Statistical and geostatistical characteristics of some formations in the Carpathian Flysch

v . 52— 1/4: 305—333, 1982 K ra k ó w 1984

Ja n A. CZUBEK, M arta BUNIAK, Jerzy ŁOSKIEWICZ, Joanna BOGACZ, Jerzy DĄBROWSKI, Andrzej LENDA, Tomasz ŻORSKI

STATISTICAL A N D GEOSTATISTICAL CHARACTERISTICS OF SOME FORMATIONS

IN THE CARPATHIAN FLYSCH

(29 Figs.)

Staty styc zna i geostatystyczna c h a r a k t e r y s t y k a n i e k t ó r y c h f o r m a c j i k a r p a c k ie g o fliszu

(29 fig.)

Jan A. C z u b e k , Marta B u n i a k, Jerzy Ł o s k i e w i c z, Joanna B o g a c z , Je- rzy D ą b r o w s k i, Andrzej L e n d a , Tomasz Ż o r s k i : Statistical and geo sta tistica l characteristics of som e formations in the Carpathian Flysch. Ann. Soc. Geol. Poloniae 52-1/4 : 305—333, 1982 Kraków.

A b s t r a c t : The aim of the paper is to establish relations b e tw e e n g e o sta tistica l im age provided by laboratory core a n a ly ses and g e o p h y s ic a l im age as represented b y field loggin g data. Once th e se relations are found it is possib le to construct calibration curves. This procedure is of sp ecia l interest in the case w h en the lo g g in g probes w e r e not p r e v io u sly standarized in calibration models.

The analysis of the applicability of formulas relating core and lo g g in g data has b e e n performed for a few Carpathian fly sch series of the upper C retaceous, P a le o ce n e and Eocene ages.

The rock matrix densities are obtained using correlations b e t w e e n rock porosities aind their bulk density. Furthermore w e obtained the shale density w hich for upper C retaceous and P a le o ce n e series w a s equal to 2.65 but for the Eocene series containing

som e carbonates the shale d ensity w a s found to be equal to 2.80.

The variogram s of porosity correspond in the high porosity regions to spherical s c h e m e w hereas in lo w porosity region w e encountered rather the de W ijsia n scheme.

The variogram s of neutron log and natural gamma logs furnished information on the shale intrinsic schem e w h ich is of spherical type but with range of ~ 1 0 meters i.e. far larger than in the porosity case.

K e y w o r d s : Carpathian Flysch, g e ostatistics, g e o p h y s ic s Jan A. ’C z u b e k , Marta B u n i a k , Jerzy L o s k i e w i c z , Joanna B o g a c z , Jerzy D ą b r o w s k i : Institute of N uclear Physics, ul. R ad zikow sk iego 152, 31-342 Kraków, Poland.

20 R ocznik PTG

Andrzej L e n d a : Institute of N u clear Physics and Techniques, A c a d e m y of Mining and M etallurgy, al. M ick iew icza 30, 30-059 Kraków, Poland.

T om asz Ż o r s k i : Institute of Exploration G eo p h y s ic s and Oil G e o lo g y , A c a d e m y of Mining and M etallurgy, al. M ick iew icza 30, 30-059 Kraków, Poland.

m anuscript received : December, 1980 accepted: January, 1981

T r e ś ć : Celem pracy jest zna lezienie z w ią zk ó w po m iędzy obrazem g e o sta ty sty c z - nym opartym na w y n ik a c h analiz próbek rdzeni w iertniczych i obrazem g e o fiz y cz n y m otrzym anym z danych profilowań radiometrycznych. Po znalezien iu ty ch za leżno ści staje się m ożliw e w y z n ac z e n ie kr z y w y c h cech ow ania. Taka procedura jest specjalnie interesująca w przypadku, g dy sondy radiometryczne nie b y ły c e c h o w a n e w o d p o ­ w iednich standaryzatorach.

A naliza sto so w a ln o śc i w zorów w ią ż ą c y ch dane z rdzeni w iertniczych i z profilo­

wania radiom etrycznego została w y k o n a n a dla kilku serii fliszu karpackiego n a le żą ­ c y c h do eocen u, paleocenu i górnej kredy.

G ęstości m atrycy skalnej otrzym yw ane b y ły za pom ocą korelacji pom iędzy poro­

w atością i ciężarem o b ję to ś c io w y m skał. Ponadto otrzym aliśm y g ę s to ś c i iłó w w y p e łn ia ­ ją cy c h pory dla u tw o r ó w górnej kredy i paleocenu , które w y n o s z ą 2,65 g/cm 3 na to ­ miast dla u tw o r ów eo c en u z a w iera ją cy ch dom ieszki w ę g la n ó w , g ę sto ść iłu była równa 2,80 g/cm 8.

W obszarze w y s o k ic h p oro w a to ści wariogram y po ro w a to ści odpow iad ają s ch em a ­ to w i sferycznem u, natom iast dla obszaru m ałych p o ro w a to ści n a p o ty k a liśm y raczej s c h e ­ mat de W ijsa. Z w ariogram ów profilowania n e u tr o n o w e g o i naturalnego p r o m ie n io w a ­ nia gamma otrzym aliśm y informację d otyczącą schematu dla zailenia, który jest typu sferycznego, ale o zasięgu 10 m, tj. dużo w ię k sz y m niż w przypadku porowatości.

