iptc 10994 a modified purcell/burdine model for … and extended to incorporate the power law p...

Copyright 2005, International Petroleum Technology Conference

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Abstract

This paper presents the development and validation of a new semi-analytical, statistically-derived model for estimating absolute permeability from mercury-injection capillary pressure data. The foundations of our new model are the classic Purcell1 and Burdine2 equations which relate absolute permeability to capillary-pressure/wetting-phase-saturation properties. We also incorporate characteristic capillary pres-sure behavior using the Brooks-Corey3 power-law model.

The final form of our proposed model allows us to compute absolute permeability as a function of effective porosity, irre-ducible wetting phase saturation, displacement or threshold pressure, and basic pore size characteristics. We tested and correlated our model using 89 sets of mercury-injection (Hg-air) capillary pressure data — including core samples from both carbonate and sandstone lithologies. In summary, we found that our model consistently yields accurate results for a wide range of rock properties.

Introduction

The fundamental relationships between pore size/geometry and basic rock properties (e.g., effective porosity, absolute permeability, etc.) are well-documented in the petroleum and petrophysics literature. Moreover, the literature is replete with models for estimating or predicting permeability from basic rock properties. Nelson4 has developed a comprehensive re-view of the literature, and he has identified five major categories of permeability models based on the physical rock attributes used in the model development: The five major model categories specified by Nelson are:

1. Petrophysical models, 2. Models based on grain size and mineralogy, 3. Models based on surface area and water saturation, 4. Well log models, and 5. Models based on basic rock pore dimensions.

In this paper, we focus on models that incorporate basic rock pore characteristics and dimensions, and specifically, pore characteristics as determined from capillary pressure data. Nelson has further classified these particular models as direct types since they not only relate rock permeability directly to the pore dimensions and connectivity, but also incorporate fundamental theories of fluid flow through porous media. Most of these direct methods — especially the early models developed in the 1940s and 1950s — use mercury-injection capillary pressure data to quantify the rock pore and pore throat characteristics.

Although the physical basis for the existing direct models is similar, we have observed varying (even inconsistent) results among the various "theoretical" models. The primary motiva-tion for our work is to "close the loop" on relating mercury-injection capillary pressure data and the simplified "bundle of tubes" permeability models proposed by Purcell1 and Bur-dine2 and extended to incorporate the power law pc(Sw) model by Brooks-Corey.3

Consequently, the principal objective of this work is to deve-lop and document a "universal" model which provides a more consistent correlation of permeability with mercury capillary pressure data for a much wider range of rock types. We begin with a review of the models developed previously — and we then document the validation of our generalization of the Purcell-Burdine-Brooks-Corey k(Hg) model.

Permeability Models—Historical Perspective

Leverett J-Function (J(Sw)): (ref. 5)

One of the first correlation models for petrophysical properties was proposed by Leverett5 who developed a relationship between wetting phase saturation and the interfacial curvature between the wetting and non-wetting fluids in the pore throats (this relationship is based primarily on a dimensional balance of the parameters (e.g., φ/k is an "equivalent length")). This concept (i.e., the "J-function") was proposed by Leverett as a dimensionless function that could be used to normalize capil-lary pressure data for a range of rock properties.

The Leverett J-function is defined as

φθσ

/cos

)( kp

SJ cw = ................................................... (1)

Where:

k = permeability, cm2 (1 D = 9.86923x10-9 cm2)

IPTC 10994

A Modified Purcell/Burdine Model for Estimating Absolute Permeability from Mercury-Injection Capillary Pressure Data C.C. Huet, Texas A&M University, J.A. Rushing, Anadarko Petroleum Corp., K.E. Newsham, Apache Corp., and T.A. Blasingame, Texas A&M University SPE Members

2 C.C. Huet, J.A. Rushing, K.E. Newsham, and T.A. Blasingame IPTC 10994

J(Sw) = dimensionless capillary pressure-saturation function σ = interfacial tension, dynes/cm θ = contact angle of incidence for wetting phase, radians φ = porosity, fraction of pore volume Sw = wetting phase saturation, fraction of pore volume pc = capillary pressure, dynes/cm2

φ/k = equivalent length, cm

Purcell Permeability Relation: (ref. 1)

In 1949, Purcell1 developed an equation relating absolute permeability to the area under the capillary pressure curve generated from mercury injection. We note that Purcell's equation assumes that fluid flow can be modeled using Poiseuille's Law where the rock pore system is represented by a bundle of parallel (but tortuous) capillary tubes of various radii. Further, the range of tube radii are characterized by the pore size distribution as computed from the area under the capillary pressure curve.

Purcell's original permeability model is given by:

wc

PairHg dSp

F.k 11

0 )cos( 6610 2

2 ∫−= φθσ .................... (2)

where:

k = permeability, md 10.66 = units conversion constant, md-(psia)2/(dynes/cm)2

FP = Purcell lithology factor, dimensionless σHg-air = mercury-air interfacial tension, dynes/cm

θ = contact angle of incidence for wetting phase, radians φ = porosity, fraction of pore volume Sw = wetting phase saturation, fraction of pore volume pc = capillary pressure, psia

We note that FP is the Purcell "lithology factor" which is used to represent the differences between the hypothetical model and actual rock pore system. The FP "lithology factor" is an empirical correction that Purcell determined for several differ-ent core samples over a range of absolute permeability values.

Rose and Bruce Study: (ref. 6)

In 1949, Rose and Bruce6 conducted a sensitivity study of rock properties and their impact on the shape of mercury-injection capillary pressure curves. They showed that the measured capillary pressure depends on pore configuration, rock surface properties and fluid properties. Rose and Bruce also found that capillary pressure curves can be used to characterize the distribution, orientation, shape and tortuosity of the pore system — as well as the interfacial and interstitial surface area. Although Rose and Bruce did not propose a perme-ability relation, they did demonstrate the use of the Leverett J-Function (Eq. 1) (with extensions of their own) to generate relative permeability-saturation profiles.

Calhoun, et al Permeability Relation: (ref. 7)

In 1949, Calhoun, et al7 showed that the Purcell1 lithology factor (FP) is inversely proportional to the formation tortuosity factor (τ). Their study also determined that the internal rock surface area could be defined in terms of the fluid interfacial tension, rock-wetting phase fluid contact angle, and the area under the capillary pressure curve. Additionally, Calhoun, et al developed a semi-empirical relationship for absolute per-

meability as a function of effective porosity, adhesive tension, capillary displacement pressure, and the value of the J-Func-tion at 100 percent wetting phase saturation. Calhoun, et al's7 semi-empirical relationship is given as:

[ ] φθσ 22012 )cos()(1

.wd

SJp

k = ...................................... (3)

where:

k = permeability, cm2 (1 D = 9.86923x10-9 cm2) J(Sw)1.0 = dimensionless capillary pressure function at Sw =1.0

σ = interfacial tension, dynes/cm θ = contact angle of incidence for wetting phase, radians φ = porosity, fraction of pore volume Sw = wetting phase saturation, fraction of pore volume pd = capillary displacement pressure, dynes/cm2

We note that Eq. 3 was validated by Calhoun, et al only for high permeability rocks.

