Clay Minerals (1994) 29, 475-490

P O R O S I T Y / D E P T H T R E N D S IN R E S E R V O I R S A N D S T O N E S : A S S E S S I N G THE Q U A N T I T A T I V E

E F F E C T S OF V A R Y I N G P O R E - P R E S S U R E , T E M P E R A T U R E H I S T O R Y A N D M I N E R A L O G Y ,

N O R W E G I A N S H E L F D A T A

M . R A M M AND K. B J O R L Y K K E *

Norsk Hydro, Research Centre, Sandsli, N-5020 Bergen, Norway and *Department of Geology, University of Oslo, PB 1047 Blindern, N-0316 Oslo 3, Norway

(Received 5 May 1993; revised 27 January 1994)

A B S T R A C T: Empirical porosity/depth trends from the Norwegian Continental Shelf are discussed in the light of models on compactional processes during burial. Linear best-fit lines are useful as a first approximation, illustrating the general decline in porosity with depth. Porosity increases non-linearly with pore-pressure, and notably higher than normal porosity is found in very high overpressure reservoirs. Hydrostatic pressure sandstones from the Haltenbanken area are on average more porous than equivalents from the northern Viking Graben. This may be related to the burial history, as generally more subsidence occurred in Late Pliocene and Pleistocene in the Haltenbanken basins. The porosity/depth data show considerable scatter, and this is to a large degree related to the initial mineralogical composition. Correlated with mineralogical analyses, the reservoir sandstones show a trend towards lower porosity with increasing clay content at shallow burial, probably due to more extensive mechanical compaction. At depths >3.0- 3.5 km, the porosity loss is highest in sandstones with low clay contents. This may be explained by a lower rate of quartz cementation in sandstones with abundant intergranular clays or clay coatings around quartz grains. These data show that pre-burial mineralogy is a very important factor in porosity prediction of sandstones, in many cases more important than the degree of overpressure.

Explorat ion in deeper parts of sedimentary basins requires improved predictions of porosity/depth relationships. These must be based on parameters related to the burial history, such as temperature and pressure, but also on predictions of the primary mineralogy and texture of sandstones. At burial depths down to 2.5-3.0 km, the porosity reduction is mostly due to mechanical compaction (Ramm, 1992) and the primary sorting and the clay content are critical factors (e.g. Scherer, 1987). At greater depth, quartz cementat ion becomes highly significant (Ehrenberg, 1990; Ramm, 1992; BjCrlyke et al., 1995; Bjcrlykke & Egeberg, 1993; Ryseth & Ramm, 1995) and porosity reduction is closely linked to processes controlling the quartz cementat ion.

Porosity/depth trends in sandstones have been presented in a series of earlier papers (e.g. Maxwell, 1964; Atwater & Miller, 1965; Fticht- bauer, 1967; Selley, 1978; Loucks etal . , 1984; Scherer, 1987; Bjcrlykke etal . , 1989, 1992; Bloch, 1991). A large number of parameters

influencing the porosity/depth gradients have been identified. Among these, preburial minera- logy and texture, pressure and temperature gradients, age and timing of petroleum emplace- ment have received most attention. However , the relative importance of these factors is not known with precision. Multivariate regression models may provide a very good fit between the original data set and the established predictive models. However , for exploration purposes such models may become impractical due to the numerous constraints involved. The purpose of this paper is, therefore, to establish models depending on three parameters , the pore-pressure, an integrated t ime-temperature index, and a ductility index. These parameters , based on both empirical experience and theoretical considerations (e.g. Ramm, 1992) are judged as very important factors controlling the porosity distribution in the subsurface. During prospect evaluation, the first two may be predicted via seismic data and basin modell ing while the last may rely on the depo-

�9 1994 The Mineralogical Society

476 M. Ramm and

sitional models for the reservoir sandstone. Fur- thermore , the possible effect of hydrocarbon emplacement is evaluated in order to address the necessity of including this parameter into the porosity prediction models.

M E T H O D S

Data included in this study are He porosity data from 110 wells from the Central Graben, south- ern and northern Viking Graben and Hal tenban- ken areas (Fig. 1), mineralogical data from X R D , and pressure data from Repea ted Format ion Tests (RFT) and Drill Stem Tests (DST), sup- plied by data from pressure composite plots. The effect of varying integrated t ime-temperature parameters is tested using burial curves based on stratigraphical data from well complet ion reports and by assuming linear geothermal gradients. The possible effect of H C emplacement is evaluated by cross plots of log-porosity vs. log-derived water-saturations.

Porosi ty

The He porosity data are frequently available from three or four plugs for each metre of core.

K. BjCrlykke

Within a single sandstone bed/interval , a wide scatter of porosity is normally obtained due to variations in mud content and carbonate cemen- tation. The arithmetic mean of all measurements is biased towards values which are too low as they normally also include data from non-producing parts of the reservoir. Instead of introducing a cut-off value, the porosity of the net reservoir, or the best part of the reservoir, can be described by the 75-percentile porosity (see Bjr et al..

1989, Fig. 7). Since this value expresses the porosity where 25% of the values are higher, it is not influenced by the lowest values.

Pressure

Compact ion of sandstones is driven by the net stess (eqn. 1) induced by the overburden load.

s e = p g h - p (1)

Se is the net stress, p the average density of overlying sediment column, g the acceleration due to gravity, h the overburden thickness and p the pore-pressure. In hydrostatically pressured sandstones, the net stress increases with burial depth. In some cases, however , the net stress

Northern Viking Graben \ . = _

: l l n

ux. :-~

r; / :

U X . , �9

m o i1 �9

r 4 s

~ l e gl �9

i

Southem Viking Graben

8W 4W o 4E 8E 12E 164E 2OrE 24E: 28E 32E 3,6E

'='I" " ,~a ," ' "st" s~-~,,, " ",~" B Brage SW Sleipner West Dr Draugen Tr T r o l l G Gulffaks Tre Trestakk �9 GG Gulffaks-y GS Gugfaks South

ZN

t N

IN

IN

y T y t i h a n s U l a

Gu Gudrunn VF Vestefrikk Gy Gyda He Heidrun Hi Hild Hu Huldra M Mime Mid Midgaard M!k Mikkel Nj Njord

SE Slelpner E a s t Sn Snorre SS Sine rbukk South

FIr. 1. Location map of hydrocarbon provinces.

Haltenbanken 7

- r,o

"r : F ' \ .

