A mathematical model of a geochemical process: Normal paraffin formation from normal fatty acids

Geochimicn

et Coamochimica

Acta, 1967, Vol. 51, pp. 1281 to 1309. Pergamon

Press Ltd.

Printed

in Northern

Irelnnd

A mathematical model of a geochemical process: Normal para

formation from normal fatty acids

KEITH A. KVENVOLDEN* Mobil Oil Corporation, Field Research (Received

29 Novenhber

1966;

accepted

and

DAN Laboratory,

in yevised

WEISER DE&W, form

Texas

9 March

1967)

Abstract-A process by which normel paraffins may be derived from normal fatty acids in sediments is represented by a mathematical model. Coupled differential equations describe changing distributions of fatty acids and paraffis as fatty acids undergo reactions involving decarbosylation and/or hydrogenation (reduct,ion). Model distributions of fatty acids end paraffis for various parameters are observed at different times during the paraffin forming process. The model duplicates many but. not all of the distributional relationships observed in

sediment,s between fatty acids and parafis

especially in the range C,,-C,,.

A MATHEMATICAL model has been developed which describes a possible role of normal fatty acids in the formation of some normal paraffins in sediments. In this model normal fatty acids serve as raw material or precursors for any or all of the following products: (1) 1ower-carbon-numbered fatty acids, (2) lower-carbon-numbered paraffins, (3) paraffins of the same carbon numbers as their precursor fatty acids. The amount of any of the three products obtained from a precursor fatty acid can be adjusted by varying certain parameters within the model. Any fatty acid formed in the transformation process reacts as did its precursor fatty acid. Normal paraffins derived from fatty acids accumulate with paraffins which were already present at the time the transformation process began ; paraffins undergo no reactions. By means of the model, distributions of fatty acids and paraffins can be observed at various times during the paraffi-forming process. The model shows how distributions of fatty acids and paraffins from a modern sediment can change with time to produce distributions of fatty acids and paraffins having some characteristics similar to those observed in ancient sediments. An inversion of the model is used to predict the distribution of normal fatty acids which formed normal paraffins in crude oils. Results coming from the model do not duplicate precisely situations found thys far in nature. However, the model does provide some insight into geochemical relationships between fatty acids and paraffins. Theories on the origin of petroleum must account for all of the numerous comUnderstanding the origin of one or a pounds that compose that complex mixture. family of petroleum compounds provides information for understanding the origin of petroleum itself. One family of compounds common in most petroleum is the normal para& hydrocarbons. These compounds must have resulted from biological precursors if petroleum was ultimately derived from living materials. Two possible precursors are originally deposited, naturally occurring n&ma1 paraffins and normal fatty acids. Naturally occurring paraffins are reasonable sources of paraffins in * Present Moffett Field,

address: California

National 94035.

Aeronautics

and

Space

1281

Administration,

Ames

Research

Center,

1283

I<.

A.

KVENVOLDEX

and D.

WEISER

petroleum because they can become part of petroleum without, any structural change. Fa,tty acids are structurally simila,r to paraffins, but they must undergo certain transformations to become paraffins. These transformations involve loss of oxygen hhrough decsrbosylation or hydrogenation (reduction). The model cleveloped in this palxr considers the effect of these transformation processes on various distributions of fatty acids and paraffins. Input d&a for the model usually are most reliable in the ra.ngc of molecules larger than C,;. This is also the range for which the model appears most al~plicable. Consequently2 in most csamples, this limited n~oloculn~ size range will be considered. In all mathematical moclels cleveloped hcrc COLAcentrations of higher-cxbon-numberccl fatty acids influence concentrations of lower-carbon-numberecl fatty acids a,nd paraffins derived t,hrough the moclcl ; tile reverse is not true. KNOWS' GENERALIZBT~~X~ Sufficient geochemical information has been collected to permit several generalizations concerning occurrences of normal fatty acicls ancl parafins. These are the generalizations which our moclel was designed to satisfy. (1) Norma.1 fatty acids a.re essential components of living organisms (~~ALSTUS. most 1MS) and, consequently, are found everywhere in biological materials; naturally occurring fatty acids have even numbers of carbon atoms in their chains. but odd-carbon-numbered fatty acids hare been found (HILDITCH and WILLIAMS, 1964). Kormal paraffin hydrocarbons also are found in biological materials but usually in low concentrations ; odd-carbon-numbered parafins are much more abundant tha,n even-carbon-numbered paraffins (GERARDE and GERARDE, 1961, 1962). (2) Fatty acids are common constituents of modern and ancient seclimcllts (COOPER. 1962). Paraffins likewise are commonly found in modern and ancient sediments (BRAS and EVANS, 1961). (3) (_‘oncentrations of fatty acids are usually higher in modern than in ancient sediments (COOPER, 1962). In contrast, concentrations of paraffins in modern sediments are usually lower tha,n in ancient sediments (BRAY and EVAKS, 1965). (1) Ratios of even- t.o odd-carbon-numbered fatty acids are higher in modern than .in ancient sediment’s (COOPER, 1962). In some ancient sediments these ratios approach 1.0 but are always larger than 1.0 (I
A mathematical

model pf a geochemical

lx-occss

1283

et ccl., 1963). Ratios of odd- to even-carbon-numbered paraffins near 14 are characteristic of crude oils (BRAY ancl Evass, 1961). A few crude oils have ratios larger than 1.15.

(MARTIN (0.90-1.15)

Normal fatty acicls arc here considered to bc tllc only- source of ne\\.ly generated normal paraffins. dltll0ugl~ other sources of paraffins possible such as from kerogen as suggested by fiBELSOS (196S). t,l 1’s 1. source will not lx cousidered. In the model normal fatty acids may react in three ways : ( 1) they may recluce clirectly to par&ins with the same numbor of carbon atoms, (2) they may lo.je CO, and oxidize to a loner carbon-numbered fatt,y acid. (3) they ma- lose (‘0, and reduce to a loner ca.rbon-ilumbcred paraffin. Thc~ loiver ci~rbon-~lumbcrcd fatty acid procluccd in (2) may react in ang or all of the three suggested ways. These three reaction pathways are the basis of our model. IVarying the inil)ort,aliw of each pathway results in different distributions of both paraffins alld fatty acids. In the model the rate constant of reaction of fatty acids and the fractions degrading are independent of carbon-chain lengths and concentrations. Detailed mechanisms of the chemical reactions that are modeled here are not laou-11. BLUJIER (l965) has shown evidence that pathway (1) takes place in secliments. and C'OOPER (1WZ) has po&~lnted pa,thways (2) and (3). Schematically t,he n~oclel, called the General Xodel. is as follow : i\lT

Carbon Ko. . . . )1 - .j - 1

‘11 - j

lhtty

S,,-j

nritl

. . . S,r-j-l

2

IL-jTl...n Zf-

S,,-j.-l

S,, = concent,ration (pmolslkg) of normal fatty 2-,, = concentration

-1 . ..+

k’.,k

-I,,-,

11 <-

K,%

s;,

(1)

acid nit,11 H carbon at,oms.

(pmols/kg) of normal paraffin with x carbon atoms.

F,, = fatty

acid fraction degrading to next lower carbon numberecl fatt,y acid.

F, = fatty

acid fraction degrading to nest lower carbon numbered paraffin.

F, = fatty- acid fraction reducing to same carbon numberecl paraffin. k = rate constant of fatty acid degrading and reducing. Note that F, + F, + F, = 1. The scheme (1) can be described by differential equations so that the concentration of any normal fatty acid and any normal paraffin can be determined at any sta,ge during fatty acid degradation and reduction. The appropriate differential

K. A.

1284

equations

arising

from

and D.