Burdine et al Permeability Relation: (ref. 2)

In 1950, Burdine et al2 extended the Purcell1 model for a bundle of capillary tubes and showed that the absolute permeability of a particular rock is a function of pore entry radii and the mercury-filled pore volume. The Burdine equa-tion is given by:

22

4

0 126

ii

ii

RX

RVn

ik ∑

== φ .................................................... (4)

where:

k = permeability, md (1 D = 9.86923x10-9 cm2) 126 = units conversion constant (Poiseuille → Darcy units) Vi = incremental pore volume filled by mercury, cm3 Ri = incremental pore entry radius, cm

iR = average pore entry radius, cm Xi = tortuosity factor (Xi=Li/Ltot), fraction Li = effective length of flow path, cm Ltot = actual length of flow path, cm

We note that the Burdine et al relation is fundamentally simi-lar (in derivation) to the Purcell relation — the interested reader is also directed to an additional reference article (ref. 8) by Burdine et al where additional detail and clarity of nomen-clature are provided for Eq. 4.

Wyllie-Spangler Permeability Relation: (ref. 9)

In 1952, Wyllie and Spangler9 developed a model using the Purcell/Burdine concept, but Wyllie and Spangler used electric log properties to determine the tortuosity factor (specifically, this is given in terms of the formation factor which is defined as the ratio of the resistivity of the formation at 100 percent wetting phase saturation to the resistivity of the formation brine).

The Wyllie-Spangler equation, which relates absolute perme-ability to mercury-injection capillary-pressure curve proper-ties, is given by

wcWS

airHg dSpFF

.k 11

0 11)cos( 6610 22

2

∫−=φ

θσ ........ (5)

IPTC 10994 A Modified Purcell/Burdine Model for Estimating Absolute Permeability from Mercury-Injection Capillary Pressure Data 3

where F is the Archie10-11 formation factor, defined by

wo

m RRaF ==

φ............................................................... (6)

where:


FWS = Wyllie-Spangler shape factor, dimensionless θ = contact angle of incidence for wetting phase, radians φ = porosity, fraction of pore volume Sw = wetting phase saturation, fraction of pore volume pc = capillary pressure, psia F = Archie formation factor, dimensionless Ro = resistivity of formation at Sw=1.0, ohm-m Rw = resistivity of formation brine, ohm-m m = empirical constant (cementation factor), dimensionless a = empirical constant, dimensionless

In Eq. 6, a is an empirical constant (a is often assumed to be 1) and m is the cementation factor (m is often assumed to be 2) Note that m is a function of pore type, pore geometry and lithology. Wyllie and Spangler also demonstrate that the tortuosity factor can be related to the formation factor deter-mined from electric log measurements (Wyllie and Spangler actually made their developments in terms of the tortuosity factor, then "converted" their result into a formulation which uses the formation resistivity factor).

Thomeer Permeability Relation: (ref. 12)

In 1960, Thomeer12 observed that a log-log plot of capillary pressure could be approximated by a hyperbola. Thomeer described the hyperbola location on the x-y coordinate system by the position of the two end-point curve asymptotes, and he defined the extrapolated asymptotes on the x- and y-axes as the displacement pressure and the bulk volume occupied by mercury at an infinite pressure, respectively.

In addition, Thomeer hypothesized that the shape of the hyperbola reflects the pore geometry, so he used the curve shape to define a pore geometrical factor. We note that Thomeer assigned the pore geometric factor a value between 0 and 10, where low values represent large well-sorted pore openings and high values represent high levels of variation in pore opening sizes. As a result of these definitions, Thomeer proposed an empirical equation relating air permeability to capillary pressure, displacement pressure, non-wetting phase saturation, and pore geometric factor.

The Thomeer model is given as:

[ ])/log(/ dcg ppF

bb e

SS −

∞= ................................................ (7)

where:

k = permeability, md pc = capillary pressure, psia pd = capillary displacement pressure, psia Sb = Hg saturation, fraction of bulk volume Sb∞ = Hg saturation at pc = ∞, fraction of bulk volume Fg = pore geometrical factor, dimensionless

Using laboratory measurements from 165 sandstone and 114

carbonate samples, Thomeer13 formulated the following equa-tion that relates absolute permeability to effective porosity, capillary displacement or threshold pressure, and the pore geo-metric factor:

233341 80683 ⎥

⎦

⎤⎢⎣

⎡= ∞−

d

b.g p

SF.k ....................................... (8)

Swanson Permeability Relation: (ref. 14)

As a follow-up effort to Thomeer,12,13 Swanson14 developed an equation to compute absolute permeability based on specific capillary pressure curve characteristics. The form of Swan-son's equation is: (using the same nomenclature as Thomeer)

2

1

a

Acb

pS

ak ⎥⎦

⎤⎢⎣

⎡= ............................................................... (9)

where:

k = permeability, md [Sb/pc]A = Hg saturation/capillary pressure "apex," fraction/psia

A = "apex" point on log(pc) vs. log(Sb) curve at which a 45o line becomes tangent

The subscript "A" (or apex) refers to the maximum ratio of the mercury saturation to the capillary pressure. Swanson hypothesized that the maximum ratio occurs at the point at which all of the major connected pore space is filled with mercury. Further, the capillary pressure at the apex reflects the dominant inter-connected pores and pore throats con-trolling most of the fluid flow characteristics.

The constants a1 and a2 in Eq. 9 represent various rock lithologies and fluid types, respectively, in the system. Swanson used regression analysis and correlated the constants in Eq. 9 with properties from 203 sandstone samples from 41 formations and 116 carbonate samples from 330 formations. The best fit of the air permeability data was obtained with

6911 389

.

