montecarlo illustrations

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Illustration of Formation Evaluation Uncertainty Characterization with Excel

MonteCarloIllustrations.xlsBe aware that Excel2007 is being used, in the Backwards Compatible mode.Excel2003 screens may appear different than those included herein.y pp ff

•The only certainty in most of our formation evaluations is the presence of uncertainty and how that issue is (or is not) addressed.

•It is in fact relatively simple to address the uncertainty question in a comprehensive,

i i f hiquantitative fashion, and to further identify where to focus time, and money, in search of an improved evaluation.

Monte Carlo with Excel Summary

•The Monte Carlo method relies on repeated random sampling to model results

•This approach is attractive when it is infeasible or impossible to compute an exact result with a deterministic algorithm.

•An advantage of Monte Carlo is that any type of distribution can be used to characterize the uncertainty distribution of any parametercha acte ize the unce tainty dist ibution of any pa amete

•Normal, log normal, etc

•The phenomena governing frequency distributions in nature often favor log-normal.

•A limitation of Monte Carlo is that special software is needed

•Not included with many commercially available petrophysics s/w packages

•The implementation of a MC Model over a large section of log data can also be very time consuming

Carbonate Petrophysics for GeoNeuraleCopyright 2009 Robert E Ballay, LLC

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•The Monte Carlo method makes use of individual probability distributions to arrive at a cumulative probability distribution

•The determination of a probability distribution is superior to the single deterministic value

•Insight is gained into the “upside” and “downside”Insight is gained into the upside and downside

•Many Oilfield analyses can be accomplished with Excel

•This eliminates both the expense of, and need to learn, commercial programs

•@Risk, Crystal Ball, etc

•Oilfield applications typically use “normal”, “log normal”, and “triangular” statistical distributions.

•Relevant Excel functions

•Rand, NormDist, NormInv, LogNormDist, LogInv, TriangleInv

•With Monte Carlo, insight is gained into the “upside” and “downside”

• + / - 1 σ will encompass ~ 68% of the distribution

• + / - 2 σ ~ 95 % of the distribution


http://en.wikipedia.org/wiki/Standard_deviation


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•Supplemental material•Relevant spreadsheets

•MonteCarloModeling.xls•MonteCarloIllustrations.xls

RollingTheDice.PDF

MonteCarloModeling.PDF

Illustrative Excel-based Monte Carlo Evaluations

•Phi(Rhob)

•Dual Porosity Cementation Exponent (Wang & Lucia)

•Sw(Archie)

•Relevant spreadsheet•MonteCarloModeling.xls•MonteCarloIllustrations.xls


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Excel MC Evaluation of Phi(Rhob)•Your bulk density log has been run across a cored, wet (Rho_fluid=1.00), limestone (Rho_ma=2.71) and correlates with compaction corrected core plug porosityas seen at right, and below•Is the observed correlation consistent

0

2

0.00

0.10

0.20

0.30

Porosity

Phi(Core)

Phi(Rhob)

with a MC simulation that is based upon expected uncertainties? •Exhibit following

0 20

0.30

y

Log vs Core

12 inch Log

4

6

Dep

th Rhob MC Exercise

0.00

0.10

0.20

2.25 2.35 2.45 2.55 2.65 2.75

Cor

e P

lug

Poro

sity

Wireline Bulk Density

8

10MonteCarloIllustratePhi(Rhob).xls

Excel MC Evaluation of Phi(Rhob)•The ‘synthetic’ Rhob is a mathematical construct based upon the 12 inch average of sequential Phi(Core) plugs and user specified Rho_ma & Rho_fld, with random noise added•Because of the averaging effect (blue

0

2

0.00

0.10

0.20

0.30

Porosity

Phi(Core)

Phi(Rhob)

Phi(Avg)

Because of the averaging effect (blue line vs purple dots), which is also present in wireline tools, the high and low porosities may not be ‘seen’ by the tool

•Hence the shortfall of Phi(Rhob) at the ‘peaks’ and ‘troughs’•Be aware that because the noise is ‘random’, the graphics will change

4

6

Dep

th Rhob MC Exercise

random , the graphics will change every time the spreadsheet is opened

•Note that if core plugs are carefully selected, to represent the entire foot, the averaging issue will be minimized•Exhibit following

8



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Excel MC Evaluation of Phi(Rhob)•The ‘synthetic’ Rhob is a mathematical construct based upon the 12 inch average of sequential Phi(Core) plugs, user specified Rho_ma & Rho_fld, with random noise added•Exhibit following

0

2

0.00

0.10

0.20

0.30

Porosity

Phi(Core)

Phi(Rhob)

Phi(Avg)

Rhob MC Exercise

Exhibit following

4

6

Dep

th8


Excel MC Evaluation of Phi(Rhob)•Your bulk density log has been run across a cored, wet (Rho_fluid=1.00), limestone (Rho_ma=2.71) and correlates with compaction corrected core plug porosity as seen below•Is the observed correlation consistent with a MC simulation that is based upon expected uncertainties?

Log vs Core

MonteCarloIllustratePhi(Rhob).xls

•Assume the Phi(Core) Rhob MC Exercise

0.20

0.30

Plu

g Po

rosi

ty

Log vs Core

12 inch Log

•Assume the Phi(Core) data is very well known, and all uncertainty is in the Rhob measurement (ie does a MC simulation of Phi(Rhob) account for the observed “scatter”).•“Scatter” in Rhob is

Rhob MC Exercise

0.00

0.10

2.25 2.35 2.45 2.55 2.65 2.75

Core

P

Wireline Bulk Density

Scatter in Rhob is about + / - 0.02 gm/cc•“Scatter” in Phi(Rhob) then about + / - 1 pu•Exhibit following

•MonteCarloIllustrations.xls includes a Phi(Rhob) model


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Excel MC Evaluation of Phi(Rhob)•Is the observed correlation consistent with a MC simulation that is based upon expected uncertainties? •“Scatter” in Rhob is ~ + / - 0.02 gm/cc; “Scatter” in Phi(Rhob) then ~ + / - 1 pu•Exhibit following

Rhob MC •Assume all uncertainty is in the Rhob measurement.Solution•Set RhoG & RhoF STD small, and find (vary) RhoB STD required to

match observed + / - 1 pu scatterMonteCarloIllustrations.xls

Excel MC Evaluation of Phi(Rhob)•Is the observed correlation consistent with a MC simulation that is based upon expected uncertainties? Yes•MC simulation matches observed RhoB scatter, with RhoB STD ~ 0.01 gm/cc which is a physically realistic value•Exhibit following

