Geological Stress State Calibration and Uncertainty Analysis

John McFarland1, Alan Morris2, Barron Bichon1, David Riha3,David Ferrill4, Ronald McGinnis5

Southwest Research Institute, 6220 Culebra Road, San Antonio, TX1Research Engineer

2Staff Scientist3Principal Engineer

4Department Director5Research Scientist

NOMENCLATURE

n Number of observed data pointsp Number of calibration parametersθ Vector of calibration parametersy Vector of observed datas Vector of observable, independent variablesε Vector of model calibration residualsG(·, ·) Functional relationship between model inputs and outputs‖σj‖ Magnitude of jth principal stressπ(·) Bayesian prior distribution function

ABSTRACT

The stress state is an important controlling factor on the slip behavior of faults and fractures in the earth’s crustand hence on the productivity of faulted and fractured hydrocarbon reservoirs. Uncertain or poorly constrainedestimates of stress states can lead to high risk both in drilling and production costs. Current methods for stresstensor estimation rely on slip vector field data, however, this information is not generally available from datasets thatare commonly used in the oil and gas industry.

This work presents an approach whereby predicted slip tendency is used as a proxy for fault displacement, whichcan easily be extracted from datasets routinely used by the oil and gas industry. In doing so, a calibration approachis developed in order to estimate the parameters governing the underlying stress state by calibrating slip tendencypredicted by the 3DStress R©software to match measured slip displacement. A Bayesian approach is employed, andseveral uncertainty sources are accounted for in the estimation process, including the impacts of limited data andcorrelated data taken from geologically similar measurement locations.

1 INTRODUCTION

Faults and fractures provide important pathways for subsurface fluid flow in many geologic settings including aquifers, geothermalreservoirs, and hydrocarbon reservoirs. They act as both conduits for and barriers to flow and are therefore the primary structuraldeterminants of aquifer and reservoir compartmentalization. In situ crustal stresses, because they control the slip behavior offaults and fractures, also exert an important control on the fluid conductivity of faulted and fractured systems. Uncertain or poorly

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

constrained estimates of stress states can lead to high risk both in energy exploration and production, and in the estimation ofreserves.

Several methods exist for the determination (inversion) of the stress tensor based on observed effects of that stress tensor usingdata sources such as earthquake focal mechanisms[1], paleostress indicators[2, 3], and microseismieity[4]. All these methods relyon knowledge of the slip vector field generated by the stress state being sought (see Ref. 5 for a review). However, slip vectorinformation is not generally available from such data sets as seismic reflection data and microseismic swarms commonly usedin the oil and gas industry.

In this paper we develop a stress inversion technique that allows the estimation of the stress state based on fault displacementdata, which can easily be extracted from data sets routinely used by the oil and gas industry. Our approach is formulated as amodel calibration problem: we make use of the 3DStress R©1 software tool that accepts the stress state as an input and has asoutput a measure of fault slip tendency. Such an approach brings a unique perspective to the problem, where the stress stateis typically estimated in an ad-hoc, trial and error procedure that requires an experienced analyst to manually “tune” the stressstate based on the observed displacement data.

Our approach goes beyond estimating a simple point value for the stress state, though. We employ a Bayesian model calibrationprocedure that develops a comprehensive representation of the uncertainty associated with the estimated stress state. Thisapproach helps us quantify the degree to which we can perturb the stress state while still maintaining a match to the fielddata. Further, the resulting uncertainty representation in the stress state can be used via uncertainty propagation to quantify theuncertainty associated with new predictions about slip tendency obtained from the model.

Section 2 will provide background on the type of geological data that we are using and how it relates to the computational modelfor predicting slip tendency. Section 3 presents the Bayesian model calibration approach. The analysis is discussed in Section 4,and we present stress state results both with and without the use of a correlation function to describe data dependencies.

2 FAULT SLIP MEASUREMENT AND PREDICTION

The earths crust is subject to stresses that are the combined result of gravitational loading and heat-transfer-driven tectonicprocesses. One type of response to these stresses is the formation and propagation of fractures. In geological terminology,shear fractures that accommodate measurable amounts of displacement are called faults. Locally, stress systems are approx-imately homogeneous, and can be considered as second rank tensors that are most commonly described in terms of threemutually perpendicular principal stresses. Some regions are subject to earthquakes, evidence that present-day in situ stressesare capable of generating fractures and causing them to slip, whereas other regions are seismically inactive. However, evenseismically “quiet” regions contain rocks that are faulted and fractured. Whether seismically active or not, faults and fracturesprovide important pathways for subsurface fluid flow in many geologic settings[6] including aquifers, geothermal reservoirs, andhydrocarbon reservoirs.