Acb

pS

k ⎥⎦

⎤⎢⎣

⎡= ....................................................... (10)

Wells and Amaefule Permeability Relation: (ref. 15)

In 1985, Wells and Amaefule15 modified the Swanson ap-proach for low-permeability or "tight gas sands." Wells and Amaefule found that by plotting the logarithm of mercury saturation (Sb, percent of bulk volume) against the square root of capillary pressure-mercury saturation ratio (Sb/pc), they could observe a well-defined minimum — which represents the inflection point of the capillary pressure curve. Conse-quently, (Sb/pc) could be calculated as the inverse of the squared minimum value. Wells and Amaefule then correlated the Swanson13 parameter with air permeabilities for 35 low-permeability sandstone samples and developed the following equations for calculating absolute permeability:

561 530

.

airA,Hgcb

pS

.k−

⎥⎦

⎤⎢⎣

⎡= ..............................................(11a)


611 221

.

airA,brinecb

pS

.k−

⎥⎦

⎤⎢⎣

⎡= ........................................... (11b)

where:

k = permeability, md pc = capillary pressure, psia Sb = non-wetting saturation, fraction of bulk volume

[Sb/pc]A = non-wetting saturation/cap. pressure "apex," fraction/psia A = "apex" point on log(pc) vs. log(Sb) curve at which a 45o

line becomes tangent

Winland Permeability Relation: (ref. 16-17)

A methodology attributed to Winland (no reference other than company) was documented initially by Kolodzie16 and extend-ed by Pittman17 where regression analysis was used to develop an empirical relationship that is conceptually similar to Swan-son.14 The "Winland" equation describes the relationship for absolute permeability, effective porosity, and a capillary pres-sure parameter (R35) as follows:

φlog864.0log588.0732.0log 35 −+= kR .................... (12)

where:

k = permeability, md φ = porosity, fraction of pore volume R35 = pore throat radius at an Hg saturation of 35 percent, µm

R35 is the capillary pressure parameter used in the Winland study — specifically, R35 is the pore throat radius (in µm) at a mercury saturation of 35 percent, where this value is a function of both pore entry size and the sorting of pore throat sizes. According to Nelson,4 the R35 parameter quantifies the largest and best-connected pore throats. We note from refs. 16-17 that other capillary pressure parameters (i.e., R30, R40 and R50 values) were considered, but the R35 capillary pressure curve parameter provided the best statistical fit.

The Winland data set includes 56 sandstone and 26 carbonate samples with permeability measurements corrected for gas slippage or Klinkenberg18 effects. This data set also includes another 240 samples of various lithologies but without permeabilities corrected for gas slippage effects. We note that the permeability computed by Eq. 12 is not the Klinkenberg-corrected permeability.

Development and Validation of New Model

The foundations of our correlation model are the classic Purcell1 and Burdine2 equations — which assume that the porous medium can be modeled as a bundle of parallel (but tortuous) capillary tubes of various radii. Further, the range of tube radii are characterized by the pore size distribution as computed from the area under the capillary pressure curve.

The classic Purcell-Burdine k-model (Eqs. 2 and 4, respect-ively) has been re-derived by Nakornthap and Evans19 — and this "redevelopment" includes considerations by Wyllie and Spangler9 and Wyllie and Gardner.20

The final form of the Nakornthap and Evans result, solved for formation permeability, is given as:

*w

cwiairHg dS

pS

n.k

21

0

332 1)(1)cos(21 6610 ∫−= − φθσω

..................................................................................... (13)

where: (written for an Hg-air system (i.e., Sw=Sair)


ω = pore throat "impedance" factor, dimensionless n = number of entrances/exits in a pore, dimensionless

σHg-air = mercury-air interfacial tension, dynes/cm θ = contact angle of incidence for wetting phase, radians φ = porosity, fraction of pore volume pc = capillary pressure, psia Sw = wetting phase saturation, fraction of pore volume Swi = irreducible wetting phase saturation, fraction of pore

volume *wS = (Sw-Swi)/(1-Swi), "effective" (or normalized) wetting

phase saturation function, dimensionless

The definition of the "effective" (or normalized) wetting phase saturation function was first proposed by Burdine2 and later utilized directly by Wyllie and Gardner.20 This definition is given as:

wiwiw*

w SSSS

−−

=1

............................................................ (14)

In this approach, we incorporate the capillary pressure curve characteristics using the Brooks-Corey3 power-law model which is given by:

λ1

1

−

⎥⎦

⎤⎢⎣

⎡−−

=wi

wiwdc S

SSpp .............................................. (15)

Where: (writing for an Hg-air system (i.e., Sw=Sair))

pc = capillary pressure, psia pd = displacement pressure, psia Sw = wetting phase saturation, fraction of pore volume Swi = irreducible wetting phase saturation, fraction of pore

volume λ = Brooks-Corey index of pore-size distribution, dimen-

sionless

Where pd is the capillary displacement (or threshold) pressure, and λ is the index of pore-size distribution. Combining Eqs. 13-15 yields the basic form of the permeability equation used in our study:

⎥⎦⎤

⎢⎣⎡

+−= − 2

1)(1)cos( 6610 2332

λ

λφθσω

dwiairHg

pS

n.k (16)

While we could not find the explicit form given by Eq. 16 in the literature, it has undoubtedly been derived as Eq. 16 is the generalized formulation used to derive the relative per-meability relations of Brooks and Corey,3 the results of which are also presented by Nakornthap and Evans.19

Nakornthap and Evans assign the ω and n parameters to address non-ideal flow behavior. To describe the ω-para-


meter, Nakornthap and Evans state:

"The ω−parameter is inserted to recognize the fact that flow through a pore of radius r overemphasizes the impedance because it ignores the larger areas available for exit-flow at either side of the constrictions formed where pores abut. Thus it may be expected that ω > 1 and that the actual magnitude of ω is a function of the average shape of pores in the medium that the model represents. ω is assumed constant for all pore sizes."

Similarly, Nakornthap and Evans describe the n-parameter, as follows:

"The numerical constant n reflects the manner in which the available interconnecting pore area is divided. In the most favorable case for flow, all the exit area is concentrated in one pore; then n = 1. It may be expected, therefore, that n > 1. It is assumed here that n is constant for all pore sizes."

Ali21 has also suggested the following concept models for representing ω and n:

φω 1

= .......................................................................... (17 )

)(11

wiSn

−= ............................................................... (18)

Substituting Eqs. 17 and 18 into Eq. 16, we can eliminate the ω and n terms directly, which yields:

⎥⎦⎤

⎢⎣⎡

+−= − 2

1)(1)cos( 6610 2242

λ

λφθσαd

wiairHgp

S.k (19)

Note that we have added an empirical parameter, α, in Eq. 19 to represent any remaining non-idealities that have not been accounted for by any other terms. If we were to attempt to utilize Eq. 19, we would likely assume α = 1, or attempt a calibration of the α-parameter for a particular data set. In fact, we did use Eq. 19 in some of our early correlation efforts as a "test function," where we plotted permeability computed using Eq. 19 versus measured permeability on a log-log plot to assess significant outlying data.