' I

NIIT/-

Central Graben

Porosity depth trends in sandstones

within the reservoir is strongly reduced by high formation pressures, and this may influence the rate of compaction. Pore-pressure data are taken from R F T or D S T where these have been available. For the remaining units, pore-pressure graphs have been consulted in order to est imate the pore-pressure. In cases where R F F or DST data were available from other units within the same well and the mud-weight data do not indicate any major shift in pore-pressure, little error is thought to be introduced. In o ther cases, the pore-pressure was approximated by correla- tion to neighbouring wells by consulting the mud- weight data in order to reveal local pressure deviations. The pressure has been recalculated in terms of mud-weight equivalents (i.e. g/cm3).

Mineralogy

Mineralogical data from --600 sandstone sam- ples were obtained by X-ray diffraction. Ran- domly orientated powdered bulk-rock samples were run on a Philips PM t700/1710 diffracto- meter with Cu-K0~ radiation. The diffractometer was connected to a PDP 11/53 computer with Philips A P D 1700 software. Peak intensities used for estimating mineral contents were those from the diffracted intensity (DI) listings given by the A P D software. Quant i ta t ive estimates of mineral composit ion were based on correlat ion to a series of external standards. Intensities of characteristic peaks for the most abundant minerals were converted to mineral contents using intensity factors presented by R a m m (1991). In addition to mineral contents, a clay index was estimated from the peak intensity of the 19.8~ peak (i.e. the 020 peak of illite and kaolinite plus the 02,11 peak of

477

chlorite) relative to the diagnostic quartz and feldspar peaks. Hence , the clay index represents an estimate of the relative content of clay minerals to stable f ramework grains. Assumed relative uncertainties in estimates of individual mineral phases are of the order 5 -20%, lowest for quartz and carbonates, and largest for individual clay minerals.

Time-temperature index (TTI )

Porosity modification subsequent to pure mechanical compaction occurs by processes involving chemical alterations. The rate of chemi- cal reactions are temperature dependent and mechanical compact ion is probably to some extent also temperature controlled (Gaut ier & Schmoker , 1989; Bloch, 1991). The Lopatin equation (Waples, 1980) is used to est imate a t ime- temperature index in order to test the effect of varying burial history on porosity/depth trends. The input for estimating TTIs is decompacted burial curves obtained from stratigraphical data in the well complet ion reports. The decompact ion is per formed using the solidity vs. depth function introduced by Baldwin & Butler (1985). The temperature history implied assumes constant linear depth profiles below the sea floor: T = 5~ + 13-Z, where [5 is the temperature gradients from Table 1.

P O R O S I T Y vs. D E P T H T R E N D S

Porosity data from about 260 cored sandstone units from 110 wells from the North Sea and Hal tenbanken area are plotted in Fig. 2. Each

Table 1. Temperature data from the Norwegian shelf. The gradients assume 5~ at the sea floor and linear gradients vs. depth. References are (A) Spencer et al. (1987); (B) Ehrenberg (1990).

Depth Temp. 13 Depth Temp. 13 Field m RSF ~ ~ Ref Field m RSF ~ ~ Ref

Ula 3400 143 4l A Agat 3400 118 33 A Mime 4050 161 39 A Tyrihans 3360 136 39 A/B Sleipner ~ 2400 93 37 A Midgaard 2100 91 41 A/B Sleipner V 3500 123 34 A Mikkel 3900 156 39 B Gullfaks 1720 72 39 A Trestakk 3700 148 39 B Oseberg 2400 99 41 A Smcrbukk 3700 143 37 B Troll 1450 68 43 A SmCrbukk S 3500 140 39 B Brage 2300 98 40 A Heidrun 2080 85 39 B

478 M. Ramm and

40

3O

g

o_

R

o 8

~ _ o~6q oo oO o ~ . ~ .~

. - _ 7 ~ ~ ~ ~ �9 �9 �9 o ~ o o

�9 o

�9 $

~J = 42.7 * 0 0069 x z �9 �9 . �9

I . . . . . . . . . . . . I ~ , "

i ooo 2ooo 3~o 4ooo sooo

D e p t h { m RSF)

FIG. 2. Porosity/depth data from Norwegian shelf sand- stones. The data points are mean depth vs. 75-percentile porosity from all He porosity measurements obtained within defined sandstone units (i.e. Formation, Member or individual sandstone bodies with homogeneous porosity distribution). Units with <15 measurements are excluded. Filled circles represent non-reservoir units comprising muddy sandstones or siltstones with low permeability. The line represents the least square regression lines for the

arenitic units.

unit represents a formal lithostratigraphic forma- t ion/member , where these show uniformly distri- buted porosity, or a sandstone unit within a format ion/member where more units (thicker than 10 m) with uniform porosity occur. The linear regression line based on all 260 units has an explained variance, r 2 = 0.62. The scatter about the line is large and porosity values varying from < 5 % to 25% are observed at depths near 4 km. The inverse linear correlation between porosity and depth is improved when units classified as texturally immature greywackes and silty sand- stones are excluded. The regression line: ~ = 42.7-6.9-Z (Z is depth in km) has an explained variance of 0.73. However , the spread about the regression line is still significant, particularly at depths >4 km, where the porosity varies from --10-25%.

The porosity/depth functions are influenced by several factors and, statistically, it is difficult to isolate the effect of single factors and processes. In the following, the porosity data will be analysed in terms of the different Norwegian shelf sub-basins, pore-pressure, subsidence history, oil/gas saturation and mineralogy.

The sandstones from the Central Graben (Fig. 3A), representing Triassic and Middle Jurassic fluvial and Upper Jurassic open marine sandstones, show an apparent 'shotgun' pattern. The linear porosity/depth trend estimated from

K. Bjarlykke

the sub-data set has a very low explained variance (r 2 = 0.19). Obviously, parameters other than depth are very important in explaining the porosity variation. A separate study of these sandstones (Ramm & Forsberg, 1991) has docu- mented that much of the porosity variation is due to variations in the mechanical stability of the sandstones. Fur thermore , some clean sand- stones, characterized by a ubiquitous microcrys- talline quartz coating, are only affected to a small extent by pressure solution and quartz cemen- tation. These sandstones are among those deviat- ing most from the general trend shown in Fig. 2.