KVEN-VOLDEN

WEISER

(1) are :

dX _. 2 = -kX,,-, dt

+ F,kX,++,

dY _. 2 = F,kX,-,+l dt Given the concentration (initial fa.tty acid) :

;

j = 1,. . . ) ?z -

+ F+X,-j;

of normal

fatty

j=l,...,n-I

acid for each carbon

(Sb)

1. number

X,(O), X,(O)> - * - >X(O) and the concentration para.ffin) :

(2d) at time

0

(34

of normal paraffin for each carbon number at time 0 (initial I’,(%

Y,(O), * * * > Y,(O)

(3b)

the problem is to find X,(t) and Y,(t) at any time t greater than zero. The solution to equation (2) (Appendix 1) is:

dr,-j(t)

=

2j y [ i=o

Y,(t) = F,.X,(O)[I

+

rT,-j(o);

X,+i-j(0)

1

e-kL;

j = 0,. . . ,n - 1

- e-kl] + Y,(O)

Wf

Cab)

j = 1,. . . , n - 1.

A dimensionless time can be defined as

r = kt and solutions, equations (4), can be expressed in terms of T rather than kt. Here -r is a time, temperature, catalytic factor for these reactions and is more than a fun&ion of real time. With this model (l), or equations (2), if initial fatty acids (3a) and initial paraffins (3b) are given and two independent parameters are specified (for example F, and F,); the amount of fatty acid and paraffin at each carbon number can be determined at any specified dimensionless time (7) from solutions shown in equations (4). A Control Data Corporation 1604 computer was used to calculate solutions to the equations for various values of 7, F, and F,.

A mathematical

modpl of a geochemical

process

1285

ILLUSTRATIONS OF THE GENERAL MODEL This model can be applied to distributions of fatty acids and paraffins in sediments. For this paper distributions of fatty acids and paraffins in sediments were obtained by methods described by KVENVOLDEN (1966c). By examining fatty acids and paraffins in modern sediments with the model, distributions of fatty acids and NORMAL

FATTY

24 22

ACIDS

-r=o

IL

20 18 16 i

23

25

27

carbon

29

31

33

35

number (4

NORMAL

PARAFFINS

10 :

8 a 7

6

=

2

4

0

17

19

21

23

25

27

carbon

29

31

33

35

number (b)

Fig. 1. Normal fatty acids and paraffins. (a) In a modern sediment (San Nicolas Basin) T = 0. (b) Derived from (a) by the General Model.

F, = 0.80, F, = 0.15, Fr = O-05; 7 = 1, 3, 5, 15, 30. paraffins in ancient sediments can be approximated. For example, concentrations of fatty acids and paraEins in two modern sediments, one from the San Nicolas Basin (offshore California) and the other from the Mississippi Delta (KVENVOLDEN , 1966c), have been used in the model. Distributions of fatty acids and paraffins at T = 0, 1, 3, 6, 15, and 30 are shown in Figs. 1 and 2. Normal fatty acids are above; normal paraffins are below. Values of the independent parameters are P, = 060,

F, = 0.05 and F, = 0.15. The figures show that’ distributions of fatty acids change from those having decided preferences of even-ca,rbon-numbered molecules (T = 0) t*o those with smooth di&ibutions of even- and odd-carbon-numbered fatty acids (7 = 3-O). -It the same t4ime the lota,l, absolute concent,ratjions of fatty a,cids diminish. Distributions of para.ffins a!so change. and total, absolut,e concent,rations of these NORMAL

FATTY

ACIDS

a 7 6 ,” $4 E :

5

3 2 1 0

I?

19

21

23

25

27

29

carbon

31

33

35

31

33

35

number

(a) NORMAL

PARAFFINS

4 ," 0 T E 1

3 2 1 0

17

19

21

23

25

27

carbon

29

number (b)

Fig. 3. Sormal fatty acids and paraffins. (a) In a modern sediment (Mississippi Delta) 7 = 0. (b) Derived from (a) by the General Model. F,, = c).Ew. F, = 0.15, F, = 0.0.5; + = 1, 3, 5, 15, 30.

molecules increase. Original distributions of paraffins have a definite dominance of odd-ca,rbon-numbered molecules (T = 0). As more paraffins are derived from fa.tty acids and a.re added to the paraffin mixtures, preferences for odd-carbonnumbered paraffins diminish. Changes in distributions and concentrations of fatty acids and paraffins observed in the model accord with the generalizations stated earlier concerning geochemical occurrences of these molecules. Starting with fa.tty acids and paraffins in modern sediments (T = 0), comparisons can be made between distributions of these molecules obtained from the model and distributions of the same molecules in ancient sediments. Three of the best comparisons are illustrated here. Only the range from C,, to C!,, is shown. For

A mathematical

model of a geochemical

process

1287

molecules smaller than C!,, little relationship has been found between distributions obtained from the San Nicolas Basin and the Mississippi Delta samples by the model and distributions in ancient sediments. This la.ck of relationship suggests tha.t additional sources of paraffins are required in the lower molecu1a.r weight range. Through the model, distributions of fatty acids and pa,ra,ffins from the San Nicolas Basin sediment (Fig. 1, 7 = 0) can be modified as in Fig. 3 to resemble cIistribntIions of fatty acids and paraffins in a sediment of C’retaceous age (Thermopolis shale. Slicrich C’ounty: \Yyoming) sliomi in Fig. 4. The tlistribut~ions of NORMAL

FATTY

carbon NORMAL 1”

b

=

'17

19

21

23

25

carbon rig.

3. Sormal

ACIDS

number PARAFFINS

27

29

31

33

35

number

fatty acids and paraffins derived from a modern sediments (San Sicolas Basin) by t,he General Model. F, = 0.80, F, = 0.15, F, = 0.05; 7 = 0.3 and 0.6

Fig. 3 were obtained by setting parameter values at P, = 0.N): F, = 0.05, F, = 0.15 and T = 0.3 and 0.8. Fatty acid distribut.ions have even-carbon-numbered molecules more abunda,nt than odd-carbon-numbered ones. C,, is most abundant in the model derived distributions while C’,, is most abundant in the shale. Paraffin distributions from the shale and model are very similar. Both have higher concent)rations of odd-carbon-numbered molecules than even ones. A comparison of total concentrations of fatty acids and paraffins in the c’,,-C’,, range from the shale and model is of interest. At T = 0 in the model the concentration of fatty acids is 128 ~~mols/kg and the concentration of paraffins is 10 pmols/kg. In the shale concentrations of fatty acids and paraffins are 142 ,umols/kg and 17 pmols,/kg respectively. With increasing r values the concentration of fatty acids in the model decrease and concentrations of paraffins increase. At 7 = 0.3 the concentration of paraffins from the model is 17 pmols/kg which is the same as the concentration of paraffins in the shale. Distributional patt,erns of pa,raffins are almost esactly the same in the

K.

1288

A.

KVENVOLDEN

and

carbon

number

NORMAL

17 Fig.

4. Sormal

19

21

23

fatty

acids

and

25 carbon

D.

WEISER

PARAFFINS

27 number

29

paraffis in Thermopolis \Vyoming.