1. INTRODUCTION

The geostatistical system of the well logging data interpretation (Bo- gacz et al. 1979) needs some statistical and geostatistical characteristics of formation parameters. Here an example of such characteristics for the Carpathian Flysch series of the upper Cretacous, Paleocene and Eocene ages is presented.

The data are coming from a region stretching from the neighbourhood of Limanowa up to Brzozow-Rymanow area. It has been divided into two distinct geographic regions:

1. Slopnice-Limanowa region shown in Fig. 1; it measures circa 12X6 kilometers.

2. Gorlice-Jaslo-Krosno-Brzozow region measuring circa 72X20 kilo­

meters.

In the second region 28 boreholes have been situated at the Potok Fold (circa 12X2 kilometers which is shown in Fig. 2). A v ery short description of the borehole sample data available for these two regions is given in Table 1. About 20 cubic centimeters has been used as the

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152 156 160 16*. X [ h m ]

Fig. 1. Słopnice — Limanowa region Fig. 1. Rejon Słopnice — Limanowa

X [ k m ]

Fig.2. Potok Fold region Fig. 2. Rejon Fałdu Potoka

standard volume for each sample taken from the core for the laboratory measurements. The x, y coordinates of each borehole have been known as well as the depth below the sea level at which each core sample has been taken out.

For some boreholes the gamma-ray, neutron-gamma and th e caliper logs have been also available in the digital form. These logging data have been also pretreated to obtain the nuclear tool responses in cpm not influenced by the logging speed, ratem eter time constant and the appa­

ratus dead time. For the natural gamma-ray log th e data have been also pre-interpreted to be not influenced by the borehole diameter, gamma- -ray absorption in the borehole fluid, and the radioactivity of the neigh­

bouring layers. From the neutron-gamma ray log the gamma-ray back­

ground has been subtracted. These nuclear well logging data have been known as the averaged values for the one meter intervals along the borehole depth

All these data, the laboratory and the logging ones, have been sub­

mitted to the statistical and to the geostatistical analysis to get some characteristic param eters for the formations listed in Table 1.

Ta b l e 1

Core data a v ailab le for the Carpathian F ly sch Region:

S ło p n ic e —Lim anowa G orlice—J a s ło —Krosno—

•—Brzozów (Potok Fold incL) Litho-stratigraphic Eocene age. Krosno layers P a le o ce n e and Upper Creta­

description (KO), and G rzybów shales ceous ages. Istebna layers and C ergow a sandstones and Czarna Rzeka shales.

(GC). Sandstones, shales, S a nd stones and shales of the

mudstones. series A, B, C and D.

N o. of boreholes 25 63

N o. of layers 82 203

N o . of sam p les for:

porosity 675 1914

dry bulk d e n sity 675 709

p erm eability — 816

o la y content 140 122

2. TREATMENT OF THE LABORATORY D A T A

The statistical distributions of the laboratory data h ave been calcu­

lated first. They have been fitted to the normal or to the log-normal dis­

tributions. Average value, the second and the third central moments

have been calculated, together with the histograms and the cumulative distributions. Here the abscissae h ave been expressed in the experimen­

tally observed standard deviation units around the average value, whereas the cumulative distribution have been presented onto the Gaus­

sian scale.

P O R O S I T Y

None of the mentioned above distributions fitted well the experim en­

tal data, if taken for the whole regions. Much better fit has been obtained when the data corresponding to the well defined litho-stratigraphic series w ere treated within the radius of about 10 km. As an example the distri­

bution of the porosity 0 for the sandstone B of the W estern part of the Potok Fold is given in Fig. 3. Here, the experimental distribution fits well the noimal distribution within the 95 per cent confidence' belt (which is given as the two dotted lines in this and in all forthcoming figures concerning the distributions). The same situation for the Słopnice

N o r r . a l d i s t r i b u t i o r . :

< X >

0.81A02E 01

l ;o r c s i t y : ! No of sets

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MI 02 0.10396E 02

r; = 13,

SIGMA 0.32243E 01

MI 03 0.8S629E 01

95.

IS: St:

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Vertical scale of histogram:

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r

* I > * 1 4 1 I X X X I I ■ t o i l I

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r i v x i x i i i i r t ii j i i x x r x j j x x i i x o x X X I I X X X I X X I X X X X I

Ti ' i ruiim nni *UK

I I X X I I I I 1 1 X I X X X X X X I X X X X X X X I X I i n n n n i u i u i n x x x x i i x i x x i i i x x x x x x i x x i i x I X x x i i x i x y x x x x i i x i x x i i J t x x x x x x x X X X l l X I I X X r X I I I X X X I X X X X X X X X X X I X x X X l I I . r X X I I X I I I X X I X X X X X X X X X X X X X I X X X x x x x r x i x i x i / x x x x x K i i x x x x x x x i x j j ; X X X X X X X X I X X X X I I X X I I K I I X X X X X X X X K X I I

X X X X

. II

X I X C 1 X I X I I X X X I X X X I X I X X I X X I I X X X

-1 ¹

-2 -1 4

[SIGUA]

Fig. 3. Porosity distribution for sandstones B from the Potok Fold

Fig. 3. Rozkład w sp ó łczyn n ik a porow a to ści dla p ia s k o w c ó w B z rejonu Fałdu Potoka.