The porosity data from the southern Viking Graben are from the Tertiary Heimdal Forma- tion, the Upper Jurassic Int ra-Draupne sand- stone, the Middle Jurassic Vestland Group and the Triassic Skagerak Formation. Most porosity data from this area cluster along the regression line. Substantially higher porosity is found in deep marine sandstones within the Draupne Format ion in the Gudrun Field (24% at 4.3 km).

The data set from the northern Viking Graben (Fig. 3C) comprises porosity data from the Jurassic Statfjord Format ion and the Dunlin and Brent Groups in the Gullfaks, Hild, Brag�9 Veslefrikk, Oseberg and Huldra Fields. Further- more, data from the Upper Jurassic Viking Group in the Troll Field and the Cretaceous Cromer Knoll Group in the Agat Field are included. The best-fit porosity/depth trend is rather parallel to the overall regression line from Fig. 2. Higher than average porosity occurs in the shallowly buried Gullfaks Field (1.6-1.8 km) and in the more deeply buried Hild, Huldra and Gul l faks-Gamma Fields (3.6-4.1 km). All these reservoirs are characterized by very high pore- pressures. In some wells from blocks 30/3 and 30/6 (i.e. Veslefrikk and Oseberg Fields), where the pore-pressure is approximately hydrostatic, the Brent Group sandstones have porosity values between 5 and 11% lower than the regression line.

The data set from the Hal tenbanken area (Fig. 3D) includes porosity data from the Lower Jurassic Bht Group, the Middle Jurassic Fangst Group and the Uppe r Jurassic Rogn Formation. The sandstones differ from those from the North Sea by being more porous at shallow to interme- diate burial depths (2-3 km). At burial below 3.5- 4 km, however , they have rather low porosity (< 15%). Hence, the best-fit linear porosity/depth

Porosity depth trends in sandstones 479

40

30

20

10

30

20

10

. . <entra I Oraben . . , o o,1.

o ~ ~ o ' ~

I ~ = 4 0 8 . 0 0 0 5 9 x Z I " e ~ 1 7 6 N = 3 0 r2 "0192 e b" A

i i

o a o �9 " �9 ~ Northern Viking Graben

�9 - . " ? ~ � 9 ~~ �9 �9 o o o ~ ~

" " C ~ 41 8 - 0.0066 X Z �9

N = 84, ,-2 = 0.670

, , , , , Southern Viking Graben

g : ~ ee O

~

J " ' 1 3 ~ = 4 0 2 - 0 0 0 6 1 x Z �9 �9

N - 22, r 2 = 0.749

, ' ~o0 o0 Haltenbanken Area

.~. ~" .,~,, o

= 46 .4 o ~ == '= ~ = o

N " 59 r 2 "0857 J �9 D 0

0 1~0 2~0 3~0 4~0 5~0 l~O 2~0 3~0 4~0 5~0

Depth (m RSF) FIG. 3. Porosity/depth data from sandstone units from: (A) the Central Graben; (B) the southern Viking Graben; (C) the northern Viking Graben; and (D) the Haltenbanken area. Solid lines are least square regression lines for the particular regions. The dashed line is the best-fit line for all arenitic units (e.g. solid line in Fig. 2). Deeply buried sandstones from the Central Graben show a poor correlation with depth and are in general more porous than sandstones in the other areas. Higher than average porosity sandstones in the southern and northern Viking Graben mainly represent highly over- pressured reservoirs. The data from the Haltenbanken area indicate a steeper gradient than the data from the other areas.

t rend is s teeper and has a h igher in te rcep t than the t rends es t imated f rom the Nor th Sea data. Most porosi ty data plot fairly close to the t rend and the expla ined var iance is high (r 2 = 0.86). Dev ia t ion f rom the t rend is observed for sand- s tones f rom block 6407/7, where the porosi ty is 3 - 11% lower than the bes t fit t rend. In the G a r n Fo rma t ion , which is the main reservoi r uni t in this area, var ia t ions in porosi ty are, according to E h r e n b e r g (1990), caused mainly by the com- b ined effects of compac t ion and quar tz cemen- tat ion. E h r e n b e r g showed tha t many of the low porosi ty sands tones at bur ia l depths >3 .5 km have been subjected to extensive chemical com- pact ion and conta in 10-20% quar tz cement , der ived mainly f rom in te rg ranu la r pressure solu- t ion and pressure solution a long stylolites. Sub- stantially h igher than normal porosi ty is observed at --4.5 km in the Tilje F o r m a t i o n in Well 6506/ 12-7, in the SmCrbukk Field. In the Tilje Forma- t ion in this area , anomalous high porosi ty has

previously been found to be p rese rved by grain- coat ing chlori te (Eh renbe rg , 1993).

P O R E - P R E S S U R E

Porosi ty /depths t rends in sands tones f rom differ- en t pore-pressure domains are shown in Fig. 4. A n overal l t rend of increasing porosi ty with increasing pore-pressure is apparen t . The regres- sion line for the sands tones within in te rmedia te pore-pressure zones (1.3-1.59 g/cm 3) deviates f rom the o the r t rends by having a lower porosi ty/ dep th gradient . This is due to the high porosi ty in some sands tones in the Centra l G r a b e n (Fig. 3A) , and, except for these sands tones , the inter- media te ly over -pressured sands tones are not significantly more porous than those with nea r hydros ta t ic pore-pressure ( < 1 . 3 g/c3). While the porosi ty is only slightly h igher in the inter- media te ly over -pressured sands tones than in the

48O

4O

30

20

10

30

20

10

M. Ramm and K. BjOrlykke

o o% PG 1.00-1.29 g/cm 3

o o

�9 ; ~ ~ o �9

me �9 ~ o

�9 ~ "o;: (~ : 44.4- 0.0000 x Z I , �9 �9 N = 107, r ' = 0,833 I �9 .A

~ PG 1.60-1.80 g/cm 3

t : . o ~) = 46.5 - 0.0076 x Z I

N : 28 , r 2 : 0.880 (~.v

PG 1.30-1.59 g/cm 3

~ * ~ o ~ o o

�9 �9

Q = 36.8 - 0.0082 X z. I N = 4 6 r : . 0 3 7 4 ] ~ " B

0 0 1~0 2~0 3~0 4~0 5~0

, , , , , , PG > 1.10 g/cm 3

~ ~ ~176 o

- 59.3 - 0.0100 X Z [ I

N, : 14, r 2 : 0.616 D I

1000 2000 3000 4000 5000

Depth (m RSF) FI~. 4. Porosity vs. depth trends for sandstones situated within different pore-pressures: (A) near hydrostatic pore- pressure; (B) intermediately over-pressured sandstones; (C) highly over-pressured sandstone; and (D) very highly over- pressured sandstones. Solid lines are regression lines for the sandstones from the actual pore-pressure domains. The dashed lines represents the best fit from all arenitic units (e.g. solid line in Fig. 2). The porosity increases with increasing pore- pressure. The low porosity/depth gradient in the intermediately over-pressured sandstones is caused by the high porosity in some Central Graben sandstones. The porosity seems to be scarcely affected when the pore-pressure is increased from 1.0

to 1.6 g/era 3, but it increases significantly as the pore-pressure approaches the lithostatic pressure gradient.

