## ABSTRACT

Calculating subsurface pressures and predicting overpressured zones, in particular for safe drilling operations, is an integrated approach based on data and assumptions from various sources. Uncertainties arise from the input data and assumptions, but also from the pore-pressure modeling workflow including shale discrimination and the definition of a normal compaction trend line. These stages are usually performed manually by an expert and are prone to subjective, human interpretation. The quantification of pore-pressure uncertainty associated with the manual modeling stages is, therefore, challenging. Algorithms were developed to account for and quantify the resulting uncertainty and enable automated user support of at least parts of the workflow, thus introducing more objectivity into the modeling steps. The first algorithm performs a statistical analysis on gamma ray logs to discriminate between shale and nonshale formations. The second algorithm calculates a series of normal compaction trend lines from porosity-indicating logs from which average pore-pressure models and uncertainty envelopes can be determined. Furthermore, the behavior of trend-line envelopes from the series of trend lines was quantified by a parameter $Q$, which turned out to become constant in the overpressure region. The algorithms were applied to 23 data sets from different regions worldwide. Pore-pressure uncertainty was identified to be in the range of up to 8% for shale discrimination and less than 20% for normal compaction trend-line setting. In addition, pore-pressure uncertainty in the overpressure zone correlated with the $Q$-factor, which can be used to estimate pore-pressure uncertainty at greater depth from while-drilling measurements in the normal compaction zone. The results also exemplify regional uncertainty variations, which imply that modeling parameters need to be adjusted for specific regions. Moreover, the examples demonstrate that automated algorithms are beneficial methods to add objectivity and reproducibility to the modeling procedure.

## INTRODUCTION

Formation fluid pressure in excess of hydrostatic (overpressure, geopressure, abnormal pore pressure) has been recognized to cause severe drilling problems such as influx of formation fluid into the wellbore, resulting in a kick. Formation pressure can be measured in boreholes but only in permeable formations such as sands. For safe drilling, understanding the complete pore-pressure profile with depth is required, and for impermeable formations such as shales, pore pressure has to be inferred from geophysical data such as seismic interval velocities for predrill studies or downhole formation evaluation (geophysical) logs for real-time while-drilling updates of predrill models (Fertl and Chilingarian, 1977; Mouchet and Mitchell, 1989).

In particular, in young sedimentary basins, the phenomenon of compaction disequilibrium (or undercompaction) is most responsible for the increase in pore fluid pressure (e.g., Dickinson, 1953; Hart et al., 1995). When sediment compacts, the fluid trapped in the sediment pores dissipates with ongoing burial. The rate of dissipation depends on the permeability of the sediment and the distance to the drainage path. In permeable sediments, the pressure equilibration takes place almost simultaneously with sedimentation, even under high accumulation rates. In fine-grained, less-permeable sediments, such as shales, equilibrium can only be maintained under slow sedimentation rates. High sedimentation rates in conjunction with low sediment permeability may lead to delayed pore water release, which results in fluid pressures higher than hydrostatic pressure and porosity will be maintained.

Apart from compaction disequilibrium, many other mechanisms that can cause formation overpressure are discussed (e.g., Osborne and Swarbrick, 1997; Swarbrick et al., 2002). These mechanisms include fluid expansion or aquathermal pressuring (e.g., Barker, 1972; Shi and Wang, 1986; Luo and Vasseur, 1992), diagenetic reactions (Freed and Peacor, 1989), load transfer (Lahann and Swarbrick, 2011), hydrocarbon generation (e.g., Spencer, 1987; Guo et al., 2011; Tingay et al., 2013), buoyancy, lateral transfer (Yardley and Swarbrick, 2000), vertical transfer (Tingay et al., 2007), and also lateral stress due to tectonic activity (e.g., Screaton and Saffer, 2005). Compaction disequilibrium is generally considered to be the most important factor causing formation overpressure, although other mechanisms can contribute significantly (Swarbrick et al., 2002).