Perhaps the most significant contribution of this work will be presentation of Eq. 16 — as this relation clearly states that permeability should be a power law function of displacement pressure, index of pore-size distribution, irreducible wetting phase saturation, and porosity. Recasting Eq. 16 as a power law correlation model gives us:

543

2)(1

2)(1

1aa

wia

ad

Sp

ak φλ

λ −⎥⎦⎤

⎢⎣⎡

+= ...................... (20)

where a1, a2, a3, a4, and a5 are correlation constants — coeffi-cient a1 incorporates all of the "constant" terms (i.e., 10.66, ω/n, and (σHg-aircos(θ))2).

The form of Eq. 20 (or a simplified modification) permits us to create other relations — specifically, we can make model sub-stitutions for other parameters (in our case, pd and λ) and create a "universal" (albeit simplified) model for permeability based solely on porosity (φ) and irreducible wetting phase saturation (Swi). This effort is presented in Appendix B.

We also utilize the power-law model form as a mechanism to correlate the displacement pressure (pd). In this case, we correlate the displacement pressure (pd) in terms of permeabil-ity, porosity and irreducible wetting phase saturation using:

432 )1(1b

wibb

d Skbp −= φ ............................................. (21)

where b1, b2, b3, and b4 are correlation parameters for the capillary displacement (or threshold) pressure.

Lastly, we correlate the "index of pore-size distribution" (λ) with permeability, porosity, irreducible wetting phase satura-tion and capillary displacement pressure, again using a power-law model. This formulation is given as:

5432 )1(1cd

cwi

cc pSkc −= φλ ......................................... (22)

where c1, c2, c3, c4, and c5 are correlation parameters for the pore geometric factor.

To calibrate the proposed power models (Eq. 20-22), we have used mercury-injection capillary-pressure data from the literature1,10,20 and industry sources. Furthermore, we have tested our new model using samples from both sandstone and carbonate lithologies. Although we did not evaluate a range of different carbonate rock types, we expect our new model to be most applicable to carbonates with an inter-granular type of porosity and not "vuggy" carbonates.

We reviewed approximately 120 data sets — but used only 89 data sets in this work. The data not used in this study were set aside for a variety of reasons (i.e., suspicious character in the capillary pressure data (e.g., "double porosity" behavior), erroneous capillary data (poor calibration, poor character), and we also used only Hg-air capillary pressure data — so air-oil, and oil-water data were set aside for later studies).

The data sets used in our correlations have the following ranges of properties:

0.0041 md < k < 8340 md 0.003 (fraction) < φ < 0.34 (fraction) 0.007 (fraction) < Swi < 0.33 (fraction) 2.32 psia < pd < 2176 psia

Results and Discussion

Estimation of pd, Swi and λ from Regression:

Our initial calibration process was performed to estimate the capillary displacement pressure (pd), irreducible wetting-phase saturation (Swi), and the index of pore-size distribution (λ) on a sample-by-sample basis using Eq. 15 (i.e., the Brooks-Corey pc(Sw) model).

We could have attempted a "global" calibration of the pd, Swi and λ parameters simultaneously with the model parameters in Eqs. 20-22. Such a process would (in concept) be more robust — i.e., coupling the calibration of the Brooks-Corey model with each of the regression models (Eqs. 20-22). However, the quality of data, coupled with the bias (human intervention) required to properly fit the Brooks-Corey pc(Sw) model to an individual sample data set, required that we perform this calibration separately. The results of the pd, Swi and λ calibra-tion, along with the input permeability (k) and porosity (φ) data for this work are summarized in Appendix A.


In Fig. 1 we present a typical pc(Sw) data-model regression to illustrate our calibration process. We clearly note that, while the data-model fit is good, human intervention is required to ensure that the model is properly applied to the data.

Fig. 1 — Example correlation of Brooks-Corey pc(Sw) model to a typical core data set for this study.

We believe that this "separate" calibration of the pc(Sw) data-model is appropriate, and we note that the majority of the effort in our correlation work focused on this particular task.

Estimation of k, pd, and λ Using Regression:

The regression setup for Eqs. 20-22 is fairly straightforward, as we used the Solver Module in Microsoft Excel22 to perform our regression work. We formulated each regression problem in terms of the sum-of-squared residuals (SSQ), sum-of-absolute relative error (ARE) and — depending on the case — these regressions were performed using the residual or absolute relative error based on the logarithm of a particular variable. A summary of our results for the k, pd, and λ regres-sions is given in Table 1.

Table 1 — Overall regression statistics for k, pd, λ — power law models.

Case

Fig.

SSQ* (ln(unit)2)

ARE (percent)

k

2

2.8534 ln(md)2

30.4243

pd

3

1.5476 ln(psia)2

24.9406

λ

4

0.8367

17.7197

* SSQ statistics based on ln(k), ln(pd), and λ, respectively.

We present the results of our permeability (k) optimization in Fig. 2. We note excellent agreement between the measured permeabilities and those calculated from Eq. 20. The opti-mized coefficients from the regression analysis of Eq. 20 are summarized in Table 2.

Table 2 — Regression summary for k (Eq. 20).

Optimized coefficients for k (Eq. 20):

Coefficient Optimized Value a1 1017003.2395 (md) a2 1.7846 a3 1.6575 a4 0.5475 a5 1.6498

Statistical summary for k (Eq. 20):

Statistical Variable Value Sum of Squared Residuals 2.8534 ln(md)2 Variance 275036.1525 md2 Standard Deviation 524.4389 md Average Absolute Error 30.4243 percent

Substituting the coefficients from Table 2 into Eq. 20, we have:

1.64980.54751.65751.7846 )(12)(1951017003.23 φλ

λwi

dS

pk −⎥⎦

⎤⎢⎣⎡

+=

..................................................................................... (23)

Fig. 2 — Permeability correlation based on mercury capillary pressure data (Eq. 20 used for regression).

We note that Eq. 23 was used to calculate the entire perme-ability range — from low permeability (tight gas sands) to un-consolidated sands. From our perspective, the generalized permeability relation (Eq. 20) has theoretical rigor (see Appendix A) and may be a "universal" permeability model — valid for different lithologies, pore systems, and pore struc-tures.

We recommend that the generalized form (Eq. 20) continue to be tested systematically. We will (again) note that care must be taken in assessing pc(Sw) suitable for such correlations. We have elected to consider Hg-air systems only for simplicity — extensions to other systems must continue systematically, with diligent data screening and a "simplest" model first mentality.