Rhob MC •Vary RhoB STD to match observed + / - 1 pu scatterSolution•RhoB STD ~ 0.01 gm/cc yields uncertainty in Phi(Rhob) of ~ +/- 2 STD

+/- 1 pu (+/- 2 STD encompasses 95% of the distribution)MonteCarloIllustrations.xls


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Excel MC Evaluation of Phi(Rhob)•MC simulation matches observed RhoB scatter, with RhoB STD ~ 0.01 gm/cc •Random noise in RhoB(Synthetic) was + / - 0.03 gm/cc square distribution

•+ / - 0.03 gm/cc random noise corresponds to ~ 2 3 STD, STD ~ 0.01 0.015 gm/cc

•Yes, the MC simulation and the synthetic Rhob are consistent

Rhob MC Solution

MonteCarloIllustrations.xls

Phi (Rhob) Reference Material Follows

Excel Monte Carlo Evaluation of Phi(Rhob)

•Porosity estimation with density log is dependent upon

•Grain density (mineralogy)

•Fluid density (mud filtrate, connate water, hydrocarbon)

•Bulk density (measured)

•If the distribution of uncertainty in each of these is estimated, the net uncertainty in Phi(Rhob) can be characterized with Monte Carlo simulation

•Exhibit following

Reference Material


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•Grain density (mineralogy) is dependent upon our ability to identify the presence and concentration of the various minerals

•In the case of a limestone - dolostone mix, if the ability to distinguish mineralogy is about 10% , the grain density (2 sigma, 95% ) range is about

• +/- 0.03 gm/ccg

•One RhoG sigma is ~ 0.0075 gm/cc


E ndPoint R hoG P ctP rsntL imestone 2.710 0.9Dolostone 2.870 0.1Mixture 2.726If mineralogy is 10% uncertain,R hoG is ~ +/‐ 0.015 gm/cc uncertain

•Note that the presence of a mineral whose density is significantly different than limestone - dolostone, would expand the uncertainty range Grain Density

E ndPoint R hoG P ctP rsntL imestone 2.710 0.1Dolostone 2.870 0.9Mixture 2.854If mineralogy is 10% uncertain,R hoG is ~ +/‐ 0.015 gm/cc uncertain

•Salt: 2.17 gm/cc

•Anhydrite: 2.96 gm/cc

•Siderite: 3.94 gm/cc

•Pyrite: 5.00 gm/cc

Grain Density

Reference Material

•From the preceding calculations we recognize the two sigma (95 %) RhoG range to be ~ 0. 03 gm/cc

•One RhoG sigma is ~ (0.03 gm/cc) / 4 = 0.0075 gm/cc


+ / 2 95 % f th di t ib ti

Excel Monte Carlo Evaluation of Phi(Rhob) Reference Material




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T & P Dependence (GeoLog)

1.10

1.12

1.14

1.16

1.18

1.20

y (g

m/c

c)

75 F / 0 psi175 F / 2500 psi225 F / 2500 psi

•Fluid density is dependent upon the relative amounts of

•mud filtrate,

•connate water

h d b

Net Fluid Density

Concentration Relations250

1.00

1.02

1.04

1.06

1.08

0 50 100 150 200 250Salinity (k ppm)

Den

sity•hydrocarbon

•Mud filtrate density is dependent upon salinity, temperature and pressure

•Salinity may be specified in terms of TDS or PPM

•Consider two salinity's: 50 kppm & 150 kppm

•At 175 F, 2500 psi, the two densities are

0

50

100

150

200

0 50 100 150 200 250Salinity (k ppm)

TDS

(mg/

l)

75 F175 F

, p ,

•Spreadsheet Salinity_Density Conversions.xls

•50 kppm => 1.011 gm/cc

•150 kppm => 1.085 gm/cc

Reference Material

•Crude density may be specified as gm/cc or API Gravity

•API Gravity is a measure of specific gravity per the American Petroleum Institute, graduated in degrees on a hydrometer which was designed so that most values would fall between 10 and 70 API gravity degrees.

Th bit f l d t bt i thi ff t i


•The arbitrary formula used to obtain this effect is:

API Gravity = (141.5 / SG at 60° F) - 131.5

•Sixty degrees Fahrenheit is used as the normal reference for measurements and tablesare published that give adjustments for temperature. (ASTM D1298)

•A heavy oil with a specific gravity of 1.0 (the density of pure water) would have an API Gravity of:

C rude Dens ity (60F )(141.5 / 1.0) - 131.5 = 10.0 degrees API

C rude ens ity (60F )gm/cc AP I0.500 1520.600 1040.700 710.800 450.900 261.000 10Reference Material


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•Fluid density is inherent in pressure profiles

•Spreadsheet Fluid_Density_PressureGradient.xls

•Fluid density calculated from first principles (preceding) should be cross-checked against any available pressure data

F luid Gradient ‐ Dens ity R elation

0.80

1.00

1.20

1.40

1.60ure Gradient

ps i/ft

ps i/m


0.00

0.20

0.40

0.60

0.00 0.20 0.40 0.60 0.80 1.00 1.20

F luid Dens ity (gm/c c )

Press

u ps i/m

Reference Material


•Fluid density is dependent upon the relative amounts of mud filtrate, connate water and hydrocarbon that are present

•Consider two hypothetical endpoints

•50 kppm mud filtrate and 50 kppm connate water

50 k d filt t d 150 k t t•50 kppm mud filtrate and 150 kppm connate water

•If mud filtrate and connate water are similar

•the net fluid density is dependent only upon the amount of (unmoved) hydrocarbon present

•If the connate water is saltier than the mud filtrate

•the net fluid density is dependent upon the efficiency of the mud filtrate f y p p ff y f fdisplacement of both hydrocarbon and connate water

•Mud filtrate invasion can be more complex than one might at first expect

•Exhibit followingNet Fluid Density

Reference Material


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Invasion and Two Fronts•In a hydrocarbon interval, the mud filtrate can push formation water ahead of it, which itself moves hydrocarbons. There can then be two fluid banks created

• the first having mud filtrate salinity • the second of connate water salinity

Reference Material

Allen, David et al. Invasion Revisited.

