spe 102093 pore perm relationship

Three Statistical Pitfalls of Phi-K TransformsPierre Delfiner, SPE, TOTAL S.A.

Summary

Phi-k transforms are used widely to predict permeability. Some ofthe difficulties of this exercise are well identified, such as thehomogeneity of the population (rock typing), the matching of coresand logs (especially depth matching), and the problem of per-meability upscaling. Not so well-known, however, are the pitfallsof a statistical and geostatistical nature that may create signifi-cant biasesalways in the same directionan underestimationof permeability.

The passage from Phi to k is performed in three steps: (1) incored wells, an exponential regression equation is established be-tween core porosity and core permeability k; (2) in uncored wells,log porosity is used instead as input to predict permeability; and(3) the same equation is sometimes used again to populate the cellsof a dynamic reservoir model in 3D, where input porosity valuesare obtained by interpolation.

The core-scale regression equation generally underestimatespermeability by at least a factor of 2. The origin of the bias lies inthe reverse transformation from logarithmic to arithmetic scale. Toavoid this pitfall, a new permeability estimator is proposed, based onthe quantile curves of the Phi-k crossplot. This estimator is data drivenand does not assume a priori any particular functional relationshipbetween Phi and k, such as an exponential-regression function.

One of the simplest diagnostic tools to check the agreementbetween log and core porosity is a crossplot of one against theother. In the absence of bias, the points are expected to be distrib-uted along the yx line. In reality, they either are or they are not,according to which variable is plotted along the x-axis. This ap-parent paradox is elucidated by bivariate regression theory andrelated to the difference of investigated volume between core andlog data.

Direct input of upscaled cell porosity into an exponential core-scale permeability transform amounts to forcing geometric perme-ability averaging, which may again lead to serious underestimationof the true upscaled permeability when heterogeneity is significant.

Introduction

Porosity-permeability correlations are often used to predict perme-ability and to populate the cells of a dynamic reservoir model. Thepassage from Phi to k typically involves three steps:

1. In cored wells, a regression equation, or transform, is estab-lished between core porosity and core permeability, or more ex-actly, between core porosity and the logarithm of permeability.

2. In uncored wells, log-derived porosity is used as input to thisequation to predict permeability.

3. The same equation is sometimes used again to distributepermeability in 3D at the scale of the cells of a reservoir model,where input porosity values are now obtained by interpolationbecause most cells are not traversed by a well.Each step has its own complexities and pitfalls. This paper willmention just a few.

In cored wells, it is important to ensure that the core measure-ments are representative and reliable and will not bias the Phi-klaw (k and K denote point and block permeabilities, respectively).Plugs with microfissures enhancing permeability should be re-moved. Likewise, oven drying may cause clay collapse and per-

meability increases by a factor of 5 to 30 (Soeder 1986). [Scan-ning-electron-microscope (SEM) images show how clay fibers canbe smashed against the pore walls (Pallat et al. 1984)]. On theother hand, very-high permeability data may be missing because ofpoor core recovery in unconsolidated sands. All these biasesshould be removed from the analysis. Of course, the porosity-permeability correlation should be established only in reservoirsections, and is generally improved by working rock type by rocktype (i.e., rocks that have a similar porous network resulting froma common geological history). As k ranges over several orders ofmagnitude, porosity-permeability data are usually plotted in semi-log scale, which, as we will see, is not without consequences.

In uncored wells, a Phi log is usually available and is substi-tuted for Phi core in the porosity-permeability equation. This isvalid if, for the rock type considered, the two measurements areconsistent, which cannot be taken for granted. The porosity mea-sured in the laboratory on core samples depends on the process ofcore cleaning and drying, but is closest to total porosity. Loganalysis provides both total and effective porosity (i.e., excludingclay-bound water in shaly formations). When comparing log-derived porosity with core porosity, one must make sure to usetotal porosity. Another source of discrepancy is the differencebetween the volume measured by a core plug and the volumeinvestigated by logs.

Finally, when comparing core and log data, be it porosity orpermeability, correct depth matching is critical because minimaldifferences in depths may modify the correspondence between logand core measurements drastically and lead to erroneous conclusions.