31

33 shale,

35 Sheridan

County,

model at T = 0.3 and in the shale. Concentrations of fatty acids derived from the San Nicolas Basin sediment through the model can never equal concentrations of fatty acids in the Thermopolis shale because this shale has a greater abundance of fatty acids than does the San Nicolas Basin sediment. At T = 0.3 the ratio of even- to odd-carbon-numbered fatty acids is too high compared to this ratio in the shale. At 7 = 0.8 the distribution of fatty acids in the model resembles in general the distribution of fatty acids in the shale (Fig. 4), but the concentration of fatty acids in the model has reduced to 107 Cbmols/kg. At this 7 value, however, the concentration of paraffins has increased to 28 pmols/kg. These observations suggest that (1) T = O-3 is the correct dimensionless time for duplicating with the model both the distribution and total concentration of paraflins in the shale ; (2) T = 0.S is the correct dimensionless time value for duplicating the distribution, but not the total concentration, of fatty acids in the shale; (3) the originally deposited fatty acids in the shale had slightly higher concentrations than do fatty acids in the San Nicolas Basin sediment; and (4) originally deposited fatty acids reacted as follows: the fraction of fatty acid forming the next lower carbon number fatty acid was about 80% ; the fraction of fatty acid forming the next lower carbon numbered paraffi was about 15% ; and the fraction of fatty acid reducing to paraffin n-as about 5%. If the parameter values are changed so that P, = O-9, P, = 0.05 and F, = 9.05 at 7 = 7 distributions of fatty acids and paraffins from the model (Fig. 6) have some aspects similar to distributions of these compounds in another sample of Cretaceous age (Mowry shale, Converse County, \+‘yoming) (Fig. 6). Although distributions are similar, total concentrations of fatty acids and paraffins in the shale are greater than

A mathematical

model

of a geochemical

1289

process

in the model. In the C17-C’35 range the fatty acid concentration in the shale of 114 pmol/kg contrasts with the fatty acid concentration from the model of 64 pmols/kg. Paraffin concentration in the shale is 65 ,umols/kg and in the model is 4% ,umols/kg. The original sediment which formed the shale likely had higher concentrations of f&y acids than does the San Nicolas Basin sediment. Fatty acids NORMAL

Q

FATTY

ACIDS

!

f---:

17

19

21

23

25

27

carbon

NORMAL

29

!

,

,

,

,

,

31

33

35

31

33

35

,

number

PARAFFINS

I-

Zkk,, 17

19

21

23

25

27

carbon Fig.

5. Normal

fatt,y

29

number

acids and paraffins derived from a modern Nicolas Basin) by the General Model.

sediment

(San

F, = 0.90, F, = 0.05, F, = 0.05; i- = 7.

‘ii--;:!,,,, 23

25

27

carbon NORMAL

Fig.

6. Sormal

fatt,y

acids

and

29

31

33

35

number PARAFFINS

carbon number paraffins in Mowy \Vyoming.

shale,

Converse

County,

in the I\iowr~- shale have ahnost a smooth distribution of even- and odd-csrbonnumbered molecules as does the distribution of fatty acids from the model. Para%ns in the shale also ha,\-e a smooth distribution with only slight predominances at most of the odd-ca.rbon-numbered positions, especially at C’,,, C’,, and Cs3 (Fig. 6). The distribution of paraffins from the model (Fig. 5) is smoother than in the previous is very smooth; for esample (Fig. 3). For molecules sma,ller than c’,, the clistribution molecules larger than C’,5 t,he distribution has predominance at c’,,, c’,,, Cz9, c’,, and result from the strong influence of concentrations of (‘33. The predominances original parttins. SufEcient parafins have not been generated from fatty acids to smooth the paraflin distribution in the c(2j-(.4Q5 region. Two aspects of the parafliu Both are quite smooth in the C’l,-C’,, region and distributions are similar, however. both hare odd-carbon-numbered preclominanccs in the C’,,-C’,, region. Although in the C’,,-C’,, region distributions from the sediment a.nd model are smooth, they are not pa,rallel. C’oncent.ra.t,ions of paraffins in the shale decrease almost monotonically with increasing carbon number I\-bile in the model concentrations of parafEns crude oils normal paraffins are distributed slowly increase from (~lli to c’,,. 111m:my Some crudes even contain normal paraffin distrias in the MowrJ- shale sample. butions with slight odd-carbon-numbered predominances especially at C’,, and larger. Model studies of fat,ty acids and paraffins in the Sau Nicolas Basin and Mississippi Delta samples ha.ve fa,ilecl to produce distributions of paraffins exactly like in crude oils. Insufficient paraflin was genera,ted in the model to provide monotonically decreasing concentrntions of paraffins at increasing carbon numbers in the range C’,,-c,,. Distributions of paraffins more like those in petroleum can be obtained from the model when fatty acids and paraffins from a sample of algal ooze (Mud Lake, Florida) (KVENVOLDES, 1966c) are used as starting materials. Paraffin distributions shown in Fig. i were obtained by setting F, = 0.9, F, = 0.09, F, = 0.01 ancl T = 03. Initial fatty acid and paraflin clistributions (T = 0) are also shown. Distributions of fatty acids and paraffins in a Permian sample (Lamar limestone, C’ulberson C’ounty, Texas) have many aspects which can be duplicated in the moclel applied to the Mississippi Delta sample (Fig. 2: T = 0) setting F, = 0.63, F, = 0.30, F,, = 0*07. and T = 1.0 (Figs. 8 and 9). The concentrations of fatty acids and paraffins (c’,&‘,,) in the Mississippi Delta sample a,re 56 CLmols/kg and Gpmols/kg reSpeCtiVely (T = 0). At 7 = I.0 the fatty acid concentration has reduced to 2B ,umols/kg which is the same as the concentration of fatty acids in the limestone. Paraffin concentration at 7 = 2.0 is 35 pmols/kg which compares well with the concentration of paraffins in the limestone of 33 pmols/kg. Distributions of fatty acids and paraffins in the limestone and the model at 7 = 2 also compare reasonably well (Figs. S and 9). Fatty acids with even-carbon-numbers are slightly dominant both in the model and the rock. The most abundant fatty acid from the model is c,, ; the most abundant. from the limestone is C,,. Paraffins ranging approximately from C,, to C,, a,re distributed in such a way that even-carbon-numbered molecules are more abundant, than odd in the model and the limestone. The most abundant paraffin in bot,h distributions is Cz6. In the model: paraffins with odd numbers of carbon atoms at Cz9 and C,, are more abundant than their even-carbon-numbered neighbors. These odd-carbon predominances result from the influence of the original paraffin distribution as mentioned earlier. Thus the model produces a

A mat.hcmatical

model

of a geochemical

NORMAL

FATTY

carbon

1291

process

ACIDS

number (kl)

NORMAL

PARAFFINS

j&++;--.-,._, 31 , 17

19

21

25

23

27

carbon

29

33

35

number

(b) Fig. 7. Normal fatty acids and paraffins. (a) In algal ooze (Mud Lake. Florida) T = 0. (b) Derived from (a) by the General Model. F,, = 0.90, F, = 0.01, P, = 0.09; T = T. NORMAL

FATTY

ACIDS

ij;;

I ! 17

19

21

23

25

27

carbon NORMAL

8. Sormal

fatty

acids (Mississippi

33

35

PARAFFINS

and paraffins Delta) by

F, = 0.63, F,

31

number

carbon Fig.

29

,

number the

derived General

= 0.07, FT = 0.30;

from Model.

a

-I- = 2.

modern

sediment

1292

K. A. KVEFLVODEN and D. WEISER

distribution of parafbns with even-carbon-numbered molecules most abundant in one range and odd-carbon-numbered molecules most abundant in another range. Parafhns in some rocks and crude oils have similar distributional features (Fig. 19). The fact that the model has duplicated reasonably well distributions and concentrations of fatty acids and paraffins in this ancient limestone leads to the following conclusions : (1) according to the model fatty acids and paraffins originally deposited in Lamar limestone would have been distributed in much the same way as are fatty acids and paraffins in the Mississippi Delta sediment; (2) a dimensionless time of 7 = 2.0 is correct for duplicating with the model both distributions and concentrations of fatty acids and paraffins. Relationships between dimensionless time and NORMAL

FATTY

ACIDS

4 53

II

17

19

21

23

25

27

carbon

NORMAL

29

31

33

35

31

33

35

number

PARAFFINS

4l,"

3-

;

2E--t 1

Io 17

IIIIIII 19

21

23

III 25

carbon

Fig. 9. Normal

IlIIIh 27

29

number

fatty acids and paraffins in Lamar Texas.

