## Abstract

The Heather oil field is located in the Northern North Sea and is operated by DNO Heather Limited. Oil is produced from sandstones of the Middle Jurassic Brent Group. Although approaching non-commercial flow rates, infill projects are under way which target unswept oil. At this late stage in field life, projects are typified by extracting as much new oil from the field as possible with minimal expenditure. These result in incomplete or degraded log data. This paper describes how the application of innovative techniques can replace data shortfalls, providing quality data without compromising budgetary constraints.

Four infill wells were drilled during 2000 and 2001. Innovative petrophysical techniques have been used to enhance the electric log data gathered. Fuzzy logic techniques have been applied to predict reservoir permeability and choose perforating intervals. Some sections of the highly deviated wells were necessarily logged in sliding mode and fuzzy logic was also used to repair the resulting degraded log curve data. Following the infill well programme, field production has increased from 5000 BBL oil per day (BOPD) to 7000 BOPD. This success was partly due to the better location of wells, particularly 2/5-H62Y, because of the improved reservoir model resulting from these petrophysical techniques.

The height above free-water level (HAFWL) was estimated across the Heather Field using the so-called FOIL function (bulk volume of water plotted as a function of HAFWL) ( Cuddy 1993). The FOIL function has given valuable insights into free-water level variation across the Heather Field and has improved water saturation modelling. A method is presented in this paper for geosteering using the FOIL function.

## Introduction

The Heather oil field is located in Block 2/5 in the Northern North Sea, approximately 90 miles northeast of the Shetland Islands (
Fig. 1). Oil is produced from sandstones of the Middle Jurassic Brent Group, at depths between 9500 feet and 11 600 feet below sea-level. Current (March 2001) production is 7000 BBL oil per day (BOPD) plus 23 000 BBL water per day (BWPD), representing a field water-cut of 77%. To date, 116 × 10^{6} stock tank barrels (STB) of oil have been produced from the Heather Field out of an estimated in-place volume of 464 × 10^{6} STB. Production is from 22 gas-lifted wells with pressure support from nine water injectors. The production facility is a single steel-jacketed platform set in 470 feet of water. Gas is imported from the western leg of the Far North Liquids and Associated Gas System (FLAGS). Oil is exported to the Sullom Voe Terminal via the Ninian Pipeline system. In addition to the main field, there are a number of satellite discoveries and prospects, also containing oil in Middle Jurassic sands. The map (
Fig. 1) indicates the two largest satellites: the North Terrace (estimated 45 × 10^{6} STB of oil in place) and West Heather (estimated 67 × 10^{6} STB of oil in place). None of the Heather satellites have yet been put on production.

At this critical stage in field life, the data acquisition aspects of the project are typified by extracting as much new data from the field as possible with minimal expenditure or delay. This inevitably results in incomplete and sometimes even degraded log data. This paper describes how the application of innovative techniques can make good any data shortfalls resulting from the commercial pressures of marginal mature field management, providing quality data without compromising budgetary constraints. DNO Heather Limited, the field operator, drilled four infill wells during 2000 and 2001. Following the infill well programme, field production has increased from 5000 BOPD to 7000 BOPD.

Innovative petrophysical techniques have been applied to enhance the log data gathered using logging while drilling (LWD):

log curve repair to improve LWD data and to fill in log data gaps;

permeability prediction from wireline and LWD data;

water saturation modelling;

proposed method of geosteering using water saturation modelling.

## The Fuzzy Logic Concept

Log curve repair and permeability prediction techniques described in this paper are based on the concept of fuzzy logic (
Cuddy 2000*a*). Fuzzy logic is an analytical statistical technique. Fuzzy logic is an extension of conventional Boolean logic (zeros and ones) that has been developed to handle the concept of ‘partial truth’, i.e. truth-values that lie between ‘completely true’ and ‘completely false’. Dr Lotfi Zadeh of UC/Berkeley introduced it in the 1960s as a means to model uncertainty (
Zadeh 1965).