Oilfield Review. July 1991

•Saturation / salinity fronts and fluid banks as water base filtrate invades water wet, hydrocarbon bearing formation.•Exhibit following Mud filtrate invasion can be more

complex than one might at first expect

Invasion & Two Fronts•Resistivity profiles for Two Front model of water base mud invading water wet, hydrocarbon charged, reservoir with various Rw & Rmf combinations•There can in general be bothThere can, in general, be both Saturation / salinity fronts and fluid banks •The fluid banks are

•Mud filtrate salinity •Connate water salinity

•Formation Resistivity (ohm-m)

Mud filtrate invasion can be more complex than

one might at first expect

is displayed along the vertical axis, and Distance from Well bore along the horizontal, for each combination

Allen, David et al. Invasion Revisited. Oilfield Review. July 1991Well Worth The Read


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•Mud filtrate invasion can be more complex than one might at first expect

•Additional considerations

Mud filtrate invasion can be complex

•In multiple hydrocarbon charged carbonates, we have found that the Phi(Rhob) which best matches core, is the estimate which takes RhoF to be that of the reservoir

diti d filt tconditions mud filtrate

•In effect, the mud filtrate invasion has been 100% efficient


Reference Material


•Iterative corrections, which are intuitively attractive, may over-correct

•Simple calculation, assuming that fluid density is that of mud filtrate at reservoir conditions, produced a better match to core in this example - exhibit following

Gas Interval of Limestone Reservoir Gas Interval of Limestone Reservoir

Reference Material

Mineralogy Well-known

Phi Estimate Determined Iteratively

Mineralogy Well-known

Phi Estimate per Mud Filtrate Density

Phi(core) with Compaction Correction



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Gas Interval of Limestone Reservoir Gas Interval of Limestone Reservoir

•Be aware that cored wells are typically drilled slower•More efficient mud filtrate penetration.

•Un-cored wells can conceivable be ‘different’.

Reference Material

Mineralogy Well-known Mineralogy Well-known

Phi Estimate Determined Iteratively

Phi Estimate per Mud Filtrate Density



Excel Monte Carlo Evaluation of Phi(Rhob)•Consider the hypothetical endpoint:

•50 kppm mud filtrate & 50 kppm connate water

•If mud filtrate and connate water are similar, the net fluid density is dependent only upon the amount of (unmoved) hydrocarbon present and the density of the brine versus hydrocarbon phase

Reference Material

•From these calculations we find the two sigma RhoF range to be ~ 0.05 gm/cc

•One RhoF sigma is ~ (0.05 gm/cc) / 4 = 0.0125 gm/cc

•Exhibit followingE ndPoint R hoF P ctP rsnt50kMud_50kWtrMudF iltrate 1.011 0.900C onnateWtr 1 011 0 000

E ndPoint R hoF P ctP rsnt50kMud_50kWtrMudF iltrate 1.011 0.700C onnateWtr 1 011 0 000

Net Fluid Density

C onnateWtr 1.011 0.000C rude 0.700 0.100Mixture 0.980


E ndPoint R hoF P ctP rsnt50kMud_50kWtrMudF iltrate 1.011 1.000C onnateWtr 1.011 0.000C rude 0.700 0.000Mixture 1.011

E ndPoint R hoF P ctP rsnt50kMud_50kWtrMudF iltrate 1.011 0.800C onnateWtr 1.011 0.000C rude 0.700 0.200Mixture 0.949


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•From the preceding calculations we recognize the two sigma RhoF range to be ~ 0.05 gm/cc



+ / 2 95 % f th di t ib ti




Excel Monte Carlo Evaluation of Phi(Rhob)•If the connate water is saltier than the mud filtrate, the net fluid density is dependent upon the efficiency of the mud filtrate displacement of both connate water & hydrocarbon

•Now there are more variations of ‘mixing’ to consider

•Two sigma RhoF uncertainty may be larger, but 0.05 gm/cc uncertainty is not

Reference Material

unreasonable for illustration purposes

•One RhoF sigma is ~ 0.0125 gm/cc


50kMud_150kWtrMudF iltrate 1.011 0.500C onnateWtr 1.085 0.300C rude 0.700 0.200Mixture 0.97150kMud_150kWtrMudF iltrate 1.011 0.400C onnateWtr 1 085 0 400

50kMud_150kWtrMudF iltrate 1.011 0.800C onnateWtr 1.085 0.000

Net Fluid Density

50kMud_150kWtrMudF iltrate 1.011 0.600C onnateWtr 1.085 0.200C rude 0.700 0.200Mixture 0.964


50kMud_150kWtrMudF iltrate 1.011 1.000C onnateWtr 1.085 0.000C rude 0.700 0.000Mixture 1.011

C rude 0.700 0.200Mixture 0.949


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•Repeatability of modern bulk density measurements is generally very good

•Repeatability does not always mean accuracy

•An erroneous value may be repeated

•For illustration purposes, the net

Reference Material

p p ,uncertainty in RhoB is taken as + / - 0.01 gm/cc

•One RhoB sigma is ~ (0.02 gm/cc)/4 = 0.005 gm/cc

•Consult your Service Company about your specific tools

Illustrative Uncertainties courtesy Stefan Calvert, BG India

Measurement Uncertainty

•From the preceding calculations we recognize the two sigma RhoB range to be ~ 0.02 gm/cc



+ / 2 95 % f th di t ib ti





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•Porosity estimation with density log is dependent upon




Reference Material

•If the distribution of uncertainty in each attribute is estimated, the net uncertainty in Phi(Rhob) can be characterized with Monte Carlo simulation

•With the considerations and assumptions preceding, the largest uncertainty is in RhoF

•Because porosity is typically 30 pu or less (ie less than the matrix volume), th fl id d it t i t i

Monte Carlo Simulation of Phi(Rhob)Phi = (RhoG ‐ RhoB)/(RhoG ‐ RhoF)Specify Parameters Below


the fluid density uncertainty is ‘discounted’ accordingly


Setting Up Excel

Other Tables will update automaticallyAttribute Mean StdRhoG 2.710 0.0075RhoB 2.300 0.0050RhoF 1.000 0.0125

Phi(RhoB) 0.240


•If the distribution of uncertainty in each of the following components is estimated, the net uncertainty in Phi(Rhob) can be characterized with Monte Carlo simulation



B lk d it ( d)

Reference Material


•As a QC device, Monte Carlo result statistics are compared to input attributes

•Monte Carlo result statistics correctly reproduce the input attributes

•Exhibit followingMonte Carlo Simulation of Phi(Rhob)Phi = (RhoG ‐ RhoB)/(RhoG ‐ RhoF)Specify Parameters Below

Cross‐check Specs Monte Carlo ResultsRhob Phi(Rhob)