limestone,

Culberson

County,

real times are not known because rates of reaction are not known. These rates probably depend more on temperature, pressure and chemical environments than on real time (KVENVOLDEN, 1966a); (3) in the process of transformation the fatty acid fraction reducing to paraffin is 30%. This high percentage is required to generate sufficient paraffins with even-carbon-numbered molecules more abundant than odd. The fatty acid fraction forming the next lower carbon numbered fatty acid is about 63th while the fatty acid fraction forming the next lower carbon numbered paraffin is 7%. DISCUSSION OF GENERAL NODEL Distributions of fatty acids and paraffins obtained from the model do not parallel distributions found in ancient sediments for at least two reasons: (1) original fatty acids and paraffins in ancient sediments likely were not distributed in exactly the same manner as are fatty acids and paraffins in the modern sediments we have studied and (2) the model in its present form does not account for all possible geochemical processes that may modify distributions of fatty acids and paraffins. One possibility not included in the present model is a scheme for addition of carbon atoms. Then lower carbon-numbered fatty acids could produce higher

A mathematical

model

of a geochcmical

process

1293

carbon-numbered paraffins and higher carbon-numbered fatty acids. Experimental results of JURG and EISMA (1964) suggest that the formation of higher carbonnumbered paraffins from lower carbon-numbered fatty acids is geochemically possible. In the present study only a limited range of fatty acids and paraffins (C’,,-C,,) has been investigated. In modern sediments fatty acids with smaller numbers of carbon atoms are common. In fact, C,, fatty acid is by far the most abundant of all fatty acids. Fatty acids smaller than C,, serving as source for some higher carbon-numbered paraffins may supply the necessary paraffins to make model derived distributions more like those found in ancient sediments and crude oils.

Mowry Natrona 0

E

shale: Co.,

I17

19

21

23

25

27

carbon

19

21

23

25

27

carbon

Fig. 10. molecules

29

31

33

35

number

29

31

33

35

number

Sormal paraffins in a shale and crude oil. Even-carbon-numbered most abundant in lower molecular weight range; odd-carbon-numbered molecules most abundant in higher molecular weight range.

Another modification of the model would permit closer duplication between distributions of fatty acids and paraffins derived through the model and distributions observed in ancient sediments. If the rate constant or fraction of fatty acid involved in reactions were dependent on the carbon number, a more general model involving a greater number of parameters could be developed. The mathematics of such a model would be the same as presented previously with F, = F,(n), F, = F,(?L), and k = k(n). The amount of fatty acid participating in reactions is a function of n which can be set for each carbon number. However, present data do not warrant the development of a more complicated model at this time. The General Model can theoretically be used to determine initial distributions of fatty acids and paraffins from distributions of fatty acids and paraffins observed in ancient sediments. In mathematical terms the problem is to find Xj(0) and Y,(O) given X,(t) and lrj(t) where t is any time. In practice the equations (Appendix 2) for solving this problem are almost useless with the present experimental data

K. A. KVENTOLDENand D.

1294

WEEISER

except for small times. The reason for this is that the equation for fatty acids has an ekt factor. If the term multiplied by eki is negative, as is often the case with these data, it continues to become more negative at an exponential rate. This effect does not permit the General Model to be used to go baclrwa,rds in “time” for present experimental data. In the General Model distributions of normal fatty acids and normal paraffin hydrocarbons can be determined for any T value. T values around 30-X are equivalent to infinite time at least for all examples studied thus far; that is, at these 7 values concentrations of fatty acids and paraffins do not change. Fatty acids have essentially disappeared, and their products cannot add significantly to the concentrations of normal paraffins. The General Model can show, therefore, the distribution of fatty acids and paraffins at infinite time, if initial distributions of fatty acids and paraffins as well as two parameters, such as F, and F,, are given. The inverse problem can also be solved and will be discussed later. SPECIAL CASES In the General Model F,, F, and F, are not zero. Several special cases of the General Model can be obtained by setting one or two of these independent parameter equal to zero. The following cases will be considered: (i) F, = 0; (ii) F, = 0; (iii) F, = 0 ; (iv) F, = F,

(v) F,=F,=O; (vi) FT=FD=O;

F,, F, # 0 F,, F, # o F,, F, f 0 = 0; F, # 0 F, f 0 F, # 0

Case (i) F, = 0 The first special case of the General Model results when F, = 0. Then F, + It involves no reduction of fatty acids. ,Schematically the Degradation Model is

F, = 1. This case is called the Degradation Model. C’arbon No. * * * n. - j - 1

n-j

n-j+l...

n-l

n

The scheme is essentially the same as proposed by C!OOPER(1962) and COOPER and BRAY (1963) to explain relationships of fatty acids and paraffins. COOPER postulated that original fatty acids degrade by means of a free radical decarboxylation in which at each step a lower carbon numbered fatty acid and paraffin are formed. Each newly-formed acid participates in the same reaction. Generated paraffins collect with paraffins originally present. Absolute concentrations of fatty acids diminish. Operation of this process on mixtures containing dominantly evencarbon-numbered fatty acids, under conditions favoring formation of more lower

A mathematical

model

process

of a geochemical

1296

carbon-numbered fatty acids than paraffins, can produce mixtures in which there is little preference for either even-carbon-numbered fatty acids or odd-carbon-numbered paraffins. The differential equations (2) become

dX, -at

-kX,

ax _. n) = -k X,+ at d ye,

-

-I- F,k X,,++l;

j = 1,. . . ) n -

= F,k Xn++l;

at

j=l,...,n-I

1.

The mathematics are similar to those for the General Model (Appendix corresponding to (4) are : JY,-,V)

j (F,kt)i x,+i-j(0)] +-

=

e-kt ;

j = 0,.

. . ,n

i=O

Yn-j(t)= F, (:$ FaiXn+i-j+l(0) +

1.

(64

[ 1 - e-kt ,io y]]

j = 1,.

yn-j(");

-

1). Solutions

. . ) ?a -

1.

(6b)

Comparison of solutions (6a) and (4a) shows that for a given initial fatty acid distribution the Degradation Model gives the same fatty acid distribution at a dimensionless time, kt, as does the General Model. The Degradation Model with F, = 0.80 applied to fatty acids and paraffins in the Mississippi Delta sample gives results for paraffins shown in Fig. 11. Fatty acid distributions here are the same as in Fig. 2 which is from the General Model where F, = 0.80. Distributions of paraffins from the two models differ in that even-carbon-numbered paraffins can be generated more quickly in the General Model than in the Degradation Model because NORMAL

PARAFFINS

6 5 $4 z3 E 2

2 1 0

17

19

21

23

25

carbon

27

29

Fig. 11. Normal parafis. a modern sediment (Mississippi

(a)

In

(b)

Derived

from

(a) by

31

33

35

number

t,he Degradation

F, = 040, F, = O-20,F,. = 0; 7 =

Delta)

T = 0.

Model. 1, 3, 5, 15, 30.

K. A. KVEXVOLDEN

1296

and D. WEISEK

the General Model includes reduction of fatty acids. Results from the Degradation Model operating on fatty acids and paraffis from the Mississippi Delta cannot produce distributions of normal paraffins as shown in Fig. 9 (Lamar limestone) where even-carbon-numbered molecules are more abundant than odd from Crs to C a6 Case (ii) F, = 0 Another case (ii) of the General Model results when F, = 0 for FT # 0. Then F, = 1 - F,. This case is called the Reduction Model. It involves production of lower-carbon-numbered fatty acids, possibly derived from /l-oxidized fatty acids through Willgerodt-type reactions ( ORR,personal communication). Paraffins may be obtained by reduction of the precursor fatty acid with no loss of carbon atoms. The model is shown schematically as CarbonNo....n--j--l Fatty

n-j

Bcid . . . S,-j--1 3 2

1 Pa.raffin . . . 1rn-j-1

n -j+1...