Science and our way of thinking is heavily influenced by Aristotle’s laws of logic formulated by the ancient Greeks and developed by many scientists and philosophers since then ( Kosko 1993). Aristotle’s laws are based on ‘X or not-X’; a thing either IS, or IS NOT. This has been used as a basis for almost everything that we do. We use it when we classify things and when we judge things. Managers want to know whether something is ‘this’ or ‘that’, and even movies have stereotypical goodies and baddies. Conventional logic is an extension of our subjective desire to categorize things. Life is simplified if we think in terms of black and white. This way of looking at things as true or false was reinforced with the introduction of computers that only use the binary digits 1 or 0. When the early computers arrived with their machine-driven binary system, Boolean logic was adopted as the natural reasoning mechanism for them. Conventional logic forces the continuous world to be described with a coarse approximation, and in so doing much of the fine detail is lost.

We miss a lot in the simplification. By only accepting two extreme possibilities, the infinite number of possibilities in between them is lost. Reality does not work in black and white, but in shades of grey. Not only does truth exist fundamentally on a sliding scale, it is also perceived to vary gradually by uncertainties in measurements and interpretations. Hence, a grey scale can be a more useful explanation than two end points. For instance, we can look at a map of the Earth and see mountains and valleys, but it is diffcult to define where mountains start and valleys end.

This is where the mathematics of fuzzy logic comes in. Once the reality of the grey scale has been accepted, a system is required to cope with the multitude of possibilities. Probability theory helps quantify the greyness or fuzziness. It may not be possible to understand the reason behind random events, but fuzzy logic can help bring meaning to the bigger picture. Take, for instance, a piece of reservoir rock. Aeolian rock generally has good porosity and fluvial rock poorer porosity. If we find a piece of rock with a porosity of 2 porosity units (pu), is it aeolian or fluvial? Since this has a low porosity value we could say it is definitely fluvial and get on with more important matters. But let’s say it is probably fluvial but there is a slim probability that it could be aeolian. Aeolian rocks are generally clean (i.e. contain little or no clay minerals) and fluvial rocks shalier (i.e. contain clay minerals). The same piece of rock contains 20% clay minerals. Now, is it aeolian or fluvial? We could say it is has an approximately equal likelihood of being aeolian or fluvial based on this measurement. This is how fuzzy logic works. It does not accept something is either ‘this’ or ‘that’. Rather, it assigns a greyness, or probability, to the quality of the prediction on each parameter of the rock, whether it is porosity, shaliness or colour. There is also the possibility that there is a measurement error and the porosity is 20 pu not 2 pu. Fuzzy logic combines these probabilities and predicts that, based on porosity, shaliness and other characteristics; the rock is most likely to be aeolian and provides a probability for this scenario. However, fuzzy logic says that there is also the possibility it could be fluvial, and provides a probability for this to be the case too.

In essence, fuzzy logic maintains that any interpretation is possible, but some are more probable than others. One advantage of fuzzy logic is that we never need to make a concrete decision. In addition, fuzzy logic can be described by established statistical algorithms. Computers, which themselves work in ones and zeros, can do this effortlessly for us.

## Application of Fuzzy Logic in Petrophysics and Field Rehabilitation

Geoscientists live with error, uncertainty and fragile correlations between data sets. These conditions are inherent to the geosciences, because of the challenge of designing and building sensors to measure complex formations in hostile environments. Even in the laboratory it is diffcult to relate a log response to a physical parameter. Several perturbing effects such as mineralogy, fluids and drilling fluid invasion can influence a simple measurement such as permeability. Conventional techniques try to minimize or ignore the error. Fuzzy logic asserts that there is useful information in this error. The error information can be used to provide a powerful predictive tool for the geoscientist to complement conventional techniques, and is now used routinely in formation evaluation (
Norland 1996;
Cuddy 2000*a*).

Many rock properties measured by electric logging tools are interrelated and can be, in turn, correlated with laboratory measurements of other properties such as permeability. If a log curve is incomplete, the missing section can be restored by comparing the incomplete log curve with other log curves over the same interval. For permeability prediction, log curves are selected that are likely to have a relationship with permeability, for example density logs, computed effective porosity (PHI), shale volume (VSH), formation resistivity (RT), photoelectric factor (PEF) and sonic log transit time (DT). A probability distribution is assigned for each log curve such that certain log curve values are linked to certain ranges of permeability. A most likely permeability can then be assigned at every depth point according to log response.