Monte Carlo Statistics


QC the Monte Carlo

Specify Parameters BelowOther Tables will update automaticallyAttribute Mean StdRhoG 2.710 0.0075RhoB 2.300 0.0050RhoF 1.000 0.0125

Phi(RhoB) 0.240

Rhob Phi(Rhob)Mean Std_Dev Mean Std_Dev2.300 0.0049 0.240 0.0048

RhoG RhoFMean Std_Dev Mean Std_Dev2.710 0.0074 1.000 0.0125

Delta Phi 0.019


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•As a QC device, the distribution of Excel random numbers, used to drive the Monte Carlo simulation, are ‘binned’ from zero to one

•With 2000 simulation performed, we expect to find Frequency ~ 200 in each of the ten bins

•The Excel Random() has approached the ideal distribution

Reference Material

() pp


Monte C arlo Dis tribution

150

200

250quen

cy

QC the Monte Carlo

0

50

100

0.00 0.20 0.40 0.60 0.80 1.00 1.20

R andom()

Freq

Random



•With the input attributes below, one is able to characterize the Phi(RhoB) estimation in both numerical and graphical formats

•With these specifications, one expects the porosity estimate to be within + / - 2 σ (~ 2pu), 95% of the time

•Exhibit following Uncertainty in Phi(Rhob)

Reference Material

f g

Monte C arlo Dis tribution

200

300

400

500

600

700

Frequen

cy

P hi(Rhob)

Monte Carlo Simulation of Phi(Rhob)Phi = (RhoG ‐ RhoB)/(RhoG ‐ RhoF)Specify Parameters BelowOther Tables will update automaticallyAttribute Mean StdRhoG 2.710 0.0075RhoB 2.300 0.0050RhoF 1.000 0.0125

Phi(RhoB) 0.240

0

100

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Poros ity


Mean Std_Dev Mean Std_Dev2.300 0.0049 0.240 0.0048


Delta Phi 0.019




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•With the input attributes below, Phi(RhoB) is expected to be within + / - 2 σ (~ 2 pu) 95% of the time

•The Best / Worst case uncertainty would yield a Delta Porosity of nearly twice as much. It is, however, unlikely that all Best / Worst values would occur simultaneously

•Exhibit following Monte Carlo vs Best/Worst

Reference Material

f g Monte Carlo vs Best/Worst

Monte Carlo Simulation of Phi(Rhob)Phi = (RhoG ‐ RhoB)/(RhoG ‐ RhoF)Specify Parameters BelowOther Tables will update automaticallyAttribute Mean StdRhoG 2.710 0.0075RhoB 2.300 0.0050RhoF 1.000 0.0125

Phi(RhoB) 0 240


Mean Std_Dev Mean Std_Dev2.300 0.0049 0.240 0.0048


Delta Phi 0 019


Phi(RhoB) 0.240 Delta Phi 0.019

Attribute Low HighRhob 2.290 2.310 0.2355 0.2425 0.2486 0.2559RhoG 2.695 2.725RhoF 0.975 1.025 0.2238 0.2305 0.2371 0.2441

Phi(Rhob) 0.2400.032

High‐Low Numerical StatisticsPhi Range for Low Rhob

Phi Range for High Rhob

LowRhoG_LowRhoF HiRhoG_HiRhoF

Max Delta Phi



Light Hydrocarbon Effects and Corrections

Don’t let your evaluation‘go up in smoke’

•Supplemental material

•LHC_Effects.pdf

•LHC_Effects_Phi.pdf

•ArchieUncertainty pdf

Reference Material

Building for size reference

•ArchieUncertainty.pdf

•Adjust Monte Carlo spreadsheet per your specific, local conditions

Quantifying Petrophysical Uncertainties.S J Adams. 2005 Asia Pacific Oil & Gas Conference. Jakarta.

•Simply calculatinguncertainty isinsufficient unless itcan be shown thatthe appliedinterpretationmodel isappropriate

31st Annual SPWLA Symposium. June 1990

appropriate.

•Good-quality coredata provide anexcellent basis onwhich to determinethe appropriateinterpretationmodel.

•These resultsenables operators tomake better data-gathering andcompletiondecisions.


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Monte Carlo Evaluation of Dual Porosity Cementation Exponent

•You have just cored a vuggy carbonate, and the field geologist reports •About half the total porosity appears to be vuggy and poorly connected

•The remainder is IG/IX•Vuggy Porosity Ratio (VPR) = 0.50

•Your porosity calculation across the upper interval estimates total porosity ~ 10 pup y pp p y p•Assuming

•m(IG/IX) = 2.0 •Std_Phi = 1 pu•Std_VPR = 0.20*VPR

•Vug fraction estimate uncertainty is large, ~ +/- 0.4*VPR•Estimate the Dual Porosity (Wang & Lucia) Cementation Exponenty ( g ) p•How does the “m” estimate change as the vuggy porosity becomes better connected?•If Rw @ FT is 0.05 ohm-m, what Ro do we expect to observe in the water leg?•If Rmf @ FT is 0.5 ohm-m, what Rxo do we expect to observe in the water leg?•MonteCarloIllustrations.xls includes a Phi(Rhob) model

“m” MC Exercise


“m” MC Solution

•Half the total porosity is vuggy & poorly connected•m(IG/IX) = 2.0 , Std_Phi = 1 pu, Std_VPR = 0.20*VPR


•Estimate the “m” Exponent•“m” ~ 2.5

•How does the “m” estimate change as the vuggy porosity becomes better connected?

•“m” will drop, approaching 1.0

•Quality Control: Exhibit following


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“m” MC Solution

•Half the total porosity is vuggy & poorly connected•m(IG/IX) = 2.0 , Std_Phi = 1 pu, Std_VPR = 0.20*VPR



•How does the “m” estimate change as the vuggy porosity becomes better connected?