The final step is to populate the cells of a dynamic reservoirmodel with petrophysical data. Because most of the cells (say,99.8% of them) are not traversed by a well, sophisticated geo-statistical techniques are used to interpolate porosity from welldata, guided by geological trends and combined, if possible, withseismic attributes. Porosity is a key parameter controlling volumein place and is generally validated by different disciplines, whereaspermeability belongs to reservoir engineers. A correlation estab-lished at the scale of a 11-in. cylindrical core plug may not beused for a 10010010-m grid cell. Direct use of the core-scalePhi-k transform leads to a permeability field that can be signifi-cantly pessimistic. It amounts to geometric averaging, whichgenerally is not the correct upscaling method in 3D. In thepresence of strong permeability contrasts, the bias between thecorrect upscaled permeability and the geometric mean can be quitelarge. The same is true for the arithmetic mean, which is evenmore sensitive to permeability dispersion. In this paper, much ismade of the need to get the arithmetic mean correct, not becauseit is necessarily the correct averaging technique, but to avoid es-timation bias. Permeability estimation and upscaling are two dif-ferent problems.

In the above, we have discussed core only as the source ofpermeability data. Wireline formation testing (WFT) also providesdynamic information in the form of mobility data. An example willshow that the methods presented here also apply. The dynamicdata that reservoir engineers seek, however, are those of welltests because they are more representative of the reservoir scale.Phi-k transforms also remain useful in this case to redistribute theglobal permeability value provided by the test vertically over thetested interval.

Phi-k TransformBias in the Standard Transform. A Phi log k crossplot based on261 plug measurements from a reference well is shown in Fig. 1.The linear regression explains 41 % of the variance of log k, which

Copyright 2007 Society of Petroleum Engineers

This paper (SPE 102093) was accepted for presentation at the 2006 SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, 24-27 September, and revised for publi-cation. Original manuscript received for review 28 June 2006. Revised manuscript receivedfor review 11 July 2007. Paper peer approved 26 August 2007.

609December 2007 SPE Reservoir Evaluation & Engineering

corresponds to a correlation coefficient of 0.64. Engineers wouldtend to agree that this is a reasonably good linear fit, given that allsamples have been thrown into the plot without consideration ofrock types and that permeabilities range more than four orders ofmagnitude. The quality of the fit is further confirmed by inspectionof the residuals in Fig. 2, which are the differences between thelogarithms of actual and predicted ks. When plotted in logarithmicscale, these residuals represent ratios of actual-to-predicted k. Theyare symmetrically distributed around 1 and show no evidence ofany remaining structure in the data. The regression has squeezedthe data almost dry.

The plot of actual-vs.-predicted k in Fig. 3 is another presen-tation of the same information, showing the consistency of per-meability estimates.

As the value of interest is k rather than its logarithm, the stan-dard practice is to exponentiate the equation for log k:

k = 0.3187 exp0.2291 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)This is the solution provided by Excel when an exponentialtrend line is selected. To our surprise, however, the k values pre-dicted by this equation underestimate permeability by a factor ofapproximately 2, as shown by arithmetic means:

mean k = 99.61md Mean kpredicted = 51.19md.

A graphical way to look at this bias is to cumulate kh values fromthe well, bottom up (Fig. 4). The cumulative kh corresponds to thetransmissivity of the interval and is representative of the flowprofile if the skin effect is constant vertically. We observe that theprofile constructed with predicted k values is approximately halfthat obtained with actual k values.

Origin of the Bias. As shown in Fig. 3, no bias occurs in thelogarithmic scale because a least-squares fit passes through themiddle of the data (when the intercept is left free). So the predictedand the actual log k have the same mean, and, therefore, the geo-metric means of k and of kpredicted coincide. (Recall that the geo-metric mean of n positive numbers is equal to the nth root of theproduct of the numbers.) Arithmetic means (sums divided bycounts), however, are related by

mean kpredicted mean k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

Fig. 2No structure left in the residuals.

Fig. 3Actual and predicted k agree in logarithmic scale.Fig. 4Pseudoflow profiles: the actual, that predicted by expo-nential regression, and that predicted by Swansons formula.

Fig. 1Porosity-permeability crossplot with linear fit by leastsquares in semilog scale.