Several, mostly empirical, approaches exist to estimate formation overpressure. Although seismic data are predominantly used in the planning phase of a well (e.g., Sayers et al., 2002), applications while drilling that are critical to ensure safety of the drilling process rely on a well-defined predrill model, geophysical data gathered downhole while drilling, and data acquired at the surface (Mouchet and Mitchell, 1989). In addition, basin modeling is used in the predrill planning phase of a well. Predrill models based on seismic data and/or data from adjacent wells can be updated during the drilling phase using real-time logging-while-drilling data, reducing the uncertainty of the pore-pressure model (e.g., Lüthje et al., 2009; Weiland et al., 2010).

The modeling procedure to quantify overpressure requires the use of various input data and involves several workflow steps. Input data and workflow steps contribute some amount of uncertainty to the final model. The uncertainty in a pore-pressure model derived from downhole logs was estimated to be as high as $\xb130%$ (Aadnoy, 2011), which can be greater than the pressure window for safe drilling. Consequently, a more precise knowledge of the uncertainty is required.

The typical workflow for pore-pressure modeling while drilling includes several steps of human interaction. The human interpretation may be responsible for the largest part of the uncertainty associated to the pore-pressure model (Matthews, 2004). Only a few studies exist to infer the contribution of single modeling steps to the uncertainty (e.g., Malinverno et al., 2004; Sayers et al., 2006), and the uncertainty associated with human interaction is rarely addressed.

Several approaches for quantifying uncertainty associated with overpressure modeling have been described. The simplest approach is to account for the uncertainty in the model by applying a safety factor to the final pore-pressure curve. Uncertainty of this safety factor can be reduced using information from a regional database containing direct formation pressure measurements (Falcao, 2002). A decision-making process is used by Dodds et al. (2001) and Dodds and Fletcher (2004) in combination with interval probability theory to quantify uncertainty. Monte Carlo methods were applied by several authors to evaluate the uncertainty associated with input data such as downhole logs and seismic measurements (Liang, 2002; Doyen et al., 2003; Malinverno et al., 2004; Moos et al., 2004).

Although uncertainties associated with input parameters such as the logging data can be modeled by Monte Carlo simulations assuming appropriate accuracy values, quantifying uncertainty associated with manually derived input data (such as the normal compaction trend-line determined manually) is challenging. Quantifying the latter requires numerous interpretations, ideally by different pore-pressure specialists, so that sufficient pore-pressure models exist for a statistical analysis. As an alternative, we replaced single modules of the pore-pressure workflow by algorithms (e.g., Wessling et al., 2011). These algorithms provide a means to replace the manual interpretation steps with a standardized routine and allow the application of statistical methods to quantify uncertainty. In this paper, we analyze a typical pore-pressure modeling workflow with respect to uncertainty associated with the individual modeling steps based on our previous work (Bartetzko et al., 2012; Tesch and Wessling, 2012). We will discuss the sources for uncertainty and highlight approaches to quantify uncertainty. In more detail, we will demonstrate approaches for quantifying pore-pressure uncertainty specifically for one workflow module. The presented method is one step toward automating wellbore stability workflows, including pore-pressure modeling.

## METHODS

### Standard pore-pressure modeling workflow

Many, mostly empirical, methods have been developed to model pore pressure from downhole measurements (e.g., Hottman and Johnson, 1965; Eaton, 1975). Most of these methods rely on the principle of compaction disequilibrium and require definition of a normal compaction trend line, which represents the decrease in porosity with increasing burial under normal (hydrostatic) depositional conditions. Deviations from the normal compaction trend are indications for increased pore pressure (e.g., Eaton, 1975; Kan and Swan, 2001). Downhole measurements such as electrical resistivity and acoustic slowness are used as indicators for porosity. Most frequently, compressional wave slowness is applied for pore-pressure prediction although the use of shear wave slowness is also discussed in the literature (e.g., Ebrom et al., 2006). A typical pore-pressure modeling workflow as it is frequently applied (e.g., Matthews, 2004; Stafford and Goode, 2008) is described in the following. A first step in the modeling workflow is to become familiar with the geologic setting and the diagenesis of the region of interest (Figure 1). Likely overpressure generating mechanisms need to be identified to apply appropriate pore-pressure modeling methods. In addition, the existence of faults, compartmentalization, buoyancy effect in sand bodies, and other geologic structures need to be extracted from a geologic model to identify potential constraints to the pore-pressure model.

