Evaluation of Accuracy of Estimating Air Permeability From Mercury- Injection Data

Abstract

Summary Published theoretical and correlative models and extensions developed duringthis study were examined to evaluate the accuracy of predicting airpermeability from mercury-injection data. A recently developed statisticaltechnique, bootstrapping, was used to compare different variables forcorrelation with permeability and to demonstrate that correlations by lithologyor permeability and to demonstrate that correlations by lithology or by fieldoften are not statistically different. Introduction The estimation of matrix air permeability from mercury-injection capillarypressure data is important when routine permeability measurements cannot beperformed (e.g., with drill cuttings or rubble core) or are of questionableaccuracy (e.g., for rocks with microfractures). This study attempts to addressthree issues:Given mercury-injection data on a single rock sample, howaccurately can we expect to estimate its air permeability?Is the degree ofestimation accuracy dependent on the permeability level of the, sample, onlithology, or on the permeability level of the, sample, on lithology, or on theformation?Do some parts of a mercury-injection capillary pressure curvecontain more information on permeability than other parts? To answer thesequestions, correlative and theoretical models in the literature and new modelsdeveloped as part of this work were evaluated by a range of statisticaltechniques. Table 1 lists the databases used in this study. The Swanson, Thompson et al., and tight-gas-sands data sets are part of the publishedliterature. Our own data set comprises a combination of service company andin-house mercury-injection measurements on 158 samples from more than 30 fieldsworldwide. Background Correlative Approach. Swanson and Thomeer have correlated parametersobtained from capillary pressure curves with air parameters obtained fromcapillary pressure curves with air permeability data. Swanson used a databaseof 116 carbonates and permeability data. Swanson used a database of 116carbonates and 203 sandstones from 74 formations to establish a correlation ofair permeability with the maximum of the parameter S nw/Pc. He permeabilitywith the maximum of the parameter S nw/Pc. He presented different correlationsfor sandstones and limestones and presented different correlations forsandstones and limestones and found that one standard error of estimate wasabout a factor of two. Thomeer found similar results in his correlation betweenair permeability and three parameters calculated from mercury-injectionpermeability and three parameters calculated from mercury-injection data. Wallsand Amaefule used Swanson's approach to establish a new correlation for tightgas sands. They said this equation gave more accurate permeability predictionsbelow 10 d. Swanson's correlation was selected for evaluation in this studybecause it is simple, it is based on a length scale that makes good intuitivesense, and an extensive database is available in the literature from Swanson'sand other work. We also examined whether the Swanson-type correlation could beimproved by using different or more information from a capillary pressurecurve. New length scales, based on entry pressure, on the Purcell and Katz-Thompson models, and on the distribution of lengths about the Swansonscale, were compared for this purpose. Theoretical Model Approach. Some of the early theoretical models representeda porous medium by a bundle of capillary tubes but paid no attention to theirregular way in which different capillary sizes are interconnected. Thissimplistic representation made it necessary for these models to containadjustable constants, which had to be varied from sample to sample to forcegood agreement between predicted and measured permeabilities. An example is thewell-known Carman-Kozeny equation, which contains the adjustable"tortuosity" factor. The cut-and-random-rejoin model, introduced by Childs and Collis-George and subsequently modified by Marshall and Millingtonand Quirk, uses an interesting mathematical technique to overcome somedrawbacks of the early bundle-of-capillary-tube models. These authors had goodsuccess with the model. Recently, Katz and Thompson, and Thompson et al. Usedpercolation theory to develop theoretical models to predict air percolationtheory to develop theoretical models to predict air permeability frommercury-injection data. They said that their permeability frommercury-injection data. They said that their theory was valid for essentiallyall porous rock, and that their predictions agreed with experimentalmeasurements within expected predictions agreed with experimental measurementswithin expected errors. The cut-and-random-rejoin and Katz-Thompson models wereevaluated in this study. These models were chosen because they wereconceptually appealing and contained no adjustable parameters. Statistical Tests - Bootstrapping. In the context of this study, thestandard error of estimate is the most important parameter for evaluating acorrelation. It is used to provide the permeability range for a givenconfidence level within which a single prediction will lie. This study providestwo measures of the standard error of estimate for each correlation. The firstis the actual value obtained for the particular data set. The second isobtained by a relatively new technique called bootstrapping. Data sets usuallyare generated by sampling a subset of a larger population. The standard errordetermined from a particular data set depends on the sample set that wasselected. Bootstrapping can be used to determine how different the resultswould have been if a different set of samples from the same population had beenselected. population had been selected. Applications of bootstrapping areillustrated in Figs. 1 through 3, which depict results from performing a linearregression of log 10 ka with log 10 Lmax. The results of the regression arepresented here only to illustrate bootstrapping and are discussed presentedhere only to illustrate bootstrapping and are discussed in detail later. Thelinear regression yields a slope of 1.58 and a standard error of estimate of afactor of 2.8. Fig. 1 shows the range and frequency of values that would beexpected if the linear regression had been performed on 1,000 sample setsbootstrapped from the Chevron data set. The 1,000 bootstrapped sample sets weregenerated by picking 158 samples at random, with replacement, from the original Chevron data set of 158 samples. Replacement allows a particular sample in thedata set to be picked more than once, and hence, many different sets of 158samples can be created from the original data set of the same size. Each solidline in Fig. 2 represents 95 % of the range of values obtained by performinglinear regressions of log 10 ka with log 10 Lmax on 1,000 sample setsbootstrapped from the Chevron data set; the solid lines differ in the randomseed used for bootstrapping. The overlap of the solid lines shows that thespread in the values of the slope and standard error for the boot-strappedsamples is repeatable. Bootstrapping does not rely on the usually unverifiableassumptions of normality and also is not restricted to statistical measuresthat can be manipulated analytically. SPEFE P. 304

Tortuosity: a guide through the maze

“…The interested reader is referred to Le Doussal (1989), Thompson (1991) Kamath (1992) Sahimi (1993, Nelson (1994) and Bernab6 (1995) for critiques. For conduction of electrical currents, Johnson, Schwartz and co-workers (Johnson et al 1986, Schwartz et al 1989) introduced a dynamic length scale that is a function of the electrical field intensity in the medium.…”

Interrelationship between Resistivity Index, Capillary Pressure and Relative Permeability

“…They also presented a fitted relationship between permeability and capillary pressure for tight gas sands with permeabilities below 0.01 md. Kamath (1988) examined several correlative and theoretical approaches to the relationship between capillary pressure curves and permeability. He concluded that the correlative methods had a greater practical value.…”

Revisiting the applications of drainage capillary pressure curves in water-wet hydrocarbon systems

“…Permeability values were derived from MICP curves prior to generating the combined capillary curves, since numerous authors defined empirical equations between Thomeer-parameters and permeability [33,38,[51][52][53][54][55][56][57][58][59][60][61]. Permeability was estimated for all of the 26 samples using Thomeer-, and Swanson-methods.…”