Many rock properties such as porosity are distributed normally. Fuzzy logic does not require a normal distribution to work as any type of distribution that can be described can be used. However, because of the prevalence of the normal distribution, it is the best distribution to use in most cases. The normal distribution is completely described by two parameters: mean and variance. Core-plugs from a particular litho-facies may have dozens of underlying variables controlling their permeabilities but the permeability distribution will tend to be normal in shape and defined by two parameters – the average value or mean and the variance or the width of the distribution. This variance (the standard deviation squared) depends on the hidden underlying parameters and measurement error. The variance about the average value, or fuzziness, is key to the method and the reason why it is called fuzzy logic.

The normal distribution is given by:

*P(x)* is the probability density that an observation *x* (for example log response) is measured in the dataset described by mean μ and standard deviation *σ*.

In conventional statistics, the area under the curve described by the normal distribution represents the probability of a variable *x* falling into a range, say between *x*1 and *x*2. The curve itself represents the relative probability of variable *x* occurring in the distribution. That is to say, the mean value is more likely to occur than values 1 or 2 standard deviations from it. This curve is used to estimate the relative probability or ‘fuzzy possibility’ that a data value belongs to a particular dataset. If a given permeability range has an electric log response distribution with a mean μ and standard deviation *σ* the fuzzy possibility that the log response *x* occurs for the given permeability range can be estimated using
Equation (1). The mean and standard deviation are simply derived from the calibrating or conditioning dataset, usually core data. This is the starting point for predicting permeability using fuzzy logic. Fuller descriptions of the method have been published (
Cuddy 2000*a*,
*b*). In practice, core-derived permeabilities are divided into a series of ranges or bins. The associated electric log responses are analysed and the probability of the responses correlating with each bin is calculated. Permeability can then be predicted in nearby wells from the same set of log curves.

Fuzzy logic, when applied during field rehabilitation, is a powerful technique. Old fields may have missing log data that can be replaced by fuzzy logic-predicted curves. The existing core database can be used to develop a field-wide permeability predictor, for use in simulation models, identification of high-permeability zones responsible for early water breakthrough, and zones worthy of infill well placement. Some examples are described in the following sections.

### Curve repair

Recent infill wells on the Heather Field suffered from bad sections of log data in the Brent Group reservoir. Geosteering required logging in sliding mode at times, which tends to degrade log quality because the tool is not rotating and does not ‘see’ all around the borehole. Unexpected overpressure was encountered in parts of the reservoir, particularly Ness Formation sands and shales, caused by injection water. To control pressures, mud weight was increased to very high densities. This, in turn, caused differential sticking and hole washouts. Fuzzy logic methods were applied to repair defective log curves. Bad sections of log are first identified interactively. In Track 1 of Figure 2 the caliper (CALI), bit size (BS) and delta rho (DRHO) curves indicate hole washouts at the top and bottom of the logged section. Tool failure has also caused a data gap between 13 500 ft and 13 600 ft. The effect on the bulk density (BAD.RHOB) and neutron porosity (BAD.NPHI) logs can be seen in Track 2. Fuzzy logic is used to find correlations between the ‘good’ sections of the logs requiring repair and other unaffected logs. In this case correlations were established between density logs and gamma ray, conductivity and sonic logs. Based on these a prediction of bulk density (FIXED.RHOBE) and neutron porosity (FIXED.NPHI) is made. The quality of the prediction is confirmed by checking the overlay in the good sections (Tracks 4 and 5) between ‘bad’ and ‘fixed’ logs. The resulting ‘fixed’ logs are displayed in Track 3. A computer-processed interpretation (CPI) based on the corrected logs is displayed in Track 6. The CPI displays VSH, PHI and bulk volume of water (FOIL).

### Permeability prediction

Knowledge of permeability, the ability of rocks to flow hydrocarbons, is important for understanding oil and gas reservoirs. Permeability is best measured in the laboratory on cored rock taken from the reservoir. However, coring is expensive and time-consuming in comparison to the electronic survey techniques most commonly used to gain information about permeability. In a typical oil or gas field all boreholes are ‘logged’ using electrical tools to measure geophysical parameters such as porosity and density. Samples of these are cored, with the cored material used to measure permeability directly. The challenge is to predict permeability in all boreholes by calibration with the more limited core information (
Cuddy 2000*a*).