P ionow a skala histogramu: l linia = 0,2®/o w szy stk ich przypadków

AVERAGEPOROSITY<0>cumulative distribution[*]

Lon-normal distribution: Porosity. Słopnice Region. Sandstone KO No of seta: N= 0 , No of samples N= 110

u

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Fig. 4. Porosity distribution for sandstones KO from the S ło pnice region

Fig. 4. Rozkład w spółczynnika p o ro w a to ści dla p ia s k o w c ó w KO z rejonu Słopnic. P io ­ n o w a skala histogramu: 1 linia = 0,5®/o w sz y stk ic h przypadków

Fig. 6. Parameters of the normal distributions of the p o r o sity for the series A in the Potok Fold region. 1 and 2 refer to the W and E parts o f the Fold, r e sp e c t iv e ly Fig. 6. Parametry normalnych rozkładów w sp ó łczy n n ik a p o ro w a to ści serii A rejonu Fałdu Potoka. Indeksy 1 i 2 odnoszą s ię o dpo w iedn io do c zęści zachodniej i w schodn iej

Fałdu

Fig. 5. Parameters of the normal distributions of the po ro sity for the sha le/sa ndsto ne series of the G orlice— Brzozów region (including the P otok Fold)

Fig. 5. Parametry normalnych rozkładów w sp ó łczy n n ik a p o ro w a to ści serii łu p k ó w i p ia s k o w c ó w rejonu G orlice—Brzozów (wraz z Fałdem Potoka)

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STANDARD DEVIATION (5 ( l n 0 )

Fig. 7. Parameters of the log-normal porosity distributions for the Słopnice region. PM,

LP and MU stand for different shaliness of the KO or GC sandstones (PI) or shales (LU) Fig. 7. Parametry logarytm iczno-norm alnych rozkład ów w sp ó łc z y n n ik a po ro w a to ści dla rejonu Słopnice. PM, LP, LM i MU oznaczają różny stopień zailenia p i a s k o w c ó w (PI)

i łu p k ó w (LU) serii KO i GC

region is given in Fig. 4. Here the porosity fits better the log-normal distribution (the histogram in this figure is given in the linear scale, how ­ ever). To be able compare the different sets of data, the estimated param eters of these distributions have been plotted in the following co­

ordinate system: expected value ( 0 ) vs. the standard deviation 0(0 ).

The 95 per cent confidence belt for each estimated value has been also plotted. These data are plotted in Fig. 5. for the litho-stratigraphic series lor the whole Gorlice-Brzozow region, whereas in Fig. 6. this situation is given for the series A only. In the last figure the A series have been split inside the Potok Fold into the western (1) and eastern (2) parts in view to be able to get some idea about the homogeneity of the data. Here, and in all forthcoming figures, the PI stands for sandstone, LU for shales, LP, PM, MU are the different types of the shaly sandstones, and XX neg­

lects all lithology or stratigraphy indexes. W hen the rectangles of the confidence belts around each poin in this kind of presentation are, even partially, overlapping, we are not able to tell (at the 95 per cent of the

confidence level) that the two sets have been taken out from different general populations (or that they are essentially different). In this respect, for example, the shales of the series A (marked as AOLU) are not es­

sentially different from the sandstones of the series D (marked as DOPI) in Fig. 5. The same is valid for the sandstones of the series A (marked as PI) taken from different parts of the Potok Fold (cf. Fig. 6). The confidence belts for the shale distributions (marked as LU) are always larger than these one for the sandstones (marked as PI) because of much poorer sample statistics.

The same situation for the Słopnice region is given in Fig. 7. for the median value of porosity vs. the standard deviation of ln0.

As we can see from these figures, each small region is more or less homogeneous with, however, big differences between the regions. M ore­

over, the accuracy of the laboratory measurements on the core samples does not permit, in many cases, to distinguish betw een the different se­

ries. For this reason it is statistically allowed, at this stage of the statis­

tical recognition of this region, to put together some sets of data belong­

ing to different series for the further considerations, because they are not statistically different. This situation can be changed, of course, if more m easurem ent data will be available.

D R V BULK D E N S I T Y

Dry bulk denstiy 8K of sandstones for the Gorlice-Brzozów region fol­

lows quite well the normal distribution (cf. Fig. 8.), w hereas for the Słopnice region this agreement is not so good. For the whole set of the 5, data for the Słopnice region the distribution is given in Fig. 9., whereas for the shale only from this region — in Fig. 10.

W hen one uses this kind of data to plot, for a given litho-stratigraphic series in a given region, the expected 8S value against the expected poro­

sity 0 value (with the 95 per cent confidence belts), the apparent mine- ralogical density 5M can be obtained according to the formula

*s = ( l - 0 ) - 5 M (2.1)

This situation for the Słopnice region is given in Fig. 11, whereas for the Potok Fold region in Fig. 12. One obtains very easily that in the first case 5M = 2.72 g/ccm, whereas for the second region is 5M =-' 2.65 a/com.