610 December 2007 SPE Reservoir Evaluation & Engineering

To explain this inequality, consider the regression model

log k = a + b + , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)where a+b is the true regression function of log k (natural log)and is a random error with zero mean and constant variance. Ifa and b are estimated with good accuracy (so that the estimatedand true regression functions coincide), the exponential trend linegives kpredictedexp (a+b ) and

k = kpredicted exp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)where exp () is a multiplicative factor that creates a bias becauseits mean value is greater than 1. Indeed, the mean of the exponen-tial of a random variable is always greater than the exponential ofthe mean (Jensens inequality). This property results from the con-vexity of the graph of the exponential function and is illustratedgraphically in Fig. 5. Each chord linking two points on the plottedline lies above the line itself. For example, consider two equiprob-able values 2 and 2; their mean is 0 and exp (0)1. Butthe mean of exp () lies on the chord linking exp (2) and exp (2)and is greater than 1.

It is possible to evaluate the bias if we assume the error to benormally distributed with mean 0 and variance 2 [i.e., exp ()is lognormal]

Eexp = exp22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)This shows that the bias is directly related to the dispersion of logk about the regression line: If there were no dispersion (i.e., aperfect functional relationship between Phi and log k), taking theexponential would cause no bias. With data from Fig. 1, the re-sidual variance is found to be 22.438 and the bias factor isequal to 3.38. This is more than the ratio of 2 that we have ob-served. A log normal correction of the bias is not appropriate inthis case.

A Nonparametric Estimator. The bias is caused by the fact thatapplying a nonlinear transformation and taking the mean are notinterchangeable operations. The solution proposed here is to use apermeability estimator based on quantiles. Unlike the mean, thequantiles of a random variable are preserved under increasingtransformations, such as logarithm or exponential. For example, ifX10 is the 10% quantile of X (i.e., 10% of the values are less thanX10), then log(X10) is the 10% quantile of log (X). A formula fromPearson-Tukey (1965) allows the estimation of the mean of adistribution by a weighted average of the 5, 50, and 95% quantiles:

Pearson-Tukeys mean = 0.185 X5 + 0.63 X50 + 0.185 X95.. . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

This formula requires no assumption on the underlying distributionand works remarkably well for all cases of practical interest. An-other estimate of the mean can be obtained by the so-called Swan-sons rule (Brown et al. 2000)

Swansons mean = 0.3 X10 + 0.4 X50 + 0.3 X90. . . . . . . . . . . . (7)

This formula is calibrated for the log normal distribution but worksunder more general conditions.

The Pearson-Tukey formula is more accurate than Swansons,but it is more difficult to estimate X5 and X95 reliably than X10 andX90 because they are more extreme. In the petroleum industry,Swansons formula is popular because of the special role played byX10 (P90) and X90 (P10) with regard to the definition ofproved and possible reserves.

In both formulas, the Xs are conditional quantiles, that is, thequantiles of the conditional distribution of k given Phi. When Phivaries, they define quantile curves.

Fig. 6 shows the X10, X50, and X90 quantile curves obtaineddirectly from the core porosity and permeability data without anyprespecified model. The plot is divided into vertical slices with5-p.u. width, and in each slice the 10, 50, and 90% quantiles arecomputed. A 50% overlap is left between consecutive slices forbetter smoothness. The width of the slice results from a tradeoffbetween the stability of quantile estimates and the ability to seedetail. The more extreme the quantile, the more data are needed.For example, to estimate the 10% quantile, it is reasonable torequire at least two points below X10, which translates into a mini-mum of 20 data points in each slice. Estimating X5 requires at least40 points.

These quantile estimates are then combined by Swansons for-mula to obtain unbiased permeability estimates (squares in Fig. 6).Finally a trend line is fitted to these points to provide an equationto compute k for any Phi in the fitted range. In practice, it isadvantageous to do the fitting of log k against Phi in arithmeticscale because it gives us the option of using polynomial trend func-tions. When this porosity-permeability transform is used, the meanpredicted value is in line with the sample mean. The bias is gone.

mean k = 99.61md mean kpredicted = 97.44md

The Swanson curve in Fig. 4 shows the improvement in the pre-dicted pseudoflow profile.