*Y,,-, z

Y,,-jil

p.

1’

I*,r-j

NORMAL

...t

Fan

.**

J7n-j+l

n -

1

X,-i

t

12

F,&

S,,

1’

1:

J-11-1

Y,,

PARAFFINS

5

0

17

19

21

23

25

27

carbon

29

31

33

35

number

Fig. 12. Normal paral%ns. (a) In a modern sediment (Mississippi Delta) T = 0. (h) Derived from (a) by the Reduction Model. F, = 040, F, = 0, F, = 0.20; 7 = 1, 3, 5, 15, 30.

The resulting equations and solutions can be obtained from (2) and (a) by simple substitution. All expressions containing Fp as a factor become 0. Figure 12 shows the distribution of paraffins from the Reduction Model with F, = 0.80 and F, = 0.20. Paraffins which were generated are distributed so that even-carbon-numbered molecules in the range C,, to C!,, are more abundant than odd-carbon-numbered molecules. A similar distribution of paraffins was obtained from the General Model with F, = O-63, F, = 0.30 and F, = O-07 (Fig. 8). With the Reduction Model it is difllcult to produce paraffin distributions with dominant odd-carbon-numbered molecules from fatty acids and paraffins distributed as in modern sediments. Fatty acids obtained from the Reduction Model have exactly the same distribution as those from the General Model where F, = 0.80 (Fig. 2).

A mathematical

model

of a geochemical

NORMAL

FATTY

1297

process

ACIDS

9

a

,” 0 z

6 5

E4

carbon (a) (b)

In a modern Derived from

Fig. sediment (a) by

Fa = 0, FT # Case (iii)

number

13. Sormal fatty acids. (Mississippi Delta) 7 = 0. cases (iii), (iv) and (v) of the General 0, F, = 1 - F,.; T = 1, 3, 5, 15, 30.

Model.

F, = 0

When F, = 0 and F, # 0, then F, = 1 - F,, and another case (iii) ofthe General Model results, In the scheme no transformation of a fatty acid to one of lower carbon-number takes place. Schematically case (iii) is: Carbon

No. . . . n-j-l

Fatty

Acid . . . X,,-+i

Paraffin. Distributions

n-j X,-j

. . Yn-,-r

of fatty

n-j+l...n-1

Y,-,

acids are described Xj(t)

n

xn-j+l.

..

X”_,

x,

Ynejil

...

I-,-,

Y,

by j=

= X,(O) e-k’;

l,...,n

(‘1

since in the differential equation (2b) fatty acids are no longer coupled. The exponential decay equation yields distributions of fatty acids which are attenuated copies of the initial fatty acid distribution (Fig. 13). In this model the concentration of each paraffin is the sum of (1) the paraffin derived from the fatty acid of one larger carbon number, (2) the paraffin derived by reduction of the fatty acid with the same number of carbon atoms as the paraffin, and (3) the paraffin initially present. The equation describing paraffin distributions is :

Yi-l(t) Y,(t)

= [F&i(O) = F,X,(O)(l

+ FJi-,(O)](l

- e-“‘)

- e-kL) + Y,(O).

This equation results from a simple Figure 14 shows paraffin distributions

+ Yiil(O),

i=l,...,n (8)

integration of (2d) after (7) is substituted. when F, = 0.1, 0.5, and O-9 for T = 0, 1, 3,

K. A. KVEXVOLDEN

1298

end D. WEISER

5, 15, and 30. Odd-carbon-numbered normal paraffins are most abundant when FT = 0.1; however, when F, = 0.9 even-carbon-numbered paraEins are most abundant. Case (iv) Fr = F, = 0 Three other casesof the General Model are possible. One of these (iv) is obtained when F, = F, = 0 ; then F, = 1. Schematically this case is Carbon No. . . . n-j-l Fatty

Acid . . . Xn-j-l

X,-j

. . Y,~-j-l

** +

Xn-j+l

F$/ Paraffin.

n-j+l...

n-j

n

zz-I

XVI

Ek/ n/

Fy YnVj

n-l

Yn-,+l.

..

Ff/’ yn-1

yn

Chemically this is a model of simple decarboxylation of fatty acids to paraffins. The equation for fatty acids is (7), the same as the previous case (iii), and distributions of fatty acids obtained are the same as in Fig. 13. Paraffin distributions can be described by Yiml(t) = F,X,(O)(l

- e--kf) + Yi-,(O);

i =2,...,n

(9)

and r-,(t) = Y,(O). These equations are from (8) when F, = 0. Distributions of paraffins resulting from the model (Fig. 15) are similar to distributions of fatty acids, but paraf3ins of oddcarbon-numbers are most abundant while fatty acids of even-carbon-numbers are most prevalent. Case (v) F, = F, = 0. Another case (v) results when F, = 1. Then F, = F, = 0. Schematically case (v) is Carbon No. . . . n-j Fatty

Acid.

- 1 n-j

n-

j+l...

. . Xnejel

X,-j

Xn-j+l.

1’

12

).

I’+*

Y,-,+I

Paraffin . . . Y,-j--l

n-l ..

...

The equation for fatty acids is (7), the same as in (iii) and (iv). fatty acids are shown in Fig. 13. The equation for paraffins is Y,(t) = F,X,(O)(l

- e--kt) + Y,(O);

X,-l

x

p.

;;

yn-1

Jrn

Distributions

of

i=l,...,n

(10)

which comes from (8) when F, = 0. Resulting distributions are similar to distributions of fatty acids.

of paraffins (Fig. 16)

A mathematical

model of a geoohemical

NORMAL

process

1299

PARAFFINS

7

Fa Fp Fr

= = =

0.0 0.9 0.1

I- = 0, I. 3, 5, 15, 30 1 0

17

19

21

23

25

27

carbon

29

31

33

35

number

6 5 Fa Fp Fr

-?4 ;3 E

= 0.0 = 0.5 = 0.5

T = 0. If 3, 5, 15, 30

12 1 '17

19

21

23

25

27

carbon

29

31

33

35

number

6 Fa Fp Fr

T=

0

17

19

21

23

25

carbon

27

29

31

33

35

number

Fig. 14. Normal pareRins. (a) In a modern sediment (Mississippi Delta) T = 0. (b) Derived from (a) by case (iii) of the General Model. 7 = 1, 3, 5, 15, 30.

= = =

0.0 0.1 0.9

0, 1, 3, 5, 15, 30

K. A. KVENVOLDEN

1300

NORMAL

0

17

19

21

23

25

and

27

17

19

21

23

31

33

35

PARAFFINS

27

25

carbon (a) (b)

29

number

Fig. 15. Xormal paraffins. In a modern sediment (Mississippi Delta). T = O. Derived from (a) by case (iv) of the General Model. F, = 0.0, F, = 1.0, F, = 0.0; T = 1, 3, B, 15, 30. NORMAL

0

WEISER

PARAFFINS

carbon

(a) (b)

D.

29

31

33

35

number

Fig. 16. Normal par&ins. In a modern sediment (Mississippi Delta). T = 0. Derived from (a) by case (v) of the General Model. F, = O-O,F, = 0.0, Fr = 1-O; 7 = 1, 3, 5, 15, 30.