low over -pressured sands tones , those s i tuated within high (1.6-1.8 g/cm 3) and very high ( > 1 . 8 g/cm 3) pore-pressure zones statistically have sig- nificantly h igher porosity. Hence , the porosi ty seems to increase non l inearly with increasing pore-pressure . A non- l inea r regression model was tes ted and the resul t ing regression model : qb = 42 .3 -7 .2 .Z 1~ + 0 .37 .PC 4"4 ( P G is pressure gradient in g/cm3), has an expla ined var iance, r 2 = 0.77. This model shows a significantly improved corre la t ion and seems reasonab le when compared to the t rends shown in Fig. 4. The non- l inear regress ion model does not , however , consider the possibility tha t the effect of pore- pressure varies with depth .

T I M E / T E M P E R A T U R E

The t e m p e r a t u r e gradients along the Norwegian Con t inen ta l Shelf are r a the r un i form (e.g.

Table 1). However , due to differences in buria l history, sands tones at similar burial depths may have reached different levels of 'd iagenet ic matur i ty ' . Character is t ic compac t ion cor rec ted burial histories in wells f rom the n o r t h e r n Nor th Sea and the H a l t e n b a n k e n area are shown in Figs. 5A,B. All s tudied sands tones are, at p resen t , at or close to the i r max imum burial depth , a l though Neogene uplift has occurred close to the coast of wes te rn Norway (i.e. nea r the Troll Field). In most cases, however , the sands tones have exper- ienced a per iod of rapid La te Pl iocene and Ple is tocene burial . This applies part icular ly to the H a l t e n b a n k e n area where rapid deposi t ion of glacial der ived detr i tus has given a subs idense of --1 km across the area dur ing the last 3 my. Calcula ted TTIs f rom 30 wells are p lo t ted against dep th in Fig. 6. The TTI values f rom the di f ferent regions indicate a log-l inear increase with increas- ing dep th , and the H a l t e n b a n k e n sands tones

200

Porosity depth trends in sandstones

Time (Ma) 200 150 100 50

Time ( M a ) ~ 150 tO0

1 Or) rr

2 e -

a 3

"e-

m

4

481

0 - o

K" . 1 h-

E v

f..

q)

,3 a ,,..

m 4

Fro. 5. Subsidence curves for reservoir sandstones from: (A) northern Viking Graben; and (B) Haltenbanken. These sandstones are now at about their maximum burial and have experienced a rapid late Pliocene to Pleistocene burial. The Haltenbanken sandstones differ from those from the Viking Graben by a larger late Pliocene to Pleistocene subsidence.

10000

1000

100

10

1

0,1

J Northern North Sea ~,,,,'~~ ~ . s

17-1 = 10 ('1 i . ~.o~11 x~ ~+.~/-

1000 2000 3000 4000 50(]4)

Depth (m RS~ Fro. 6. TTI vs. depth of sandstones from 30 wells from the northern Viking Graben ((2)) and Haltenbanken area (+). The TTI increases approximately log-linearly with depth and the slope is about constant in the two areas. At a particular depth, sandstones from the Haltenbanken Area have a lower TTI than those from the northern Viking Graben, and this difference is due to differential burial history as indicated in Fig. 5. The difference between the TTI in the northern Viking Graben and in the Haltenban- ken sandstones is shown to correspond to -550 m differential burial depth. The TI'I from the different regions follow general trends that can be expressed as: log(TFI) = A + 0.0011-Z where A is -1.8, -2.2 and -2.4 in the northern Viking Graben, Central Graben and Haltenban-

ken/southern Viking Graben, respectively.

have, at a par t icular dep th , a significantly lower TI ' I than the n o r t h e r n Nor th Sea sandstones . O n average, the difference cor responds to a differ-

ence in burial dep th of --550 m. The sands tones f rom the Centra l G r a b e n have TTIs in t e rmed ia t e be tween the e n d - m e m b e r s f rom the H a l t e n b a n - ken and no r the rn Viking G r a b e n sands tones , while the sou thern Viking G r a b e n sands tones plot with the H a l t e n b a n k e n sandstones .

The t rends indica ted in Fig. 6 are used to approx imate TI ' I values for the o the r sands tones f rom the different regions, and in Fig. 7A the porosit ies in sands tones having pore-pressure gradients < 1.4 g/cm 3 are p lo t ted against TI ' I . The porosi ty data may be fitted with a l inear best-fit line, with a high expla ined var iance (r 2 = 0.82). Closer inspect ion of the porosi ty vs. T] ' I distribu- t ion indicates , however , tha t a non- l inear approx- imat ion may fit the data even more closely. The data f rom Fig. 7A are in Fig. 7B p lo t ted against an envisaged dep th axis es t imated f rom t h e TTI vs.

dep th t rend in the no r the rn Viking G r a b e n sands tones (Z ' = 1.6 + 0.91 - log (TTI)) . The porosi ty data seem to follow an exponent ia l poros i ty /depth t rend pr ior to a ' no rmal i zed ' burial equal to 2.5 kin, af ter which the gradient changes and the porosi ty in mos t sands tones undergoes a per iod of increased reduct ion dur ing burial to 3 .5-4 km. A few of the deeply bur i ed sands tones , however , plot nea r the con t inua t ion of the exponent ia l t rend , indicat ing tha t these sand- s tones have not been subjected to the same per iod of increased porosi ty reduc t ion as the others .

482 M. R a m m and K. BjOrlykke

40

&"

20

e-

0

0,01

40

v 30

o 20 O.