An important value for subsequent modeling is the overburden gradient (OBG, or lithostatic pressure of the overburden), which is calculated by integrating a depth-based density log. If the density has not been acquired, a pseudodensity log derived from a correlation between other formation evaluation logs such as acoustic velocity and rock bulk density may be used to calculate the OBG. Often, the density log is not available in the shallow sections, and it thus has to be extrapolated to the surface.

Pore-pressure modeling is restricted to fine-grained sediment layers, i.e., mostly shales. Therefore, in the next step, shale layers are separated from other lithologies. Different methods exist to identify shale layers, e.g., analyzing the gamma ray log or using neutron log-density crossplots. Because the definition of the shale layers is subjective, it is a workflow element with human interaction.

Some workflows include an optional step in which the resistivity and/or acoustic slowness data in the shale layers are filtered to further reduce scattering that is associated with bad hole conditions, recorded noise, etc. In the examples presented here, we applied a moving average filter over 99 data points.

Within the depth interval that is considered to be normally pressured, a normal compaction trend line will be defined in the next step using the (filtered) resistivity and/or acoustic data in the shale layers. Several soil mechanical and mathematical descriptions of the compaction trends exist (for example, linear, exponential, power law, and polynomial), but they show a large variability (e.g., Mondol et al., 2007). Therefore, normal compaction trends are usually defined manually by an interpreter, which is another element of human interaction in the workflow. We developed an algorithm based on a regression analysis that defines the depth point at which overpressure starts and calculates the normal compaction trend line for the overlying interval (Wessling et al., 2011).

Depending on the geologic structures, multiple trend lines are sometimes drawn to represent significant changes of compaction trends. Multiple trend lines are necessary at least whenever the original sedimentation process is disturbed. For example, unconformities represent an altered state of sediments deposition so that a cut or change in the compaction behavior needs to be expected. In addition, faults represent a significant alteration of the original sedimentation sequence, and multiple trend lines need to be used for the pore-pressure analysis. One example is given by Freitag et al. (2004), in which the authors use two compaction trend lines above and below a fault, respectively. The trend lines are parallel shifted by each other and, therefore, represent the same compaction trend with depth, but they include a shift between the compacted sediments above and below the fault.

Throughout this document, all pressure gradients are given in units equivalent to mud density in $g/cm3$ instead of a pressure unit. Pressure gradients are calculated by dividing the pressures through the specific gravity and depth, which is useful if the drilling fluid density is evaluated against the pressure window for safe drilling operations.

A final step in the pore-pressure modeling workflow is the calibration of the model against calibration data such as formation pressure tests, experienced influx of formation fluid into the borehole, kicks, differential sticking, or connection gas. Calibration is conducted by adjusting parameters (primarily the normal compaction trend line and/or the Eaton exponent) so the pore-pressure model fits the magnitude of the formation pore pressure at the appropriate depth locations. Information obtained from previously drilled offset well data is frequently used as predrill calibration sources, which then need to be verified during the drilling operation.

