Use of Uncertainty Analysis in Evaluating Hydrocarbon Pore
Volume in the Rainbow-Zama Area Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 K. C. G. Pritchard, SPE-AIME, Chevron Standard Ltd.
Introduction Oil was discovered in the Rainbow Lake area of north well is high. As a result, development wells have been ern Alberta in 1965 and the play extended north to the drilled only where the seismic data show the feature Zama Lake area in 1966. By the end of 1968 some to be large. There are few off-reef wells, and many of 150 separate pools had been discovered, with total the features are one-well pools, penetrated only by a oil in place approximating 2.1 billion bbl, of which crestal well. 800 million bbl is estimated as recoverable under the In Alberta, producing capability exceeds market producing mechanisms now in effect. The play is by demand. For this reason, a production allowable is no means exhausted and could extend both to the established for each well in the Province by the Alber northwest and to the southeast. It is conceivable that ta Oil and Gas Conservation Board. This allowable is the Rainbow-Zama area of northern Alberta will ulti prorated on the basis of the recoverable reserves. mately be found to hold 5 billion bbl of oil, 3 billion Initially, the Conservation Board assigned Rainbow of which will be recovered through various recovery Zama allowables by assuming that the wellbore pa schemes. rameters were representative of the 160-acre drilling The features exist as pinnacle reefs that grew on a spacing unit. Seismic data indicated the features to be common platform.5 There is an extreme range in the essentially circular, although seismic estimates of the size of the reefs: the hydrocarbon pore volume varies exact reef flank position and side-slope angles proved from 500,000 to 500,000,000 STB. In general, the to be unreliable. Fig. 1 is a schematic of the four gross larger features have been found in the Rainbow area. models that could be deduced from the seismic data The seismograph has proven to be particularly suc and initial wellbore evidence. The range of uncertainty cessful in the Rainbow-Zama area. Modem tech is extreme. For example, the feature could conceivably niques, including common-depth-point stacking, have vary between a cone and a cylinder, with the conical located the crestal portion of the pinnacles with almost volume being one-third the cylindrical volume. The 100 percent success. The areal extent of the reefs and most likely geometric configuration is that of the the size-shape relationship are not so easily defined by frustum of a cone with side slopes of 30° to 45 0. seismic methods. A few wells, drilled on development Most of the available acreage in the Rainbow-Zama locations in the flank positions, either did not en area was competed for under bonus bid strategy. Bids counter the reef or encountered it below the oil-water were generated on the postulated reserves of an indi contact. This presents a paradoxical situation in which vidual feature, along with the best estimate of the the risk associated with an exploratory well is essen allowables to result from such reserves, overlaid on tially nil and the risk associated with a development a most reasonable decline in production capability.
A technique, including Monte Carlo simulation and history matching, for evaluating hydrocarbon pore volume in an environment of uncertainty is demonstrated through a case history study of a specific Rainbow-Zama pool in Alberta.
NOVEMBER, 1970 1357 Hence, it becomes increasingly important to evaluate reef -building organisms: stromatoporoids, stachyodes, accurately the hydrocarbon pore volume and the re and Thamnapora-type corals. covery mechanism of an individual pool or field. The The Muskeg formation is formally defined as the fact that this study is concerned with the former, does anhydrite and dolomite unit between the base of the not imply that recovery factor is less important than Watt Mountain and the top of the Keg River (Law, oil in place. 1955). In the off-reef position it may be subdivided into four units: a basal anhydrite member, a lower Exploration Concepts carbonate member (the Zama dolomite), an upper Geology anhydrite member and an upper carbonate member Following is a description of the stratigraphic col (often referred to as the Sulphur Point formation). A umnl, 2 of the Rainbow-Zama area. It is included here fifth and older member, the Black Creek salt, where to indicate the unique geological conditions that exist not removed by solution, is found in remnant form in in the area and to provide an appreciation for the the Rainbow area, but thus far has been absent in the reservoir and production problems associated with the Zama wellbores, although it is recognized on seismic exploitation of Rainbow-Zama reserves. Fig. 2 is a cross-sections. schematic of the stratigraphic section. The Black Creek salt is believed to have occupied The Chinchaga formation is approximately 300 ft a reef flank position and to have had a maximum