~I 10 i n

0 0

OO = o ~ * ~ ~ I Jv ., ,~,=0.,= I

o o + ~ . ^~_

o Halenbanlen Area o " �9 ~ ~ . ~ : ~ o

�9 N VIIdng Graben

§ SVNng Graben

A

0,1 1 10 100 1000

TTI

' ~ + I ..... I I) = 2s- o.o13 x (z-=soo) I

~ ~ � 9 1 4 9 1 4 9 � 9 1 4 9

B

1000 2000 3000 4000 5000

N o r m a l i z e d B u r i a l D e p t h ( m R S F )

FIG. 7. (A) Porosity vs. TTI; (B) porosity vs. 'thermal maturity corrected depth' of normal and low over-pressured (PG <

1.4 g/cm 3) sandstones. For those sandstones where burial curves were not available, TTI were estimated from the general TI'I trends vs. depth trends indicated in Fig. 6. The 'thermal maturity corrected depth' is calculated using the northern Viking Graben TTI vs. depth trend (i.e. log (TTI) = -1.8 + 0.0011.Z). The linear trend in (A) and the composite

exponential and linear trends in (B) are discussed in the text.

The trends indicated in Fig. 7B are probably caused by different processes operat ing at differ- ent burial depths (Ramm, 1992). At depths <2.5 km, porosity loss is mainly by mechanical com- paction. At this stage, the porosity/depth gradient may be considered a function of existing porosity and achieves an exponential form (Wood, 1989). Below 2.5 km, porosity loss by pressure solution and quartz cementat ion normally becomes more important , and these processes may effectively reduce the porosity towards 10% during subse- quent burial (Ramm, 1992). The actual depth where the increase in the porosity/depth gradients occurs may vary with burial history and tempera- ture gradients. Thus, case studies from the southern Viking Graben (Ramm & Bang, 1991) and the Cod Terrace area (Ramm & Forsberg, 1991) indicate increased porosity/depth gradients at 3.0 km and 2.8 km, respectively.

H Y D R O C A R B O N S A T U R A T I O N

Log-derived average porosity in the net reservoir sandstones from different formations in the North Viking Graben are plotted against water satu- ration in Fig. 8. Linear regression statistics on the porosity vs. water-saturation, porosity vs . depth and porosity vs . depth and water saturation are also indicated. In four of the five data sets, a slightly decreasing porosity with increasing water saturation is indicated, supporting the hypothesis that porosity reduction is re tarded by hydro- carbon emplacement . In the Ness Formation, however, the porosity is shown to increase with increasing water saturation. Fur thermore , in the Etive Formation, the explained variance is not improved by including the water saturation in the porosity vs. depth model. The dependence of porosity on water saturation is most pronounced

Porosi ty depth trends in sandstones 483

40

30

20

10

30

20

10

30

20

Tarbert Formation

. . . . : . : . . � 9

l o l

~) �9 20 .3 - 0 .032 X Sw, r 2 �9 0 .063

(~ = 32 .0 - 0 .0048 x Z , r z = 0 ,413 A ~) = 34 ,3 - 0 .004g x Z - 0 .038 X S w, r a = 0 .503

J i

Etive Formation

�9 �9 �9 % �9 : � 9

0 = 21 .2 - 0 ,032 x Sw, r 2 = 0 .048

0 = 36 .4 - 0.0061 x Z , . . ~ = 0 .626

�9 36 .5 - 0 ,0060 x Z - 0 .019 x S w, r ~ �9 0 ,625 C

Statfjord Formation

Ness Formation

�9 ~ o �9

: 18.5 + 0 .018 x S,,,, ~ = 0 .023

(~ : 33.3 - 0.00S0 X Z, ~ = 0 .809

~) = 32 .2 - 0 ,0050 X Z- + 0 ,017 X S w, t ~ �9 0.(;28

�9 ~ ~ �9

�9 o o l ~

B

Oseberg Formation

~) = 23 .4 - 0 .052 x S,,, r 2 = 0 .270

= 30.8 - 0 .0044 x Z , # ,, 0 .313

�9 38 .0 - 0.00S0 x Z - 0 .080 x Sw, r : = 0 .665 D 20 40 60 80 lOO

Water Saturation (%)

�9 . . 1

1 0 ~) = 21 .7 - 0 .048 x Sw, r~ = 0.1

�9 El 0 ~ = 3 3 ' 5 " 0 "0044 x 7 - " 0"020 X S * ' r~ = 0"696 i ~ . ]

20 40 60 80 lOO Water Saturat ion (%)

FIG. 8. Log-porosity vs. water saturation in five important reservoir sandstones in the Northern Viking Graben. The porosity and water saturation are averaged over the net-reservoir parts of the formations, i.e. all intervals with higher than cut-off permeability, which is 1 mD in the shallow (oil) reservoirs and 0.1 mD in the deep (gas) reservoirs. Regression statistics on the porosity vs. (1) water saturation, (2) depth and (3) depth and water saturation are also indicated. The solid lines are the porosity vs. water saturation regression lines. In four of the five formations, the porosity increases with increasing hydrocarbon saturation, but in the Ness formation, no such trend is indicated. In the Etive Formation, the

explained variance is not improved by including the water saturation in the porosity-depth model.

in the Oseberg Format ion. In this case, however , several of the wells with high water saturation and low porosity comprise more distal sandstone facies than some of the high porosity wells within the Oseberg Field, and it is not obvious whether the porosity difference is related to differences in

pore-fluid composi t ion or to mineralogical and textural differences. There is also a significant negative correlat ion be tween porosity and water

saturation in the Statfjord Format ion. In this case, all wells with low water saturat ion and high porosity are from the Tampen Spur area where the pore-pressure is higher than in the Horda Platform area. Hence , the present data sets do not provide any conclusive observat ions on the role of hydrocarbon emplacement . Nevertheless , the porosity is obviously more strongly related to depth than water saturat ion and it is likely that

484

porosity reduction continues after oil emplace- ment, maybe at a slightly retarded rate.

M I N E R A L O G Y

The dependence of porosity on bulk mineralogy is tested by estimating correlation coefficients between the deviations from the general porosity vs. depth and pressure gradient t rend (A~ = He-~ - (42.3 - 7.2 - Z + 0.37 pG4"4), where Z is burial depth in km and P G the pressure gradient in g/cm 3) and mineralogical parameters (Table 2).