### Automated identification of shale layers

To automatically define shale layers, we use a simple approach based on the gamma ray log. During a manual shale discrimination, cutoff lines are drawn on a gamma ray log, and only depth points with gamma ray values higher than the cutoff lines are considered in the further modeling process. We developed an algorithm based on the statistics of the gamma ray log to identify the shale layers. The algorithm subdivides the data set into intervals of identical length that overlap each other. In the example in this study, we used three overlapping lines that overlap each other by two-thirds. For each interval, the 80th percentile of the frequency distribution of the gamma ray values is determined. This value gives the cutoff value for the specific depth interval. As the depth intervals overlap, there are three cutoff values for each depth point and thus three lithology classifications (shale or nonshale) for each depth point. In the example given here, only depth points with all three classifications indicating shale were accepted as shale. Other rules are possible, such as when two out of three classifications indicating shale are accepted as shale, but they should be assigned with a lower confidence.

We compared our algorithms with results from manual shale identification. Each analyst performs the process slightly differently, so we used two different manual strategies or approaches: One approach referred to as “detailed” in the following uses a large number of individual cutoff lines that follow moderate variations in the gamma ray. The other manual approach is referred to as “major units.” In this approach, the well is subdivided into major lithological units and one cutoff line is defined for each lithological unit.

### Quantification of uncertainty associated with trend line variations

To overcome the challenge of subjectivity by manual trend line interpretation, we propose a method to automatically calculate the compaction trend line in a way that an uncertainty can be assigned to the pore-pressure model associated with the variability in the trend line definition. However, the application of automated algorithms is also affected by subjectivity as long as relevant analysis and processing parameters need to be predefined by the user and cannot be measured or derived from theory. An automated algorithm thus needs to be designed in such a way that the influence of users on the parameters is minimized.

In the following, we propose a method for the automatic calculation of compaction trend lines from porosity-indicating logs. The basic idea is to generate a series of normal compaction trend lines by linear regression on porosity-indicating logs over automatically alternating depth intervals. As the linear normal compaction trend is observed on semilogarithmic plots, linear regression is conducted on the logarithm of the shale discriminated porosity-indicating logs. The series of trend lines is then used for two different uncertainty quantification methods. The alternation of the depth intervals is essential for this method because the start and end points of the interval for regression need to be defined by the user. Asking different users to define one depth interval for the normal compaction trend line will probably result in significantly different automatic analysis results, which are again a consequence of subjective user interaction. In this paper, we restrict our method to using linear regression functions to fit the porosity-indicating logs in the normal compaction zone, although other functions (such as exponential, power-law, or polynomial fits) may also be useful.

For uncertainty quantification methods, two intervals need to be defined from which the series of trend lines are generated: a start interval containing *n* data points and an end interval containing *m* data points (see Figure 2, left, for nomenclature). Both intervals need to reside within the normal compaction zone. As a first step, a linear regression analysis is performed over the interval beginning at the first data point from the start interval ($i=1$) and ending at the first data point from the end interval ($j=1$), yielding the normal compaction trend line $TL1,1$. Step $n$ of the analysis defines the interval for linear regression from data point $i=n$ to $j=1$, giving $TLn,1$. The final linear regression analysis is performed for $i=n$, $j=m$ to obtain trend line $TLn,m$ (see step $n*m$ in Figure 2). This approach gives a series of $n*m$ trend lines.

The series of normal compaction trend lines is subsequently used to define a “trend-line envelope” and to calculate the uncertainty of the pore-pressure model associated with trend-line variations within the trend-line envelope.

Two different methods are introduced here to derive the trend-line envelope and the uncertainty of the pore-pressure model as a result of variations in the normal compaction trend line: a statistical and a geometrical approach. A trend-line envelope from the statistical approach is derived by calculating the average normal compaction trend line and its standard deviation from the series of trend lines (see Figure 3, left track). Then, the statistical approach calculates one pore-pressure model using equation 1 or 2 for each of the $n*m$ trend lines of the series that has been calculated by the linear regression, which likewise results in a series of $n*m$ pore-pressure models. This series is then statistically analyzed to derive an average pore-pressure model and its standard deviation ($\xb1\u2009\u20091\u2009\u2009s$; see Figure 3). Of course, a statistically sound result requires a sufficient amount of trend lines and pore-pressure models in the series. This is achieved by setting the start and end intervals for linear regression in a way that at least 50 data points reside in either of the two intervals, yielding more than 2500 trend lines and models.