2.1 Slip Tendency Analysis

Slip tendency analysis is based on the premise that the resolved shear and normal stresses on a surface are strong predictorsof both the likelihood and direction of slip on that surface[7]. The method has been used successfully to characterize fault slip[8]

and fault slip directions[9, 10]. Fractures in high slip tendency orientations are, in many cases, better flow conduits than fracturesin low slip tendency orientations[6, 7, 11, 12]. The effect of stress anisotropy is greatest when the effective stress conditions ona fault or fracture approach those required for slipthe so-called critical stress[13−15]. Thus, preferential fluid flow through faultand fracture pathways is more pronounced the greater the differential stress and the greater the area of faults and fracturesexperiencing high slip tendency[7, 12, 16, 17]. Analysis of the effects of in situ stresses on existing faults and fractures is extremelyimportant to optimizing hydrocarbon production from fractured reservoirs, designing and interpreting “hydrofracs” to stimulatehydrocarbon production, designing and operating fractured rock geothermal fields, and predicting the effects of undergroundCO2 sequestration. In addition to the fault and fracture network, the in situ stresses are key inputs to this analysis.

13DStress R©is a commercial software tool developed at the Southwest Research Institute

2.2 Stress Inversion

Several methods exist for the determination (inversion) of the stress tensor from the effects of that stress tensor using datasources such as earthquake focal mechanisms[1], paleostress indicators[2, 3], and micro-seismicity[4]. All these methods rely onknowledge of the slip vector field generated by the stress state being sought. However, slip vector information is not generallyavailable from such data sets as seismic reflection data and microseismic swarms commonly used in the oil and gas industry.Our stress inversion technique uses slip tendency as a proxy for fault displacement, a measure that can easily be extracted fromdata sets routinely used by the oil and gas industry. If it can be assumed that the fault displacements were all generated by thesame stress state, then this type of data provides information about that stress state. Different stress states would be more orless likely to have produced the observed pattern of fault displacements. Thus the process of using observed displacement datato estimate the underlying stress state in conjunction with a software tool that predicts slip tendency is an example of an inverseproblem, also known as a model calibration problem (model calibration will be discussed in Section 3).

2.3 Geological Field Data

For the purpose of developing an inversion technique, the most useful natural data set is one that incorporates detailed faultslip data from a relatively small volume of rock, and for which the assumption that fault slip was geologically synchronous isappropriate. The Canyon Lake Spillway Gorge in Comal County, Texas contains faults that are amenable to this analysis andcan be mapped at the required level of detail. Several sub-horizontal bedding-parallel pavements adjacent to the Hidden Valleyfault[18] are cut by networks of small-displacement (≤ 1 m) normal faults (Figure 1). The surfaces of these faults are clearlyvisible and are commonly decorated with slickenline indicators in the form of grooves, ridges and swales, or fibrous calcite. Highresolution stratigraphic mapping[19] and close inspection of faulted strata permit precise measurements of displacement parallelto slickenlines on exposed fault surfaces. Approximately 348 slip surfaces were identified and measured (strike, dip, and rakeusing the right-hand rule, and displacement), and their locations recorded using a real-time kinematic global positioning system.

Figure 1: Measuring orientation and slip data on small faults at Canyon Lake Gorge. Measurement sites are temporarilymarked with tape then surveyed using a real-time kinematic global positioning system (staff with antenna).

3 BAYESIAN MODEL CALIBRATION AND UNCERTAINTY QUANTIFICATION

We use the term model calibration to refer to the process of estimating or tuning unobservable model parameters in order toimprove agreement between model output and experimentally observed responses. Note that in most cases of interest, therelationship between the model inputs and outputs is only observable by exercising a computational simulation, meaning that themodel can not be manipulated analytically or inverted to directly solve for the unknown parameters. As such, model calibrationis a numerical process, much like numerical optimization.

