Generation of a Synthetic Vertical Profile of a Fluvial Body

INDIANA U. PAUL 'OTTER INDIANA GEOLOGICAL SURVEY F' BLAKELY I BLOOMINGTON, IND.

ABSTRACT depositional processes. From the petroleum engi- neer's viewpoint it seems reasonable to believe Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021 Any stratigraphic section or bedding sequence that the synthetic generation of rock properties and can be synthesized if there is a transition procedure their distribution in a reservoir should be relevant from one lithology or bedding type to another, and in the study of reservoirs. if thickness distributions of the different lithologies Any stratigraphic section or bedding sequence are known. Stratigraphic sections of a fluvial can be generated provided there is a transition sandstone body were synthesized with five bedding procedure from one lithology or bedding type to types: cross- bedding, massive beds, parting another, and provided the thickness distributions of lineation, ripple mark and mudstone. The transition the different units are known. The transition procedure from one bedding type to another used procedure involves random processes that are dependent, Markovian random processes which have either independent or dependent. If the depositional a memory that extends one step backward in the process is independent, previous deposition will depositional process. As observed in nature, median have no influence on present deposition. However, grain size and wave thickness (cross-bedding if it is dependent, past deposition will influence and ripple mark) decline upward in the synthesized either present or future deposition. Such a dependent sections as proportions of the different bedding depositional process can be thought of as having a types change. Grain size and permeability were memory that extends backward in time through one also incorporated into the sections. By changing or more pulses of deposition. A process with a the transition procedures, bed thickness distribu- memory can be described by a Markov process. tions, rate of upward decline or sand wave height Because the concept of memory or dependence and length, different types of sections can be appears to be in accord with our understanding of synthesized, thus making it possible to model many many depositional processes, Markov processes different sedimentation problems. were used to synthesize the bedding sequences of this study (see Appendix). INTRODUCTION The above methods are perfectly general and are This paper describes a general method for appropriate for any stratigraphic section or bedding synthesizing stratigraphic sections and bedding sequence: bedding types in a beach deposit, an sequences of sedimentary, metamorphic or igneous evolving carbonate bank or the changing lithologic origin. Synthetic is of interest for fill of a thick geosyncline sequence. We chose to several reasons. close correspondence between synthesize a vertical profile of a fluvial sandstone real and synthetic sections suggest that the factors body because its characteristics were well docu- used in the synthesizing model may indeed be the mented, much was known about fluvial processes and fluvial-deltaic sandstone bodies constitute an correct ones, thus giving the investigator a check important class of petroleum reservoirs. on his assumptions. Rapid, inexpensive simulation of many stratigraphic sections permits one to CHARACTERISTICS AND ORIGIN OF synthesize a rock body (sandstone or carbonate FLUVIAL CYCLE reservoir) or, on a larger scale, the fill of a sedimentary basin. Harbaughl gives an example of The fluvial cycle has been well documented in mathematical simulation of a carbonate basin. He recent years by ~ersier,2Allen3-5 and ~isher.6.~ simulated the basin in the hope that improved Deposits from fluvial cycles range from 10 to 150 ft prediction would follow better understanding of the or more in thickness and are characterized by a "fining upwards": coarse with occa- Original manuscript received in Society of Petroleum Engineers office Jan. 13, 1967. Revised manuscript of SPE 1778 received sional conglomerates grade upward into medium- to May 12, 1967. Publication authorized by permission of State fine-grained sandstone, and hence into siltstone and Geologist, Indiana Dept. of Natural Resources, Geological Survey. @ Copyright 1967 American Institute of Mining, mudstone. The dominant sedimentary structure is Metallurgical, and Petroleum Engineers. Inc. inclined bedding: thick cross-beds in the lower half '~eferencesgiven at end of paper.