These data can be now compared with the mineralogical composition of the investigated rocks. This is done in Fig. 13, w here for the Słopnice region (the KO and GC series) and for the Potok Fold (series C: shales COLU and sandstones COP!) the distribution of data, in the triangle system: quarz + feldspar, carbonates and clay minerals, is done. A ssu­

ming the mineralogical densities 5M for quarz + feldspar, carbonates and

Cumulativedistribution [*]Cumulativedistribution fo]

Normal distribution: Dry bulk density: Potok Fold. Sandstone, IX-PI No of sets: N= 98 • No of samples N = 621

<X> MI 02 SIGMA. MI 03

0.23962E 01 0.28036E-01 0.167^5E 00 -0.45086E-02

...ł "fi

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Vertical scale of histogram 1 line = .2 %

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/V II

J IV V X X X I X I

. XI X I IX X I I X

I X X X I X X 1 1 1 X I XI X I I X H I I X I X I X I X I I I I X X I X I I I I X I I I X X IX X X I X X X I X X I I X X X X X X V I I X X I t l V I H I X I X I

J, H I I X I I I X I I 1 I I I X X ( I X

4 . xixxxixx m i x ^{x i i} i i ix i i

X X X X I I X X X I X I I I I X I I X I I I I I I

J ...-...

• J, X X I I K X X X I I I X X X V I X X X I I I X I I I I I I I I X

- I X X I I X X I I X X X I I X I I X I I I I I X I I I I I I I I I I

• « • X , i i i i i i i i i i i i n i i i i i i i i x i n i i m i i i i

X X X X X X I I I X l I I I I I I I l I I I I I I I l I I I t l X I I I I X I l K X l I I X I X I X l l l l I I I X I X l l I I l I l l l I I I l I l I I l

-I

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[s i g m a]

Normal distribution: Dry bulk density: Słopnice Region. XZ-XZ No of sets: Na 79 ^, No of samples F* 675

<X> MI 02 SIGMA MI 03

0.26585E 01 0.51629E-02 0.71&53E-01 -0.56929E-03

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cr.t?. ««J • . , i x i i n 1 1 i n i n 1 1 * * ; * 5

,.j « II IX 1IX II III III II 171 II III =

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X X i r v I I

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1 X I I X X IX

X X 11 I I I X X

I I I 1 1 I I I XX

1 I I X X I I I I 1 1

X I I X I 1 1 I I I 1 1 1 1 1 XI I X I 11 I I I X I X I I I XI I X I I I I I I I I I XI I XI 1 1 1 1 1 1 1 1 1 X 1 I I IX I X I I I 1 1 I I I xxx ^{1 1}

I X XI xxx 1 1 I I I I I I II 1 X 1 XX I X X I I I I I 1 X 1 I I

♦ ♦ ^♦

* ♦ ^{’ «1'} «

’o ’ *

Fig. 8. Dry bulk d e n s ity distribution for the sandstones from the w h o le G orlice—Brzo­

z ó w region

Fig. 8. Rozkład ciężaru o b ję to ś c io w e g o szkieletu dla p i a s k o w c ó w c a łe g o rejonu Gor­

l ic e — Brzozów. P io no w a skala histogramu: 1 linia — 0,2®/b w s z y s tk ic h przypadków

Soraal distribution! Dry bulk density* Slqpnloe region.

8h*le-XX. XX-L0

He of setsi ? , Ho of aaaples 61

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.266441 01

HI 02

0.79«51S-02 0.89359»-01^SIGMA.

MI 09

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.11.

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Vertical scale of bistOfgrsB 1 line =» .5 %

of all «rents

*e«eeeeee*eeeoe«*e*

I

■ I 1

IS 1I I I I I I

1111 IIII IIII

i i

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iiiniii iii jiiiiim i|i

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[BIGMAj

Fig. 10. Dry bulk d e n sity distribution for the sh a les from the Sło pnice region Fig. 10. Rozkład ciężaru o b j ę to ś c io w e g o szkieletu dla łupkó w z rejonu Słopnic

Fig. 9. Dry bulk d e n s ity distribution for the core sam ples taken from the S ło p n ic e region Fig. 9. Rozkład ciężaru o b ję to ś c io w e g o szkieletu dla próbek rdzeni w z ię ty c h z rejonu

Słopnic. Pionow a skala histogramu: 1 linia = 0,2% w s z y s tk ic h przypadków

AVERAGE POROSITY < $ > [ % ]

Fig. 11. Correlation b e tw e e n the e x p e c te d v a lu e s of dry bulk density and porosity for the S ło p n ic e region

Fig. 11. Korelacja pom iędzy wartościam i o c z ek iw a n y m i ciężaru o b j ę to ś c io w e g o .szkie­

letu i w sp ó łczy n n ik a p o ro w a to ści dla rejonu Słopnic

clay minerals as being 2.65, 2.71 and 5sh (unknown), respectively, it was easy to obtain that for the Słopnice region 5sh = 2.80 g/cm3 with the average (quarz 4- feldspar + carbonates) rock matrix mineralogical d en ­ sity 5M — 2.67 g/ccm. For the Potok Fold, by the same manner, it was obtained 5sh — 2.65 g/cm3. It should be noted here that when one w ants to get these data from the direct correlation between the 5S and 0 data for each sample, the result is not so clear. The examples are given in Figs 14 and 15. Here the 6S vs 0 crosspiots are given for the Słopnice and for the Goriice-Brzozów regions together with the regression lines. One can obtain, by the simple overlay of Eq. (2.1) onto these crossplots, the ap ­ parent densities &M equal to 3.4 or 2.2 g/cm3 which is, of course, not ad­

missible for this kind of lithology. This effect is probably due to the low accuracy and/or confidence which can be paid to the laboratory data obtained from the core samples. Our approach based on the' expected values of the marginal distributions for the joint distribution of these data is much more convenient, and the eventual inaccuracies in the