We also correlate the capillary displacement pressure with per-


meability, porosity and irreducible wetting phase saturation — the results of which are shown in Fig. 3 using a power law correlation model (the regression summary for this case is given in Table 3). Although we have developed more complex models for the correlating the displacement pressure, we believe that Eq. 24 is an excellent "general" model. We also note that Thomas, Katz, and Tek23 proposed a similar formulation, where this model is also plotted on Fig. 3 for comparison.

Table 3 — Regression summary for pc (Eq. 21).

Optimized coefficients for pc (Eq. 21):

Coefficient Optimized Value b1 640.0538 (psia) b2 0.8210 b3 -0.5285 b4 0.8486

Statistical summary for pc (Eq. 21):

Statistical Variable Value Sum of Squared Residuals 1.5473 ln(psia)2 Variance 110928.0679 psia2 Standard Deviation 333.0587 psia Average Absolute Error 24.8721 percent

Substituting the coefficients in Table 3 into Eq. 21, we have:

0.8486-0.52850.8210 )1( 640.0538 wid Skp −= φ .......... (24)

Fig. 3 — Displacement pressure (pd) correlation based on mercury capillary pressure data (Eq. 21 used for re-gression).

In our effort to correlate the index of pore-size distribution (λ) with permeability, porosity, irreducible wetting phase satura-tion and displacement pressure, we found less conformity in the resultant correlations. We believe that this behavior is due to the character of the index of pore-size distribution — recall that this parameter is an exponent in the Brooks-Corey pc(Sw) relation (Eq. 15). We have observed that Eq. 15 is relatively unaffected by the λ-parameter (i.e., the model is relatively

insensitive to the λ-parameter).

In addition, we believe this insensitivity may make it more difficult to estimate the λ-parameter initially from pc(Sw) data than correlating the λ-parameter against other variables. Regardless, our characterization and correlation of the λ-parameter was less successful than our correlation of perme-ability (k) and capillary displacement pressure (pd).

We present the correlation of the λ-parameter using a power law model in Fig. 4, and we present the regression summary for this case in Table 4.

Table 4 — Regression summary for λ (Eq. 21).

Optimized coefficients for λ (Eq. 22):

Coefficient Optimized Value c1 0.00980 c2 -0.6341 c3 0.3792 c4 -0.6835 c5 0.6698

Statistical summary for λ (Eq. 22):

Statistical Variable Value Sum of Squared Residuals 0.8367 Variance 0.0395 Standard Deviation 0.1988 Average Absolute Error 17.7197 percent

Substituting the coefficients in Table 4 into Eq. 22, we have:

0.6698-0.68350.3792-0.6341 )1( 0.00980 dwi pSk −= φλ . (25)

Fig. 4 — Pore geometric factor (λ) correlation based on mer-cury capillary pressure data (Eq. 22 used for regres-sion).

Our correlation of the λ-parameter yielded the weakest results (in terms of a graphical comparison (Fig. 4), not in terms of a statistical regression). The results shown in Fig. 4. clearly show weak (if not poor) conformance of the model to the data (i.e., the blue circle symbols, relative to the red dashed line


(perfect correlation)). To better understand (but probably not quantify) this deviation, we have also constructed a "non-parametric" correlation of the λ-parameter using the methods given in ref. 24.

A non-parametric correlation is the optimal statistical rela-tionship for a given data set on a point-by-point basis — any parametric (i.e., functional) correlation which yields better statistical metrics than the corresponding non-parametric cor-relation has "over-fitted" the data (i.e., fitted the errors in the data). Our non-parametric correlation of the λ-parameter for this case is shown by the green square symbols on Fig. 4. The relative similarity of the data in Fig. 4 suggest that our non-parametric correlation and our correlation using a power law model are comparable — which validates our use of the (relatively simple) power law model for this case.

As closure for this discussion regarding the correlation of the λ-parameter, we believe that the very basis of the λ-parameter (it is an exponent), coupled with the quality of data used to define the λ-parameter are the causes of the relatively weak correlation of the λ-parameter shown in Fig. 4. Based on the non-parametric correlation for this case, we do not recommend additional efforts to improve the parametric correlation. But, we do suggest recasting the problem so that permeability is directly related to the various measurable rock properties, including porosity (φ), irreducible wetting phase saturation (Swi), and displacement pressure (pd). We also recommend use of some parameter other than the index of pore-size distribu-tion (λ) to represent the "curvature" of the capillary pressure curve. Finally, we would also comment that Eq. 25 (i.e., the power law correlation for the λ-parameter) is probably more than sufficient for practical applications.

Summary and Conclusions

Summary:

Using the relations of Purcell,1 Burdine,2 Brooks and Corey,3 Wyllie and Spangler,9,20 and Nakornthap and Evans19 we have developed a base model to correlate permeability from mer-cury capillary pressure data. Our base model for permeability is given by Eq. 16:

⎥⎦⎤

⎢⎣⎡

+−= − 2

1)(1)cos( 6610 2332

λ

λφθσω

dwiairHg

pS

n.k (16)

Generalizing Eq. 16 into a correlation form yields Eq. 20:

543

2)(1

2)(1

1aa

wia

ad

Sp

ak φλ

λ −⎥⎦⎤

⎢⎣⎡

+= ...................... (20)

It is relevant to note that Eq. 16 suggests (under the assump-tions of a "bundle of capillary tubes," Darcy's law, and other constraints which are related to how the capillaries are con-nected) that we can consider permeability to be a power law function of φ, Swi, pd, and λ. We recognize this simplicity, but we also suggest that Eq. 16 (or Eq. 20) should be a good starting point for the correlation of permeability.

Summarizing, we achieved the following power law correla-tions in this work:

k = f(φ, Swi, pd, and λ): Fig. 2

1.64980.54751.65751.7846 )(12)(1951017003.23 φ

λλ

wid

Sp

k −⎥⎦⎤

⎢⎣⎡

+=

..................................................................................... (23)

pd = f(φ, k, and Swi): Fig. 3

0.8486-0.52850.8210 )1( 640.0538 wid Skp −= φ .......... (24)

λ = f(φ, k, Swi, and pd): Fig. 4

0.6698-0.68350.3792-0.6341 )1( 0.00980 dwi pSk −= φλ . (25)

The results of our modeling efforts suggest that the correlating properties of the porous media (k, φ, Swi, pd, and λ) are not specifically dependent upon lithology — but rather, these properties uniquely quantify the fluid flow behavior of the porous medium. In that sense, we see this work as a generalized correlation for flow in porous materials — including soils, filters, sintered metals, bead packs, and porous rocks. As we noted earlier, we believe that this work is applicable to carbonates with an inter-granular type of porosity — not to cases of "vuggy" carbonates.