thick and is basically an anhydrite unit. The contact thickness of 260 ft. It is removal of the Black Creek Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 between the Chinchaga and the overlying Keg River salt by solution that has resulted in the collapse fea formation is marked by an abrupt change from an tures that are detected by seismic tests. This salt solu hydrite to carbonate. tion commenced during late Elk Point time (that is, The Lower Keg River platform has a uniform thick after most of the Muskeg formation was deposited), ness of 150 to 160 ft over the area. It is basically a and continued throughout the rest of the Devonian brown argillaceous calcilutite (or lime mud) with crin period. oid fragments and thin-shelled brachiopods. It is upon The lower anhydrite unit onlaps the reef slopes, but this platform that the individual reefs have grown. in the Zama area appears to be absent over the reef Generally, the platform has poor porosity and permea crests. Thus its thickness varies from 80 ft in the oif bility, but in some portions of the Rainbow area, there reef position to zero in the reef crestal position. This is sufficient flow in the platform to provide an active unit is easily identified on the density log because of water drive and pressure communication between in the high density exhibited by anhydrite. It is this lower dividual features. anhydrite that forms the cap rock over the Keg River The Keg River reef member is an organic reef with reef features, except where it is absent in the crestal associated detrital carbonates. It has grown as a multi positions. tude of pinnacles on the Keg River platform. The Dip meters run in flank wells detect bedding dips diameter of these pinnacles varies from a few miles up to 45 0 in this unit. This is taken as evidence that in the Rainbow area to several hundred feet in the the Black Creek salt was removed after deposition of Zama region. Vertically, the reefs grow from about this lower anhydrite unit, and the measured dips are 100 ft to a maximum of 800 ft. The vertical dimension the result of collapse. seems greater in the Rainbow area; a maximum of 350 The Zama dolomite is a brown, laminated, medium ft of thickness appears normal for the Zama area. The crystalline, sucrosic dolomite. It overlies the Keg River reef is often dolomitized and the basal 40 ft is con reef and when the lower anhydrite unit is absent in sistently a dolomitized crinoidal calcarenite. Organi the crestal position the Zama member forms a con cally, the Keg River reefs are composed of normal tinuous reservoir with the Keg River (Fig. 3). The Zama member exhibits good intergranular and vuggy porosity and permeability, which deteriorates towards the off-reef position. Since few off-reef wells have INITIAL CONSERVATION BOARD MODEL been drilled to date, the precise off-reef properties of the Zama dolomite are not known. The possibility 160 ACRES of off-reef permeability, however, leads to the in triguing suggestion of pressure communication be tween features through the Zama dolomite. The upper anhydrite member consists of micro crystalline anhydrite interbedded with fine crystalline dolomite. This unit forms the rock capping the Zama INITIAL CHEVRON STANDARD MODELS dolomite (and of course the Keg River reef where the two are commingled). It is found to be thicker in reef flank positions (about 450 ft) and thinner in reef-crest positions (about 300 ft). This indicates that the Black (-5 Creek salt was undergoing solution during deposition 0 of this member. CYLINDRICAL8 FRUSTUM OF A CONE CONICAL The Sulphur Point formation (or Bistcho member, MOST OPTIMISTIC MOST LIKELY MOST PESSIMISTIC after McCamis and Griffith2) is approximately 80 ft Fig. I-Gross models of Keg River reefs as suggested by thick. It overlies the Keg River reefs and has generally seismic and well bore evidence. been found to be gas bearing where reefs are present.