AVERAGE POROSITY < 0 > t % 1

POTOK FOLO REGION:

— O CO-LU ---- ■ CO-PI

SŁOPNICE REGION:

*9 £ £ * £ 4? <9 # CLAY MI N E R A L S (<yb) Csh

Fig. 12. Correlation b e t w e e n the e x p e c te d v a lu e s of dry bulk d e n s ity and po ro sity for th e G orlice— Brzozów region

Fig. 12. Korelacja pom iędzy wartościam i o c z ek iw a n y m i ciężaru o b ję to ś c io w e g o s z k ie ­ letu i w sp ó łc z y n n ik a po ro w a to ści dla rejonu G orlice— Brzozów

?•!

1 \t 11 1» || i« 14 ł« i* it It Ił II IT It II 10 t ł > < 1 1 1 »*»

11 ft |t 7 1* 1* SI |l II |l It 1i M II • Il > I l l ł 1

Fig. 14. Correlation b e tw e e n thy dry bulk density and p o r o sity for the sam ples of the Słopnice region. The tw o regression lines are drawn

Fig. 14. Korelacja pom iędzy ciężarem o b ję to ś c io w y m szkieletu a w sp ó łc z y n n ik ie m p o ­ r ow atości dla próbek z rejonu Słopnic. Zaznaczono obie linie regresji

Fig. 13. M in eralogical com p osition of the series KO and GC for the Sło pnice region and CO— PI and CO—LU for the G orlice—Brzozów region in w e ig h t per cent of th e

dry sample. The range of occurrence for e a ch series w a s defined arbitrarily Fig. 13. Skład m ineralogiczny serii KO i GC dla rejonu Słopnic oraz serii CO— PI i CO— LU dla rejonu G o r l i c e —Brzozów w procentach w a g o w y c h suchej próbki. Z asięg

w y s t ę p o w a n ia p o s z c z e g ó ln y c h serii z o sta ł u s t a lo n y arbitralnie

KN9

•rla d® X)

£>

fi

Fig.

Fig.

P o r o s i t y 0

15. Correlation b e tw e e n the dry bulk d ensity and porosity for the sam ples of the

Gorlice—Brzozów region. The two regression lines are drawn

15. K orelacja pom iędzy ciężarem o b ję to ś c io w y m szkieletu i w s p ó łc z y n n ik ie m p o ­ row atości dla próbek rejonu G orlice— Brzozów. Zaznaczono obie linie regresji

analysis results are increasing the confidence belts only. W hen the het- erogenesity of the core samples is the cause of these discrepancy, our method eliminates it even completely.

CLAY C O N T EN T

The statistics for the clay Csh (per weight) determinations was v e ry poor for both regions (cf. Table 1.1.). Probably for this reason th ey follow rather well the normal distributions (within the 95 per cent confidence belt valid for this number of data!). An example of these distributions is given in Figs 16 and 17.

Normal distribution: Shale content? Potok Fold. XX-XX

No of seta: N= i8 y No.of samples Ns 122

< X >

0.J7934E 02 0 . 4 1 1 6 0 E 0 3^{UI 02}

SIGMA 0.20288E 02

MI 0}

0.13655E 04

ao

•H+>

.o♦dtl

-po

.60'

.s?

.3;. .11..5*

• 8? ^.

.M

* 5 .-

*>.•.

■ A . w .SH ^. .u.31'.

►©

-pa>

O

\ S \ J

11 ,

.+?

• J5 ^.

Pig. 2.14. Shale content distribu­

tion for the samples taken from the Gorlice-Brzozow region.

-5

i t I X » X

XX XX %i

XX XIXX XX

., „ XX XX * *

X X X X X X I X

-5 -1 _{3 r 4 ,}

[s i g m a]

Normal distribution! Shale content] Slopnico Region. H - Z 2 No of setst H= , Ko of samples Na 140

<C> MI 02 SIGMA MI OJ

0.J90J6E 02 0.67229B 02 0.8199JE 01 0.84108E 02

Fig. 16. Shale content distribution for the sam ples taken from the G orlice— Brzozów region

Fig. 16. Rozkład zawartości iłu w próbkach w zięty ch z rejonu G orlice—Brzozów. P io ­ now a skala histogramu: 1 linia = 0.5fl/o w szy stk ich przypadków

P E R M E A B I L I T Y

W e have performed the correlation between the porosity and the logarithm of the permeability for the core sample data obtained for the Gorlice-Brzozow region. An example for such correlation is given in Fig.