It is interesting to superimpose the straight-line fit of Fig. 1(exponential regression function) on the quantile curves of Fig. 6.We observe that it is close to the X50 curve. As linear-regressionestimates the mean of log k given Phi (conditional distribution), weconclude that the mean and the median of log k are close, whichwould suggest that log k given Phi has a symmetric distribution,such as a normal one (i.e., k log normal). But if this were true, theX50 curve should be halfway between X10 and X90, which is not thecaseX50 being closer to X90, especially for larger values of Phi.We also observe that the X90 curve levels off, indicating a kind ofcapping of permeability. As noted earlier, the log normal model isnot appropriate for the present data setthis is not a problem

Fig. 5The exponential of the expected value is less than theexpected value of the exponential.

Fig. 6Estimated quantiles X10, X50, X90, Swansons mean(black squares), and exponential-regression function.


because we do not need a distribution model to compute Swan-sons mean.

Mobility Data. In Fig. 7, mobility data (k/) from three differentwireline tools have been placed into a plot against vertically av-eraged reservoir porosity (to avoid depth mismatch between for-mation tester and porosity logs). The first use of formation testersis to measure pressure. Mobility is estimated with drawdown andprovides a good qualitative indicator, but quantitative results aremuch affected by wellbore conditions and thin-scale heterogeneities.With such dispersion, a statistical approach to data is required.

The same procedure as before is applied, using vertical slices of1-p.u. width and no overlap, and considering only porosity data inthe 6 to 20 p.u. interval. This time, however, trend lines are fittedfirst to the X10, X50, and X90 quantile curves and then Swansonsmean is computed. Linear trend lines provide good fits in semilogscale. The overall linear-regression curve coincides with the X50curve. The X10 and the X90 curves are located symmetrically oneither side of the X50 curve, which would be consistent with a lognormal distribution of mobility, though with a dispersion increas-ing with porosity. Considering only the data in the 6 to 20 p.u.interval, the following mobility means are obtained:

Actual = 60.84. Exp regression = 11.18. Swanson = 41.47.The actual mean is the largest because it is affected by a few largeoutliers. For example, if the top 1% of mobility values are removed(16 values), the actual mean decreases by 35% to 39.73. Theexponential regression is clearly underestimating mobility. Swan-sons mean provides a better description of the data.

Discussion. The proposed procedure differs from a straight mul-tiplicative correction equalizing the means of actual and predictedvalues. Fig. 8 shows the hypothetical multiplier to apply to theexponential-regression estimate to get Swansons mean: It is farfrom a constant. It is an increasing function of porosity in the caseof the mobility data and a bell-shaped function varying between 1and 3.4 in the case of core data. A constant multiplier is found inthe special case of proportional quantiles, meaning quantile curvesthat are parallel when plotted in logarithmic scale. This occurs, forexample, in the constant-variance log normal error model of Eq.5but in both examples presented, the variance did depend on Phi.

It would not be a good idea to attempt to match the samplemean permeability because it may be dominated by a few highpermeabilities. The sample mean is nonrobust. Likewise, a directdetermination of the regression function by computing permeabil-ity arithmetic means in porosity slices is not a good idea. Bycontrast, the median and other quantiles, if not too extreme, arerobust against the presence of a few outliers. Our proposed esti-mator inherits this robustness.

Statistically this estimator is nonparametric in the sense that itavoids having to specify a priori a particular functional relation-ship between Phi and k, such as an exponential regression. Thedata are allowed to speak for themselves. For example, curvilinearrelationships may be fitted between Phi and log k, as in Fig. 6. Theonly significant assumption is the validity of Swansons formulafor the mean.

Core Porosity vs. Log Porosity

We now have an improved equation to predict core permeabilityfrom core porosity. Can it be used with log porosity?

Phi-core-Phi-log Regression. One idea that comes to mind is tocross plot log porosity against core porosity to check if they are inagreement. We expect the points to be distributed along the yxline. Fig. 9 shows such a plot with 943 points and its associatedlinear trend line. The regression explains a fraction of varianceequal to R20.8433, corresponding to a correlation coefficient of0.918 between core and log porosity. This can be considered asvery good agreement.