•“m” will drop, approaching 1.0

2

1

•Quality Control: 1) Random centered on 200 counts per bin. 2) Monte Carlo population statistics match specification statistics

FF = Ro / Rw = Cw / Co = 1 / [ ]

mip = 2.0, Φv & av held constant across range of Φip

FF[Archie(m=2)] = 1/ Φt2 = 1/[Φip + Φv] 2

Dual-porosity Model / What If Characterizations Dual Porosity / Type 1

10

100

1000

Form

Fac

tor

av=1

av=10

av=100

av=1000

“m” MC Solution

p

10.01 0.10 1.00

Total Porosity [ Phi(v)=0.05 ]

av=1000

Archie (m=2)

Dual Porosity / Type 1

3

4

nt

Φip + Φv = 6 pu

Phi(v) = 0.05 as Phi(Total) varies


•The Monte Carlo results are consistent with the spreadsheet results of the Dual Porosity “m” Exponent Model Module

0

1

2

3

0.01 0.10 1.00

Total Porosity [ Phi(v)=0.05 ]

Cem

ent E

xpon

en

av=1

av=10

av=100

av=1000


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•Estimate the Dual Porosity (Wang & Lucia) Cementation Exponent•“m” ~ 2.5

•How does the “m” estimate change as the vuggy porosity becomes better connected?•“m” will drop, approaching 1.0

•If Rw @ FT is 0.05 ohm-m, what Ro do we expect to observe in the water leg?

“m” MC Solution

f @ , p g•Ro = .05 / (0.10^2.5) ~ 15.8 ohm-m

•If Rmf @ FT is 0.5 ohm-m, what Rxo do we expect to observe in the water leg?•Rxo = .50 / (0.10^2.5) ~ 158 ohm-m



•If Rw @ FT is 0.05 ohm-m, what Ro do we expect to observe in the water leg?•Ro = .05 / (0.10^2.5) ~ 15.8 ohm-m

•If Rmf @ FT is 0.5 ohm-m, what Rxo do we expect to observe in the water leg?•Rxo = .50 / (0.10^2.5) ~ 158 ohm-m

•The above result is related to a quick look methodology for Rw estimation

“m” MC Solution

q gy f

•See Quick Look module for additional details

•Taking the ratio of Archie’s equation, in the invaded and un-invaded zones yields

Swn / Sxo

n = ( Rw / Rmf ) * ( Rxo / Rdeep )

•In the water leg, Sw 1 and Sxo 1 to yield

Rw = Rmf * ( Rdeep / Rxo ) = 0.50 * (15.8 / 158) = 0.05w mf ( deep xo ) ( )

•Rw estimate independent of porosity


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•Archie’s cementation exponent reflects the tortuosity of the brine filled pore system

•In a general way, we expect the following relative relations between “pore type”, “m” exponent and “BVW”.

•As compared to IG/IX porosity, vuggy porosity that is not connected requires a higher Archie “m”, to reflect the more tortuous pore geometry.


g , p g y

•Small pores (chalk) do not necessarily infer an “m” that is larger than the “m” of large (relatively well-connected) IG/IX pores.

•Small pores (chalk) will exhibit a larger BVW than larger (relatively well connected) IG/IX pores

•Vuggy porosity, in the case of relatively large vug pore bodies, will tend to exhibit a smaller BVW than both IG / IX porosity, and chalk.p y,


Generic Relations

•The general, relative effect on Sw(Archie) of the above considerations, is then

•In the water leg, unrecognized vuggy porosity evaluated with an “m” ~ 2.0 can appear as “pay”

•The more tortuous pore geometry will result in an increased resistivity, at a specific porosity.


p p y

•In the hydrocarbon interval, unrecognized vuggy porosity evaluated with an “m” ~ 2.0 may have an optimistically low Sw, at a specific porosity & resistivity.

•Interpretation of the same interval, with an appropriately higher “m”, will yield a higher Sw if everything else is the same.

•In the hydrocarbon interval, relatively large vugs will exhibit a lower BVW than IG/IX porosity. p y


Generic Relations


7/28/2009

23

•At a specific porosity, above the transition zone, we then expect vuggy porosity to correspond to higher resistivity than IG/IX porosity, for two reasons

•The pore system is more tortuous

•The BVW is likely lower.

E hibit f ll i



Generic Relations

•Once the attributes which affect “m” are recognized, one is faced with the question of how to better identify and evaluate intervals for which vuggy porosity may, or may not, be present.


•Sw(Resistivity Ratio), which is independent of both porosity and

t ti t d tcementation exponent, compared to Sw(Archie) with an “m” of 2.0 (or locally appropriate value), is an option

•QuickLook_Ballay.pdf

•Chapter V: Combining Water Saturation by Ratio Method, Moveable Hydrocarbon Index, Bulk Volume

How to recognize & evaluate vuggy porosity

y ,Water and Archie Water Saturation (found with Google, pub details n/a)



7/28/2009

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•When NMR & Image Logs are available porosity can be potentially partitioned

•Modern (NMR & Image) techniques

•Case Study of Permeability, Vug Quantification and Rock Typing in a Complex Carbonate. N.Gomaa, A. Al-Alyak, D. Ouzzane, O. Saif, M. Okuyiga, D. Allen, D. Rose, R. Ramamoorthy, E. Bize. SPE Annual Technical


y g , , , y,Conference and Exhibition. San Antonio, Texas, Sept 2006

•In the case of legacy data (ie no NMR – Image), the historical [Phi(Dt) vs Phi(D - N)] protocol can be investigated

•If the porosity partition can be determined, “m” can be modeled with Wang & Lucia’s Dual Porosity model and Sw(Archie) calculated

•Exhibit followingf g


•…..continued…. how to better identify and evaluate intervals for which vuggy porosity may, or may not, be present.

•Archie independent options

•If the formation water is sufficiently salty, LWD Neutron Capture Cross-section

If t t li it i l ti l f h


•If connate water salinity is relatively fresh

•The dielectric tool (in combination with supplemental porosity and resistivity tools), can be used to deduce “m” in the flushed zone, and this “m” then used to interpret the deep resistivity measurement

•C/O log

•NMR may allow Fluid Type ID (water or hydrocarbon)

•Pressure profile for fluid gradient and identification

•Others?



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25

•We propose a generalized dual-porosity model to calculate variable cementation exponents for carbonates with heterogeneous vuggy pores.

•The constant av may be used to characterize the connectivity of different types of vuggy pores:

•an av greater than 100 for separate-vug dominated carbonates,


v g p g ,

•an av less than 20 for touching-vug-dominated carbonates,

•and an av of 1 for well-connected planar fractures.