Conclusions:

The following conclusions have been derived from this work:

1. The permeability (k) can be successfully correlated to the porosity (φ), capillary displacement pressure (pd), irreducible wetting-phase saturation (Swi), and the index of pore-size distribution (λ) using a theoretically defined power law correlation model.

2. The capillary displacement pressure (pd) can also be correlated using a power law model to the permeability (k), porosity (φ), and irreducible wetting-phase saturation (Swi). This observation confirms the fundamental work proposed in ref. 23.

3. The correlation of the index of pore-size distribution (λ) is somewhat problematic — the λ-parameter may be only weakly defined. The pc(pd,Swi,λ,Sw) model given by Brooks-Corey (ref. 3) is robust and suitable for this work, but — we find that the model can be relatively insensitive to the λ-parameter (i.e., a different combination of the pd,Swi, and λ-parameters can yield equi-probable correla-tions of pc. This is an issue that is most likely related to the quality and character of the capillary pressure data.

Recommendations:

The following recommendations are proposed:

1. Consideration of more complex correlation models for:

k = f(φ, Swi, pd, and λ) pd = f(φ, k, and Swi) λ = f(φ, k, Swi, and pd)

Our experience with non-parametric regression (ref. 24) as applied to this work suggests that the proposed power law models are sufficient, and we would warn against "over-fitting" data in this work with excessively complex data models.

2. Extension of the results of this work to liquid-liquid and gas-liquid systems.


Acknowledgements

We would like to express our thanks to Anadarko Petroleum Corp. and Apache Corp. for providing portions of the core data — and for permission to publish results from this study.

Nomenclature

Variables are defined where a particular equation is given.

Superscripts, Subscripts, or other Characters:

air = air Hg = mercury SSQ = sum-of-squared residuals ARE = sum-of-absolute relative error

References

1. Purcell, W.R.: "Capillary Pressures-Their Measurement Using Mercury and the Calculation of Permeability Therefrom," Trans. AIME, 186 (1949), 39-48.

2. Burdine, N.T.: "Relative Permeability Calculations from Pore Size Distribution Data", Trans. AIME, (1953), 198, 71-78.

3. Brooks, R.H., and Corey, A.T.: "Properties of Porous Media Affecting Fluid Flow." J. Irrig. and Drain. Div. ASCE (1966) 92: 61-88.

4. Nelson, P.H.: "Permeability-Porosity Relationships in Sedimen-tary Rock," The Log Analyst (May-June 1994), 38-62.

5. Leverett, M.C.: "Capillary Behavior in Porous Solids," Trans, AIME 142 (1941), 341-358.

6. Rose, W. and Bruce, W.A.: "Evaluation of Capillary Character in Petroleum Reservoir Rock," Trans. AIME, vol. 186 (1949), pp 127-142.

7. Calhoun, J.C., Lewis, M. and Newman, R.C.: "Experiments on the Capillary Properties of Porous Solids," Trans., AIME (1949) 186, 189-196.

8. Burdine, N.T., Gournay, L.S., and Reichertz, P.P.: "Pore Size Distribution of Petroleum Reservoir Rocks", Trans. AIME, (1950), 189, 195-204.

9. Wyllie M.R. and Spangler M. B.: "The Application of Electrical Resistivity Measurements to the Problem of Fluid Flow in Porous Media," Research Project 4-G-1 Geology Division Report No. 15 (March 1951) Gulf Research and Development Company.

10. Archie, G.E.: "Electrical Resistivity Log as an Aid in Determin-ing Some Reservoir Characteristics," Trans. AIME (1942) 146, 54-62

11. Archie, G.E.: "Introduction to Petrophysics of Reservoir Rocks," Bull., AAPG (1950) 34, 943-961.

12. Thomeer, J.H.M.: "Introduction of a Pore Geometrical Factor Defined by the Capillary Pressure Curve," Trans., AIME (1960) 213, 354-358.

13. Thomeer, J.H.M.: "Air Permeability as a Function of Three Pore-Network Parameters," JPT (April 1983), 809-814.

14. Swanson, B.F.: "A Simple Correlation between Permeabilities and Mercury Capillary Pressures," JPT, (Dec. 1981), 2488-2504.

15. Wells, J.D. and Amaefule, J.O.: "Capillary Pressure and Per-meability Relationships in Tight Gas Sands," paper SPE 13879 presented at the 1985 Low Permeability Gas Reservoir held in Denver, CO, May 19-22.

16. Kolodzie, S., Jr.: "Analysis of Pore Throat Size and Use of the Waxman-Smits Equation to Determine OOIP in Spindle Field, Colorado," paper SPE 9382 presented at the 1980 Annual Fall Technical Conference of Society of Petroleum Engineers, Sept. 21-24, 1980.

17. Pittman, E.D.: "Relationship of Porosity and Permeability to Various Parameters Derived from Mercury Injection-Capillary

Pressure Curves for Sandstone," AAPG Bull., vol. 76, No. 2 (February 1992) 191-198.

18. Klinkenberg, L.J.: "The Permeability of Porous Media to Li-quids and Gases," paper presented at the API 11th Mid-Year Meeting, Tulsa, OK (May 1941); in API Drilling and Product-ion Practice (1941) 200-213.

19. Nakornthap, K. and Evans, R.D.: "Temperature-Dependent Relative Permeability and Its Effect on Oil Displacement by Thermal Methods," SPERE (May 1986) 230-242.

20. Wyllie, M.R.J. and Gardner, G.H.F.: "The Generalized Kozeny–Carman Equation: Part II," World Oil, (1958), 146(5): 210–228.

21. Ali, L., personal communication with T. Blasingame (1995). 22. Microsoft® Office Excel 2003, Microsoft Corporation (1985-

2003). 23. Thomas, L.K., Katz, D.L., and Tek, M.R.: "Threshold Pressure

Phenomena in Porous Media," SPEJ (June 1968) 174-183. 24. Breiman, L. and Friedman, J.H.: "Estimating Optimal Trans-

formations for Multiple Regression and Correlation," Journal of the American Statistical Association (September, 1985) 580-598.

25. Timur, A.: "An Investigation of Permeability, Porosity, and Residual Water Saturation Relationships for Sandstone Reser-voirs," The Log Analyst , Vol. 9, No. 4, 8-17.