1358 JOURNAL OF PETROLEUM TECHNOLOGY Because there is scanty off-reef control, it is not known whether off-reef locations are productive or not. The FT. SIMPSON SHALE Sulphur Point contains a lower sucrosic dolomite unit (about 60 ft) which resembles the Zama dolomite and MUSKWA SHALE an upper limestone member (about 20 ft) of medium grained, brown, pelletoidal calcarenite. SLAVE PT. The Watt Mountain formation varies from 10 to 30 ft and is a green pyritic shale. It is thinnest over WATT MTN. the Keg River reefs and forms the cap to the Sulphur Point reservoirs. SULPHUR POINT The Slave Point formation is about 200 ft thick ~ and is a transgressive carbonate sequence. The lower ANHYDRITIC ~ (,!) portion of the formation is generally algal, grading a... w MBR. upwards into "cabbage" stromatoporoids and amphi ::..:: ::..:: .-J (/') pora. Fair to good porosity and permeability are w :::::> ZAMA deVeloped in the Slave Point, and it is often gas bear a:: :!: 1------/ w ( ing in the Zama-Rainbow area. In geological age it is a... a... equivalent to the very prolific Beaverhill Lake forma :::::> a:: BLACK ) K~i; w Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 tion to the south and east. CREEK RIVER > The Fort Simpson shales cap the Slave Point reser a:: REEF (,!) voirs. In the Zama-Rainbow area, the lowermost Fort 0:: U~ w (,!) ::..:: Simpson beds are termed the Muskaw shales. These w are black, bituminous, laminated and highly radio ::..:: L. KEG RIVER active. They are easily identified on the gamma ray t-=a... log. ::..:: CHINCHAGA .-J As has been indicted above, there are four pros w pective zones in the Zama-Rainbow area: the Keg -.J River reef, the Zama dolomite, the Sulphur Point formation and the Slave Point formation. The re Fig. 2-Stratigraphic section. mainder of this paper deals primarily with the Keg River reef and somewhat with the Zama dolomite as 600 the two are commingled in the Zama Keg River CC pool. 500 Seismography The reflection seismograph has been uniquely suc cessful in locating the Keg River pinnacle reefs. The 400 .,...: technique used is known as common-depth-point u.. 3 stacking. ,4 In concept the technique is simple, but I it could not be used routinely until the advent of the ~ 300 <.!) digital computer. The idea is to select a number of I.J.J surface shot-point and geophone locations so that = Reefoid a signal will be received at each of these locations from a common point in the subsurface. The most common configuration is referred to as 600 percent stacking. This provides for six independent travel 100 SUBPLATFORM paths to the one common depth point. The resulting seismograms are then digitized (or more commonly Genera Ily Tight Limestone recorded on digital tape) and summed via computer. o The technique is illustrated schematically on Fig. 4. CHINCHAGA The in-phase pulses amplify and the out-of-phase Fig. 3-Schematic of a typical Zama reef. pulses cancel, providing the composite seismograph shown on the bottom of Fig. 4. If the seismic program has been properly designed, the in-phase pulses rep I I I resent the common depth point; the out-of-phase PRIMARY MULTIPLE pulses represent multiples and random noise. ~ - As can be imagined, the surface positioning of shot 'A points and geophones is extremely critical to this type 1;:3: ~t:: of seismic program. Hence-initially, at least-con K tinuous interpretation of field data and adjustment of CONVENTIONAL f1 ~ ~ r-- ~ field technique are necessary. In the Zama-Rainbow STACKED TRACE V I.. - ..... MULTIPLE RESIDUAL I area, very close line spacing is required, and once a I I I I I I I pinnacle has been detected a series of radiating seismic Fig. 4-Schematic showing the concept of lines is run across it to delineate the feature areally. common-depth-point stacking.
NOVEMBER, 1970 1359 NW Stacked seismic profiles across Rainbow-Zama fea 6-32 2-32 7-32 10-32 12-33 12-28 2Jf.SE • • •• • -<>- tures suggest that the Keg River reef exists as the FEET frustum of a cone, the crestal area is well defined, and 800 ~ome estimate can be made as to the areal extent and slope of the reef flank. Because of the success of wells 600 drilled in the crestal position, it was thought that seismic data could also accurately map the reef areal extent. However, as pressure-production history be came available and the oil in place was computed by material balance, the volumetric estimates of oil in --200 place based on seismic data proved to be optimistic. Geologic Models lOWER KEG RIVER MEMBER As drilling progressed in the Rainbow-Zama area, DATUM: CHINCHAGA FORMATION interpretation moved from the realm of geophysics o 30iXJ into that of geology. The reef facies proved to be IwawI J.""