainty. The method allows for the inclusion of both seismic and well data, and allows both visualization of the well-conditioned fault uncertainty and simulations of realistic fault geometry to honor all the input data. The fault uncertainty model, along with the methods used for well conditioning and stochastic simulation of fault surfaces, is presented in Sect. 2, and in Sect. 3 results from the application of the methods on a simple fault model are shown. Issues that might arise when using the methods in reservoir modeling workflows are discussed in Sect. 4. The work presented in this paper is an extension of the work presented in Røe et al. (2010), and describes methods implemented in the Havana fault modeling tool (Norwegian Computing Center 2013).

## 2 The Fault Surface Uncertainty Model

### 2.1 Fault Parameterization

### 2.2 Prior Probability Distribution for Fault Surface Location

### 2.3 Conditioning to Well Data

Two types of well data are used for conditioning: well picks, which are observations of the fault surface in the well, and well path points, which are points along a well path that do not hit the fault. Well picks may optionally be assigned an observation uncertainty. For each well data point \((x_i, y_i, z_i)\) a corresponding normal distribution \(N\left( \mu ^{*}(x_i, y_i), \sigma ^{*}(x_i, y_i)\right) \) is created in the transformed domain.

#### 2.3.1 Well Picks with no Uncertainty

#### 2.3.2 Well Picks with Uncertainty

#### 2.3.3 Well Path Points

#### 2.3.4 Well Conditioning of Fault Surfaces and Fault Uncertainty Envelopes

### 2.4 Unconditional Fault Simulation

### 2.5 Conditional Simulation

## 3 Results

As can be seen from the results, the method presented gives realistic-looking fault surface realizations that honor all the input data without showing any bulls-eye effects near the well picks. As noted in Røe et al. (2010), the stochastic simulation algorithm is efficient and well suited for use in Monte Carlo methods.

## 4 Discussion

### 4.1 Handling of Fault Truncations

The extent of the faults is specified with a fault tip polygon. Depending on how this polygon is defined near truncations, the faults might be moved so that they do not touch, eliminating the truncations. Faults might also be moved in such a way that new truncations are introduced. In the authors’ opinion, this is a desirable attribute of the model since it allows examination of the uncertainty in reservoir compartmentalization, as introduced by the specified fault uncertainties. To be able to obtain the desired truncations, a set of alternative truncation rules for faults that have intersecting fault uncertainty envelopes must be specified. This specification can either be done manually by examining all possible intersections and specifying a truncation rule for each of these, or more automatically based on rules specifying the truncation hierarchy, by ordering the faults according to fault age. For workflows that rely on unchanged topology, rejection sampling can be used. If the topology of the simulated realization does not match the topology of the base case model, the realization is rejected and an alternative realization is simulated. Another approach is to establish uncertainty envelopes and fault tips in such a way that no topology-changing realizations are possible. This can be done by ensuring that the fault uncertainty envelopes only intersect near existing truncations, and that the truncated parts of the fault tips are so large that they encompass the whole uncertainty envelope of the truncating fault.

### 4.2 Fault Length Uncertainty

There is a significant uncertainty in fault lengths, since parts of faults will not be visible in the seismic data. Therefore, a fault tip usually is generated based on a fault displacement gradient. The length of the fault and the resulting fault tip are closely connected to the displacement field associated with the fault. A realistic stochastic model for this is important to analyze the impact of faults on reservoir performance. The fault uncertainty envelopes presented in this paper closely follow the base case seismic interpretation. However, such envelopes could be extended to also encompass the uncertainty in fault tip location.

### 4.3 Updating the Reservoir Simulation Grid

The algorithms listed here update the structural model. However, in most workflows the reservoir simulation grid must be updated. In general, the only way to accomplish this is to rebuild the entire grid. As shown in Seiler et al. (2010), it is possible to update the gridded fault traces based on the differences between the base case structural model used to build the reservoir simulation grid and the simulated realization. Based on these modified fault traces, the rest of the grid then can be updated. This works for grids with simple fault geometries, and for cases where there are no changes to the topology of the model.

## 5 Conclusions

The algorithm presented here enables the rapid stochastic simulation of realistic fault surfaces. The parameters needed are the fault uncertainty envelope and a variogram describing the smoothness of the fault surfaces. Critically, the simulated fault surfaces can be constrained by well observations and along well paths, without any significant increase in the time needed to generate a realization. Using this fast simulation of realistic fault surfaces, Monte Carlo methods can be employed to estimate probability distributions for reservoir volumes or the probability for different fault scenarios. The method can also be used in well-planning workflows where it is important to estimate and update the uncertainty of faults bounding geological targets.

## Notes

### Acknowledgments

The authors thank several personnel at Statoil Research Center in Trondheim for constructive discussions, and especially Oddvar Lia for tireless testing of and significant input to the method. The authors also thank colleagues at NR for valuable input to the project and Roxar ASA for cooperation and support regarding the fault format and structural model in RMS. The research presented in this paper was funded by Statoil. The writing of this paper was funded by the Research Council of Norway.

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