18 for the sandstones of the C series in the Potok Fold. This correlation is rather poor but significant. A very similar picture was obtained for the other litho-stratigraphic series in this region.

» 1 ) A a • 11 1ft 1? '3 9 I 1 1 1JWPM

Fig. 18. Correlation b etw een the porosity and the logarithm of the permeability for the samples of the sandstone C in the Potok Fold. The two regression lines are drawn Fig. 18. Korelacja pom iędzy w sp ó łc z y n n ik ie m po r o w a to śc i a logarytm em w spó łczy nnika przepuszczalności dla próbek p ia s k o w c ó w C Fałdu Potoka. Zaznaczono obie linie re­

gresji

Fig. 17. Shale content distribution for the samples taken from the Słopnice region Fig. 17. Rozkład zaw artości iłu w próbkach w zięty ch z rejonu Słopnic. Pionowa skala

histogramu: 1 linia = 1%> w sz y stk ic h przypadków

21 R o c z n i k P TG

V A R I O G R A M S

The variograms y(d) (Matheron, 1965) of the porosity and of the bulk density have been calculated. Because of the well-to-well shortest dis­

tance of the order of about 1 km, the vertical variograms had some acceptable behaviour only. W e have calculated them as the average for all boreholes in a given region, excluding from the computation some layers for which we already knew (from the previous calculations) their strange behaviour.

In the Slopnice region the porosity 0 followed m ore or less th e de Wijs scheme:

Y(d) = 3.a.[ln(d/l) + 3/2] (2.2) with a = 8.10- 6 [%2 • 10-4] and 1 = 7 cm,

whereas for the Gorlice-Brzozow region this variogram was of the sphe­

rical type:

Y(d) = C • {*3/21' <d/a) ~ (1/2)' (d/a)’ d | a (2.3) with C = 1.5 10—3 [%2 • 10“ 4] and a = 1.5 m.

The examples of these variograms are given in Figs 19 and 20, w here for the variograms given by Eqs (2.2) and (2.3) the confidence belts of

Fig. 19. V ertical variogram of porosity for the Słopnice region. The ex perim ental data marked by h e a v y points ( • ) h a v e b e e n obtained from less than 50 pairs of data Fig. 19. P io n o w y wariogram w s p ó łc z y n n ik a p o r o w a to śc i dla rejonu Słopnice. Dane do­

św iadczalne zaznaczono pełny m i kółkam i ( 0 ) z o s ta ły otrzymane z m niej niż 50 par d a n y c h

Fig. 20. V ertical variogram of porosity for the G orlice—Brzozów region. Porosity d a t a

are taken from the core sam ples

Fiy. 20. P io n o w y wariogram w sp ó łc z y n n ik a po r o w a to śc i dla rejonu G orlice— Brzozów.

W sp ó łcz y n n ik p o ro w ato ści b y ł m ie r zo n y na próbkach rdzeni

± ov have been calculated. The standard deviation ov has been obtained as the fluctuations of the so called local variogram according to the Ma- theron's idea (Matheron, 1965).

The variograms for the 5S values have been of the de W ijs type for both regions with the a coefficients of the order of 1.1 10-4 g2/cm6 for the Słopnice region, and a = 6.5 10~4 g2/cm 6 for the other one, but the agreem ent with the experimental variograms was rather poor one.

The spherical variograms of the porosity shown in Fig. 10 has been used to recalculate the porosity variance "seen" by the borehole neutron tool. Here the knowledge of the function F(v) in the three dimensional space was needed for the spherical variogram (Matheron, 1965)

F(v) = — J J ^{Y ( i - x} ’ ) . d x . x \ (2.4) V2 v v

I w here v is the volume of the sample, the elementary volumes dx and dx

— > — *

are located at the two extermities of the vector h = x — x' and they are walking independently each other inside the volume v. This sextuple integral can be reduced to the triple one which in the case of the v ario­

gram of the type h* and for the rectangular parallelepiped (for the volume v) of the dimensions a ^ b ^ c is of the form:

F^ (a,b,c) =

=■---- --- J‘ J J (a - x).(b — y).(c — z).(x* + y 2 + z!)x/2.dx.dy.dz (2.5) a*b2c2 o o o

21*

which for the spherical variogram given by Eq. (2.3) finally gives:

Fsph(a,t) = C

-= a.[0.5 - t2.(0.5 lnt - 2.8556905 • 10-1) + t8 • 2.4072624 ■ 1 0 -1 - t 4 • 3.54166 • 10~2 +

t6 • 2.1577381 ■ 10“* — t 8 • 4.6378412 • 10“4 + ...] —

a3 • [0.05 + t2 • 8.333317 • 10~2 + t 4 • (4.6203835 • 10-E — 7.083333 • 10“ 2 • lnt) + t 5 • 3.1561269 • 10~2 -

t 6 • 4.3154739 • 10-3 + t 8 • 2.3189507 • 10“ 4 - ...], (2.6) where b = c

t = b / a S

yo.5

( a - 2 - 1) (2.7) and the sizes of the parallelepiped are expressed in the units of the range a of the spherical variogram (cf. Eq. (2.3)), thus a ^ 1. The plot of the function given by Eq. (2.6) is given in Fig. 21. The, so called, linear equi­

valents for the parallelepipeds in the spherical scheme can be obtained from these data

Fig. 21. Plot of the function Fsph (a, t) for the spherical variogram, g i v e n by Eq. (2.6).