Fig. 7Mobility data from wireline formation testing. Estimatedquantiles X10, X50, X90 (squares), fitted curves, and Swansonsmean. RDTTM (Halliburton, Houston, 2007) is 396 points; RFTTM(Schlumberger, Paris, 2007) is 653 points; and MDTTM (Schlum-berger, Houston, 2007) is 634 points.

Fig. 8Ratio of Swansons mean to exponential-regression es-timates as functions of porosity.

Fig. 9Log porosity vs. core porosity.


However, this regression line y0.824 x+1.61 is not the onewe expect. We may be tempted to find explanations for thisapparent bias, perhaps faulty laboratory procedures (such ascore cleaning), or interpretation issues (e.g., bulk-density interpre-tation parameters).

Now look at Fig. 10. It is exactly the same plot as that shownin Fig. 9 except that the x- and y-axes have been interchanged. R2is the same, but the equation of the regression line is different andnow has a slope of about 1. No correction is needed.

What happened? What is the physical meaning of having twodifferent answers with the same data set? Which one is correct?

The Two Regression Lines. First, we observe in Table 1 that coreand log porosities have the same mean value of approximately 9.4p.u. There is no overall bias. Core data, however, display a greaterdispersion than log data.

To explain the existence of two different regression lines, weappeal to bivariate regression first principles, which are illustratedgraphically in Fig. 11 for a bivariate normal distribution. Theellipse represents the shape of the scatter plot, and also of prob-ability contours. From this cloud of data points, we can define tworegression lines, one giving the mean of all y values for a fixed x(the line through the points of contact of the vertical tangents to theellipse), and the other giving the mean of all x value for a fixed y

(the line through the contacts of horizontal tangents). These tworegression lines are different. Their slopes are not reciprocal toeach other, but their product is equal to the square of the correla-tion coefficient:

0.824 1.0235 = 0.8433.Therefore, if one of the regression lines has a slope of 1, the otherregression line necessarily has a slope less than 1. As we will seenext, the difference of scale between cores and logs explains thedifferent slopes. Both plots are correct, but the one to use is Phi-core against Phi-log because it allows us to check the agreementbetween porosities. If they agree, the regression is yx, so that thebest estimate of Phi-core to be entered into the core-scale perme-ability equation is Phi-log itself.

A Simple Geostatistical Model. We now propose a simple geo-statistical model explaining the regression equations. It is based onthe notion of support, which is the rock volume over which aphysical quantity is measured or defined (Chils and Delfiner1999). It is difficult to specify exactly the volume investigated byporosity logs; moreover, it depends on the logging tool and on theformation properties themselves. Surely this volume is much largerthan the volume of a plug (say 1,000 times). A larger supportmeans a smaller dispersion, which explains why Phi-log has alower standard deviation than Phi-core (Table 1). We may regardthe log measurement as a spatial average of plug measurements, orequivalently, we may regard the measured plug as a randomsample from the population of all the plugs that can be takenwithin the volume seen by the logs. The mean value of Phi-coreover all these possible locations is equal to Phi-log.

E -core |-log) = -log. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)This theoretical model (slope1, intercept0) is consistent withthe experimental regression parameters (slope1.023, inter-cept 0.175). Furthermore, Eq. 8 implies the following addi-tive model:-core = -log + , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

where is a random component, independent of Phi-log andwith mean equal to 0. It represents the effect of small-scale het-erogeneities that can affect plug measurements and is completelyunpredictable from Phi-log alone. If Eq. 9 is correct, the differencePhi-corePhi-log should be distributed symmetrically about 0, beuncorrelated with Phi-log, and be positively correlated with Phi-core. This is exactly what we observe in Figs. 12 and 13.

The notion that Phi-log is a spatial average of Phi-core providesa physical background to the regression of Fig. 9 that surprised us.A slope less than one pulls log-porosity estimates toward themean: Estimated Phi-log is lower than Phi-core for core porositiesgreater than the mean, and greater than Phi-core for core porositiesless than the mean. This regression effect reproduces the aver-aging effect of the logs.