We formulate the calibration problem in a general sense by first formalizing the relationship between the model inputs and outputsas follows:

y = G(θ, s), (1)

where y is the model output (also known as the response or dependent variable), θ is a p-dimensional vector of unknowncalibration parameters, s is a vector of observable independent variables (sometimes referred to as covariates), and G(·, ·)represents the functional relationship defined by the computer simulation.

Consider that the calibration parameters are to be estimated using n experimental observations y = (y1, . . . , yn)T of the de-pendent variable(s) that correspond to the values of the independent variables s1, . . . , sn. The statistical model that relates thepredicted and observed values of the response is then written as

yi = G(θ, si) + εi, i = 1, . . . , n, (2)

where the εi are the random error terms (the model of Eq. (2) is referred to as a nonlinear regression model).

Thus, the objective of the calibration analysis is to obtain estimates of the calibration parameters θ1, . . . , θp based on the experi-mental measurements y1, . . . , yn. Perhaps the most straightforward approach for tackling the problem is to formulate the analysisas a nonlinear least-squares problem (see Ref. 20). However, there is an important feature of model calibration problems that issometimes overlooked, which is that there may be a wide range of model calibration parameters that provide comparable fits tothe observed data. In the literature, this is sometimes referred to as the problem of non-uniqueness. In fact, the degree to whicha range of parameters may fit the data is related to the amount of data available, and the range generally decreases as moredata become available.

In these situations we say that the parameters are subject to uncertainty, meaning that the “true” values of the calibrationparameters are not known. When performing a calibration analysis, it is important to be aware that the best fitting (e.g. least-squares estimator) value of the calibration parameters is not necessarily the only feasible value. Several techniques are availablefor quantifying the degree of uncertainty (or conversely confidence, as in confidence intervals or confidence regions) that existsin estimated calibration parameters (refer to Ref. 20 for a description of classical approaches). For this work, we adopt aBayesian approach, which allows us to develop a comprehensive representation of our complete state of knowledge regardingthe unknown calibration parameters (Ref. 21 discusses differences between classical and Bayesian uncertainty quantificationapproaches).

Bayesian analysis is based on the single equation known as Bayes’ theorem:

f(θ | y) =π(θ)f(y | θ)Rπ(θ)f(y | θ) dθ

, (3)

where as above, θ contains the variables being estimated (calibration parameters), y contains observed data, π(θ) is known asthe prior distribution for the unknowns, f(y | θ) is the likelihood of observing the data given a particular value of the unknowns,and f(θ | y) represents the posterior state of knowledge about the unknowns. The goal is to compute the posterior distributionfor the unknowns, which is actually a probability density function that captures the complete state of knowledge about the un-knowns. This posterior density function can be used to quantify the “most likely” values of the unknowns, associated uncertainty,correlations among components, etc.

The prior distribution for the unknowns, π(θ), is also a probability density function2, and it is used to capture all information aboutthe unknowns that is available before taking into account the data y. The prior distribution can be used to incorporate parameterconstraints by defining π(θ) = 0 wherever θ is not a feasible parameter vector.

2In most cases, the prior distribution is allowed to be an improper probability density function, meaning that its integral may not converge.[25]

In fact, the reference prior distribution that we employ in Eq. (5) is improper.

The last element of Bayes’ theorem is the likelihood function, f(y | θ), and it serves the purpose of measuring the agreementbetween the data and predictions (note that the likelihood function is viewed as a function of θ; the observations y hold fixedvalues). The specific form of the likelihood function is based on the joint probability density function used to model the errorrandom variables, ε1, . . . , εn. The most common assumption is that the errors are independent and identically distributed andhave a normal distribution with zero mean and some unknown variance, σ2. In this case, the likelihood function is given by

f(y | θ, σ2) =

nYi=1

1

σ√

2πexp

»− (yi −G(θ, si))

2

2σ2

–. (4)

Note that independence for the error terms implies independence for the experimental observations y1, . . . , yn. Whether or notthis assumption is appropriate will depend on the application and should be carefully considered. If the measurements are notindependent, then different error models may be evoked. Dependent errors are typically modeled using the multivariate normaldistribution, in which case an error correlation matrix is also required. The correlation matrix would usually be constructed by firstformulating a correlation function in terms of the independent variables, si, . . . , sn, and additional correlation parameters, sayφ. Ideally one would treat the correlation parameters as additional objects of Bayesian inference, however when the number ofobservations n is large, this may not be feasible, in which case an alternative approach is to first derive point estimates (perhapsusing maximum likelihood) for the correlation parameters, and then to treat them as knowns for the remainder of the analysis.