SEPTEMBER, 1967 become thinner upward and pass into ripple mark caps the cycle. The conglomerates commonly consist and parting lineation. Also present are some of fragments of peat, coal, shale or clay ironstone. seemingly massive beds which, in reality, are Casts and carbonized impressions of logs in the probably either thinly laminated or ripple marked,8 basal part of the cycle, as well as rootlets and and a few mudstone or shale partings. Mudstone finely macerated plant debris in the overlying Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021

-. 0 90 I80 270 (mm) THICKNESS ORIENTATION BEDDING (cm) (degrees) TYPES MEDIAN GRAIN SIZE CROSS-BEDDING

EXPLANATION

~.:v -0 . - Cross- bedded sondstone

Floser structure: Thin beds of ripple Massive" sondstone marked siltstone ond fine-grained - .... sondstone interlominoted with shale. PIG. 1 - UPWARD DECLINE OF GRAIN SIZE AND THICKNESS OF CROSS-BEDDING IN SANDSTONE, MANSFIELD FORMATION, SOUTHWESTERN INDIANA. 22

244 SOCIETY OF PETROLEUM ENGINEERS JOURNAL siltstones and mudstones, are common. Although in fluvial sandstone bodies. other may be present, they Figs. 2 and 3 link depositional processes to the are not present in great abundance. Most fluvial empirically observed fining upwards fluvial cycle. sandstones are faunally barren, but concentra- This linkage provides the rational in using Markov tions of vertebrate remains are not uncommon processes to generate synthetic sections. in Mesozoic and later sandstones. Fig. 1 shows a vertical profile of the Pennsylvanian Mansfield METHODS formation that illustrates a fairly typical fluvial The upward change in kinds and thickness of cycle. Excellent documentation of the vertical sedimentary structures in a sandstone body were profile of an ancient fluvial sandstone body is approximated by dividing the sandstone body into given by Ilewitt and M0r~an.9 three parts: a lower, dominantly cross-bedded zone, Although there are some differences in details, a middle rippled and horizontally bedded zone and the fluvial succession is fairly well established and an upper zone transitional to mudstone. The has been related to four different facies. 101117677 following assumptions were made: (1) the deposi- These are (1) poorly bedded basal thalweg tional process is Markovian with a one -step that may contain granules, pebbles and even dependence, (2) the sandstone body consists of cobbles, (2) cross-bedded sands of point bars, (3) Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021 five bedding types or states: cross-beds (S1), massive rippled and horizontally bedded fine sands and silts, beds (S2), parting lineation (S3), ripple (wavy) usually of the inner flood plain, and (4) backwater bedding (S4) and mudstone (S5), (3) thicknesses of silts and clays. Origin of this fining upwards the different bedding types are assumed to have a sequence is shown in Fig. 2, a block diagram of a log-normal distribution and (4) the upward decline at a bend. Basal thalweg sands in current competence is reflected in sand wave usually constitute only a small portion of the total thickness, which decreases linearly upward. The section and may be indistinguishable from the last assumption implies that most of the sandstone point bar sands, almost all of which predominate. body was principally deposited in the lower flow Some parting lineation may be found in any of the regime. four facies, but it is best developed in the rippled The dynamics described in Figs. 2 and 3 provide and horizontally bedded fine sands and silts. These the justification for the assumption that the four facies are deposited simultaneously in geo- depositional process is Markovian with a one-step graphically distinct areas and are superimposed in dependence. Average hydraulic conditions at the a regular manner as the stream migrates laterally. sediment interface in the channel change but slowly The velocity distribution within the channel at as deposition proceeds; hence, there is a strong the stream bend provides more insight into the probability of a continuity or dependence in the origin of the fining upward (Fig. 3). The highest resultant types of bedding and sedimentary structures velocity is proximal to the outside of the bend. The deposited. This implies a memory. Such-a memory inset graph of Fig. 3 shows a plot of the decline of is, moreover, fully consistent with short-term, day velocity along the bottom, from the thalweg at X to to day variations in discharge. It is these short-term the inside of the bend at Y. This approximately perturbations that introduce the random, probabilistic linear decline in velocity shoreward is accompanied elements in most sedimentary processes. Of course, by an upward decline of both sand wave height (cross-bedding and ripple mark thickness) and grain size because both are correlated with each other and discharge12-14 at least in the lower flow regime. Hence, as one moves along the bottom from the thalweg to the inside of the meander bend, both sand wave height and grain size decline in response to declining velocity and thus produce the upward decrease of bed thickness and grain size observed