Comparison of Empirical Models for Calculating the Vuggy Porosity & Cementation Exponent of Carbonates from Log Responses. Fred Wang & Jerry Lucia

If the porosity partition can be determined, “m” can be modeled


•Pore geometries control the interrelationship of petrophysical properties. •The three most important pore geometry characteristics are

•amount and types of pores or shape•interconnectedness of pores (tortuosity)•size of interconnecting pore throats


“m” and the Vuggy Porosity Ratio

•Various “m” exponents as a function of vuggy porosity ratio

•If VPR = Phi(vug)/Phi(total), then•Phi(vug) = VPR*Phi(total)•Phi(ip) = Phi(total) - Phi(vug)

Petrophysical Characterization of Permian Shallow-Water Dolostone. M H Holtz, R. P. Major. SPE 75214, 2002http://www.beg.utexas.edu/mainweb/presentations/2002_presentations/holtz_spe0402ab.pdf

“Vuggy” porosity tends to increase “m”

•The “Myers” trend is relatively flat, as the ‘vugs’ are connected. The Myers data can serve as a ‘test’ of the Dual Porosity “m” Model (exhibit following) Supplemental Material


7/28/2009

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•This indicates that equation 30 can be used to


If the porosity partition can be determined, “m” can be modeled

model reservoirs with separate vugs, touching vugs, and fractures using appropriate values of av.

Comparison of Empirical Models for Calculating the Vuggy Porosity and Cementation Exponent of Carbonates from Log Responses. Fred P. Wang and F. Jerry Lucia

•The Myers data can serve as a ‘test’ of the Dual Porosity “m” Model

Supplemental Material

Cementation Effects on “m”

•Wyllie and Gregory constructed lab formations with varying degrees of cementation

•Cementation (reduction in pore throat size) corresponds to an increase in “m”

•The baseline of unconsolidated spheres has “m” ~ 1.1

•The various cemented bead packs have similar slopes with “m” ~ 4

Various Lab-induced Cementations

Lab characterization of “m” range

M R Wyllie and A R Gregory: Formation Factors of Unconsolidated Porous Media: Influence of Particle Shape and Effect of Cementation, Petroleum Transactions of the AIME 198 (1953): 103-110. Schlumberger Technical Review, Volume 36 Number 3

g



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Cementation Exponent and Porosity Type

•Focke and Munn investigated both laboratory and wellbore “m” measurements, within the context of thin section determined pore geometries

•Thin Section ‘a’ : Rock Types 1 and 2 d t i t l i tcorrespond to intergranular grainstone,

limestone and dolostone

•Thin Section ‘b’ : Rock Type 3 is sucrosicdolostone with intercrystalline porosity

•Rock Types 1, 2 and 3 exhibit similar resistivity responses and have been combined into a single Group g p


Cementation Exponents in ME Carbonate Reservoirs. J W Focke and D Munn, SPE Formation Evaluation, June 1987

IG / IX PorositySupplemental Material

•Rock Types 1, 2 and 3exhibit similar resistivity responses (above 5 pu)and have been combined into a single Group with

•Limestone and dolostone grainstone, and sucrosic dolostones.S b l f t diff t

Laboratory Measurements


into a single Group with ‘m’ ~ 2.0

•The porosity < 5 pubehavior is discussed in more detail by authors separately (they specifically note that this is not a ‘fracture effect’)

•Symbols refer to different wells and reservoirs

is not a fracture effect ).


IG / IX PorositySupplemental Material


7/28/2009

28


•Thin Section ‘c’ : Rock Type 4, the moldiclimestone grainstone, represents a diagenetic inversion

•the original porosity (between the grains) was filled with cement,

•the original grains were dissolved to form the current porosity

•The porosity is very poorly interconnected

•‘m’ ranges from about 2.0 to 5.4

V P it


Vuggy Porosity



•Thin Section ‘d’ : Rock Type 6 is mudstones and chalks with matrix porosity

•No significant moldic, vuggy, fracture or fissure porosityf f p y

•A constant ‘average m’ of 2.0 is reasonable for this Rock Type

Mudstone & Chalk

Cementation Exponents in Middle Eastern Carbonate Reservoirs J W Focke & D Munn, SPE Form Evaluation, June 1987



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Mudstones & chalk with matrix porosity and no significant moldic, vuggy, fracture or fissure porosity

Moldic limestone with permeability less than 0.1 md

Small poresdo not

necessarily mean high

‘m’


Increased porosity does not always

mean lowered ‘m’

Cementation Exponents in Middle Eastern Carbonate Reservoirs J W Focke & D Munn, SPE Form Evaluation, June 1987


Critical BVW•The Kansas Geological Survey has summarized Bulk Volume water for a variety of reservoirs and pore types

•BVW=Constant values in Texas carbonates, displayed as a cumulative frequency of values reported per Reservoir, illustrate variations that can be expected from one reservoir to the next.

• Chalk, as one would expect for very small pores, has a high irreducible BVW

http://www.kgs.ku.edu/Gemini/Help/PfEFFER/Pfeffer-theory4.html#bvw_pickett

Cumulative frequency plots of irreducible bulk volume water for reservoirs by Texas carbonate formation.

Bulk Volume Water and Pore Type



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Critical BVW• Vuggy porosity tends to have a lower BVW(Irr), than does Intercrystalline / Ingergranular

•Due to the vuggy pore bodies(usually) being larger than the IX/IG pore bodies

•Relatively larger pore bodies infer relatively less surface / volume

•S/V ~ (4 π r2)/(4 π r3/3) ~ 1/r

•If the rock is water wet, BVW will decrease as S/V decreases Cumulative frequency plots of irreducible bulk

f ihttp://www.kgs.ku.edu/Gemini/Help/PfEFFER/Pfeffer-

theory4.html#bvw_pickett

volume water for reservoirs by pore type.



Pore Body Size

•Bill Guy (KGS) has identified Critical BVW’s based on his experience in Kansas.

•Lansing and Kansas City oolitic g yporosity has lower BVW than does the intergranular porosity

•Chat macro-porosity has lower BVW than does chat micro-


porosity

•Chat: pseudo-brecciated cherty carbonate. Petroleum Potential of SE Kansas, NE Oklahoma & SW Missouri. KGS




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31

John Doveton Comments

•As to LKC oolitic porosity having a lower BVW than intergranular porosity, I will leave Bill Guy to answer that one, although the probable answer will be the bigger the pores, the smaller the BVWi.

•This was first pointed out by Buckles, a petroleum engineer who worked for Imperial Oil in Calgary, so that BVW's were often known as Buckles' numbers.

•Buckles published data for a variety of Canadian reservoirs and contrasted low BVWi’s in Devonian reefs with higher BVWi’s in clastics. He explained the phenomenon in terms of internal surface area which, of course, is a function of pore size.

•The Ellenburger and Austin Chalk data are taken from a book on Texas reservoir data.

Th d i t ti l it d t t k f i i di f


•The vugs and interparticle porosity data are taken from reservoirs in an appendix of a book by Chilingar et al (1972).