The process of actually computing the posterior distribution, f(θ | y), is fairly involved. Numerical sampling techniques such asMarkov Chain Monte Carlo simulation are widely used; details are given by Ref. 21. However, once the posterior distribution isobtained, it provides a comprehensive, quantitative representation about the state of knowledge of the unknowns. Refs. 21–24provide detailed case studies of the use of Bayesian inference in this fashion.

4 ANALYSIS

4.1 Introduction

The stress state is an important controlling factor on the slip behavior of faults and fractures in the earth’s crust. However, datasets commonly available in the oil and gas industries do not establish a direct connection with the stress state. Instead, thesedata sets typically capture slip displacements at various locations on one or more faults. Commercial software packages arecurrently available to predict slip tendency based on a given stress state, but the analyst is required to specify the stress state.In practice, this often gives rise to an ad-hoc “tuning” process, in which the stress state is manually (and subjectively) adjusted inorder to obtain some level of agreement between the slip tendency predicted by the simulation and observed slip displacementsmeasured in the field.

We will present a rigorous approach in which the tuning process is formulated as a model calibration problem, as discussed inSection 3.

4.2 Approach

The objective of the Bayesian calibration approach is to construct the posterior distribution for the unknowns, based on someobserved data. Here the unknowns are the set of parameters that describe the stress state, and the observed data consist ofmeasurements of fault displacement. Referring to Eq. (2), the first step is to formalize definitions for the observed data, y, thecalibration parameters, θ, the independent variables, s, and the performance model G(·, ·).

As mentioned above, the data y simply consist of n measurements of actual surface displacements at various locations on oneor more faults. The independent variables, s, that are observable and associated with each measurement are the strike and dipangles that define the fault orientation. Note that for each displacement measurement there are corresponding measurementsof the strike and dip angles for that location.

The vector of calibration parameters, θ, contains the full set of parameters necessary to define the stress state. This stress stateconsists of the orientations and magnitudes of three principal stresses. Section 4.2.1 below provides a discussion of how wechoose to represent the stress state for the purpose of calibration, in order to comply with certain conventions and constraints.

In addition to the calibration parameters, the error variance, σ2, is also treated as an object of Bayesian inference, so that itsposterior distribution is obtained as well. The prior distribution that we use is the non-informative reference prior[25]

π(θ, σ2) ∝ 1

σ2, (5)

which is independently uniform in the calibration parameters and log σ2.

The performance model, G(·, ·), defines a predictive relationship between the calibration parameters, the independent variables,and the response. In this case the performance model accepts as inputs the stress state as well as the fault orientation (definedby the strike and dip angles) and returns a predicted fault displacement. This application presents an additional challenge,though, which is that the actual computational model does not predict real displacement but instead slip tendency. This necessi-tates the formulation of a linking function that converts slip tendency to something that is comparable with actual displacement.This is discussed in more detail in Section 4.2.2, but we note that the model operator G(·, ·) is actually two functions, the sliptendency analysis engine and the conversion from tendency to displacement, so that the values returned by G(·, ·) have thesame units as the measurements y.

4.2.1 STRESS STATE PARAMETRIZATION

As mentioned above, the in situ stress state consists of the orientations and magnitudes of three principal stresses: σ1, σ2, andσ3. As such, it is necessary to chose a set of calibration parameters that completely define the stress state.

We first consider the parameters that define the orientation of the stress state. The orientation of the stress state is constrainedsuch that the three principal stresses remain mutually orthogonal. As such, there are three degrees of freedom associated withthe stress state. We choose a coordinate system that defines the orientation in terms of an initial state and three global, orderedrotations. The initial stress state has the first principal stress vertical and the second and third in the horizontal plane, with thesecond directed towards north and the third directed towards east. The first rotation is referred to as the azimuth, and it is aleft-handed (clockwise as seen looking down on the horizontal plane) rotation about the vertical axis. The azimuth defines thecompass direction of the second principal stress. The second rotation is referred to as the plunge, and it defines a right-handedrotation about the axis of the second principal stress. The third rotation is referred to as the tilt, and it defines a right-handedrotation about the axis of the third principal stress.