The second proposed method extracts the two normal compaction trend lines exhibiting the largest and smallest slopes, respectively, out of the series of $n*m$ trend lines. These two extreme trend lines define the trend-line envelope from method 2 and are then used to calculate pore-pressure models using equation 1 or 2; see Figure 4.

## RESULTS

### Identification of shale layers

Figure 5 compares the algorithm for shale layer identification with the two manual approaches explained above. Each analysis creates a shale data subset, and the three shale data subsets from the three analyses each comprise a different amount of data. Although the algorithm yields the smallest data set with only 16% of the original data comprised, the detailed approach still includes 19% of the data of the original data set and the major units approach contains the largest amount of data (43% of the original data set).

In Figure 5, only a slight difference is visible in the resistivity data of the three shale data subsets, either not filtered or filtered. Significant variances, however, occur when pore pressure is calculated from these subsets and differences in pore pressure of up to $0.1\u2009\u2009g/cm3$ or 8% occur.

### Trend-line uncertainty and the $q$-factor

The $q$-factor has been introduced to characterize the expected pore-pressure uncertainty in the overpressure section from variations of the normal compaction trend lines. An example is given in Figure 6, showing the resistivity log, the average normal compaction trend line, and the trend-line envelopes derived from methods 1 and 2 (Figure 6, left). Note that the envelope from method 1 is nonlinear and does not cross at $\u223c825\u2009\u2009m$, in contrast to the trend lines derived from method 2. Nevertheless, all trend lines exhibit an approximately linear behavior with depth on a semilogarithmic scale. It is thus suspected that $q$ is more or less constant at greater depth for both methods, as can be observed on the right of Figure 6. The introduction of the constant $Q$ in equation 5 thus seems legitimate. For analyzed trend lines from method 1, $q$ is negative within the upper depth interval, above which the enveloping trend lines are closest to each other, which is due to the decreasing envelope size with depth. Below the “tipping point” at $\u223c825\u2009\u2009m$, $q$ becomes positive and eventually approaches a constant value of $Q=0.0003/m$. The quantity $Q$ calculated from trend-line envelopes from method 2 delivers a constant value of $Q=0.00058/m$ below the tipping point because the envelope is here defined by linear boundaries.

Different data sets (resistivity and acoustic slowness as porosity-indicating logs) from the regions of the Asia Pacific, the Gulf of Mexico, South America, the North Sea, and Canada were available to calculate the $Q$-factor. In general, the $Q$-factor turns out to be smaller for acoustic logs compared to resistivity logs for methods 1 (Figure 7) and 2 (Figure 8). For the Asia Pacific region, the $Q$-factor from acoustic logs shows little variation in magnitude (between $1\xd710\u22125$ and $2\xd710\u22125$; see Figure 7), in contrast to the $Q$-factor from resistivity logs. This observed minor variation cannot be transferred to the Gulf of Mexico and North Sea, and the number of data sets analyzed from South America and Canada is not sufficient. Note that no rule seems to exist if the $Q$-factor is analyzed from the acoustic compressional (dtc) and shear (dts) slowness, respectively. A comparison of the $Q$-factors between methods 1 and 2 shows that method 2 delivers larger $Q$ (compare $Q$ from Figures 7 and 8).

Another result from the comparison of $Q$-factors is that their values are to some extent similar within the specific regions but vary by orders of magnitude between the different regions. The largest uncertainties are observed from the North Sea data sets, and the smallest uncertainties exist in the Gulf of Mexico region. This difference is probably due to the similar lithology within fields or basins, but significant lithology variations exist between different basins worldwide. The proposed methods for quantifying trend line uncertainty and the resulting $Q$-factor primarily mirror the log variability in the normal compaction zone, which is in turn a response to lithologies.