..",! extremely complex and so it became necessary to Fig. 5--Geological model o·f the Rainbow construct composite geological models that would take Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 Keg River "A" pool.' into account the spatial distribution of the various ZAMA DOLOMITE CONTROL WELLS facies in an individual reef. I 23 4 5 The most elaborate reef models have been con ~ ~., -9- 600 9.8 9.716 3:0 POROSITY structed by Banff Oil. 5 Fig. 5 shows the Banff model 11.0 of the Rainbow Keg River "A" pool. This interpreta UPPER tion is based on five wellbores that penetrated the reef 500 ANHYDRITE and two off-reef wells. As can be seen from Fig. 5, Banff has established 14 separate lithofacies. Since most of the reef features in the area have been pene 400 trated by only one well bore, it is necessary to extrapo late this spatial variation of lithofacies to the one-well pools. The Chevron Standard approach6 was somewhat different. Several wells were theoretically relocated as if they occurred in a single reef. The crest-flank rela tionship for each well was based on seismic data, 4.0 4. 5. 5.6 NIL POROSITY biofacies and lithofacies, thickness of the Keg River, I ~3r7.7 ~ I 2,9 I 100 • ••• ••• -9- and thickness and presence of anhydrites. Fig. 6 is the SUBPLATFORM I 234 5678 9 Chevron geologic model of the Keg River and over KEG RIVER REEF CONTROL WELLS lying Zama dolomite. Porosity in the Zama dolomite o CHINCHAGA deteriorates in the direction of the reef flanks; Fig. 6 shows 10 percent porosity in the reef crest position, Fig. 6--Chevron composite geological model deteriorating to 3 percent in the reef flank position. (modified after Ref. 6). In the Keg River reef, Chevron has established four lithofacies. Lithofacies I is referred to as "simple '" 25 '" 50 u.J u.J layered". This unit ranges from thinly laminated to ~ 20 ~40 thick-bedded; it contains lump-pellet micrite with :2 :2 < < '" 15 '" 30 minor corals, ostracods and stromatoporoids. oLA.. oLA.. Lithofacies II is termed "complex massive". It is a:: 10 a:: 20 u.J u.J nonbedded medium to very fine lump-pellet spari ~ 5 1 ~ 10 I :::> micrite with heterogeneous fossil assemblages includ z :::> 1 ,I Z 0 J.1 ,l ing tabular, encrusting stromatoporoids, simple corals, 0 6 7 8 9 10 o 0.1 0.2 0.3 PERCENT FRACTION OF PORE SPACE crinoids, bryozoa, gastropods, tentacultes, ostracods, AVERAGE POROSITY WATER SATURATION and algal tubes with stromatactis texture. '" 25 Lithofacies III is a transition from the complex u.J ~20 massive to the true reef and has been termed "near ~ reef wash" or "detritus". This contains angular frag '" 15 oLA.. ments of stromatoporoids and corals in a fine-sand a:: 10 to fine-gravel matrix of fossil and rock fragments. u.J n ~ 5 r I Lithofacies IV is the "true reef" and is made up of :::> rL Z II h bulbous stromatoporoids and colonial corals. 00 5 10 15 20 01.0 1.1 1.2 1.3 As can be seen from Figs. 5 and 6, these lithofacies THOUSANDS OF ACRE - FEET RES. BBlS.! S.T. BB L. ROCK VOLUME INITIAL OIL FM. are arranged more or less radially throughout the reef, VOL. FACTOR with the better effective porosity existing in the rim Fig. 7-Machine-generated histograms; input parameters regions of the reef and separated from the central core to the volumetric equation. by a zone of poorer reservoir characteristics.
1360 JOURNAL OF PETROLEUM TECHNOLOGY Three oil-base cores had been taken in the Rain The Zama Keg River CC Pool bow-Zama area. Plots of water saturation vs porosity General Information were prepared, and most pessimistic, most optimistic, The seismic data show the feature to be essentially and most likely curves were drawn through the maze radial and to cover an area of about 200 acres. The of points. A cross-correlating feature of the Monte discovery well was drilled by Chevron Standard at Carlo simulator was used so that three porosity-water the location LSDI3-SEC34-TI17-R5-W6. This lo saturation relationships were put into the simulator. cation was predicted to be in a crestal position by Recent work has shown that a lithologic split-out is seismic information; and subsequently the seismic possible so that a connate water-porosity relationship interpretation was confirmed by the wellbore. Texaco can be obtained for subunits (for example, the Zama Canada drilled a development well in LSD4-SEC3- dolomite and the Keg River reef). It is also possible T118-R5-W6 to the north of the Chevron discovery. to incorporate permeability into a three-parameter, Chevron geophysics suggested this well would pene "Kozeny-like" equation. However, these correlations trate the reef flank, which it did. The Zama dolomite were not available at the time the study was under and the Keg River reef are commingled in the 13-34 taken, so only the correlations as shown were used. well but are separated by 77 ft of dense anhydrite in Several PVT analyses were available over the area. the 4-3 well. These suggested that the initial oil formation volume