Next consider the parameters that govern the magnitudes of the principal stresses. The first thing to note is that the simulationthat predicts slip tendency is not a function of the absolute stress magnitudes, but only their relative magnitudes. We take this tomean that there are only two “real” degrees of freedom associated with the stress magnitudes, assigning a nominal value of 100to the first principal stress.

In practical applications, the type of stress state may be known beforehand based on knowledge of the predominant fault type,earthquake activity, or other geological information. For the present work, the stress state is what is referred to as a normalfaulting environment, meaning that the largest principal stress (here referred to as the first principal stress) is approximatelyin the vertical direction, and the intermediate (second) and least (third) principal stresses lie approximately in the horizontalplane. In fact, the approximate compass direction (azimuth) of the second principal stress can also be inferred based on thepredominant strike angle associated with the measurement data. Because the stress magnitudes are governed by differentparameters than the orientations, we want to introduce some type of constraint to ensure the correct ordering of magnitudesbetween the first, second, and third principal stresses.

It should be noted that theoretically, such a constraint could be evoked in terms of a prior distribution that only has support wherethe constraint is met. This is a valid strategy, but we have found that the problem is better conditioned (that is, the Markov ChainMonte Carlo sampling algorithm is able to explore the parameter space more efficiently) if we instead choose a parametrizationthat itself takes care of the constraint. As mentioned above, we assume a nominal value for the magnitude of the first principalstress:

‖σ1‖ = 100 (6)

Given the constraint that ‖σ1‖ ≥ ‖σ2‖ ≥ ‖σ3‖, we define a parameter (say, θ1) governing the second magnitude in terms of a

fraction reduced from the first magnitude. Formally, we have3:

θ1 =‖σ1‖ − ‖σ2‖‖σ2‖

(7)

The parameter governing ‖σ3‖ is defined similarly as a fraction reduced from ‖σ2‖:

θ2 =‖σ2‖ − ‖σ3‖‖σ3‖

(8)

4.2.2 COMPARISON FUNCTION FOR MODEL OUTPUTS AND FIELD MEASUREMENTS

As mentioned above, the computational model has as output slip tendency as opposed to real displacement. Slip tendency isa measure that mostly takes values between zero and one, with zero indicating a low propensity for the surfaces to slip andone indicating a high propensity for slip. Further, we do not necessarily expect a linear relationship between slip tendency andslip magnitude. This is because the physical process of two surfaces slipping past each other is something that happens slowlyover time. In fact, different faults or fault locations may have “started” slipping at different points in time. What this means isthat a surface with a high slip tendency may have either a large actual displacement (started slipping early) or a small actualdisplacement (started slipping late). However, a surface with a low slip tendency should have little or no actual displacement.

We will let y denote the slip tendency measure that is output from the computational model. To simplify the formulation, insteadof directly deriving a transformation from y to G(·, ·), we construct an error measure, y −G(·, ·), which is equal to ε from Eq. (2).The error measure intends to capture the following features:

1. We expect a positive relationship between tendency and displacement

2. Large displacement with small tendency should be penalized more than small displacement with large tendency.

We propose the following error measure (a.k.a. penalty function) based on these considerations:

ε =

8<:2ד

yymax

− y”

if yymax

> y,

−0.5דy − y

ymax

”2

otherwise,(9)

wherey =

y − 0.2

0.8− 0.2, (10)

and ymax is the maximum observed real displacement (the termy

ymaxserves to normalize the observed displacements). Note

that the constants 2, 0.5, 0.8, and 0.2 could all be adjusted, but we have found that these are reasonable values for ourapplication.4

The basis for Eq. (9) is that the ideal state is that in which each normalized displacement measurement ( yymax

) equals thecorresponding normalized prediction (y): in these cases the corresponding penalty or error measure is zero. Deviations from thisideal state are penalized based on whether the actual displacement is more or less than “expected.” If the actual displacementis more than expected (the first case in Eq. (9)), the penalty is more severe, and if the actual displacement is less than expected(the second case) the penalty is not as severe.