### Pore-pressure uncertainty

Representative values to characterize the uncertainty of the pore-pressure models were calculated from equation 8 for the overpressured section. Uncertainty estimation from method 1 yields relative pore-pressure uncertainty of less than 20% for the data sets used for investigation. About two-thirds of the data sets show uncertainties even below 10% (Figure 9). As we have observed from the $Q$-factors, the pore-pressure uncertainty varies between different regions. Data sets from multiple wells drilled in one Asia Pacific basin clearly show that the uncertainty is always lower (below 5%) for acoustic logs compared to resistivity-derived uncertainties (below 20%). Data sets from the Gulf of Mexico and the North Sea do not consistently show this behavior. In single wells, the uncertainty from acoustic logs turns out to be larger than the uncertainty from the resistivity logs.

## DISCUSSION

In this study, we analyzed the modeling workflow with respect to the uncertainty that contributes to the total uncertainty of the pore-pressure model, particularly to those modeling steps that involve human interaction (i.e., shale discrimination and trend-line definition). The uncertainty of a pore-pressure model derived from geophysical well logs not only originated from the uncertainty inherent in the geophysical measurement itself, but it had contributions from several sources as shown in Figure 1. Whereas accuracy of downhole resistivity measurements is between 0.4% and 4% and accuracy of the acoustic measurements is between 2% and 5%, Aadnoy (2011) assumes the uncertainty in a pore pressure model derived from logs as being considerably higher than the accuracy in direct measurements (30% compared to 2%, respectively).

Other pore-pressure uncertainty quantification methods, e.g., using Monte Carlo simulation (e.g., Malinverno et al., 2004; Moos et al., 2004), assign an uncertainty or probability to the normal compaction trend-line parameters and the porosity-indicating logs. In contrast, the approach proposed in this paper calculates uncertainty of the trend line based on the variability of the porosity-indicating log. The first method proposed above gives the trend-line envelope as a statistical means for trend-line variations, whereas the second approach gives the extreme value envelope that can be obtained from the input data. Results from the statistical approach in turn can be used as input parameters for a Monte Carlo simulation.

The process of shale discrimination creates a subset of data that will be the basis for further modeling. This modeling step is usually carried out manually. Thus, the experience of the analyst and the knowledge about local geology are important. Moreover, the type of lithology and the contrast in the lithology indicating log (e.g., gamma ray) between shale and nonshale rock are essential. A minimum thickness for the individual shale layers is required to ensure that the shale is able to maintain pressure. For example, in the case of thin alternating layers, the minimum thickness of the shale layers may not be given. Different approaches for shale discrimination (manual or using the algorithm) result in shale data subsets of different size, i.e., depending on how the discrimination criterion is applied a larger or smaller number of data points is classified as shale. As shown in Figure 5, most of the shale points are plotted on top of each other, but each data subset contains some outliers that are not included in the other data sets. Although these differences still appear to be minor in the shale resistivity data sets, they can be increased when the pore pressure is derived from resistivity. To some extent, differences between the shale resistivity data sets are increased by the filtering step that is carried out prior to calculating pore pressure. As the shale data subsets contain a different number of data points for the same depth interval, they have different data densities. Furthermore, sampling in the shale data subsets is not regular because sandy layers are excluded and data points only occur in shale layers. Applying a standard filter length defined by several data points results in a different degree of smoothing of the shale data subsets, depending on their data density and causes differences in the calculated pore-pressure curves.