Direction of migration Ripple bedding and parting lineation \

-~o~L~"~~~~~"~~~~~~4 812 16 20 24 28 32

Hundreds of Feet

L~hannel center lag deposit, thin FIG. 3 - VELOCITY DISTRIBUTION (FT/SEC) IN and discontinuous CURVING WILLOW CUTOFF OF MARSHALL POINT REACH OF MISSISSIPPI RIVER (MILE 448.5).23 NOTE FIG. 2 - DIAGRAM SHOWING POINT BAR AND ASYMMETRY. IN LOWER RIGHT CORNER, PLOT OF DEVELOPMENT OF SAND FACIES (CIRCLES SHOW MEAN VELOCITY (FT/SEC) AGAINST DISTANCE FROM RELATIVE GRAIN SIZE OF SAND FACIES). THALWEG POSITION X TO INNER SHORE Y.

SEPTEMBER, 1967 as our insight into sedimentary processes grows, distribution, P is percentage of reduction desiredi longer memories may be more appropriate than the by the time height H is reached, H is limiting height one-step dependence used here. Certainly there is (length) of section and h is current height. little geologic evidence to support the idea that For example, if a reduction of 70 percent were successive pulses of sedimentation are independent desired (if total height were 200) and if current of one another; transition matrices based on height were 50, then t'= [1- 0.7(50/200)]t = 0.825t. lithologic transitions in ancient sediments show a A linear reduction of 70 percent was assumed. The one-step dependence to be a reasonable model following procedure was used to generate a section. (Potter and Blakely, in preparation). 1. Choose randomly an initial state i which then Because no actual data were readily available, specifies a particular row of the transition matrix P. values for the transition probabilities pii were 2. Choose randomly the thickness t of the ith assumed for each of the transition matrices in the lithology from its frequency distribution. three zones using the data in Table 1. These values 3. Reduce t to t : seemed to be reasonable based on both our 4. Select the following lithology j in accordance experience and as judged by outcrop and core with the probabilities of the pij's of the ith row, all descriptions of both modern and ancient fluvial of which sum to one.

deposits. In addition, these values yielded 5. Now, let i = j. Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021 equilibrium proportions of bedding types comparable 6. Return to Step 2. to those found in actual sections. The equilibrium Fig. 5 illustrates the process but omits the proportions were obtained by iteration (128 times) thickness conversion, Step 3.. The process was of the transition matrices (Appendix). computed on a CDC G-15 computer. Blakely and Thickness distributions of the different bedding Potter15 describe the program. Fig. 6 details a types are shown in Fig. 4. Parting lineation and portion of the resultant section. Empirical correla- mudstone were given a common distribution. The thickness of cross-bedding was modeled from actual data, but other thicknesses were assumed. Bed thickness of cross-bedding and ripple mark was linearly reduced to simulate the upward decline in sand wave height. This linear reduction is a good approximation to the linear decrease in velocity observed along the bottom from one thalweg to the edge of the point bar (Fig. 3). Thickness was reduced by the equation t ' = [l - P (h/H)]t, where t' is adjusted thickness, t is randomly selected thickness from the log-normal

TABLE 1 - TRANSITION PROBABILITIES*

Log Thickness (cm) FIG. 4 - LOG- NORMAL PROBABILITY PLOT OF THICKNESS DISTRIBUTIONS OF DIFFERENT BED- DING TYPES.