Case (vi) F, = 1, Fr = 0 When F, = 1, then F, = 0, and F, = 0, and a final case (vi) of the General Model results. In this case fatty acid distributions tend to smooth with increasing values of T, but no paraffins are produced. The original distribution and concentration of paraffins remains unchanged as the fatty acids disappear. This is not what has been observed when paraffins from modern and ancient sediments have been compared. Equations for this case can be obtained by proper substitution in (4a), (4b), and (4~). The General Model and cases (i) and (ii) are most useful in trying to explain relationships of fatty acids and parafiins in sediments. Cases (iii), (iv) and (v) are

A mathematical

model of a geochemical

process

1301

less useful because the fatty acids obtained during the transformation process are attentuated copies of the original fatty acid distributions. If the original distribution has a predominance of even-carbon-numbered fatty acids, this predominance remains as the fatty acids disappear. In sediments distributions of fatty acids tend to smooth out with time so that even- and odd-carbon-numbered fatty acids are about equally abundant. Case (vi) is also of little use because no paraffins are produced. INVERSIONS

An inversion of the General Model can be used to find the distribution of initial normal fatty acids if distributions of normal paraffins at initial and indite times are given, and if two parameters such as F, and F, are specified. An inversion of the Degradation Model or other special case models can also be used for the same purpose. From solutions (4) at infinite time, x,-j(

co) = 0 ;

F7xnP)

j = 0,. . . ,n -

= Yn(~)

-

1

(114 (lib)

Y,(O) j-l

J’S,-j(O) = [Yn-j( 03) - J’n-j(O)1- (Fp + FctJ’r)& J’,“X,+i-j+,(O); j = 1,. . . , n -

Equations (11) can be inverted (Appendix For F, # 0,

(12a)

- yn+i-,+,(O)l; where K = F’,/FT which is the ratio of the fraction parafiin to the fraction of fatty acid reducing to paraffin. which is the Degradation Model (case i) yn-j(

m)

-

-

j

=

1,. . . ,n -

1

(12b)

of fatty acid degrading to For F, = 0, (F’, + F, = l),

yn-j(0)

= (1

(llc)

3) to get the following:

Y,( 00) - J’“(O) Fr

X,(O) =

-%I-j+,(O)

1.

F,)

+FL

yn-j+l(

co)

-

y7+j+,(0)1,

a

j = 1,.

. .,n

-

1.

(12c)

Equations (12) give the initial fatty acid distribution when final and initial paraffin distributions and two parameters (FT and F,) are known. This inversion can be used to determine the initial distribution of fatty acids that produced paraffin distributions found in crude oils according to the model. In their studies MARTIN et al. (1963) reported distributions of normal paraf&s over broad molecular weight ranges (C,-C,,) for several crude oils. The inversion of the model has been applied to their data for the Uinta Basin oil as shown in Fig. 17. The parameters are set at F, = 0 and F, = 0.9. An additional assumption is that

1302

K. A. K~ENVOLDEN

and

D. WEISER

there are no initial parafhns. This assumption not only is convenient but reasonable. Studies of fatty acids and paraffins in modern sediments (KVENVOLDEN, 1966c) show that absolute concentrations of fatty acids are usually at least an order of magnitude greater than absolute concentrations of paraffins. Therefore, paraffins generated from fatty acids usually mask most initial paraffins. Thus initial paraffin concentrations were set at 0 for this illustration. A further assumption was made that the distribution of paraffins in Uinta Basin oil represents the end product of total NORMAL FATTY ACIDS FROM INVERSION DEGRADATION MODEL (CASE i) F. = 0.9, Fp = 0.1. F, = 0

NORMAL

OF

PARAFFINS FROM UINTA BASIN CRUDE UINTAH COUNTY. UTAH

OIL DATA FROM. MARTIN. WINTERS AND WILLIAMS c IP63I

OS -

Fig.

17. Normal

fatty

aoids Basin

predicted crude oil

as precursors (Degradation

of normal Model).

paraffins

in Uinta

fatty acid degradation ; i.e. that infinite time was reached. The fatty acid distribution (Fig. 17) shows distinctly higher concentrations of even-carbon-numbered molecules compared with concentrations of odd ones. In this respect the model derived distribution and distributions in modern sediments are similar. Although there is this similarity, the model derived distribution (Fig. 17) does not copy fatty acid distributions in the modern sediment samples studied here (Figs. 1, 2 and 7, 7 = 0). Another special case of interest is the inversion of the Reduction Model (case ii). In this case F, = 0 (P, # 0), and therefore, K = 0 and F, + Fr = 1. Substituting into equations (12), (12a)

X,-,(O) =

YTI-j(=J) - Y,-,(O) l-FF,

j=l,...,?&-1.

(13)

The right side of equation (13) for the inverted Reduction Model equals the right side of equation (12~) for the inverted Degradation Model. This equality means that the concentration of initial fatty acid of carbon number p + 1 of the inverted Degradation Model equals the concentration of the initial fatty acid of carbon

A

mathematical

model

of 8 geochemical

1303

process

number p of the inverted Reduction Model. Initial distributions are just shifted copies of one another. For data from the Uinta Basin oil with F, = 0 and F, = O-9 and again the assumption of no initial paraffins, the distribution of fatty acids resulting from the inverted Reduction Model has odd-carbon-numbered molecules in greater abundance than even-carbon-numbered molecules. This hind of distribution is not acceptable because we believe that initial fatty acid distributions should contain dominantly even-carbon-numbered molecules as in living organisms and modern sediments. Inversions of the other special cases of the General Model can be found in a similar way from considerations of equations (11). CARBON PREFERENCE INDICES Carbon preference indices have been used to describe distributions of normal paraffins and fatty acids. BRAY and EVANS (1961) and COOPERand BRAY (1963) defined a measure of relative abundances of odd- to even-carbon-numbered normal paraffins and called this measure a carbon preference index (CPI). A measure of the relative abundance of even- to odd-carbon-numbered normal fatty acids was defined by KVENVOLDEN (1966a). To distinguish between distributions of paraffins and fatty acids he used CPI, for the carbon preference index of normal paraffins and CPI, for the carbon preference index of normal fatty acids. Expressions for CPI, and CPI, are:

CpI c’p1

P

(14) = 1 Y&l + Y,(t) + . . . + Y,-z(t) Y&) + y,(t) + * - * + Yn&) i 1 Y,.&) 5 YzV) + Y&) + . . * + Y&t) + y4w + y,(t) + - . - +

A

= ! z(t) 2 1x,(t)

+ x4(t) + * * * + LIV) + x,(t) + . . . + X,-,(t)

+ x,(t) W)

+ x,(t) + -w)

+ . . . + X,-,(t) + . * . + x,(t)

I

(15)

where n is an odd number. These measures have been useful in characterizing distributions of paraffins and fatty acids, but they do not reflect many of the interesting features sometimes exhibited by these distributions. For example, some sediments and crude oils have been found in which even-carbon-numbered paraffins are dominant in one range of carbon numbers and odd-carbon-numbered paraffins are dominant in another range. Figure 10 shows two such cases. CPIp~cZO-c,,~values for these distributions are l-01 for the sediment and I.00 for the oil. Such values do not reflect the fact that paraffins of these samples have both even- and odd-carbon-preferences depending upon the carbon-number range considered. Rather the values may suggest that paraffins in both the sediment and crude oil have equal abundances of odd- and even-carbon-numbered paraffins everywhere in the range C,,-C,,. Limiting values of CPI, and CPI, can be determined for the Degradation Model (case i). This case is of special interest because it applies to the scheme originally proposed by COOPERand BRAY (1963) for the degradation of a single fatty acid of carbon number n. With the following initial conditions X,(O) = Y,(O) = 0,

j = 1,. . . , n -

and -uO)

f 0

1,

(16)

K. A. KVENVOLDENand D. WEISER

1304

then at infinite time in the Degradation Model

(17)

,,I,=$+&).

This equation is the same as that obtained by COOPERand BRAY. For F, between zero and one, CPI, is greater than one. CPI, equals one when P, is one. For the same initial conditions (16), from equation (6a), the fatty acid distribution is : Xi(t)

= [(r?'J

X,(O) e&l,

i = 1, . . . , 72.