In this study, two methods of evaluating uncertainty factor ranged from 1.14 to 1.27 reservoir bbljSTB. Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 have been used. Monte Carlo simulation, which has The value measured on a sample taken from the been adequately described elsewhere, 12,13 in general, 13-34 well is 1.20 reservoir bbljSTB, so this was consists of placing a range and a type of distribution selected as the most likely Boi for the Keg River CC on the various parameters that occur in an evaluation. pool. These parameters are grouped in the form of an A sensitivity analysis of the random data shows the algebraic equation, and this equation is then solved oil in place to be strongly dependent on the rock many times by computer on sample parameters that volume. Results of the volumetric computation are have been generated by a Monte Carlo random tech shown as Figs. 8 and 9. Fig. 8 is the histogram of oil nique. The range and distribution of the solution is in place as it was solved by the machine, and Fig. 9 then presented in statistical probability format. is the cumulative probability curve of the same data. History matching has also been discussed else As can be seen, the oil in place ranges from about 2.5 where. 10,11,14 In sophisticated history-matching tech to 7.0, with a modal value of 3.2 million STB. niques, the match function is minimized in a logical systematic fashion. The two most common techniques Oil in Place from Steady-State Material Balance are the method of steepest descents and the method of Fig. 10 shows the pressure-production history of the least squares. History matching is similar to Monte Zama Keg River CC pool until the time the steady Carlo simulation in that a multitude of cases are state material balance evaluation was undertaken. evaluated via computer and the input parameters are Bubble-point pressure was measured on a bottom assigned a range. However, it is not a random process. hole sample from the 13-34 well and found to be Cases that do not satisfy the match criterion are re 1,055 psig. If a pool has undergone a steady-state jected and parameters are selected for the next itera depletion, the pressure-production history above the tion in such a way as to allow the match function to bubble point should exhibit a straight line. A straight pass through a minimum. line can be fitted to the first three pressure points of the Keg River CC pool. However, the points of Aug. Volumetric Hydrocarbon Pore Volume 20, 1967, and Sept. 16, 1967, exhibit a line displaced Hydrocarbon pore volume is calculated from the fol upwards but with the same slope as the original line. lowing well known equation: In the Keg River CC pool, a zone of poor porosity and permeability precludes hydraulic communication (1) with the Keg River platform, no gas cap exists, and the field was undersaturated at discovery. Hence it The Monte Carlo simulation technique was used to was felt that the steady-state material balance should evaluate this equation as it applied to the Keg River be applicable to this pool. CC pool. Fig. 7 shows histograms of the input vari The apparent pressure anomaly was rationalized as ables as they were generated by the machine. Con follows. The 4-3 well penetrated the Keg River reef sidering the wide variation in the possible geometric during the second week of July, 1967. A formation configuration of the pool (Fig. 1), it was felt that rock pressure of 2,100 psi was expected, but because of volume (V R) could reasonably vary from 5,800 to withdrawals from the 13-34 well the reservoir pres 16,000 acre-ft. A most likely rock volume based on sure was only 1,500 psi. Thus, the drilling mud the truncated cone concept (but ignoring the over weight overbalanced the formation pressure and lost lying Zama dolomite) was 6,450 acre-ft. Footage circulation resulted. No good estimate of the volumes weighted porosities from several well bores in the area of drilling mud lost to the formation is available, but showed a range in porosity from 7.3 to 9.3 percent. it is conceivable that it could have been great enough The footage weighted porosity from the two wells in to recharge the formation. The last two pressure the CC pool was 8.2 percent, and hence 8.2 percent points, therefore, exhibit a continuing straight-line was selected as the most likely porosity to apply to decline but are displaced upwards from the original the pool. line.