The definition of the normal compaction trend line is the other modeling step involving human interpretation; thus, the result and its uncertainty will largely depend on the experience and regional knowledge of the analysts. Different analysts may yield very different results (e.g., the example in Matthews, 2004). We calculated the normal compaction trend line using a regression. In this case, different factors add to the uncertainty in the trend line, including the uncertainty in the geophysical data set itself (e.g., resistivity or acoustic data), the quality of shale data set, the filtering process, and the length of the depth interval included into the regression analysis. Variation of each of these factors will cause changes in the trend line. To quantify the uncertainty introduced by the selection of the depth interval over which the trend line is calculated, we used the approach of calculating an array of trend lines with systematic variations in start and end points. Again, the proposed methods are useful to quantify pore-pressure uncertainty associated with the definition of the normal compaction trend line, yet without putting the analysis results into a geologic context. The presented methods are thus considered as supportive functions that aim at guiding users through the pore-pressure modeling procedure. The user still plays a crucial role in the procedure. The proposed algorithms are able to automatically apply functions to well log data, but they cannot produce the interpretation of the results and associated assumptions by the users.

The relevance of calculating a series of trend lines instead of using a single trend line for pore-pressure calculation becomes obvious from the log variability shown in Figures 3 and 4 within the normal compaction zone. Whereas an acceptable fit is obtained within the normal compaction zone for all trend lines, this “slight” difference in the normal compaction zone has significant influence on the calculated pore-pressure model at greater depth. In another study, Bartetzko et al. (2012) show that the average pore-pressure uncertainty in the overpressure zone is three to four times larger than the average pore-pressure uncertainty in the normal compaction zone, if the proposed methods for calculating a series of trend lines are applied.

A comparison of methods 1 and 2 for calculating the trend-line envelopes shows that both methods deliver comparable results. However, due to the different philosophy behind the methods, they should be used depending on the assumptions and expectations of the pore-pressure engineers. Method 1 (trend-line envelope as a statistical means) should be applied if undercompaction can be assumed to be the main overpressure generating mechanism so that the determination of normal compaction trend lines from porosity-indicating logs is reasonable. In that case, the assignment of a statistical uncertainty to the pore-pressure model is useful. However, method 2 should be applied if the origin of overpressure is unclear and the pore-pressure model from the normal compaction trend line can become erroneous. In that case, method 2 (extreme boundaries for a normal compaction trend line) is useful as guidance for pore-pressure engineers by restricting the manual change of the normal compaction trend line to stay within the envelope or otherwise trigger advice to include other pressure generating mechanisms into the pore-pressure modeling procedure.

Of particular interest is the propagation of the uncertainty associated with the variation of the normal compaction trend line with depth. The variations in the trend lines are caused by slight variations of the porosity-indicating logs in the normally compacted zones, i.e., due to variations in lithology or rock properties. We introduced $Q$ to quantify how the uncertainty caused by these small variations propagates with depth and thus affects uncertainty of the pore-pressure model in the overpressure region. As $Q$ can be calculated while drilling is still taking place in the normally compacted zone, it represents an early indicator on uncertainty propagation to greater depth. If a sufficient amount of wells has been drilled in a specific region so that a correlation between the pore-pressure uncertainty and $Q$ can be derived, then a real-time (while-drilling) application may be implemented. The application would calculate $Q$ after reaching the overpressure zone and use the correlation from historical wells to estimate the pore-pressure uncertainty that is expected at greater depth. As an example, a correlation between $Q$ and the relative pore-pressure uncertainty has been found from the analysis of acoustic logs from the Asia Pacific region (Figure 10). More correlations are required, and more wells need to be analyzed to obtain a better overview of the applicability of the proposed method, so the presented correlation is shown for illustration rather than for application.