As discussed in Section 3, the most common statistical model for the residuals is the normal distribution model with expectationzero. As Eq. (9) is broken into two case statements, we would not expect the residuals to have a symmetric distribution aboutthe origin, and the analysis results confirm that they do not. Nevertheless, we will still apply a normal distribution model forthe residuals. From an implementation standpoint such a formulation is preferred, both because of its more intuitive behavior

3Note that internally to the MCMC algorithm we actually work with log θ1, which enables the sampler to explore values on the entire realline, since θ1 ≥ 0.

4In fact, here 0.2 represents the smallest slip tendency for which we expect any actual measured displacement, and 0.8 is the slip tendencythat we expect to correspond with the largest measured displacement.

and because the use of a likelihood function based on a skewed distributions can create convergence problems for the MCMCsampler. From a theoretical standpoint, the principle of maximum entropy provides some justification for the use of the normaldistribution5, even though the residuals are not expected to be symmetric about zero.

4.2.3 ADDRESSING CORRELATIONS AMONG MEASUREMENTS

As discussed in Section 3, the Bayesian inference procedure requires the formulation of a statistical model for the residuals,εi, i = 1, . . . , n. The usual approach is to assume that the residuals are independently and identically distributed, each followinga normal distribution with zero mean and some (possibly unknown) variance. However, whether or not the assumption that theresiduals are independent is appropriate deserves careful consideration, and such a decision will need to be made on a case bycase basis.

The important factor to consider is whether or not the individual measurements are related to one another. For example, ifthe measurements are taken at closely spaced intervals in time (or similarly closely spaced locations in space) from the sameexperiment, then it is likely that the assumption of independence is not appropriate. This consideration is particularly important,because from an uncertainty standpoint, assuming independence is non-conservative.

For the current work, it is not clear beforehand whether or not the assumption of independence will be appropriate, and in factwe will compare results obtained but with and without such an assumption. Recall that the n field measurements are all takenfrom the same geographical region. It is possible that certain measurements may have been taken at locations that are closeenough together (possibly on the same fault) that these measurements are not actually providing independent information aboutfault displacement (picture a geologist measuring fault displacement at one location and then moving one centimeter along thefault and taking another measurement of displacement).

The first challenge is to decide how the correlations among measurements will be characterized. The natural mechanismis through the formulation of a correlation function, defined in terms of some observable coordinates associated with eachmeasurement (perhaps a subset of the independent variables s). Two approaches were considered: (a) a correlation functiondefined in terms of the northing and easting (coordinates defining the geographic location of each measurement), and (b) acorrelation function defined in terms of the strike and dip angles of each measured fault (which are the also the elements ofs). The first approach may seem more intuitive, particularly to non-geologists, but it admits the possibility of recognizing twogeographically “close” measurements as being correlated, even though they may be on completely separate faults. The secondapproach makes more sense from a geological perspective, as the strike and dip angles are what define a particular fault, andin conjunction with a stress state, what govern the slip tendency.

Taking the second approach, the correlation function is formulated as6

c(yi, yj) = exp

"−„

strikei − strikej

φ1

«2

−„dipi − dipj

φ2

«2#, (11)

where φ1 and φ2 are the correlation lengths associated with the strike and dip angles, respectively. As discussed in Section 3,the full Bayesian approach would treat the correlation lengths as unknowns. For the current analysis, though, such an approachis not practical because it involves the inversion of the n × n correlation matrix at each step in the Markov Chain Monte Carlosimulation. A more practical approach is to estimate the correlation lengths a priori and then treat them as known constantsfor the remainder of the analysis. This can be done using the common “two-stage” estimation procedure[20] in which a pointestimate for the calibration parameters θ is first obtained under the assumption that the residuals are independent, and then amaximum likelihood procedure is used to obtain point estimates for the correlation lengths.

The likelihood function for the unknowns is now based on the multivariate normal probability density function. If we denote then× n correlation matrix by R, then the likelihood function is given by

f(y | θ, σ2) = (2π)−n/2 `σ2n |R|´−1/2

exp

»− 1

2σ2εTR−1ε

–, (12)

where ε is the vector of residuals, ε1, . . . , εn.5The normal distribution is a maximum entropy distribution among all continuous distributions with known mean and standard deviation.

As such, its use introduces a certain amount of conservatism into the results.6Note that because the model G(·, ·) is deterministic, the correlation among the residuals is the same as the correlation among the field

measurements.