(18)

These values are substituted into equation (15), the definition of CPI,. As t approaches infinity the first term in equation (15) goes to zero, since the denominator contains a higher power oft than the numerator. The second term goes to infinity, since there the numerator contains a higher power of t. Therefore, CPI, goes to infinity as t goes to infinity. This conclusion appears contrary to observation of fatty acids in sediments where older rocks tend to contain fatty acids with smaller CPI, values than do younger sediments (KVENVOLDEN, 1966c). Although CPI, values go to infinity in the model they first pass through a minimum. Therefore CPI, values obtained at t values less than t (minimum CPI,) behave as do CPI, values obtained from fatty acid distributions in sediments. In equation (17) CPI, for large time values is always greater than or equal to one for the very special case of no initial paraffins and only one initial fatty acid at carbon number n [equation (IS)]. A similar result is obtained with the Degradation Model (case i) for the most general initial conditions (3a) and (3b) (Appendix 4). With a minor assumption on the size of YZ( co) - Y,-i( co), CPI, is greater than or equal to one if and only if E, is greater than or equal to 0, - C where, E, = X,(o)

(n odd)

+ -X&o) + . . . + X,-,(o),

(194

0, = X5(0) + X,(O) + * * - + X,(O), c=P

+FJ (1 _ jl ) (To Q

-

TE)

+ (1 TF

(lgb) 0

) (Y3(a)

-

Y3WL

T, = Y&J) + Ye(O) + . . . + Y,-,(o),

(1W

T, = Y&I)

(19e)

and + YJO) + . . . + Y&I).

Essentially E, is the sum of the initial even-carbon-numbered fatty acids, 0, is the sum of the initial odd-carbon-numbered fatty acids, T, is the sum of the initial even-carbon-numbered paraffins: and To is the sum of the initial odd-carbonnumbered paraffins. An examination of equation (19c) shows that C is positive and small compared to E, and 0,. Thus in all cases where even-carbon-numbered initial fatty acids (E,) dominate odd-carbon-numbered initial fatty acids (O,), CPI, must be greater than one for the Degradation Model. This means that a more general model is necessary to explain examples where CPI, is less than one. In Fig. 9, for example, CPI, equals 0.90 which is explained by application of the General Model (Fig. 8).

A mathematical

model

of a geochemical

process

1305

CONCLUSIONS

1. A postulated geochemical process has been studied by means of a mathematical model. 2. This process produces normal paraffin hydrocarbons from normal fatty decarboxylation and/or hydrogenation (reduction).

acids by

3. Differential equations for a General Model and six special cases are developed and solved. Calculations of solutions were performed on a CDC 1604 computer. 4. In the General Model fatty acids serve as source for lower carbon numbered fatty acids and paraffins. Fatty acids also serve as source for parafEns which have the same carbon number as the fatty acids. Paraffins derived from fatty acids add to paraffins present initially. In the limit fatty acids completely disappear. 5. The General Model duplicates many geochemical relationships observed in sediments between fatty acids and paraffins especially in the range C,,-C,,. a. Distributions of fatty acids with initially high ratios of even- and odd-carbonnumbered molecules as in modern sediments change to distributions with lower ratios as in ancient sediments. Absolute concentrations of fatty acids decrease and eventually disappear. b. Distributions of paraffins with initially high ratios of odd- to even-carbonnumbered molecules as in modern sediments change to distributions with lower ratios as in ancient sediments. Absolute concentrations of paraffins increase until all fatty acids have disappeared. 6. None of the special cases explains the experimental data as well as the General Model. Two special cases are cited. a. The Degradation Model (case i) applied to fatty acids and paraffins in modern sediments cannot produce distributions of paraffins with even-carbon-numbered molecules more abundant than odd. This kind of distribution has been noted in some ancient sediments and crude oils. b. The Reduction Model (case ii) requires that initial fatty acid distributions have odd-carbon-numbered molecules in greater abundance than even-carbonnumbered molecules in order to produce results observed in most ancient sediments. However, initial fatty acid distributions as required for this model have not been observed in living organisms or modern sediments. 7. The General Model in its present form does not generate results which duplicate precisely distributions of fatty acids and paraffins observed in sediments. However, in the molecular weight range considered here (C,,-CB,) there is similarity between model-derived distributions of fatty acids and paraffins and distributions of these kinds of molecules in sediments. Processes similar to those modeled here likely occurred during the geochemical history of sediments. Acknowle~l~entents-~~0 were used to calculate to results of Appendis type reactions.

arc grateful to J. R. FINGER, who wrote the computer programs which and plot the illustrations, to R. G. WOOD for helpful discussions leading 4, and to W. L. ORR, who suggested that (case ii) could model Willgerodt-

1306

A. KVENVOLDEN

K.

and D. WEISER

REFERENCES ABELSON P. H.

(1963) Organic geochemistry and the formation of petroleum. Proc. Sixth World Petroleum Congress sect. 1, 397-407. BLUMERM. (1965) Organic pigments; their long term fate. Science 149, 722-726. BRAY E. E. and EVANSE. D. (1961) Distribution of n-paraffins as a clue to recognition of source beds. Cfeochim. Cosmochim. Acta 22, 2-15. BRAY E. E. snd EVANS E. D. (1965) Hydrocarbons in non-reservoir source beds. Bull. Am. Ass. Petrol. Beol. 49, 248-257. COOPER J. E. (1962) Fatty acids in recent and ancient sediments and petroleum reservoir waters. Nature 193, 744-746. COOPER J. E. and BRAY E. E. (1963) A postulated role of fatty acids in petroleum formation. Cfeochim. Cosmochim. Acta 27, 1113-l 127. GERARDEH. W. and GERARDED. F. (1961) (1962) The ubiquitous hydrocarbons. Ass. Food and Drug Ogicials of U.S. 25 and 26, 1-47. GRAHAMD. W. (1965) The separation and characterization of isoprenoid acids from a California petroleum. PhD Diss., Univ. Calif., Berkeley; Univ. Microfilms 65-8174, Ann Arbor, Mich. HILDITCH T. P. and WILLIAMS P. N. (1964) Tlhe CIAemical Constitution of Natural Fats (4th edition). Wiley. JURGJ. W. and EISMA E. (1964) Petroleum hydrocarbons: generation from fatty acids. Science 14, 1451-1452. KVENVOLDENK. A. (1966a) Molecular distributions of normal fatty acids and paraffis in some Lower Cretaceous sediments. iVature 209, 573-577. KVENVOLDENK. A. (1966b) Normal fatty acids in sediments. Am. Oil Chem. Sot., 57th Annual Meeting, Los Angeles, Calif., April 24-27. (To be published in J. Am. Oil Chem. Sot.) KVENTOLDENK. A. (1966c) Evidence for transformations of normal fatty acids in sediments. Third International Meeting on Organic Geochemistry, London, England, Sept. 26-28, 1966. (To be published in Advances in Organic Qeochemistry III.) MARTIN R. L., WINTERSJ. C. and WILLIAMS J. A. (1963) Distributions of n-paraffins in crude oils and their implications to origin of petroleum. Nature 199, 110-114. RALSTONA. W. (1548) Fa.tty Acids and Their Derivatives. Wiley.

APPENDIX~-SOLUTION From

equation

(2a)

with

initial

conditions

(3a)

S,(t) Substituting

from

equation

(Al.l)

into

d-Y,-, dt which

has the

and

= X,(O)

equation

=

eekt.

(2b)

-kS,-,

(Al.l)

for j

= 1,

+ F&Y,(O)

= [X,-l(O)

emkL,

+ F,k.Y,(O)t]

(A1.2)

epkt.

(A1.3)

general j 1

(F&t) 7

1

x,+&j(o)

This equation (A1.4) can be checked by substitution the same as equation (4a) which was to be proved. From equations (Bc), (Al.l), and initial condition Y,(t) is equation

(4b).

= F,.&(O)

[l

-

j=O,...,n

e+,

i=O

This

(3b)

solution X,-,(t)

In

TO GENERAL MODEL

into

equation

(2b).