Appendix A: Summary of Data Used in This Study

Table A-1 — Summary of data used in this study

Input Data pd, Swi and λ Calibration Results

No. φ

(fraction) k

(md) Swi

(fraction) pd

(psia) λ

(dim-less)1 0.125 0.187 0.010 123.28 0.534 2 0.162 0.184 0.008 435.11 0.850 3 0.129 0.020 0.020 580.15 0.600 4 0.068 0.004 0.007 1667.93 0.679 5 0.104 0.012 0.010 725.19 0.689 6 0.108 0.025 0.010 725.19 0.689 7 0.084 0.014 0.010 725.19 0.935 8 0.062 0.006 0.010 696.18 0.982 9 0.089 0.024 0.010 696.18 0.738

10 0.085 0.013 0.010 1174.81 0.896 11 0.117 0.057 0.020 391.60 0.680 12 0.111 0.036 0.017 435.11 0.575 13 0.079 0.028 0.020 522.14 0.754 14 0.123 0.018 0.008 1015.26 0.661 15 0.073 0.019 0.020 638.17 0.815 16 0.166 0.046 0.008 797.71 0.960 17 0.050 0.005 0.060 942.75 0.914 18 0.084 0.127 0.020 319.08 1.359 19 0.166 0.166 0.020 275.57 1.189 20 0.083 0.041 0.020 435.11 0.906 21 0.075 0.016 0.010 580.15 0.704 22 0.071 0.006 0.010 1667.93 1.625 23 0.066 0.017 0.010 652.67 1.116 24 0.069 0.007 0.010 1232.82 1.238 25 0.086 0.068 0.010 406.11 1.165 26 0.066 0.031 0.010 478.62 1.175 27 0.086 0.029 0.010 725.19 1.341 28 0.071 0.018 0.010 797.71 1.570 29 0.095 0.080 0.010 435.11 1.194 30 0.076 0.087 0.020 319.08 1.120 31 0.258 814.000 0.070 7.25 1.280 32 0.207 434.000 0.200 5.80 0.800 33 0.204 82.300 0.030 7.25 0.424 34 0.214 303.000 0.080 9.43 0.980 35 0.209 210.000 0.091 10.15 0.800 36 0.265 8340.000 0.030 2.90 1.511 37 0.235 438.000 0.080 8.70 1.130 38 0.320 868.000 0.150 8.70 1.718 39 0.335 4570.000 0.090 5.08 1.637 40 0.272 296.000 0.120 14.50 1.050 41 0.266 250.000 0.120 8.70 0.610 42 0.287 640.000 0.140 7.25 1.659 43 0.046 0.019 0.010 435.11 0.958 44 0.092 0.061 0.010 377.10 0.736 45 0.067 0.054 0.010 391.60 1.170 46 0.106 0.339 0.015 145.04 0.798 47 0.051 0.076 0.010 174.05 0.750 48 0.071 0.070 0.010 290.08 0.860 49 0.075 0.128 0.010 188.55 0.636 50 0.077 0.089 0.010 246.56 0.830

Table A-1 — Summary of data used in this study (cont'd)

Input Data pd, Swi and λ Calibration Results

No. φ

(fraction)k

(md) Swi

(fraction) pd

(fraction)λ

(dim-less)51 0.056 0.054 0.010 290.08 1.003 52 0.088 0.069 0.010 333.59 0.760 53 0.116 0.178 0.010 246.56 0.848 54 0.127 0.191 0.010 246.56 0.850 55 0.089 0.085 0.010 319.08 0.903 56 0.056 0.070 0.010 224.81 0.897 57 0.091 0.037 0.010 435.11 0.748 58 0.083 0.042 0.010 435.11 0.920 59 0.069 0.033 0.010 507.63 0.943 60 0.103 0.057 0.010 362.59 0.750 61 0.043 0.046 0.010 261.07 0.945 62 0.044 0.089 0.010 188.55 0.890 63 0.039 0.057 0.010 217.56 1.212 64 0.115 0.016 0.010 1276.33 0.848 65 0.167 0.027 0.010 2175.57 1.050 66 0.371 14.600 0.090 52.21 0.759 67 0.265 11.500 0.050 72.52 1.400 68 0.247 3.800 0.190 75.76 1.011 69 0.220 116.000 0.330 14.50 1.500 70 0.133 48.000 0.030 8.70 0.707 71 0.132 467.000 0.030 2.90 0.753 72 0.125 174.000 0.050 7.25 1.400 73 0.110 351.000 0.030 2.32 0.587 74 0.148 117.000 0.060 14.50 1.061 75 0.109 16.600 0.020 18.85 0.982 76 0.136 72.200 0.020 8.70 0.844 77 0.126 16.500 0.020 23.21 1.015 78 0.153 209.000 0.020 13.05 1.200 79 0.260 170.000 0.320 11.17 1.159 80 0.250 950.000 0.290 4.64 0.816 81 0.137 0.027 0.060 725.19 0.913 82 0.123 0.014 0.020 1377.86 0.959 83 0.039 0.013 0.010 652.67 0.459 84 0.057 0.015 0.050 710.68 0.585 85 0.126 0.272 0.010 319.08 0.653 86 0.159 0.469 0.020 159.54 0.528 87 0.126 0.326 0.010 145.04 0.521 88 0.133 0.352 0.010 145.04 0.536 89 0.126 0.112 0.020 188.55 0.556


Appendix B: Comparison with Timur’s Permeability Model

Introduction to Timur's model for permeability:

Timur25 proposed a generalized equation for permeability as follows:

Cwi

BTimur

SAk φ

= ......................................................... (B-1)

where: (A, B, C are generalized constants)

kTimur = Timur correlation for permeability, md φ = porosity, fraction of pore volume Swi = irreducible wetting phase saturation, fraction of pore

volume

Eq. B-1 can be evaluated in terms of the statistically deter-mined parameters A, B, and C. Timur applied a reduced major axis method of regression analysis to data obtained by laboratory measurements conducted on 155 sandstone samples from three different oil fields from North America. Based both on the highest correlation coefficient and on the lowest standard deviation, Timur chose the following result for per-meability.

2

441360

wi

.Timur

S.k φ

= .................................................. (B-2)

Derivation of Timur's formulation using models of perme-ability, capillary displacement pressure, and the index of pore-size distribution:

Our approach to the derivation of Timur's base relation (Eq. B-1) is to note that in a general form, Timur's base relation can be written as:

χβαφ wiTimur Sk = .....................................................(B-3a)

where α, β, χ are generalized constants. Our goal in this parti-cular proof is to provide a specific combination of relations that, upon combination, yield the form given by Eq. B-3a (or at least a result that is an identical form).