NOVEMBER, 1970 1361 ~ 25 Above the bubble point, in a depletion-type res ervoir without water influx, oil in place may be calcu lated from the steady-state material balance equation:
20 N = NpBo ~ P"'"" (2) ...J ~ Ce t:J.p BOi ' Q.. I- ~ oct 15 where (I) (1- Sw)co + Sw Cw + Cma "'- C = (3) 0 e (1 - Sw) 0:: 10 LU ~ c:c Since the Monte Carlo simulator was selected as the ~ l- evaluation technique, and it was not known which :::> z 5 input parameters would most influence the computa tion, all parameters were allowed to vary. Those that were functions of other parameters were evaluated by I I I I I I assigning a variable range and distribution to those 00 I 2 3 " 4 5 6 7 8 9 ~ other parameters. For example, the van der Knaap9
MILLIONS OF STOCK TANK BARRELS correlation shows rock compressibility to be a func Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 tion of porosity, overburden pressure and internal Fig. 8-Histogram; volumetric oil in place. pressure. Overburden pressure, in tum, is a function of rock density and depth. As can be imagined, handling the van der Knaap rock compressibility correlation presented a difficulty. 100 It was overcome by defining the rock compressibility as 90 '\ Cma = 10 logA , (4) 80 1\ >- and expressing the equation of the van der Knaap S 70 :\ lines as iii ca 60 \ o log A = Intercept - slope X log 3,000 (5) a:: Pnet 0.. 50 \ .... and ~ 40 1\ 1362 JOURNAL OF PETROLEUM TECHNOLOGY TABLE I-INPUT PARAMETER DATA TO THE MONTE CARLO SIMULATOR FOR NONDEPENDENT VARIABLES Distribution Variable Minimum Mode Maximum Type Depth, ft 5,050 5,100 5,150 Triangular Rock density, psi/ft 0.90 1.00 1.05 Uniform Initial pressure, psig 2,133 2,139 2,144 Uniform 1,319 1,322 1,325 Uniform Porosity, fraction 7.3 B-2 9.3 Normal ON compressibility, voi/Vol/psi 6.37 x 10-" 7.67 X Hi" 9.20 x 10-' Normal Water compressibility, vol/vol/psi 2.4 x 10-" 2.7 X 10-" 3.0 X 10-" Triangular Oil produced, STB 47,000 47,200 50,885 Triangular Rock volume, acre·ft 5,800 6,450 16,000 Triangular Initial oil formation volume factor 1.14 1.20 1.27 Normal Slope (for van der Knaap computations) 0.570 0.609 0.624 Triangular TABLE 2-INPUT PARAMETER DATA TO THE MONTE CARLO SIMULATOR FOR INTERDEPENDENT VARIABLES Water Saturation, Normal Distribution Porosity Water Saturation 7.0 7.5 8.5 9.0 9.5 Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 8.0 -- Minimum 0.057 0.053 0.050 0.047 0.044 0.042 Mode 0.186 0.173 0.162 0.153 0.144 0.137 Maximum 0.272 0.254 0.238 0.224 0.211 0.200 Rock Compressibility (van der Knaap), Triangular Distribution Porosity Intercept 5.0 8.0 10.0 Minimum 7.0 x 10-6 4.20 X 10--6 3.35 x 10-6 Mode 7.6 x 10-6 4.36 X 10-" 3.52 x 10-" Maximum 8.3 x 10-' 4.80 X 10--6 3.75 x 10-6 11 shows the simulator-generated histograms of the seen, the oil in place varies from 3.8 to 5.1 million major input variables to the steady-state material bal STB, with a modal value of 4.2 million STB. ance equation. Scatter diagrams were made for most of the variable Oil in Place by tbe Unsteady-State Material Balance combinations. The two of most interest show the On Nov. 2, 1967, a pressure point was taken that van der Knaap rock compressibility to be mainly a caused a revision in the straight-line interpretation of porosity function for this problem, whereas the ma the pressure-production data. Fig. 14 shows that all terial balance oil in place is a function of effective the data can be easily fitted with a smooth curve; the compressibility. No trends were apparent from any inclusion of the Nov. 2, 1967, pressure point makes of the other scatter diagrams. it untenable for a straight line or a displaced straight Fig. 12 is the histogram of the oil in place as cal line to be fitted to the data. culated by the material-balance equation (MBE), and In an undersaturated reservoir a curved pressure Fig. 13 is the cumulative probability curve. As can be production history is generally thought to indicate an 100 r-- 90 ~ \ en 40 80 w >- .....J - ~ 70 \ ...J a; ~ 30 (i!j 60 \ o NOVEMBER, 1970 1363 active aquifer. However, in the Keg River CC pool tD = 6.323 X 10-3 k 2 (11) a zone of poor porosity and permeability at the base en the next re/rw specified and reruns the entire com en... 1 0: JULY 5 1967 putation . Q. 1600 ~ 0: UNSTEAOY STATE A minor complication exists in that the pressure g 8AlANfE 0: I~ ~MATERlAi drop used in the Hurst-van Everdingen equation is ~... 