The sizes of the parallelepiped are in the units of the range a of the spherical vario- gram

Fig. 21. W y k r e s funkcji Fsph (a r t) dla wariogramu sfe r y c z n e g o w g wzoru (2.6). Rozmiary prostopadłościanu są podob ne w jed no stka ch z a sięg u a wariogramu sferycznego

3. LOGGING D A T A TREATMENT

Using th e deterministic approach (Zorski, 1979, Czubek and Zorski, 1979) the gamma-ray and the neutron-gamma ra y logging data w ere re­

calculated to obtain the logging deflections free of the influence of the apparatus and of the borehole conditions.

The crossplot of the neutron and gamma-ray logging deflections give and information about the lithology in the shaly-sandstone sequence, as it is shown in Fig. 22.

Fig. 22. Crossplot of the neutron and gamma-ray lo g deflections for sa ndsto nes in the Potok 13 borehole. The lith o lo g y identification z o n e s delim inate this part of the

borehole as the s h a ly sandstone of different porosity

Fig. 22. Z esta w ien ie w skazań profilowań n e u tr o n o w y c h i gamma dla p ia s k o w c ó w Otworu Potok-13. Strefy identyfikacji litologiczn ej określają tę c z ę ś ć otworu jako za-

ila n y p ia s k o w ie c o zm iennym w sp ó łc z y n n ik u p o ro w a to ści

The variograms of the 1..^ (natural radioactivity) and I ny (neutron- -gamma log deflections) intensities give the information about the vario­

grams of the shale content and that of the porosity. An example is shown in Figs 23 and 24 for the sandstone of the B series in the Sobniów 23 bo­

rehole (Potok Fold).

A n interesting information can be obtained from th e correlation be­

tw een the gamma-ray and the neutron-gamma variograms for a given formation. Such correlation for the sandstone of the C series in th e Sob­

niów 23 borehole is given in Fig. 25. Here, for the distance d lower than

K>‘

6 ^—

- /

/ ' \

4 —

<Z

X<

o

3 —

t —

I I

S0BN10W 23 W E L L S A N D S T O N E 0 R92 5 -16555 m

' I 1 I 1 I

8 10 12 14 I I

18 I

16 18 20 d [m]

Fig. 23. Variogram of the natural r a d io a ctiv ity for the sa ndsto nes of the B series in the S o b n ió w 23 borehole

Fig. 23. W ariogram naturalnej prom ieniotw órczości dla p i a s k o w c ó w serii B w otw orze S o b n ió w 23

/

\ /

\

SOBNIOW 23 W E L L S A N D S T O N E B 1492.5 -1655.5 m

| ' | I | 1 I

8 10 12

I ' I 1 I

14 16 18

1 I

20 d [m]

Fig. 24. Variogram of the neutron-gamma, ray lo g deflectio ns for the sandstones of the B series in the S o b n ió w 23 borehole tale«ft for the same w e l l sectio n as in Fig. 23 Fig. 24. Wariogram w skazań profilow ania neutron-gamma dla p i a s k o w c ó w serii B

w otw orze S o bn ió w 23 w z ię ty c h dla teg o sa m eg o odcinka otworu jak na Fig. 23

/ ljJ d , [ CpM2 ]

Fig. 25. Correlation b e tw e e n the Yi-oo (d) anc* Vin (d) variograms. This correlation has the p hysical m eaning for the distances d lo w e r than the range a of the spherical vario- yram. Here, for the logging data in the sandstones of the C series in the S o b n ió w 23

borehole, the range a has been estim ated as a = 20 m

Fig. 25 Korelacja po m iędzy wariogramami Yi-oo (d) * Yin (<*)• Ta korelacja ma fizy czn y se n s dla o d le g ło ś c i d m niejszy ch niż zasięg a wariogramu sfery czneg o . Tutaj, dla p ia s ­ k o w c ó w serii C w otw orze S o b n iów 23 zasięg wariogramu dla w y n ik ó w profilow ań z o­

stał o k r e ślo n y na a = 20 m

th e range a = 20 m of the spherical variograms Yi.loc(d) and Yin„(d), the linear relationship can be found:

VI„,(d) = B + S-v,,„(d), (3.1) w here the nature of the constants B and S is that:

B/S = 6.25 • o2(0) • K3, (3.2)

w here o2(0) is the variance of th e porosity "seen'' b y th e neutron probe, and K is the calibration factor for the gamma-ray log when th e shale content C 'h (per weight of the rock "in situ" is determined from this kind of log:

i w ==K. + K -c ;h . (3.3) Using all this statistical and geostatistical information th e statistical calibration curves (Bogacz et al.( 1979, Czubek et al., 1977) have been found. Thus, the volume shale content Vsh and th e rock porosity 0 from the gamma-ray and neutron-gamma ra y logs have been obtained. Here

the three different shale contents have been carefully taken into account:

Csh — the shale content per weight of the dry rock, Cjh — the shale content per weight of the natural, in situ rock (when the pore space fulfilment is also taken into account), and Vsh — the volume shale content.