In the complete data set, there is a negative correlation between A~p and carbonate and clay content, and there is a positive correlation between quartz content and Adp. When samples with more than 10% carbonate are excluded,

M. Ramm and K. Bjcrlykke

similar trends are observed, and the correlation between A~ and clay content becomes more pronounced. Hence , the porosity/depth distribu- tion is related to the clay content and clay-rich sandstones lose porosity more rapidly during burial than clean sandstones. Nevertheless , the correlation between the clay content and porosity is rather complex and the following three obser- vations should be noticed: (1) The correlation between the Clay Index and porosity is stronger at shallow (<3.5 km) than at deep burial (>3.5 km); (2) at depth greater than 3.5 kin, there is no systematic correlation be tween clay content and porosity in clean sandstones (C1 < 0.2); and (3) at deep burial, there is a positive correlat ion between quartz content and porosity in muddy sandstones (C1 > 0.2), but a negative correlation

Table 2. Correlation coefficients between mineralogical parameters from XRD and A~, i.e. the difference between the He porosity and the non-linear porosity vs. depth and pore-pressure trend (A~ = q) - (42.3 - 7.2.Z l-~ + 0.37.pG44)). Mineralogical parameters are the total contents of the individual clay minerals (chlorite, micas and kaolinite), quartz and total content of carbonates (i.e. calcite + dolomite + ankerite + siderite), and also the relative peak intensities of feldspar (3.24 lk and 3.19 tk) and total clay (4.5 A) to quartz (4.26 A). The clay index (CI) is the content of total clays relative to the

total content of stable framework grains (i.e. quartz + feldspar). Numbers in parentheses are the significance levels.

Total Carbonate <10%

CI > 0.2 CI < 0.2

All All Z < 3500 Z > 3500 All Z < 3500 Z > 3500 All Z < 3500 Z > 3500 N=590 N=497 N=259 N=238 N=158 N=65 N=93 N=339 N=194 N=145

Depth 0.048 -0.094 -0.031 -0.171 -0.016 0.110 0.167 -0.114 -0.082 -0.235 (.25) (.037) (.62) ( .0084) (.040) (.38) (.11) (.036) ( . 0 2 6 ) (.0044)

Chlorite -0.103 -0.193 -0.363 -0.071 -0.169 -0.232 -0.079 0.014 0.072 0.008 (.012) (.00013) ( .0000) (.28) (.034) (.063) (.45) (.80) (.32) (.92)

Illite -0.194 -0.319 -0.404 -0.245 -0.147 -0.18l -0.107 0.001 -0.060 -0.071 (.0000) ( .0000) ( .0000) (.0000) (-.066) (.15) (.31) (.99) (.32) (.40)

Kaolinite -0.241 -0.350 -0.407 -0.340 -0.340 -0.289 -0.516 0.087 0.032 0.133 (.0000) ( .0000) ( .0000) ( .0000) ( .0000) ( . 020 ) (.0000) (. 11) (.66) (. 11)

Quartz 0.429 0.217 0.285 0.187 0.346 0.238 0.417 -0.199 -0.092 -0.303 (.0000) ( .0000) ( .0000) ( .0038) ( .0000) ( . 057 ) ( .0000) ( .0002) (.20) (.0002)

K-feldspar/ -0.002 0.123 0.063 0.173 0.029 0.138 0.037 0.178 0.125 0.258 Quartz (.96) ( .0006) ( . 3 1 ) ( .0076) (.72) (.27) (.72) (.001) (.82) (.002)

Albite/ -0.013 0.027 0.027 0.027 0.027 -0.138 -0.022 0.153 0.114 0.203 Quartz (.75) (.54) (.72) (.60) (.42) (.27) ( .84) ( .0047) (.11) (.015)

Total -0.617 -0.188 -0.190 -0.190 -0.175 -0.160 -0.222 0.010 -0.047 0.068 Carbonate (.0000) ( .0000) ( .0037) ( .0000) (.028) (.20) (.33) (.85) (.51) (.42)

Clay/ -0.494 -0.590 -0.425 -0.462 -0.553 -0.435 -0.432 -0.062 -0.163 0.068 Quartz ( .0000) ( .0000) ( .0000) ( .0000) ( .0000) ( .0000) ( .0000) (.25) (.023) (.42)

Clay -0.374 -0.526 -0.618 -0.461 -0.507 -0.605 -0.483 -0.125 -0.228 0.006 index (.0000) ( .0000) ( .0000) ( .0000) ( .0000) ( .0000) ( .0000) ( . 0 2 2 ) ( .0014) (.95)

Porosity depth trends in sandstones

between quartz content and porosity in clean sandstones (CI < 0.2). Fur thermore , it is noted 40 that at burial depths greater than 3.5 kin, the 7~0 porosity in the clean sandstones (CI < 0.2) decreases more rapidly than the porosity in ~ 2o muddy sandstones (e.g. correlation coefficients ,0 between Aep and depth, Table 2).

A qualitative model for the dependence of o - porosity on depth and clay content may be defined according to these observations. During burial towards 2.5-3.0 km, porosity reduction occurs mainly by mechanical compaction (Ramm, 1992) and the porosity/depth trend depends on the clay to stable f ramework grain ratio. Below 2.5 kin, clean sandstones lose porosity by chemi- cal compaction while sandstones with CI > 0.1 are less affected by chemical compaction as the clays prevent quartz precipitation. Quanti tat ively, these relations are tested with an exponential regression model (eqn. 2)

r = A - e - (~ '+( l~c0z) (2)

3where A, cr and 13 are regression coefficients. A is related to the initial porosity at zero burial depth, er is a f ramework grain stability factor for clean sandstones (CI = 0) and [5 is a factor describing the sensitivity towards increasing clay index. The regression analyses were run on a reduced sample set f rom which sandstones with pressure gradients > 1.5 g/cm 3, samples with more than 10% carbonate and samples with C1 < 0.1 at 'norma- lized' burial depths greater than 2.5 km were excluded. The resulting exponential porosity/ depth trends are illustrated in Fig. 9. Most of the deeply buried sandstones with C1 lower than 0.1 plot near a linear porosity/depth trend. This trend intercepts with the CI < 0.1 exponential trend at about 2.5 km and has a slope ~ - 1 3 % / k m . Thus, eqns. 3a and 3b represent a close approximation to the overall effect of depth and mineralogy on the porosity/depth distribution.