Data Screening. The difference of support may not matter inmassive beds but cannot be overlooked in the presence of hetero-geneities. We have to be sure that a core sample used to establishthe transform is statistically representative of a thick enough ho-mogeneous formation for the logs to read the same porosity as thecores. A simple procedure to clean the data set is to keep only thesamples for which Phi-core and Phi-log are in agreement within agiven tolerance, to be taken as narrow as possible without elimi-nating too many data points. Experience shows that this improvesthe porosity-permeability correlation.

Fig. 10Core porosity vs. log porosity.

Fig. 11Bivariate regression: the two regression lines.


Alternatively, procedures have been proposed to scale up coredata so that they can be represented as a pseudowell log, and regres-sion equations are established at well-log scale (Worthington 2004).

Upscaling to Reservoir ScaleThe grid used to build a numerical model of the flow in the res-ervoir is relatively coarse. The permeability K to be assigned toeach cell of this coarse grid is a complicated function of the per-meabilities k defined at a smaller scale. How to best define thisfunction is the upscaling problem. It is a difficult problem with ahuge literature (Noetinger and Zargar 2004, Renard and de Marsily1997). The purpose of this last section is just to emphasize thedanger of applying the core-scale Phi-k law on upscaled Phi toevaluate the reservoir-scale K.

Examples of Upscaling Bias. The porosity assigned to a grid nodeof a reservoir model, usually obtained by interpolation, may beconstrued to represent the average cell porosity. If we plug thisupscaled cell porosity in Eq. 3 to calculate an upscaled cell per-meability K we get

log K = a + b ave = ave log k. . . . . . . . . . . . . . . . . . . . . (10)

Therefore, entering an upscaled porosity in a core-scale exponential-regression function amounts to geometric permeability averaging.

Matheron (1967) proved that, in a macroscopically isotropicporous medium, geometric averaging cannot be the correct upscal-ing method in 3D. In the case of high permeability contrasts, the

bias between the correct upscaled permeability and the geometricaverage can be quite large.

To illustrate this point, let us consider three intervals in anactual sandstone reservoir. These intervals are vertically stratified,and the flow is assumed to be parallel to the stratification, so thatarithmetic permeability averaging is the correct upscaling method.The intervals are discretized in 20 points, and permeability iscomputed point by point from porosity by the exponential formula

k = 0.0409 exp0.4203 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

We wish to compare the value obtained by arithmetic averagingalong the vertical with the result obtained by entering upscaledporosity in Eq. 11, that is, the geometric average. The ratio of thesetwo values represents the multiplicative correction factor to beapplied for history matching if permeability is derived from up-scaled porosity. For the homogeneous interval in Fig. 14 (poros-ity11.5 p.u.), the correction factor is 1.12 (5.75/5.14). In Fig. 15,the interval is heterogeneous (porosity 11.85 p.u.) and the correc-tion factor climbs to 5.2 (31/5.9). Fig. 16 shows a strong porositytrend. The porosity is 14.8 p.u. and the correction factor ratio reaches8.1 (166/20.5). The bias in these last two cases is significant.

One may object that Eq. 11 is an exponential-regression for-mula, and we have seen that these are biased. Would the upscalingbias disappear with a nonparametric permeability estimator? Theanswer is no. The upscaling bias is of another nature. Indeed,consider the special case of a multiplicative correction for Eq. 11,when quantile curves are parallel. Then all permeabilities are mul-tiplied by a constant, and this does not change the arithmetic-to-geometric permeability ratio.

A simple and common remedy to the effect of vertical hetero-geneity is to subdivide the interval, thus reducing permeabilitycontrasts. For example, if we divide the interval of Fig. 16 into justtwo layers, the upper five samples and the lower five, and computelayer permeabilities from layer porosities by Eq. 11, we find anaverage K115.2 md and the correction factor falls from 8.1 to 1.44.

These examples do not imply that arithmetic permeabilityaveraging is the correct solution to upscaling, but certainly geo-metric averaging is not. Using upscaled porosity as input to acore-scale Phi-k transform may result in severe underestimation ofblock permeability.

Power Averaging. An upscaling formula often used in 3D ispower averaging,

KV = 1|V |V kx dx1, . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)where V is the upscaled domain of volume | V | and is anaveraging exponent between 1 and 1. The values 1, 0,and 1 correspond to harmonic, geometric, and arithmetic av-eraging, respectively. Theoretical calculations justify the value1/3 for a 3D homogeneous and isotropic porous medium withlog-normally distributed permeabilities (Matheron 1967, Noet-

Fig. 12Histogram of -core -log.