The model based model for permeability for this work is given in the form of a generalized correlation as:

543

2)(1

2)(1

1aa

wia

ad

Sp

ak φλ

λ −⎥⎦⎤

⎢⎣⎡

+= ...................... (20)

Clearly, Eq. 20 is almost in the "Timur" form in terms of the porosity (φ) and irreducible wetting phase saturation (Swi) — however, we note that because we use (1-Swi), then our final model written in the "Timur" form should be:

δβφα )(1 wiTimur Sk −= ......................................... (B-3b)

For simplicity, we will use a form of Eq. 20 that is written in terms of λ, rather than λ/(λ+2). This modification will not seriously affect the character of the correlation given by Eq. 20, and will provide the algebraic form that should mimic our rendering of the Timur correlation (i.e., Eq. B-3b).

The "modified" formulation of Eq. 20 (i.e., the permeability correlation) is given as:

5432 )1()(1aa

wiaa

d Spak φλ −= ............................... (B-4)

As discussed in the body of this work, the generalized correla-tion proposed for the capillary displacement pressure (pd) is given by:

432 )1(1b

wibb

d Skbp −= φ ............................................ (21)

Lastly, the index of pore-size distribution (λ) is represented by the following generalized correlation as:

5432 )1(1cd

cwi

cc pSkc −= φλ ........................................ (22)

We first need to substitute Eq. 21 into Eq. 22 and reduce Eq. 22 into a form that only contains φ, k, and Swi. Making this substitution yields:

)()()(11

11

11

1

5445225335

5453525432

5432432

5432

)1( )(

)1()1(

)1()1(

)1(

cbcwi

cbccbcc

cbwi

cbcbccwi

cc

cbwi

bbcwi

cc

cd

cwi

cc

Skbc

SkbSkc

SkbSkc

pSkc

+++ −=

⎥⎦⎤

⎢⎣⎡ −−=

⎥⎦⎤

⎢⎣⎡ −−=

−=

φ

φφ

φφ

φλ

................................................................................... (B-5)

We now substitute Eq. 22 into Eq. B-4 to reduce Eq. B-4 into a form that only contains φ, k, Swi, and λ. This substitution gives us:

3424225322

5434232222

5432

432

5432

)()(11

11

11

1

)1( )(

)1()1(

)1()1(

)1()(

abaawi

baabaa

aawi

abawi

babaa

aawi

aabwi

bb

aawi

aad

Skba

SSkba

SSkba

Spak

λφ

φλφ

φλφ

φλ

++ −=

−⎥⎦⎤

⎢⎣⎡ −=

−⎥⎦⎤

⎢⎣⎡ −=

−=

................................................................................... (B-6)

As an intermediate result, we raise Eq. (B-5) (λ) to the power of a3, which yields:

)()(

)(11

)()(

)(11

)()()(11

5434352323

53333353

35443522

3533353

354452253353

)1(

)(

)1(

)(

)1( )(

cbacawi

cbaca

cbacaaca

acbcwi

acbc

acbcaca

acbcwi

cbccbcca

S

kbc

S

kbc

Skbc

++

+

++

+

+++

−

=

−

=

⎥⎦⎤

⎢⎣⎡ −=

φ

φ

φλ

x

x

)()(

)(11

5434352323

533333533

)1(

)(cbaca

wicbaca

cbacaacaa

S

kbc++

+

−

=

φ

λ

x


We now substitute the previous result into Eq. (B-6) which gives us the "composite equation", which is defined as:

[ ]

[ ]

⎥⎦⎤

⎢⎣⎡ −

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡=

⎥⎦⎤

⎢⎣⎡ −−

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡=

−

−=

−=

+++

+++

+

++

++

++

+

++

+

++

++

)(

)(111

)(

)()(

)()(1111

)(

)()(

)(11

)()(11

)()(11

45434342

52352322

3532

5333332

54343424

52323225

5332

5333332

5434352323

53333533

424225322

3424225322

)1(

)1()1(

))((

)1(

)(

)1( )(

)1( )(

acbacabawi

acacbaba

acaa

cbacaba

cbacawi

baawi

cbacabaa

caaa

cbacaba

cbacawi

cbaca

cbacacaa

baawi

baabaa

abaawi

baabaa

S

cba

k

SS

bcba

kk

S

kbc

Skba

Skbak

x

x

x

x

x

x

x

x

φ

φφ

φ

φ

λφ

where this form reduces to:

[ ]

⎥⎦⎤

⎢⎣⎡ −

⎥⎦⎤

⎢⎣⎡

=

+++

+++

+++

)(

)(111

)(-1

45434342

52352322

35325333332

)1(

acbacabawi

acacbaba

acaacbacaba

S

cbak

x

x φ

Or, solving for k, we have

[ ]

)(-1

)(

)(-1

)(

)(-11

111

5333332

45434342

5333332

45434342

53333323532

)1(

cbacaba

acbacaba

wi

cbacaba

acbacaba

cbacabaacaa

S

cbak

++

+++

++

+++

+++

−

=

x

x φ

Upon final reduction, we obtain:

δβφα )(1 wiSk −= .................................................... (B-7)

where:

[ ] )(-11

111 53333323532 cbacabaacaa cba +++=α .................. (B-8)

)(-1

)(

5333332

45434342

cbacaba

acbacaba

++

+++=β .............................. (B-9)

)(-1

)(

5333332

45434342

cbacaba

acbacaba

++

+++=δ ............................ (B-10)

In this work we have tuned Eqs. B-4, 21, 22, and B-7 to our database, and the results of this tuning exercise yields:

1.64230.53281.1472-1.7828 )1()( 4158064.175 φλ wid Spk −=

.................................................................................. (B-11)

0.8486-0.52850.8210 )1( 640.0538 wid Skp −= φ .......... (24)

0.6698-0.68350.3792-0.6341 )1( 0.00980 dwi pSk −= φλ . (25)

-11.05314.9250 )(1 15896.2440 wiSk −= φ ................. (B-12)

Eqs. B-11, 24, 25 are combined as prescribed by Eqs. B-7 through B-10, and the results are plotted with the tuned Timur relation (Eq. B-12) in Fig. B-1. We note good agreement, the points are identical, indicating that our algebraic exercise is correct.

Fig. B-1 — Comparison of tuned Timur relation (Eq. B-1) to the combination solution (Eqs. B-7 to B-10) for the data set used in this work.

This exercise proves that the Timur formulation can be de-rived from a fundamental formulation, albeit the relation must be tuned to a particular dataset.

iptc 10994 a modified purcell/burdine model for … and extended to incorporate the power law p...

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