0 0 Given the i th Bed 0

Randomly Randomly Sample Sample Transition Matrix Thickness to Select Distribution a j th Bed of i th Bedding Type

now let i = j

Randomly Randomly Sample Sample Transition Matrix Thickness to Select Distribution of New i th 0 0 Bedding Type 0 0 K *S crass-bedding; S2, massive beds; S,, parting lineation; S 4, :;pple mark; and S5, mudstone. FIG. 5 - SCHEME OF GENERATION PROCESS.

246 SOCIETY OF PETROLEUM ENGINEERS JOURNAL lations between thickness of cross-bedding and deviate. Of course, porosity can also be simulated grain size and permeabilities made it possible to if desired. incorporate both of the latter into the profile. Grain size and cross-bedding bed thickness data RESULTS AND SIGNIFICANCE from the Pennsylvanian Mansfield formation of Fig. 7 shows the resultant vertical profile which southern Indiana were used to obtain a regression equation from which to predict grain size, given closely resembles published examples of fluvial thickness of cross-bedding. These data yielded the ones9.l7 in that bedding thickness in the lower cross-bedded zones shows a weak, irregular, upward regression equation loglOy = 2.13 + 0.190 loglOx, decline that accelerates near the top and conse- where y is grain size, microns; and x is cross-bed and ripple mark thickness, cm. quently reduces both permeability and grain size On the basis of data from the Lower Cretaceous as the sandstone passes upward into mudstone. Bentheimer sandstone in ~erman~,l~the following The transition matrices of Sections A through C in regression equation was obtained for permeability Table 1 were used in the top, middle and basal zones. and grain size: loglo y ' = - 2.1007 + 2.221 loglox ', where y ' is permeability, md; and x ' is grain size, Repeated synthesis with rhe same transition

microns. We wish to emphasize that this somewhat matrices will yield sections with closely comparable Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021 circuitous route could be avoided if permeability equilibrium proportions of bedding types. However, and grain size could have been directly generated because there is a probabilistic element in the rather than being introduced into the model via model, neither the sequence of bedding types nor fluctuations of bed thickness. the thickness of the cross-beds and ripple marks To better simulate actual measured profiles, it will be the same; consequently, grain size and was necessary to use, for both permeability and permeability will vary as well. This is equivalent grain size, the standard error of estimate of the to generating new sections. above regression equations when predicting their, Sections can be altered by changing transition values. The standard error of estimate S of a matrices, by changing rate of reduction of sand regression equation is a measure of the reftbility wave height, by changing thickness distributions of the regression of y on x. By using the standard and by changing length of section. Table 2 summa- error of estimate, the variability of the regression rizes these possibilities and suggests the geologic is incorporated in the predicted values of y so that factors that they represent. Although primarily they no longer are precisely predicted by the formulated with quartzose sands in mind, an independent variable. This procedure makes both the grain size and permeability curves follow TABLE 2 - MODEL VARIABLES AND SUGGESTED GEOLOGIC INTERPRETATION bedding thickness in a general way rather than precisely and thus better simulates reality. This Variable Interpretation

was done by the following formula. y* = y + Sy,(R), Length of section Complex, but related to profile of where y* is adjusted for the error of estimate, y is equilibrium, and hence relative the computed value using the regression equation rates of subsidence and depasi- tian. and R is a randomly chosen value of a normal Sand wave height Correlates with stream discharge in lower flaw regime, greater dis- LOWER ZONE UPPER ZONE charge producing higher sand waves; analogous interpretation for marine currents. Cross-bedding (%) Vertical change of sand May increase, decrease or hove no wave height gradient. Increases upward in re- gressive marine shelf and beach "Massive" (S2) sands (greater turbulence of shoaling water). Decreases up- ward in fluvial and tidal sands i.n Parting lineation (S3) response to lateral migration of thalweg (lesser turbulence an in- side of paint bar) and in trans- Ripple bedding (S,) gressive marine shelf sands (lesser turbulence as shelf deepens off shore). Mudstone (S,) Thickness distributions Complex, but correlates with grain of bedding types size and discharge, usually ' thicker beds representing more rapid deposition at higher dis- charge. Transition probability, An element of the transition matrix pi j that controls the probability of passing from a particular bedding type i to i and is thus related to sequential development of hydraulic conditions at interface. Transition matrix, P Controls over-all depositional FIG. 6-DETAILS OF SYNTHETICALLY GENERATED process as expressed in kinds SANDSTONE BODY USING INDICATED TRANSITION and proportions of different bed- MATRICES AND THICKNESS DISTRIBUTIONS OF FIG. 4. ding types.