Fig. 13Phi-core/Phi-log is uncorrelated with -log, and positively correlated with -core.


Fig. 15Heterogeneous interval, (a) estimate from upscaled Phi and (b) arithmetic permeability averaging.

Fig. 14Homogeneous interval, (a) estimate from upscaled Phi and (b) arithmetic permeability averaging.

Fig. 16Porosity and permeability trend, (a) estimate from upscaled Phi and (b) arithmetic permeability averaging.


inger and Haas 1996). The averaging exponent may be related toa global anisotropy ratio (Noetinger and Haas 1996) involving thevertical-to-horizontal permeability ratio kv/kh and the correlationlengths ratio Lv/Lh. Given the stratified nature of sedimentary ge-ology, the exponent is probably closer to 0.7 than to 0.33.

In order to compute the integral (Eq. 12), we may divide thecell V into N small cells, each with porosity i, and proceed inthree steps as follows:

i ki ki KV. . . . . . . . . . . . . . . . . . . . . . . . . . . (13)At first sight, it seems obvious that a more accurate result will beobtained by starting from the unbiased permeability estimates (ki)computed by Swansons formula as per Eq. 7. A closer look,however, reveals that the unbiased quantity should be k ratherthan k itself. A better procedure is therefore

i ki KV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)It turns out that a direct estimation of k is no more difficult

than the estimation of k itself. Indeed, the power function is order-preserving so that the nth quantile of X is the th power of thenth quantile of X. Therefore, Swansons estimate of the mean of Xis given by

Swanson mean X 0.3 X10 + 0.4 X50

+ 0.3 X90. . . . . . . . . . . . . . . . . . . . . . . (15)In order to get an appreciation for the numerical difference

between the results of the two procedures, we have applied themto the permeability and the mobility data of Section 1, with valuesof ranging from 0 to 1. The power average is computed in threedifferent ways: directly from the data, from Swansons mean es-timate raised to the power , and from Swansons direct estimate

of k as in Eq. 15. If we interpret the permeability data as those ofsmall cells in a large block, the data average is the upscaled blockpermeability, and we can use it as a reference (the true value) togauge the other estimates. The results are presented in Tables 2and 3.

These tables show that a direct estimation of k is clearlypreferable from both a bias and an accuracy point of view. Startingwith an unbiased estimate of k leads to possibly serious overesti-mation of the block average. For example, when 0.5, the over-estimation is 32% in Table 2 and 67% in Table 3 (Ratio 1). Thebias increases with decreasing . In terms of accuracy the directestimator is generally within 10% of the true value (Ratio 2).

It should be mentioned that before computing Table 3, we haveremoved the 16 largest mobility data from the data set of 1,598values because these, as noted earlier, have a disproportionateinfluence on data averages. Robustness is always an issue withpermeability data, and from this point of view, it is advantageousto work with quantiles.

ConclusionsDespite their apparent simplicity, Phi-k transforms have pitfalls ofa statistical nature that may cause a serious underestimation ofpermeability. These sources of bias have been analyzed and solu-tions have been proposed to avoid them.

The first source of bias is the core-scale equation to computepermeability from core-plug porosity. Permeability is usually plot-ted in logarithmic scale, and the fit obtained is transformed back toarithmetic scale by taking the exponential of the fit. In the ex-amples presented, this back transformation generates a bias of afactor of two or five. The solution proposed is to estimate perme-ability in each porosity class by a linear combination of quantiles,such as in Swansons formula. Unlike the mean, this estimator is


robust against outliers. Furthermore, it does not assume any priorfunctional relationship between Phi and k; the structure of thescatter plot determines the shape of the transform function.

The difference between porosity measured on plugs and thatcomputed from logs is a second source of bias. We have discussedone aspect, the difference of investigated volume between logs andplugs, and have shown how it can explain finding a slope of thePhi-core against Phi-log regression less than unity, without involv-ing any bias. A recommendation is to select permeability measure-ments that are representative of an interval thick enough to have agood match with logs.