4.3 Results

For the first analysis, the observations are assumed to be independent, so that the likelihood function of Eq. (4) is employed. Thestress state is estimated based on n = 329 measurements of real slip displacement7, with corresponding observations of thestrike and dip angles (s) for each fault. The posterior distribution is constructed using Markov Chain Monte Carlo sampling witha multivariate Gaussian random walk proposal density to obtain 20,000 samples from the posterior distribution8. The proposalcovariance matrix is automatically adapted based on the history of the chain[26], and additional re-scaling of the step sizes is alsoused to achieve an efficient exploration of the parameter space. After convergence, excellent mixing properties were observed,as shown in Figure 2.

Figure 2: Markov Chain for three parameters after convergence.

The posterior mode of the estimated stress state is described in Table 1. Recall that the stress state is derived from the fivecalibration parameters used in the Bayesian inference process (see Section 4.2.1). The stress state is displayed graphically inthe stereograph plot of Figure 3. The space of the stereograph plot is defined over the strike and dip coordinates, so it representsa visualization of the slip tendency as a function of fault orientation.

TABLE 1: Posterior mode for estimated stress state treating observations as independent

Principal Stress Magnitude Azimuth Plungeσ1 100 30◦ 70◦

σ2 56 52◦ -19◦

σ3 31 140◦ 7◦

The most likely (mode) stress state from the posterior can also be visually compared with the observed data. This is donein Figure 4, which plots the orientations of the measured faults on top of the slip tendency stereograph. The measured faultsare broken into three categories: high, medium, and low displacement. For good agreement between the predictions andobservations, we expect the high measured displacements to correspond to regions of high slip tendency, the medium measureddisplacements to mostly also correspond with regions of high slip tendency, and the low measured displacements to mostlyappear in regions of moderate slip tendency. Visually, the stress state corresponding to the posterior mode achieves excellent

7Some of the 348 measurements mentioned in Section 2.3 were not used in the analysis because they contained incomplete information.8After convergence, 80,000 samples were generated, and every fourth sample was stored.

Figure 3: Stress state corresponding to posterior mode assuming independent observations. The colored contoursrepresent slip tendency as a function of the strike and dip (fault orientation).

agreement between predicted and observed slip. Notice that the high-displacement measurements have a fault orientation thatcorresponds to roughly the highest predicted slip tendency.

(a) High Displacement (b) Medium Displacement (c) Low Displacement

Figure 4: Comparison of predicted slip tendency (at the posterior mode stress state, assuming independent observations)with measured fault displacements. Each field measurement corresponds to a dot on the stereograph plot.

The correlations among the stress state parameters are tabulated in Table 2. Note that Table 2 lists ‖σ2‖ and ‖σ3‖ as opposedto the actual parameters used in the calibration analysis, θ1 and θ2: while virtually no correlation is seen between the stressmagnitudes, a very strong negative correlation of -0.93 occurs between the parameters θ1 and θ2. Otherwise the correlationsamong the parameters are mostly minor, with the largest being 0.54 between the azimuth and tilt angles. This relationship isdepicted graphically by the joint confidence regions for these parameters in Figure 5.

The analysis was also repeated using a more realistic statistical model for the residuals (and hence the observations). For thissecond analysis, a correlation function was developed for the observations in terms of the fault orientation (see Section 4.2.3).This correlation function is given in Eq. (11) and aims to capture the idea that measured displacements on faults with very similarorientations might not provide statistically independent information.

TABLE 2: Observed correlation matrix among stress state parameters

‖σ2‖ ‖σ3‖ Azimuth Plunge Tilt‖σ2‖ 1. 0.1 0.26 0.49 0.01‖σ3‖ 0.1 1. 0.15 0.23 0.01

Azimuth 0.26 0.15 1. 0.1 0.54Plunge 0.49 0.23 0.1 1. -0.03

Tilt 0.01 0.01 0.54 -0.03 1.

Figure 5: Joint confidence regions for the global azimuth and tilt angles.

Note that the correlation function introduces two additional parameters, φ1 and φ2, which are the correlation lengths for the strikeand dip angles, respectively. As discussed in Section 4.2.3, the large number of field observations makes it computationallyprohibitive to treat the correlation lengths as objects of Bayesian inference. Instead, a two-stage estimation process is used inwhich the residuals from the posterior mode of the preceding analysis are used to estimate the correlation lengths in a maximumlikelihood framework. Doing so, the correlation lengths are estimated as 15.2◦ and 0.7◦ for the strike and dip angles, respectively.