= 45 - e ( (0.23 + (0.27. Cl)) . Z

C l > 0 . 1 o r Z < 2 . 5 (3a)

dO = 2 5 - 1 3 . ( Z - 2 . 5 )

C l < 0 . 1 a n d Z < 2 . 5 (3b)

Some deeply buried, low overpressured, high porosity sandstones represent major exceptions from the model. In the present data set, all such deviant samples have, by studies using a scanning electron microscope, been found to represent

485

1000 200o 3o00 ~o00 50G0

Normalized Burial Depth (mRSF)

Fro. 9. Model on the effect of varying mineralogy in low overpressure reservoirs (e.g. eqn. 3a and 3b). During shallow burial, porosity loss is mainly due to mechanical compaction (approximated by eqn. 3a), and the porosity increase with decreasing clay + mica content (i.e. decreas- ing ductility). At burial below 2.5 km, clean sandstones lose porosity mostly via pressure solution and quartz cemen- tation (approximated by eqn. 3b). The depth axis is 'TTI-

normalized' as in Fig. 7B.

sandstones containing well developed coatings on mineral grains (i.e. chlorite or microcrystalline quartz coatings). These samples are scarcely affected by chemical compaction and quartz cementat ion, and the general porosity-depth model defined above apparently does not apply in such cases. When these samples were excluded from the data set, a good fit was found between observed values and the porosity values calcu- lated as by eqn. 3 (r 2 = 0.71).

P O R O S I T Y M O D E L

A composite coupled mechanical-chemical com- paction model is compared with the empirical data in Fig. 10. The exponential trends are represented by eqn. 3a, and the linear t rend at burial >2 .5 km is represented by eqn. 3b. The data indicate that the porosity distribution in the sandstones from the Hal tenbanken area is similar to the distribution in the Viking Graben sand- stones, although offset by a depth range similar to the difference in thermal maturity (e.g. Fig. 6). According to the model , low porosity (<10%) must be expected in reservoir sandstones buried beneath 4 kin, but preservat ion of porosity may be facilitated by: (1) very high pore-pressures; (2) short residence t ime at deep burial (i.e. rapid late Neogene subsidence); or (3) petrographic factors unfavourable to quartz precipitation. Quartz cementat ion may be hindered by intergranular clays or diagenetic coatings on the detrital quartz

486

4O

v

(/) o o

o_ 20 r--

�9 10 LO

40

v . _ ~ 3 0 (/} o. o ~. 2O

L) (3.

�9 10 Lo

0

4O

v

if) 9 o

O. 20 . J

0 O.

10 r~

M. Ramm and K. BjCrlykke

"'"""'~,."" ' o

�9 e " - e

A I " ~ . . . . .

0 1000 2000 ~ 4000

Burial Depth (mRSF)

' " ~ 0 ~ 0

...,o, ..... .~

B " ~ ...............

0 1000 2OOO 3000 4OOO 500O Normalized Burial Depth (mRSF)

"",,._ ~ ' ~ o O o + ~o,, ,,,"- , ~ - - ~ - + ~ + + ~ o

0 PG: 1.60-1.80 " ' ' " ' - . . . , = ~

~ ~ " ............ ..'~.... [] o

C ~ ~ "K . . . . . . . . . . . . .

0 1000 2000 3000 4000 5000 Burial Depth (mRSF)

FIG. 10. Coupled mechanical and chemical compaction model related to the present data set, Exponential lines refer to equation 3a and represent porosity loss by mechanical compaction of clean (CI = 0; solid lines) and wacky (CI = 1; dashed lines) sandstones. The straight lines represent the porosity loss by chemical compaction. (A) Arenitic sandstones from Haltenbanken (O), Northern Viking Graben ([]) and muddy sandstones (O). All data represent low overpressure (< 1.4 g/cm 3) reservoirs. (B) same as (A) plotted against a normalized depth axis (i.e. the burial depth for the Haltenbanken sandstones is reduced by 550 m, e.g. Fig. 7B). (C) porosity depth data for over-pressured sandstones and sandstones with

chlorite or microcrystalline quartz coatings on detrital grains.

Porosity depth trends in sandstones

grains. However, the presence of both detrital and authigenic intergranular clays reduces per- meability, and diagenetic grain coatings are probably the most important petrographic factor that may preserve good reservoir quality to deep burial (i.e. Ramm & Forsberg, 1991; Ehrenberg, 1993).

D I S C U S S I O N

About 70% of the variance of the porosity distribution in the subsurface can be related to burial depth alone. The depth dependence reflects the fact that increasing burial depth is generally followed by increasing effective stress and increasing temperature. However, variations in geothermal and pore-pressure gradients also influence the effective stress and temperature at a given depth. Porosity loss by both mechanical and by chemical compaction will be retarded by high pore-pressures, but theoretical modelling of pres- sure solution indicates that compaction is little retarded at low to intermediate over-pressure (Ramm, 1992). This is consistent with the present observations as the effect of pore-pressure is shown to increase exponentially as the pore- pressure approaches the lithostatic pressure gradient. At deep burial, highly overpressured sandstones seem to be significantly more porous than normally pressured sandstones (Fig. 10C). Similar observations were made in a study of the Garn Formation in the Haltenbanken Area (Ehrenberg, 1990). In this case, sandstones with pore-pressures corresponding to pressure gradi- ents ~>16.9 MPa/km were found to have porosity values higher than sandstones with near hydro- static pore-pressure.

Differences between the porosity/depth trend in the Viking Graben and the Haltenbanken area correlate with differences in burial history and TTI. If there is a causal relationship, it follows that the rate of porosity reduction is time- and/or temperature-dependent. The stress within the grain framework, which is the driving force for the compaction of sandstones, increases linearly with depth as long as the pore-pressure gradient remains constant. However, Gautier & Schmoker (1989) argued that the effect of increased stress is controlled by temperature as chemical bonds, grain ductility and pressure solution/precipitation processes are temperature dependent. Tempera-

487

ture dependent pressure solution on grain edges may be a part of the shallow to intermediate compaction of the sandstones, and may be necessary in order to allow re-orientation of brittle grains (FUchtbauer, 1967). Hence, Gautier & Schmoker (1989) and Bloch (1991) stressed that porosity depends on thermal maturity rather than burial depth. Mechanical compaction following an incremental increase in net stress may occur by instantaneous deformation followed by a period of slower adjustments (i.e. time-dependent primary consolidation) towards an equilibrium porosity. Further changes in porosity are thereafter related to creep deforma- tion of the mineral grains (Jones & Addis, 1985). Hence, the higher porosity in many shallow Haltenbanken sandstones compared to North Sea sandstones may indicate that the Haltenbanken sandstones have not equilibrated after the rapid Neogene subsidence (BjCrlykke et al. , 1989).