Permeability is commonly measured in the laboratory on cored rock taken from the reservoir. However coring is expensive and time-consuming in comparison to the electronic survey techniques most commonly used to gain information about permeability. In a typical oil or gas field all boreholes are ‘logged’ using electrical tools to measure petrophysical parameters such as porosity and density. Only a few wells are cored. Core samples are used to measure air permeability under a low confining pressure by invoking Darcy’s law from flow-rates and pressure drops. The challenge is to predict permeability in all boreholes by calibration between electric log data and the more limited core information.

Permeability is a very diffcult rock parameter to measure directly from electrical logs because it is related more to the aperture of pore throats rather than pore size. Although the resistivity log is somewhat related to pore throat size, most electrical logs are mainly affected by porosity. Determining permeability from logs is further complicated by the problem of scale; many well logs have a vertical resolution of typically 2 ft compared to the 2 inches of core plugs. In addition to these issues, there are measurement errors on both the logs and core. When you add these problems together, it is surprising that predictions can be made at all. The mathematics of fuzzy logic provides a way of not only dealing with errors but also using them to improve the prediction. This is an improvement on conventional methods that ignore or try to minimise the error.

First, the core permeability values are scanned by the fuzzy logic program and divided into ten (or more) equal bin sizes on a logarithmic scale. That is to say that the bin boundaries are determined so that the number of core permeabilities in Bin 1 represents the tenth percentile boundary of the permeability data. Bin 2 represents the twentieth percentile boundary and so on. In this example, there are ten divisions in the data but there is no reason why there could not be twenty or more. Each of these bins is then compared to the electrical logs. The log data associated with levels in the well corresponding to Bin 1 (very low permeability) are analysed and their mean and standard deviation calculated. In this way, not only is the average or most probable log value associated with Bin 1 calculated, but also some idea is gained of the uncertainty in the measurement as calculated by the standard deviation or fuzziness. Again, PHI and VSH are the best and first logs to try. Fuzzy logic asserts that a particular PHI value can be associated with any permeability but some are more likely than others.

This logic is clarified using
Figure 3. For simplicity, it shows only five bins that represent each of the five familiar decades for logarithmic permeability. The diagram shows only two axes (porosity and volume of shale) whereas the technique can use an unlimited number of bins in *n*-dimensional space. The mean value of PHI and VSH for each permeability bin is represented by the point at the centre of each cross. For instance, for core permeability greater than 100 mD the average PHI and VSH are 26 pu and 12% respectively. The vertical and horizontal lines through each point represent the error bar or standard deviation (fuzziness) of data in that bin. The error bars are different for each bin. The resulting permeability line is field specific and is ‘S’ shaped when shown without scatter. A real cross-plot of log data would show considerable scatter about this curve. A single curve predictor would predict different permeabilities depending on whether PHI or VSH was taken as the predictor. Take a log depth that has a PHI of 23 pu and VSH of 30% (
Fig. 3): a porosity-only predictor would estimate a permeability of 10–100 mD by extrapolating the point vertically; the volume-of-shale-only predictor would give a permeability of 0.1–1 mD by extrapolating the point horizontally.

Fuzzy logic, in contrast, can deal with ‘shades of grey’. The point at PHI 23 pu and VSH 30% would be compared to all permeability bins. Knowing the mean and standard deviation of each bin, the fuzzy possibility that the point lies in that bin can be calculated. This is done separately for PHI and VSH. Their fuzzy possibilities are combined to predict the permeability for that log depth with its associated fuzzy possibility or ‘greyness’. Figure 4 shows typical results of this analysis where each of the ten permeability bins has an associated fuzzy possibility. The highest fuzzy possibility is taken as the most probable permeability for that combination of log measurements (bin number 6, 277 mD). A predicted permeability is calculated as the weighted mean of the two most probable bins (bins 5 and 6). The fuzzy logic method is better than deterministic algorithms such as multiple linear regression as it gives better predictions at the extremes. It is important for reservoir modelling that the highest and lowest permeabilities are correctly implemented rather than just the average values.

The program uses any number of permeability bins with any number of input curves. The distribution of bin boundaries depends on the range of expected permeabilities, as described above. The number of bins depends on the number of core permeabilities available for calibration (the statistical sample size). A reasonable sample size is around 30. Consequently, the number of bins is determined so that there are at least 30 sample points per bin. For a well with 300 core permeabilities, it would be appropriate to use 10 permeability bins. Vertical permeability can simultaneously be predicted by simply comparing the core vertical permeabilities with the logs in a similar manner. ‘Blind testing’ between wells is used to test the predictive ability of the technique. In blind testing, the technique is calibrated in a cored well and ‘blind-tested’ on logs in a second cored well to see how well the predicted permeability fitted the actual core permeabilities in the second well.