These correlations reflect a negligible amount of correlation associated with the dip angle and a small amount associated withthe strike angle. The issue is somewhat complicated in this application, though, because of the bimodal nature of the faultorientation. The strike angle measures the compass angle of the fault orientation, with the convention that an angle betweenzero and 180◦ indicates a fault dipping to the west, and an angle between 180◦ and 360◦ denotes a fault dipping to the east.Faults separated by strike angles of 180◦ are known as conjugate sets and are equally likely to occur (this is readily apparent fromFigure 4(b)). As such, defining the correlation function in terms of a difference between two strike angles may not be completelyappropriate. Nevertheless, we demonstrate such an approach here and leave the possibility of more rigorous approaches forfuture work.

Having estimated the correlation lengths, we treat them as fixed constants for the remainder of the analysis. They are then usedto construct the correlation matrix for the observations, R, which is used in the likelihood function of Eq. (12). Because of therelatively short correlation lengths, we do not expect the effect of incorporating correlations to be significant, and the resultsconfirm this expectation. The posterior mode, listed in Table 3, is very similar to the previous result, and in fact the two are notvisually distinguishable.

Normally, we would expect the introduction of correlations to have a significant impact on the resulting uncertainty in the posteriordistributions, because correlated observations provide less information than independent observations. However, in this case,the correlations are small enough that the effect is not large. Figure 6 shows the marginal posterior distribution for the global

TABLE 3: Posterior mode for estimated stress state using correlation function

Principal Stress Magnitude Azimuth Plungeσ1 100 32◦ 68◦

σ2 55 53◦ -20◦

σ3 30 140◦ 7◦

azimuth angle (which is also the azimuth angle associated with the second principal stress). Here we see only a small difference,and in fact the posterior obtained with the correlation model shows less uncertainty (while this is the opposite of what is expected,the uncertainty in other parameters increased, as expected). Figure 7 plots joint 95% confidence regions for the magnitudes ofthe second and third principal stresses (recall that the magnitude of the first is arbitrarily set at 100). Again, only a small changeis seen, and it is in fact statistically insignificant at the 95% confidence level. The posterior standard deviations are listed inTable 4.

Figure 6: Comparison of marginal posterior distributions for the azimuth angle for independent and correlated observations.

TABLE 4: Posterior standard deviations from analyses treating observations as independent and correlated

‖σ2‖ ‖σ3‖ Azimuth Plunge TiltIndependent 2.3 0.49 2.4 2.2 0.95Correlated 2.4 0.50 2.3 2.1 0.97

5 CONCLUSIONS

In industries such as oil and gas, software tools are used to predict slip tendency for faults in the earth’s crust, which aids inenergy exploration and production as well as reservoir estimation. The stress state is an important factor in the slip behavior andit is required as an input for such tools. The stress state may be estimated from available geological data, but previous methodsfor doing so have typically involved subjective tuning of the stress state, requiring expert input and judgement.

This paper presents a new approach for stress state estimation from field data (also referred to as stress state inversion) byemploying calibration analysis methodology. A Bayesian approach is presented, whereby a best-fitting estimate of the stressstate is obtained in addition to a comprehensive representation of the accompanying uncertainty based on the amount of fielddata available.

Figure 7: Comparison of 95% joint confidence regions for the magnitudes of the second and third principal stresses forindependent and correlated observations.

The stress state estimation process is demonstrated using actual geological field data obtained from the Canyon Lake SpillwayGorge in Comal County, Texas. Two analyses are considered, one which aims to model the correlations present among measure-ments from similar fault surfaces, but no significant differences are seen between the results. The stress state obtained is shownto produce excellent agreement between the predicted and observed fault slip. Several quantitative and graphical summaries ofthe uncertainty associated with the estimated stress state are presented as well. However, the uncertainty representation maybe most useful when the stress state is used to obtain new predictions, in which case slip tendency confidence bounds could beeasily derived via uncertainty propagation.

ACKNOWLEDGEMENTS

The research reported in this paper was funded as part of an Internal Research and Development project at the SouthwestResearch Institute.

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