•Wherever possible, I plot up reservoir and rock-type data, so that I don't have to make generalizations


Cementation Exponent “m” and Dual Porosity Systems

•Dual (carbonate) porosity systems present a challenge to Archie’s equation

•Consider a system of ‘large’ and ‘small’ pores

•If the large and small pores are present as laminations

•The small pore lamination can ‘short circuit’ the current flow

•Analagous to laminated shaly sand problem

•Will cause the ‘net’ resistivity of the measured interval to be too low

•Sw(Archie) as calculated by routine methods will be too high

•As discussed and illustrated by Griffiths et al, following exhibit

•Modification of the cementation exponent will not compensate

Laminations



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•There is a good overlay between the Archie and Σ saturations

• in the gas interval

• the top lobe of the oil layer.

•High in the oil layer, the buoyancy of the oil is able to disrupt the continuous t h th t t f ti " h t "water phase that creates formation "shorts."

•The discrepancy between the resistivity-derived and Σ - derived oil volumes increases with increasing water saturation

•More "shorting" of the resistivity measurements, lower in the oil column.

Estimating Sw with a volume measurement R. Griffiths, A. Carnegie, A. Gyllensten, M. T. Ribeiro, A. Prasodjo, and Y. Sallam. World Oil, October 2006

Laminations


Laminations and Triaxial Resistivity

Laminations

•Review of historical induction measurements•Discussion of triaxial measurements Supplemental Material


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Additional Material on Lamination EffectsVector Resistivity

Measurement


R ti b t ti lRatio between vertical & horizontal resistivity versus Bulk Sw

Laminations


Grain size lamination effects on calculated Sw

Vector Resistivity Measurement


LaminationsSupplemental Material


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Quick Look Vuggy Porosity Sw Evaluation Kansas City - Lansing, Anadarko Basin

•The Anadarko Basin is one of the deepest basins on the North American continent

•Encompasses roughly•Encompasses roughly 35,000 square miles in western Oklahoma, the northern Texas Panhandle and Southwestern Kansas.

•It contains over 40,000 feet of potential hydrocarbon bearing sediments

http://www.dutcherco.com/anadarkobas.html

bearing sediments

Quick Look Evaluation of Vuggy Porosity


Quick Look Sw Evaluation of the Kansas City - Lansing

Anadarko Basin

• Take “n” = 2, and define the Moveable Hydrocarbon Index as follows

Sw / Sxo = [( Rw / Rmf ) * ( Rxo / Rdeep )] ^ (1/2)p

• Schlumberger (1972) guidelines are that if the ratio of Sw / Sxo > 1.0 no hydrocarbons were moved during invasion.

• True regardless of whether the zone contains hydrocarbons.

• Exhibit following

CHAPTER V: COMBINING WATER SATURATION BY RATIO METHOD, MOVEABLE HYDROCARBON INDEX, BULK VOLUME WATER AND ARCHIE WATER SATURATION. Author, date and publication n/a.

f g

No Hydrocarbons Moved



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Quick Look Sw Evaluation of the Kansas City - Lansing, Anadarko Basin

• Take “n” = 2, and define the Moveable Hydrocarbon Index as follows

Sw / Sxo = [( Rw / Rmf ) * ( Rxo / Rdeep )] ^ (1/2)

• Whenever Sw / Sxo < 0.7 for sandstones or Sw / Sxo < 0.6 for limestone, moveable hydrocarbons are indicated.

• Sxo > Sw / 0.6 => 1.67 * Sw

• If a carbonate reservoir has a Moveable Hydrocarbon Index < 0.6, you can conclude• hydrocarbons are present

• Although not necessarily in commercial quantities• the reservoir has enough permeability so that hydrocarbons have been movedduring the invasion process by mud filtrateduring the invasion process by mud filtrate.



Hydrocarbons Moved



• Ratio water saturation is calculated with the assumption Sxo = Sw1/5

Sw2 / Sxo

2 = [ Sw / Sw1/5 ] 2 = [ Sw

4/5 ] 2 = Sw8/5 = ( Rw / Rmf )* ( Rxo / Rdeep )

Sw (Ratio) = [(Rw/Rmf ) * (Rxo/Rdeep )]^(5/8) = [(Rw /Rmf )* (Rxo /Rdeep )]^(0.625)

• Remember the assumptions in calculating Sw(ratio).Remember the assumptions in calculating Sw(ratio).

• If Sw(Archie) ~ Sw (Ratio) the assumption of a step-contact invasion profile is correct and all calculated values (Sw, Rt, Rxo, and di) are correct (reasonable).

• Sw(Archie) calculated independently per routine Archie equation



Sw(Resistivity Ratio) & Sw(Archie)



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• Kansas City- Lansing Formation, northwestern shelf of the Anadarko basin.

• Moveable Hydrocarbon Index calculated according to

MHI S /S S t[(R /R )/(R /R )]


MHI=Sw/Sxo = Sqrt[(Rxo/Rt )/(Rmf /Rw )]

• Quick Look water saturation calculated according to

Sw(Rat)=[(Rw /Rmf )*(Rxo /Rdeep )]^(0.625)


Resistivity LogsCHAPTER V: COMBINING WATER SATURATION BY RATIO METHOD, MOVEABLE HYDROCARBON INDEX, BULK VOLUME WATER AND ARCHIE WATER SATURATION. Author, date and publication n/a.



• Kansas City- Lansing Formation, northwestern shelf of the Anadarko basin.

MHI=Sw/Sxo = Sqrt[(Rxo/Rt)/(Rmf/Rw)]

Sw(Rat)=[(Rw /Rmf )*(Rxo /Rdeep )]^(0.625)


•The Rxo/Rt quick look evaluation at 4,810 and 4,900 feet (top two arrows) suggest a wet zone

•The Rxo/Rt quick look evaluation at 4,924 to 4,932 (lower two arrows) are interpreted as indicating the presence of hydrocarbons.y


Resistivity LogsCHAPTER V: COMBINING WATER SATURATION BY RATIO METHOD, MOVEABLE HYDROCARBON INDEX, BULK VOLUME WATER AND ARCHIE WATER SATURATION. Author, date and publication n/a.



7/28/2009

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• The zone at 4810’ has good porosity and low Archie water saturation.

• The Moveable Hydrocarbon Index (Sw/Sxo = 0.61) is greater than 0.60

• The Ratio Method water saturation is high (53 percent) .