Finally, the upscaling of permeability to reservoir scale hasbeen discussed to warn against entering an upscaled porosity intothe Phi-k transform, because this usually leads to serious underes-timation of block permeability. The proposed quantile-based esti-mator can also be used to improve permeability power averaging.

There are, of course, more sophisticated ways of estimatingpermeability than Phi-k transforms. Typically, an electrofacies da-tabase is established from logs, including electrical images andderived textural features, using statistical techniques (e.g. cluster-ing, neural networks, and k-nearest neighbors) and driven by coredescription (Knecht et al. 2004). This database is used as a lookuptable to fetch the relevant permeability measurements. Phi-k trans-forms, however, continue to be used.

Nomenclature

E(X) expected value of Xpopulation meanE (k |) conditional expectation of k given Phi

k point permeability(ki) estimate of k at small cell i

(k)i estimate of k at small cell iK block permeability

Xn n-th percentile of the distribution of X random error (mean 0 and constant variance) viscosity2 variance of ei porosity at small cell i exponent of permeability power averaging

Acknowledgments

The author is indebted to Martin Santacoloma and Bernard Lebonfor providing examples and insight, to Bruno Lalanne for review-ing the manuscript, and to Grard Massonnat for fruitful discus-sions on upscaling. Thanks go to the management of Total forpermission to publish this paper.

ReferencesBrown, G.C., Hurst, A., and Swanson, R.I. 2000. Swansons 304030

Rule. The American Association of Petroleum Geologists Bulletin 84(12): 18831891.

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Knecht, L., Mathis, B., Leduc, J.P., Vandenabeele, T., and Di Cuia, R.2004. Electrofacies and Permeability Modeling in Carbonate Reser-voirs using Image Texture Analysis and Clustering Tools. Petrophysics45 (1): 2737.

Matheron, G. 1967. Elments Pour une Thorie des Milieux Poreux. Paris:Masson.

Noetinger, B. and Haas, A. 1996. Permeability Averaging for Well Tests in3D Stochastic Reservoir Models. Paper SPE 36653 presented at theSPE Annual Technical Conference and Exhibition, Denver, 69 Octo-ber. DOI: 10.2118/36653-MS.

Noetinger, B. and Zargar, G. 2004. Multiscale Description and Upscalingof Fluid Flow in Subsurface Reservoirs. Oil & Gas Science and Tech-nologyRev. IFP 59 (2): 119140.

Pallat, N., Wilson, J., and McHardy, W.J. 1984. The Relationship BetweenPermeability and the Morphology of Diagenetic Illite in ReservoirRocks. JPT 36 (12): 22252227. SPE-12978-PA. DOI: 10.2118/12978-PA.

Pearson, E.S. and Tukey, J.W. 1965. Approximate Means and StandardDeviations Based on Distances Between Percentage Points of Fre-quency Curves. Biometrika 52 (34): 533546.

Renard, P. and de Marsily, G. 1997. Calculating Equivalent Permeability:a Review. Advances in Water Resources 20 (56): 253278.

Soeder, D.J. 1986. Laboratory Drying Procedures and the Permeability ofTight Sandstone Core. SPEFE 1 (1): 16. SPE-11622-PA. DOI:10.2118/11622-PA.

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Pierre Delfiner is scientific advisor to the director of geo-sciences of Total in Paris. E-mail: [email protected]. Hespecializes in decision and risk analysis, probabilistic reservesestimation, and geostatistical modeling. Delfiners recent workincludes prospect evaluation methodology, appraisal ofblocks with multiple objectives, aggregation of gas resourcesfor LNG projects, and assessments of the value of informationof 4D seismic for reservoir monitoring. He served as a ReviewChair for SPE Reservoir Evaluation & Engineering. Previously,Delfiner was with Schlumberger Wireline, in charge of process-ing and interpretation software development, notably forma-tion imaging. Before that, he was research associate in Profes-sor G. Matherons team at the Center for Geostatistics of theEcole des Mines de Paris, developing Kriging and stochasticmodeling methods for the petroleum industry. Delfiner holds anengineering degree from the Ecole des Mines and a PhD de-gree in statistics from Princeton University.


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