The first one is measured in the laboratory on the core samples, the second one by the gamma-ray log, w hereas the third one influences the

neutron log deflections.

4. TREATMENT OF THE LOG DERIVED PARAMETERS

From the quantitative interpretation of nuclear logs, using the statis­

tical calibration curves, the shale content Vsh and the rock porosity 0 are obtained. These values can be treated again b y the statistical and geostatistical ways. Here the statistical distributions of Vsh are very like those for the I .^v alu es. The log derived porosities 0 fit quite well the distributions still found from the core sample data analysis.

The variograms of the Vsh values show the same behaviour as the va- riograms of the I.iocdata. The variogram of the log derived porosites 0 is

SOBNIOW 23 W E L L S A N D S T O N E 3

1492.5 -1655 5 m

*0

| t | i | i | i | i | i | i | i | i | i |

0 2 4 6 8 10 12 14 16 18 20

d[m]

Fig. 26. Variogram of the log derived porosities for the sandstones of the B series in the S o b n iów 23 borehole taken for the same w e ll section as in Figs 23 and 24 Fig. 26. W ariogram dla w s p ó łc z yn n ik a po r o w a to śc i z interpretacji profilowań dla p ia s­

k o w c ó w serii B w otworze S o b n ió w 23 w z ię ty dla tego s a m eg o odcinka otworu jak na

Fig. 23 i Fig. 24

Fig. 27. Fit of the beta distribution to the e x p e r im en ta lly o b se r v ed m arginal distribution of the log derived porosities

Fig. 27. D op aso w a n ie rozkładu beta do o b se r w o w a n e g o d o ś w ia d c z a ln ie rozkładu brze­

g o w e g o w sp ó łc z y n n ik ó w poro w a to ści otrzym anych z interpretacji profilow ań

30

co20

UJ>

e rUl

ffiZ 13

10

%h>«Q3«*

S w fc)*Q0M2

R(* -W:

Q951 < -0,9 38 <-0,923 9 5 % c o n l .b r l t

“ I--- 1---1--- POTOK - 13 WE UL CD-LU 1230,5 - 1515,5 m

m > 17,866 I T i - 3 0 , 9 4 9

r ( l « m *r>)

? ( m ) n i * n ) * N * 78 6

1,1270855 10

F I T T E D DISTRIBUTION

r

p j/ MARGINAL DISTRIBUTION

V O B S E R V E O EXPERIM ENTALLY

I

10

j . . V -

20 30 4 0 50

SHALE CONTENT

60 ⁷⁰ JL

80

Fig. 28. Fit of the beta distribution to the ex p e r im en ta lly o b se r v ed marginal distribution of the log de r iv e d shale content

Fig. 28. D op asow an ie rozkładu beta do o b s e r w o w a n e g o d o św ia dcza ln ie rozkładu brze­

g o w e g o zailenia otrzym anego z interpretacji profilowań

shown in Fig. 26 for the sandstones of the B series in the Sobniow 23 b o re­

hole. W hen one compares this variogram with the variogram of the In,.

values given in Fig. 24 for the same borehole section, the net difference is visible. The log derived porosity 0 has the spherical variogram (but regularized over the rock volume "seen" by the neutron tool) which is essentially similar to this one from Fig. 20 obtained for the core samples fiom this region.

Because all the variograms for the log derived shale content and poro­

sity have the spherical shape, the obvious conclusion is th at th ere is no vertical trend for these data, at least along the borehole sections investi­

gated here

The crossplot of the log derived shale content with the porosity shows again a v ery distinct lithology effect. For the pure shale sections this correlation follows v ery well the straight line with the negative slope, whereas for the shaly sandstones the data are dispersed inside the triangle which sides are defined by the pure shale of different porosity, pure sandstone of different porosity and hard shaly sandstone.

W hen the normal distributions for the shale content Vsh and for the porosity 0 have been found, we have been quite conscious of the ambi­

guity of these results. Both parameters, being limited inside the [0; 1]

section, cannot follow exactly the normal distribution. Just in this case the beta distribution of the first kind seems to be m ore appropriate. Thus, we have tried, starting from the assumption that both Vsh and 0 follow the joined bivariate beta distribution (within the given litho-stratigraphic series), to find the param eters of these distributions. These param eters have been derived from the marginal distributions of the Vsh and that of 0 . These marginal distributions should also be of the beta kind, and really, they are. An example is given in Figs 27 and 28, but their p aram e­

ters do not fit the same bivariate beta distribution (compare, for example, the sum of (1 + m + n) on both figures) which should have the form:

b(0 ,v s„ [ l,m,n) = *3* , • 0 1- ' ' Vsh”- ‘ • (1 - 0 - V*,)”-«. (4.1)

T ( l) • r ( m ) • 1 »

For such distribution the correlation coefficient o betw een the 0 and Vsh variables is given as:

<?(0,Vsll) = - 1 / ---l-' m - — • = (4.2.a) f (n + 1) • (m 4- n)

= _ ( 0 ) - .( y » >_____ , (4.2.bi

I (1 — <0)) - (1 — (Vsh>)

where ( 0 ) and (Vsh) correspond to the expected values. Thus, the corre­

lation coefficient (?(0,Vsh) is always negative for the bivariate beta dis­

tribution.