SEPTEMBER, 1967 MUDSTON E BEDS SAND WAVE THICKNESS MEDIAN GRAIN SlZE PERMEABILITY PER -+ METER I (cm) (microns) 0 3 6 9 EO 20 40 60 80 100 80 160 240 320 I I I I MUDSTONE 1 C------RIPPLE BEDDING m I Mudstone ,-

CROSS-BEDDING Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021 Massive beds / Mudstone

FIG. 7 - SYNTHETIC VERTICAL PROFILE OF FLUVIAL SANDSTONE BODY. NOTE UPWARD DECLINE OF PERCENTAGE OF SANDSTONE, THICKNESS OF CROSS-BEDDING AND RIPPLE MARK, GRAIN SIZE AND PERMEABILITY. BASED ON TRANSITION MATRICES A (BOTTOM), B (MIDDLE) AND C (TOP) OF TABLE 1 AND THICKNESS DISTRIBUTIONS OF FIG. 4. MUDSTONE BEDS -Z SAND WAVE THICKNESS MEDIAN GRAIN SlZE PERMEABILITY PER - METER % (cm) (microns) (md)

RIPPLE BEDDING I.. Mudstone b E

FIG. 8 - SYNTHETIC VERTICAL PROFILE (COMPARE WITH FIG. 7). BASED ON TRANSITION MATRICES C (TOP), A (MIDDLE) AND D (BOTTOM) OF TABLE 1 AND THICKNESS DISTRIBUTIONS IN FIG. 4.

248 SOCIETY OF PETROLEUM ENGINEERS JOURNAL essentially similar but somewhat more complicated (Fluvials)", Eclogue ~Gol.Helvetiae (1959) Vol. 51, table could be constructed for carbonates. 854-893. By changing the above factors, one can radically 3. Allen, J. R. L.: "Henry Clifton Sorby and the Sedi- alter the resultant profile. For example, Fig. 8 mentary Structure of Sands and Sandstone in Relation to Flow Conditions", Geologie en Mijn. (1963) Vol. shows another profile in which the basal part 42, 223-228. consists mostly of mudstone interbedded with ripple 4. Allen, J. R. L.: "Fining Upwards Cycles in Alluvial bedding, the middle zone consists principally of Successions", Liverpool and Manchester Geol. J. cross-bedding and the upper part is transitional to (1965) Vol. 4, 229-246. mudstone. The transition matrices (Sections A, B 5. Allen, J. R. L.: '", Proc., Amer. Soc. The foregoing approach assumes that permeability Civil Eng., Hydraulics Div., Paper 1331 (1957). and porosity are dependent upon rock properties Simons, D. B., Richardson, E. V. and Nordin, C. F., Jr.: "Sedimentary Structures Generated by Flow in such as median size, sorting, clay content, etc. Alluvial Channels", Primary Sedimentary Structures This assumption certainly seems reasonable and is and Their Hydrodynamic Interpretation, Soc. Econ. attractive to geologists because it permits them to Paleontologists and Mineralogists, Spec. Pub. 12 vary the geologic factors that control the primary (1965) 34-52. depositional features of a reservoir. The geologic Scheidegger, Adrian and Potter, P. E.: "Bed Thick- understanding of such geologic factors appears to ness and Grain Size: Cross-Bedding", Sedirnentology (1967) in press. be more advanced, at least in some environments, Blakely, R. F. and Potter, P. E.: "Computer Program than our understanding of permeability variations in for Synthetic Generation of Stratigraphic Sections" a reservoir. Nonetheless, it might well be profitable (1967) in preparation. to challenge this assumption and work directly with von Engelhardt, Wolf: Der Porenraum der Sedimente, permeability and porosity, if possible. If this path Springer-Verlag, Berlin, Ggttingen and Heidelberg should be followed, it seems likely that Markov (1960) Fig. 99. processes will play a role. Noorthorn v. d. Kruit, J. L..and Lagaaij, R.: "Dis- placed Faunas from Inshore Estuarine Sediments in ACKNOWLEDGMENTS the Haingoliet (Netherlands)", Geologie en Mijb. (1960) Vol. 39 (N.S. 22) 711-723. We are indebted to W. A. Pryor, U. of Cincinnati, Kemeny, J. G. and Snell, J. L.: Finite Markov Chains, for his help with the construction of Fig. 2. J. L. D. Van Nostrand Co., Inc., Princeton, N. J. (1960) 71. Doob, U. of Illinois, assisted us with the use of Fisz, Marck: Probability Theory and Mathematical Statistics, John Wiley and Sons, Inc., New York Markov processes. (1963) 250-322. Parzen, Emanuel: Stochastic Processes, Holden Day, REFERENCES Inc., San Francisco. 1. Harbaugh, J. W.: "Mathematical Simulation of Marine Kemeny, J. G., Mirkil, H., Snell, J. L. and Thompson, Sedimentation with IBM 7090/7094 Computers", G. L. : Finite Mathematical Structures, Prentice-Hall, Kansas Geol. Survey Computer Contr. 1 (1966). Inc., Englewood Cliffs, N. J. (1959) 399. 2. Bersier, Arnold: lc~eq~enceDetritiques et Divagations Potter. Paul Edwin: ''Sand Bodies and Sedimentary