1400 not the MBE pressure drop. In essence, the material 0: <..> ~;1967 t balance uses an incremental pressure drop, which may SEPT, 16, 1967 S 1200 "" be defined as en NOY~ (15) 10 20 30 40 50 60 70 80 CUMULATIVE PRODUCTION (sTB X 10 3 ) The Hurst-van Everdingen equation uses a differen Fig. 14--Pressure-production history to Nov. 2, 1967. tial pressure drop defined as 1364 JOURNAL OF PETROLEUM TECHNOLOGY could be used to evaluate oil in place. At this juncture (16) it is significant to note that the range of input variables is quite small, and indeed several could have been Hence, the influx must be recomputed at every time handled as determinants. Finally, as more pressure step. For example, at Time Step 3, production history became available, the steady-state No = B [~P3 QWl + ~P2 QtD2 + ~Pl QtD3] material balance was abandoned in favor of an un steady-state concept, and the Monte Carlo simulation (17) was abandoned in favor of a history-matching routine. and at Time Step 4 Considerable space has been devoted here to for mulating and discussing the problem at hand, whereas No = B [~P4 QtDl + ~P3 QW2 + ~P2 QtD3 less space has been devoted to the actual problem (18) solution. This is significant. Often in the past a ;'canned" computer program has been taken and the That is, the first pressure drop ~Ph acts over the problem forced to fit it. However, to obtain a satis entire Time t4 so the dimensionless influx QtD has to factory solution, the problem must first be thoroughly be reevaluated at Time t4 • defined and only then the method selected to solve it. Table 3 lists the only real solutions that satisfied If sophisticated mathematical solutions, random num the match-function criterion. However, several of ber generators or systematic history-matching tech Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 these solutions may also be screened by what is pres niques are not warranted, they should not be used. ently known of the Zama Keg River CC pool geom For example, to use Monte Carlo simulation to esti etry, through seismic surveys and the two existing mate reef volumes via the volumetric equation from wells that have penetrated the anomaly. Evidence seismic data alone would have been meaningless. A from these sources suggests that 2.0 and 90 ft are the range and most likely value of oil in place could be most reasonable values of dimensionless radius and estimated more accurately than could the individual boundary thickness, respectively, along with a bound parameters that are required to solve the volumetric ary permeability of 0.65 md. The hydrocarbon pore equation. volume corresponding to this is 6.0 million STB, made It is important that operations personnel be aware up of 4.2 million in the central portion of the reef and of techniques that will help them with their problem 1.8 million in the porous rim regions. solving, even though the final method of solution can best be selected by the individual, depending upon Conclusion the specific problem involved. Monte Carlo simula As can be seen from the Zama Keg River CC pool tion and systematic history matching are only two case history, the evaluation of oil in place progressed such tools available for evaluating uncertainty. They through a process of evolution. Initially, only seismic should be used liberally in problem solving but not data were available and at that time only a very in indiscriminately, and they should not be considered exact estimate of oil in place could be made. Then as an end in themselves, providing an infallible answer. as more wells were drilled in the area and geologic models were formulated, a more meaningful assump Nomenclature tion of facies type and porosity distribution could be B = influx conversion constant used in estimating hydrocarbon pore volume in the Bo = oil formation volume factor, reservoir various seismic features. However, it was not until bbl/STB the Zama Keg River CC pool was drilled that any meaningful evaluation of uncertainty could be under Boi = initial oil formation volume factor, reser taken. It was at this point that realistic ranges and voir bbl/STB most likely values could be assigned to the input Ce = effective compressibility, vol/vol/psi parameters and the volumetric oil in place could be Cma = rock compressibility, vol/vol/psi