1.

Equation

(A1.4)

(A1.4)

(3b) -

eekt]

+

Y,(O).

(A1.5)

is

A mathematical From

equation

model

of a geochemicel

-

1307

(2d) t

y,-j(t)

process

y,-,(O)

I

= F,k

xn-j+l(T)

dT

+ F,

X,-j(T)

s0

j = 1,. . . , n - 1.

dt,

s0

(A1.6)

Define: I Ii(l)

=

us emku du.

(A1.7)

s0 Integrating

the right

side

of equation

(A1.7)

Ii(t)

by

parts

repeatedly i

= &,

(kt)l

1 - emkt 1

7

I=0 Now substituting (A1.8) gives

from

equation

+

Equation

(Al.9)

is equation

(Al.4)

and

Yj(t),

the

was

(Al.6)

and

using

the result

of equation

~-BACKWARD X,(O)

= X,(t)

ekt

S,(O)

(Al.9)

to be proved.

is to find

X,,-,(O) and,

(Al.@

j = 1,. . . , n - 1.

which

problem

. I

equation

y*j(");

(4~)

APPENDIX Given Xi(t)

into

gives

= [X,-,(t)

IN TIME

and

-

Y,(O).

F,kS,(t)t]

From

equation

(Al.4)

ekt

in general, j z

=

s,-j(")

y,-,(O)

=

Y,-,(t)

Sn+i-j(t)

i!

C i=o This is equation (Al.4) forward or be&ward To Ford Y,(O),!first

1

( -F,kt)i

with -t substituted in time. Xj(0) is found, and j-1 - F, 2 F,“X,+,-j+,(O) ( i=O

j =o,...,n

ekt,

for

t. Thus

then,

from

equation

(Al.4)

equation

(Al.9)

APPENDIX (11),

for

F,

can be used

to go

or (4c),

j = 1, . . . n - 1.

n+i-j(0)

equations

or (4a)

(A2.1)

I)

- Frj. FaiS From

-1.

Q-INVERSION

OF INFINITE

(A2.2)

TIME

+ 0,

X,(O) = Y,(a) - Y,(O) F, ’

1 ll[ Fr 1’ 1 - Y,-,(O) Yn-1( co) y,-l(O) -y,-,(O) =CY?Lz(~) (K + J'a) F, Fr 1 1 1 -~,L-,(O)

=

yn-*(aI

-

~‘,4(0)

_ (K

+ F

)

FT

+ K(K

+ FJ

Y,(~)

-

F,

Y,(O)

’

Y,(m)

-

Y,(O)

K.

1308 and,

A.

KVENVOLDEN

and

D.

WEISER

in general, y,-j(

Xnej(0)

a)

-

j-l

yn--j(O)

- (K + F,) 2

= F,

which For

F,

is equation = 0, from

(12b) which was equations (ll),

w)

yn+i-j+l(

(-K)’

-

yn+i-j+l(“)

1”,

i=o

1

1

to be proved.

S,(O) = y+l ( w) - I’,-,w l-FF,

’

1 u[ (l---p,)1’ co) Y,-3(O) _ F y,-!A a) l',-,(O) S,-,(O) = (1 (I (1 --F,) --F,) 1’ 1 [ 1 0[ (1- F,,) 1 S,-,(O)

Y,,(m)

=

- Y,-,(O) (1 - F,)

_ F

Y,-l(c@I) - E’,-,(O)

_

l’n-j+l(

I-&

and,

in general,

Y,J-~(

s,,-j+l(c’)

K)

=

(1

[

which

is equation

(12~)

which

was

where

F

w,

-

yn-j+l(“)

~-~ARBON

PREFERENCE

INDEX

:

12 is an odd

0,

= Y3( WI

-I- I',(

q/

=

i

number.

equation

174(w)

Then

C’PI, From

l-,-j(O)

F,,)

to be provc*d.

-APPENDIX Define

-

-

from

= 1 2

CD) -I- . . . +

Ys(co)

I',-,(

+ . . . + Ynel(

equation

[

+

(A4.1)

cc),

(A4.2)

0 -1! E, I .

(A4.3)

(14) 0,

I.&

co),

+

Y&w)

-

Y,+(a)

(6b) Yj(w)

= F,

u-j-l 2

F,‘-Yj+i+l(O)

+

j=l,...,n-I.

(A4.4)

+ Yj(0) - F,Yj+,(O).

(A4.5)

Y,(O),

i=O

Equation

(A4.4)

implies

Y&a)

= F&yj+I(O)

+ F,Yj+I(a)

Define:

Substituting equations (A4.6),

E,

= X,(O)

+ -Y&O)

+ . . . + X,-,(O),

(A4.6)

0,

= X&O)

+ X,(O)

+ . . . + X,(O),

(84.7)

TE

= Y,(O)

+

Y,(O)

+ . . . +

ycl(o),

(A4.8)

To

=

+

Y,(o)

+ . . . +

Y,&o).

(A4.9)

equation

(A4.1)

Y,( ~0) from equation (A4.2), (A4.9) and

Y&W

(A4.5) (A4.8)

0, = F&

into

+ FaE, + To - F,T,.

and

using

the

definitions

from

(A4.10)

A mathematical

model of a geochemical

Similarly substituting Yi( co) from equation (A4.5) into equation from equations (A4.7), (A4.1), (A4.8) and (A4.9)

1309

process

(A4.2) and using the definitions (A4.11)

E, = F,O, + F,O, - F,Y2(m) + TB - F,To i- Pay,(O). Substituting

0, from equation (A4.10) into equation (A4.11) and simplifying, one obtains 1 (A4.12) Ev = (1 - P,2) {FpOx + F,F,E,. + F,[Y&O) - Y,(m)] + TE(l - Fu2)}.

Similarly, obtains

substituting

E, from equation

0, = (1 yF

(A4.11)

2) {Fp~, + F,F,O,

a

into equation

+F,~[Y~(o)

(A4.10)

and simplifying,

- y,(m)] + ~,(i - F,?)).

Now substitution for Ev and 0, from equations (A4.12) and (A4.13) into equation simplification gives E + F,O, + a E, + F,O, + a\ “I,’ =; 0 : $7uE .+/9+;~+O,+F,E,+~j’ z where Fo2[Y2(0) - Y,(m)1 + TotI - F,‘) o:= (1 - F,) F,[Y,W

8=

- Y,(m)1 +T,(l (1 - F,)

- Fa2)

CPI,

%

(A4.3) and (A1.14)

(A4.15)

(A4.17)

- Y,-l(m)l.

to 0, + F,E, + B, then from equation

If 7 is small compared

(~4.13)

(d4.16)

and Y = (1 + F,)[Y,(=3)

one

(A4.14)

E, + F,O, + a 0, + F,E, + B '

(A44.1S)

This assumption is equivalent to redefining CPI, and ignores possible small effects arising from a very large difference in the concentration of the smallest and the largest even-carbon-numbered paraffin. In all of the esamples considered in this study, the assumption is satisfied. Assume CPI, 2 1. From equation (A4.18) -5 + F,Oz 0, + F,E, which

can be simplified

+ a > 1 + /9 = ’

to get B-a

(-44.19)

Defme:

B-a -’ Substituting

for p +

a

from equations

= (1 - F,)'

(85.16)

and (A5.15)

c=~(T,-T~)+$[Y,(m)-YIT,(~)~. P 1, Now Y2( m) 2 Y(0); and, in general, T, > T, as occurs sediments. Thus C is positive, and E, 2 0, - C, when CPI, 2 1. If equation CPI, 2 1. Therefore: CPI,

and simplifying (84.20) in living

organisms

and modern (-44.21)

(A4.21) is assumed instead, then a similar argument shows that B 1 if and only if E, 2 0, - C which was t,o be proved.