The framework grain stability of sandstones depends on their grain composition. The reduc- tion in pore-space caused by plastic deformation of sandstones containing different proportions of ductile and non-ductile grains was addressed by Rittenhouse (1971). Reduction in pore-space of regularly packed sandstones was, after collapse of the ductile grains, shown to be close to a continuous function of the content of ductile grains. If, however, the ductile grains constitute <5% of the grains, then bridging between non- ductile grains may prevent collapse. This theoreti- cal approach, considering geometric packing of spheres, can probably not be directly applied to natural sandstones containing various propor- tions of clay minerals. However, the presence of clays within ductile fragments, pelletoids, and clay strings facilitates mechanical compaction of the sandstones (Ramm, 1992). Authigenic, inter- granular clay minerals may, however, not affect the framework stability to the same degree. Table 2 indicates a strong negative correlation between clay content and porosity, when the CI is >0.2. In cleaner sandstone (CI < 0.2), however, this dependence is less obvious. Hence, the porosity vs. clay content distribution indicates a similar relationship to that expected from the relations between the content of ductile grains and porosity (Rittenhouse, 1971). A similar result has also been obtained in a closer study of deeply buried sandstones from the Cod Terrace area (Ramm & Forsberg, 1991).

488 M. Ramm and

The unfavourable steep porosity/depth gradi- ent below 2.5-3 km is prominent in many clean sandstones and reflects porosity loss by chemical compaction (Ramm, 1992). More than 15% porosity beneath 4 km therefore requires retarded chemical compaction. Small variations in sandstone composition, such as chert and chlorite coatings on detrital grains, which may not be easily detectable by XRD, may play a major role in the late diagenetic modifications (e.g. Ramm & Forsberg, 1991; Ehrenberg, 1993). In fact, among the deeply buried (>4 km) sand- stones included in the present data set, porosity values >15% were encountered only in sand- stones from highly over-pressured reservoirs and in sandstones with chlorite and/or chert coatings on framework grains.

In this study, porosity is not found to be much higher in hydrocarbon-saturated than in water- saturated reservoirs. In some cases the porosity may be slightly higher above the hydrocarbon- water contacts, but it is not always obvious whether this difference is due to primary differ- ences in sandstone facies and petrology or due to differences in diagenesis caused by early hydro- carbon emplacement. In the cases where the hydrocarbon-filling has continued up to present, the hydrocarbon-water contact may have con- tinuously moved downward, and a clear porosity change should not be expected, even if the hydrocarbon emplacement retards diagenetic processes like quartz cementation.

Porosity loss by mechanical compaction should not be expected to be affected by hydrocarbon emplacement. Furthermore, chemical compac- tion and quartz cementation probably continue after hydrocarbon emplacement as the residual water may provide a sufficient path for diffusion of silica from the points of stress-induced silica dissolution within grain contacts and stylolites towards the sites of quartz precipitation, although the process may be slightly retarded as indicated by Saigal & Bj~rlykke (1992). In this respect, it is worth noticing that petroleum inclusions within quartz overgrowths are commonly observed within the oil zones (e.g. Walderhaug, 1990; Ehrenberg, 1990; Nedkvitne et al., 1993), show- ing that oil emplacement normally does not halt quartz precipitation. Furthermore, in a recent diagenetic study on the permeability distribution in the Trestakk Field in the Haltenbanken area (Ehrenberg & Boassen, 1993), no differences

K. Bjorlykke

were found between the degree of diagenetic alteration, including compaction, quartz precipi- tation and illitization, between a well near the oil- filled crest and a second well located within the water zone. If the assumption is correct, that hydrocarbon emplacement does not strongly affect the porosity evolution, allochemical dia- genetic processes including precipitation or dis- solution from advective pore-water are probably less important than isochemical processes which include compaction and chemical adjustments towards more mechanically and chemically stable fabrics and mineral compositions. Hence, forma- tion of secondary porosity during deep burial, e.g. due to feldspar dissolution during illitization of kaolins, may be important in redistributing the porosity. The total porosity may, however, not be significantly affected as long as dissolved solids are not transported out of or into the sandstone by advective pore-water flow.

C O N C L U S I O N S

( I ) Linear regression of porosity vs. depth may be used as a first approximation during porosity prediction. Variations in pressure, temperature and burial history accounts for some porosity variation but the initial mineralogical and textural composition is probably the most important factor.

(2) At a given burial depth, there is a positive correlation between porosity and pore-pressure. This relation is, however, non-linear and signifi- cantly higher than normal porosities which only occur within highly over-pressured sandstones (pressure gradients >1.6-1.7 g/cm3).

(3) North Sea sandstones have reached a greater 'diagenetic maturity' and lower porosity than Haltenbanken equivalents at the same depth. Different porosity/depth gradients between the two areas may be due to a higher Pliocene/Pleistocene subsidence in the Halten- banken area.

(4) For burial <2.5-3 km, there is a clear negative correlation between porosity and clay content. Mechanical compaction is probably the most important process down to this depth and the porosity gradients therefore depend on the framework grain stability which is mainly a function of the ratio of clays plus micas to stable framework grains.

Porosity depth trends in sandstones

(5) Chemical compact ion and quartz cemen- ta t ion are i m p o r t a n t porosi ty-reducing processes at bur ia l depths > 2 . 5 - 3 km and at great bur ia l depths , the porosi ty in c lean sands tones decreases more rapidly than the porosi ty in more clay-rich sandstones . This can be expla ined by the inhibit- ing effect of quar tz cemen ta t i on by the clay minerals .

(6) Due to the porosi ty loss by chemical compact ion , mos t clean sands tones have < 1 0 - 15% porosi ty at bur ia l dep ths > 4 km. Higher porosi ty is found where chemical compac t ion is r e t a rded by ex t reme pore-pressures or by coat- ings on quar tz grains.

A C K N O W L E D G M E N T

Early parts of the research on which this study is based were conducted when the first author was employed at the University in Oslo, and was supported by Norsk Hydro and by VISTA, a research cooperation between Statoil a.s. and the Norwegian Academy of Science and Letters. Porosity, pore-pressure and stratigraphical data have been provided by the Norwegian Petroleum Directorate, Norsk Hydro and Statoil a.s. Norsk Hydro is gratefully acknowledged for permission to publish the present study.

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