evaluated. Later, as some pressure-production history became available, the steady-state material balance Co = oil compressibility, vol/vol/psi Cw = water compressibility, vol/vol/psi F = history-match function, psi TABLE 3-UNSTEADY-STATE MATERIAL BALANCE, h = thickness of zone contributing to influx, REAL SOLUTIONS ft Ratio of Effective External to Permeability Thickness Total k = effective boundary permeability, md Internal at Internal at Internal Hydrocarbon Radius Boundary Boundary Pore Volume. m = number of pressure-production points re/rw (md) (tt) (STB) used in history match 2.00 0.52 100.0 6.2 X 10· 2.00 0.65 90.0 6.0 X 10' n = identifying index of pressure-production 2.00 0.98 80.0 5.8 X 10' ' data points 2.00 2.29 70.0 5.6 X 10' . N = hydrocarbon pore volume, STB 2.50 0.81 70.0 6.7 X 10' 2.50 1.22 60.0 6.3 X 10' N e = oil influx, reservoir barrels 2.50 1.85 50.0 6.0 X 10' . N p = cumulative oil produced, STB 2.50 5.90 40.0 5.6 X 10' . 3.00 4.78 30.0 5.8 X 10' P = formation pressure at any time, psi NOVEMBER, 1970 1365 Pk = calculated pressure at Point n Facies Relationship, Zama Area, Alberta", Bull. of Cdn. Petro Geology, (Dec., 1967) 15, No.4, 434. Pne! = net effective pressure, psi 3. Marr, J. D. and Zagst, E. F.: "Exploration Horizons Po = observed pressure at Point n from New Seismic Concepts of CDP and Digital Process ing", Ray Geophysical Div., Mandrel Industries Inc. p ovb = pressure due to overburden, psi 4. Zagst, E. F.: "Horizontal Stacking Improves Seismic f:::.p = pressure drop, psi Data", Oil and Gas J. (Aug. 16, 1965) 97-104. 5. Langton, J. R., and Chin, G. E.: "Rainbow Member Qt = dimensionless influx Facies and Related Reservoir Properties, Rainbow Lake, Alberta", Bull. of Cdn. Petro Geology (March, 1968) 16, r e = external radius, ft No. 1, 104. re/rw = ratio of external to internal radii, dimen 6. Chevron Standard Ltd.: "Zama Field, Oil-in-Place Sub sionless mission, Keg River, G, H. T and 0 Pools, Sulphur Point C Pool", filed with the Alberta Oil and Gas Conserva r w = internal radius, ft tion Board (Jan. 31, 1968) Figs. 7 and 8. 7. van der Knaap, W.: "Nonlinear Behavior of Elastic Sw = water saturation, fraction of pore space Porous Media", Trans., AIME, (1959) 216, 176-187. t = real time, days 8. van Everdingen, A. F. and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems in tD = dimensionless time Reservoirs", Trans., AIME, (1949) 186, 305-324. V R = rock volume, acre-ft 9. Craft, B. C. and Hawkins, M. F.: Applied Petroleum Reservoir Engineering, Prentice-Hall Inc., Englewood Downloaded from http://onepetro.org/JPT/article-pdf/22/11/1357/2226229/spe-2584-pa.pdf by guest on 30 September 2021 8 = angle over which influx is occurring, de- Cliffs, N. J. (1959) 205. grees 10. Brill, J. P., Bourgoyne. A. T. and Dixon, T. N.: "Nu merical Simulation of Drillstem Tests as an Interpreta p. = viscosity of oil, cp tion Technique", J. Pet. Tech. (Nov., 1969) 1413-1426. ~ = summation symbol 11. Fletcher, A. R. and Powell, J. J. D.: ·"A Rapidly Con vergent Descent Method for Minimization", The Com T = weighting factor for Point n puter Journal (July, 1963) 6, No.2, 163-168. 1> = porosity, fraction of rock volume 12. Walstrom, J. E., Mueller, T. D. and MacFarlane, R. C.: "Evaluating Uncertainty in Engineering Calculation", J. Pet. Tech., (Dec., 1967) 1595-1603. Acknowledgment 13. Stoian, E.: "Fundamentals and Application of the Monte Permission granted by the management of Chevron Carlo Method", paper presented at the 16th Annual Standard Ltd. to publish this paper is gratefully Meeting of CIM, Calgary, Alta. (1965). acknowledged. 14. Spang, H. A., III: "A Review of Minimization Tech niques for Non-Linear Functions", SJ.A.M. Review (Oct, 1962) 4, No.4, 343-365. JPT References 1. Hriskevich, M. E.: "Middle Devonian Reefs of the Rain Original manuscript received in Society of Petroleum Engineers bow Region of Northwestern Canada, Exploration and office July 7, 1969. Revised manuscript received June 29, 1970. Exploitation", Proc., Seventh World Pet. Cong., Mexico Paper (SPE 2584) was presented at SPE 44th Annual Fall Meeting, held in Denver, Colo., Sept. 28-0ct. I, 1969. © Copyright 1970 City (1967) 3, 733. American Institute of Mining, Metallurgical, and Petroleum Engi· 2. McCamis, J. G., and Griffith, L. S.: "Middle Devonian neers, Inc. 1366 JOURNAL OF PETROLEUM TECHNOLOGY