The value $Q$ also provides information to compare uncertainty resulting from different porosity-indicating logs (such as acoustic logs and resistivity logs) within one well or from the same porosity-indicating logs from different wells. The value $Q$ is obviously affected by the log variability. The curve variability affects the uncertainty associated with the series of trend lines calculated over the various depth intervals. Larger variability causes a larger uncertainty, and thus a trend-line envelope will strongly broaden with depth in the overpressure region. As discussed above, $Q$ is also affected by the filtering process as this controls the variability of the shale data set. Strong filters (i.e., large filter coefficients) will result in a stronger smoothing of the data and cause a reduced variation of the trend-line parameters slope and intercept. Comparing $Q$-factors from different regions showed that the values can be similar for a specific region but may vary between different regions (Figures 7 and 8). In particular, $Q$-factors are largest for the data sets from the North Sea and smallest for the data from the Gulf of Mexico. This is probably because the North Sea shows much more variability in sedimentation patterns and has been more greatly affected by tectonic processes in its geologic history than the Gulf of Mexico. A regional $Q$ could be determined as soon as several wells are drilled within one region and can be applied to determine uncertainty in the predrill pore-pressure model prior to drilling the next well. In addition, uncertainty for pore-pressure models calculated from resistivity data is higher than for acoustic data for wells from the Asia Pacific basin, whereas data sets from the Gulf of Mexico and North Sea do not show similar trends. Despite the large number of data sets included for the Asia Pacific region, the wells are from a relatively restricted area, which explains the similar trends in uncertainty for resistivity and acoustic measurements. In contrast, wells from the Gulf of Mexico and North Sea are from a larger area and lithological and geologic variations are more important.

We observe higher $Q$ values in pore-pressure models based on resistivity measurements than in those calculated from acoustic logs, although in general, the accuracy of while-drilling resistivity measurements is higher than the accuracy of acoustic measurements. Resistivity and acoustic measurements are sensitive to porosity, but porosity is not the only factor influencing the measurement. In particular, resistivity measurements are sensitive to even small changes in temperature and salinity. Moreover, clay minerals are a major constituent of shale and contribute to electrical conductivity due to electrical conduction at mineral surfaces. Therefore, even small variations in clay mineral content and type affect the resistivity log.

A comparison of $Q$ values from different porosity-indicating logs of the same well gives information on the log delivering the lowest pore-pressure uncertainty associated with trend-line variations. In addition, $Q$ can be used to compare the expected pore-pressure uncertainty from trend-line variations between wells from different locations within the same region and for similar geologic conditions. A large $Q$ value denotes a larger pore-pressure uncertainty compared to logs showing a small $Q$ value.

## CONCLUSION

Modeling pore pressure from downhole geophysical logs requires a complex multistage and multiparameter workflow. Each of the different data sources and modeling steps contributes to the uncertainty of the final pore-pressure model. Although the uncertainty contributed by the geophysical measurements can be quantified knowing the measuring devices and processing procedures, the uncertainty inherent in the geologic model and manual processing steps is more difficult to quantify and has a larger impact on the overall uncertainty than the geophysical measurements. In particular, the manual processing steps are considered to be a source of large uncertainties that are difficult to quantify. By replacing some of the manual processing steps by algorithms, the subjective interpretation is replaced by a standard calculation. This allows for applying statistical processes to investigate the uncertainty contribution of different parameters and to quantify uncertainty from the individual modeling steps.

In conclusion, replacing manual processing steps by algorithms enables reducing the subjectivity of the human interpretation steps. More objective modeling results are thus achieved if the algorithms are used as automated steps within the pore-pressure modeling workflow. They can also be used to assist the analyst, for example, by giving a trend-line envelope of the most probable and extreme trend line that can be derived from the porosity-indicating log.

Automation adds value to the pore-pressure modeling workflow because manually difficult activities (such as drawing large amounts of trend lines or asking many interpreters for a statistically sound uncertainty analysis) can be easily conducted to calculate uncertainty of the final pore-pressure models from the different processing steps. Modeling results can thus be reproduced by automated algorithms, so that the final model becomes more transparent for modelers and interpreters.

## ACKNOWLEDGMENTS

The authors acknowledge the financial support of the conducted work by the Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU, Germany), grant no. 0325073. We also acknowledge the reviewers of the manuscript, in particular, M. Tingay and S. Bordoloi, for their time and efforts spent to support us with fruitful comments and discussions.