Fuzzy logic was applied to predict reservoir permeability in Heather Field wells, to help determine production index and choose perforating intervals. An example from the Heather Field is shown ( Fig. 5). Track 4 shows the comparison between core-derived (CKHL) and fuzzy-predicted permeabilities in one of the cored Heather wells (K and KP). K is the permeability predicted from log data using core data from this well only. KP is the predicted permeability using core data from a number of other wells instead. KP is the blind test. Input log curves were RHOB, NPHI, conductivity (COND), DT and VSH. Note that in this case PHID is the calculated log porosity. Both K and KP curves fit the observed core permeability data very well except between 11 770 ftbrt and 11 775 ftbrt, where some of the electrical logs are adversely affected by the Rannoch shale.

### Water saturation modelling

In the Heather Field, oil is trapped in a tilted fault block structure, dipping down to the northwest ( Fig. 6). Structural closure is at approximately 12 800 ft true vertical depth subsea (ftTVDss). This is deeper than the base Brent Group reservoir in any of the Heather wells and an oil–water contact is not observed in the main field. The Brent Group reservoir ranges between 125 ft and 370 ft thick, with an average thickness of 224 ft. There is a continuous oil column in most wells in the Heather Field, except in the cemented zones indicated on the figure. Free-water levels were observed in satellite oil pools and in a separate fault compartment (known as B Block) in the extreme east of the main field ( Fig. 6).

The free-water level (FWL) in an oil or gas field is the elevation of the oil–water or gas–water contact free of capillary effects, i.e. the level of the fluid contact in a very wide borehole. It can be defined as the depth in the reservoir where the pressure in the oil phase is equal to the pressure in the water phase. In the reservoir, the fluid contact as indicated by electric logs may be considerably shallower because of capillary effects in the reservoir rock giving 100 percent water saturation above the free-water level. Water saturation (SWE) versus height above FWL (HAFWL) functions can be used to describe the initial fluids in place in the reservoir model and investigate reservoir compartmentalization. This method is derived from capillary pressure theory and therefore is based on the FWL rather than the oil–water contact (OWC; Cuddy 1993).

If SWE alone is plotted as a function of HAFWL, the shape and position of the curve is dependent on porosity and permeability. This dependence can be reduced by plotting bulk volume of water (BVW) against HAFWL. BVW is the product of water saturation and porosity, referred to here as FOIL. The FOIL–HAFWL function is known as a FOIL function. Where the FOIL function is largely unaffected by rock type, it is a powerful water saturation tool for complex lithologies ( Cuddy 1993).

FOIL–HAFWL plots of wells in the Heather Field and its satellites indicate there is a good correlation between the BVW or FOIL and HAFWL that is independent of porosity. The plots also indicate some hitherto unrecognised complexities. In order to establish the position of the FWL, FOIL data are first plotted in TVDss (
Fig. 7). Only net reservoir data more than one metre from a bed boundary are displayed and used for line fitting. The data within one metre of electrofacies bed boundaries are eliminated to reduce data that could be affected by the different vertical logging tool responses. Data that are non-net (as indicated by a PHI cut-off of 10 pu) are not used because the rock matrix in these intervals compute water saturations that are greater or equal to 100 saturation units (su). The FWL lies where data become asymptotic to the FOIL axis. In this case (
Fig. 7) the FWL is at approximately 10 730 ftTVDss, a depth that matches the structural spill point of the North Terrace satellite. Following identification of the FWL elevation, the FOIL data are replotted in terms of HAFWL (
Fig. 8). A curve can be fitted to the data points using appropriate curve-fitting algorithms and the equation of the line determined. A simple function describes the correlation: BVW=aH^{b}, where BVW is SWE*PHI=FOIL; SWE is effective water saturation; PHI is porosity; H is HAFWL (height above the free-water level); and a, b are constants.

The form of this function was verified by plotting the logarithm of BVW against the logarithm of HAFWL
(Equation 2). This is the form of the straight-line equation *y*=*mx*+*c*.