SEPTEMBER, 1967 Environments: A Review", Bull., AAPG (1967) 337- say that a sequence of events forms a Markov 365. process; if for any i, j, n = 1, 2, 3, . . , the proba- 23. Krumbein, W. C.: "FORTRAN IV Computer Programs bility that an event E will occur at a given trial, if for Markov Chain Experiments in Geology", Computer specified events have occurred at the preceding Contribution 13, Kansas Geological Survey (1967) 38. trials, is a number which depends on E and on the APPENDIX last preceding event, but not on the other preceding events. Thus, P ij= P(E. /E i) = P(E~/E~,Eh, Eg . . . .). I Below are briefly outlined the Markov processes For a three-state system of cross-bedding (S1), used in this paper. More detailed but still elementary ripple mark (S2) and parting lineation (S3), the accounts can be found in Kemeny and Snell,18 transition matrix P is Fisz, 19 Parzen20 and m rum be in.^^ If one considers a three-state system of cross- bedding .(S1), ripple mark or wavy bedding (S2) and parting lineation (Sg), the probabilities of the three different states are p1 = N1 /N, P2 = N2 /N and p3 = N3 /N, where N1, N2 and N3 are the number of different events (bedding types) of the three states, where pl is the probability of a-cross-bed fiollowing and N is their sum. If these long-term probabilities a cross-bed, p12 is the probability of ripple mark Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021 do not change (remain stationary) during the following a cross-bed, P3.2 is the probability of depositional interval, the probabilities of depositing ripple mark following parting lineation, etc. Each a different lithology at each step of the process are row of this matrix sums to one, since something very nearly pl, p2 and p3 if the number of beds in always has to follow any given lithology. the section is large. This is true whether the Iterating or raising the transition matrix to depositional process is Markovian or not. successively higher powers gives the transition The concept of conditional probability is central probabilities of going from lithology i to j in 2, 3, to Markov processes. Conditional probability is 4 . . . ., up to n steps. If all the elements pijn of defined as P (E~/E~)= P (E1E2 )/P (E2 ), where the some power of the transition matrix of a Markov symbol P(E1/E2) is read "the probability of an chain are positive, as they commonly are, the chain event El given the event E2" and the symbol is called regular and the powers p of the transition P (El E2) is read "the probability of El and E2", matrix approach a limiting matrix T with the follow- i.e., their joint occurrence. If the events El and E2 ing properties: (1) all its elements are positive, are independent, then P (E1/E2) = P (El), which (2) each row is the same and sums to one and shows that the probability of El given E2 depends (3) each row is a vector which gives the equilibrium only on the probability of El alone; in short, El is proportions of the different states, in the sense completely independent of E2. that if the process is started off with probability If the depositional process consists at each step Pi for state j, then after n steps the probability will of independent trials such as these, it does not still be pi for state j. For a proof see Kemeny have a memory. If, on the other hand, transitions et aL21 and Kemeny and Snell?8 The equilibrium from one bedding type to another depend on proportions can also be obtained directly by solving conditional probabilities, the process has a memory a set of simultaneous