As the dataset is reasonably linear with log axes a power function should correctly describe the correlation. The constants a and b are determined by least squares regression, with BVW being the independent or predicted variable. The constants a, b and the FWL can also be determined simultaneously using genetic algorithms or conventional techniques. Typical values for a and b are shown on Figure 8. The classical water saturation curves for the field are easily derived from this function by dividing bulk volume of water by porosity: SWE=BVW/PHI.

FOIL curves were constructed for other areas of the Heather Field ( Figs 9 and 10). It was expected that all main field data would indicate a FWL at the structural spill point of 12 800 ftTVDss. Surprisingly, at least three different FWLs were observed in the main part of the field. The FOIL plot for Northwest Heather ( Fig. 9) does indicate that the FWL downflank is at or near the spill point. Crestal wells ( Fig. 10) may share a common FWL with what had previously been assumed to be the isolated faulted B Block. Mid-field wells, appear to have a FWL at around 11 550 ftTVDss. The mid-field FWL coincides with the OWC on the original petro-physical study. This OWC had been chosen before any wells had been drilled in Northwest Heather and was based on the early observation that there was no moveable oil in the cemented zone down-dip of the mid field area.

If the FWL elevation is known, SWE can be calculated using the FOIL function. It is instructive to perform this calculation and compare the result with SWE determined from the resistivity logs ( Fig. 11). This is a useful test of the FOIL function. In the Heather Field example shown, there is generally good agreement between the SWE curve computed from resistivity logs and the water saturation curve (SWH) computed from the FOIL function. Where there are differences, these may give us important information about the formation response. The FOIL function assumes initial reservoir conditions and vertical connectivity between beds, and this may not always be the case. The example well ( Fig. 11) features a swept zone (numbered 1 on the figure), thin sands (2) and thick clean sands (3). In the swept zone, the FOIL function predicts SWE values that are too low, although these may represent original saturations. In thin sands, the FOIL function may be giving a better assessment of water saturation. This is because SWH is computed from the porosity log, whereas SWE is computed from the resistivity log. Porosity logs typically have better vertical resolution than resistivity logs and, in the case of an oil-bearing zone, oil-in-place is missed by the resistivity log. In thick clean beds the two water saturations coincide.

### Proposed method of geosteering using water saturation modelling

In appropriate circumstances, the FOIL function can be used for geosteering by determining the distance from the FWL. The method is applicable in horizontal wells where it is desirable to maintain a certain distance above the FWL to avoid water coning. The FWL is determined in offset wells and resistivity log data are extracted from a 3D geological model. The resistivity response is modified for tool type and hole angle and a resistivity/height profile is backed out via the FOIL function. An example is shown ( Fig. 12) of a resistivity model (Rt) for a horizontal borehole in a thin oil-bearing sand. The Rt response in the overlying shale is coincidentally the same as in the water leg. The response of different resistivity curves can be compared to the model to determine which depth of investigation will be least affected by the influence of adjacent beds. In this case the best match is between the red curve (R55P) and the modelled Rt curve.

## Conclusions

Fuzzy logic is a powerful predictive tool with many rock property prediction applications. Permeability prediction by fuzzy logic allows better choice of perforating intervals and can be applied to model building to map permeability, although it is still reliant on a good core permeability database. Permeability prediction is useful to complement current technology and to gain insight to older wells without core and extensive logging programmes. When LWD data are of lower quality than wireline data due to drilling rate and borehole effects, curve repair using fuzzy logic can bridge the quality gap.

The FOIL function is a simple and effective way of describing fluid saturations and can easily be applied in geological and reservoir engineering simulation models. Saturation modelling with the FOIL function has many uses including understanding of structural complexity and use in geosteering to maintain a required distance from fluid boundaries.

The methods described here use basic log datasets, such as porosity and density, which are cheap and easy to obtain, rather than depending on new and expensive logging technology. Over recent years, oil exploration has suffered due to erratic and often low oil prices. Oil producing countries are now struggling to meet demand and there is an urgent need to find new reservoirs and make effcient use of existing resources. Fuzzy logic can make an important contribution to this endeavour.

## Acknowledgments

The authors wish to acknowledge the contributions made by staff of Helix-RDS Limited and DNO Heather Limited.