equations.21 and is a dependent Markovian process. Dependence The last of the above properties is especiaily may extend backward one or more steps. Here we useful because it permits one to see what long-term are primarily concerned with only one-step depend- proportions of bedding types a particular transition ence. If the number of states produced by the matrix will ~ield.Based on matrix iteration of 128 depositional process is finite, as is usually true, times the equilibrium proportions in the different the corresponding probability process is called a states of the four transitions, matrices of Table 1 finite Markov process. were (A) 0.53, 0.21, 0.05, 0.06, 0.015, (B) 0.05, The transitions from one state to another in a 0.01, 0.18, 0.53, 0.24, (C) 0.00, 0.00, 0.05, 0.06, Markov process are given by a transition matrix 0.89 and (D) 0.05, 0.04, 0.04, 0.47, 0.40. The of probabilities. The individual elements P.. of a relation between transition matrix and long-term l! transition matrix are called transition probabilities equilibrium proportions is not a unique one, however, and give the probability of state j (mudstone for because different transition matrices can yield the instance), if the preceding state was i (parting same equilibrium proportions. lineation). The probabilities are approximately the "empirical probabilities" Nij /N where Nij TERMS AND CONCEPTS is the number of pairs ij and Ni is the total number Absorbing state: A state that once entered of i's. Each Pij is a number between 0 and 1. If cannot be left. pii is 0, the transition from states i to j cannot Conditional probability: The probability of one occur; if pii is 1, the transition is certain to occur. event given another. If pii = 1, i cannot be left. Finally, the symbol Event: The outcome of an experiment. piin is used to indicate the probability of going Finite Markov process: A Markov process with from state i to state j in n steps. only a finite number of states. With these definitions, we may more formally

SOCIETY OF PETROLEUM ENGINEERS JOURNAL Fluvial cycle: A sequence of events through which Memory: The influence of past on present depo- one pulse of the fluvial process moves (character- sition or of present on future deposition. May extend ized by fining upwards). Cycle may be complete or one or more steps backward. interrupted. State: A particular lithology or bedding type. Independence: Two events El and E2 are Stationarity: Same as equilibrium; transition independent if the probability of E 1 given E2 probabilities do not change with time. A steady- equals probability of El alone. state process. Independent trials process: A probability process Transition probability pij: The probability of without a memory. Probability of deposition depends going from state i to state j. only on the long-term frequencies of the different Stochastic process: Processes that involve the lithologies. mathematics of random events, either independent Iteration: The process of raising a matrix to or dependent. successively higher powers. For a transition matrix, Transition matrix P: A matrix whose elements each power corresponds to an additional step in are transition probabilities. Each row sums to the process. one. Controls the over-all depositional process Markov process: Loosely speaking, any probabil'ity as determined by kinds and proportions of bedding

process with a memory of one step. or lithologic types. *** Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/243/2152790/spe-1778-pa.pdf by guest on 30 September 2021

SEPTEMBER, 1967