Th l l ti i di t th t th b t

From Density-Neutron.

• These calculations indicate that the zone may be wet.





• The zone at 4810’ has good porosity and low Archie water saturation. • The Moveable Hydrocarbon Index (Sw/Sxo = 0.61) is greater than 0.60 and the Ratio Method water saturation is high (53 percent) . These calculations indicate that the zone may be wet.

h l l i hi h f h i di h i i h hi h b lk


• The calculation which further indicates the zone is wet is the very high bulk volume water value (0.095)

• Exhibit following• This BVW is based upon Φ * Sw(Archie), with “m” = 2.0, and will increase if Sw(Ratio) is used for the calculation (ie the zone will look even “wetter”)

CHAPTER V: COMBINING WATER SATURATION BY RATIO METHOD, MOVEABLE HYDROCARBON INDEX, BULK VOLUME WATER AND ARCHIE WATER SATURATION. Author, date and publication n/a.Supplemental Material


7/28/2009

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Critical BVW

• The calculation which further indicates the zone is wet is the very high bulk volume water value (0.095)• Carbonate reservoirs with a bulk volume water greater than 0.04 are gusually wet• Exhibit following

Cumulative frequency plots of irreducible bulk f iB lk V l W t


volume water for reservoirs by pore type.Bulk Volume Water


Critical BVW

• Carbonate reservoirs with a bulk volume water greater than 0.04 are usually wet• The zone at 4810 feet is oomoldicwith high porosity and high “m”g p y g• Exhibit following

Cumulative frequency plots of irreducible bulk volume water for reservoirs by pore type



volume water for reservoirs by pore type.

From petrography



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• The interval 4932 => 4936 feet has good to fair porosity, and low Archie water saturations.

• The Moveable Hydrocarbon Index (0.47 => 0.46) is low

• The water saturation by the Ratio Method (39 to 38 percent) is low


The water saturation by the Ratio Method (39 to 38 percent) is low




• The interval 4932 => 4936 feet has good to fair porosity and low Archie water saturations. Both the Moveable Hydrocarbon Index (0.47 => 0.46) and the water saturation by the Ratio Method (39 to 38 percent) are low.

• The bulk volume water value at 4,932 (0.037) and 4,936 (0.032) are both lower than h i i l ff i f 0 04the critical cut-off point of 0.04.

• Our evaluation suggests this zone should be oil-productive.




7/28/2009

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• The interval 4932 => 4936 feet has good to fair porosity, and low Archie water saturations.

• The Moveable Hydrocarbon Index (0.47 => 0.46) is low, as is water saturation by the Ratio Method (39 to 38 percent).

From petrography

• The bulk volume water value at 4,932 (0.037) and 4,936 (0.032) are both lower than the critical cut-off point of 0.04.

• Our evaluation suggests to us that this zone should be oil-productive.



Intergranular / Intercrystalline Porosity



• The interval 4,932 => 4,936 feet has good to fair porosity, and low Archie water saturations.• The Moveable Hydrocarbon Index (0.47 => 0.46) is low, as is water saturation by the Ratio Method (39 to 38 percent). • The bulk volume water value at 4 932 (0 037) and 4 936 (0 032) are both lower than• The bulk volume water value at 4,932 (0.037) and 4,936 (0.032) are both lower than the critical cut-off point of 0.04.

REB comments: If BVW is calculated with Sw(Ratio), it will rise above the 0.04 critical value in the upper (but not lower) zone ?? Without core (or cuttings), how do we eliminate the possibility of vuggy porosity in Upper Zone?



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• A similar evaluation approach is described in the literature for the oomoldicSmackover reservoirs

• These reservoirs are similar to the oomoldic (4810 feet ) zone in the Lansing-Kansas City

Mit h ll T i (1983) l d th t th S k t b l• Mitchell-Tapping (1983) concludes that the Smackover cannot be properly evaluated by the standard Archie technique.

• When Moveable Hydrocarbon Index, water saturation Ratio Method, and bulk volume water are used, then correct judgments about the productive potential can be made.

• Always remember the underlying assumptions, and importance of locally specific guidelinesg


•Hydrocarbon moveability factor(HCM): derived from the shallow and deep resistivity data.

•For HCM less than 0.75,

Quick Look Evaluation in Egypt, with HC Moveability

hydrocarbon is moveable

•For HCM greater than 0.75, the hydrocarbon is immovable.

•When HCM is less than 0.25, the moveable hydrocarbon is gas

•When HCM greater than 0.25 and

G.M. Hamada. Hydrocarbon Moveability Factor: New Approach to Identify Hydrocarbon Moveability and Type from Resistivity Logs. Emirates Journal for Engineering Research, 9 (1), 1-7 (2004)

less than 0.75, the moveable hydrocarbon is oil.

•Field examples have been analyzed with the HCM factor.



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Monte Carlo Evaluation of Sw(Archie)

•You have just drilled a new carbonate well, and performed an Sw(Archie) evaluation with the following parameters

•“a” = 1, Rw = 0.05 ohm-m, Rt = 10 ohm-m•Phi = 20 pu, Std_Phi = 1 pu (95% confidence range is 18 pu 22 pu)•“m” = 2.0, Std_m = 0.10 (95% confidence range is 1.8 2.2)•“n” = 2.0, Std_n = 0.20 (95% confidence range is 1.8 2.2)

•What is the Upper deterministic Sw limit?•What is the Lower deterministic Sw limit?•What are the 95% Upper and Lower Monte Carlo Sw limits?•MonteCarloIllustrations.xls includes an Sw(Archie) model

“Sw” MC Exercise


•“a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification

•This simulation is approximating an Sw interpretation for which the


p fporosity, “m” & “n” estimates are each subject to individually specified uncertainty

•Porosity (for example) is described by a Gaussian distribution, centered on 20 pu with a standard deviation of 1 pu

•“a” = 1, Rw = 0.05 ohm-m, Rt = 10 ohm-mf p


“Sw” MC Solution

•Phi = 20 pu, Std_Phi = 1 pu •95% confidence range is 18 pu 22 pu

•“m” = 2.0, Std_m = 0.10 •95% confidence range is 1.8 2.2

•“n” = 2.0, Std_n = 0.20 •95% confidence range is 1.8 2.2


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•With the specifications at right, Monte Carlo results are as follows

• Sw(mean) = 0.356

• σ (Sw) = 0.038



•There is a 95% likelihood that Sw is contained within + / - 2 σ 600

Monte Carlo Distribution

Sw

•What are the 95% Upper and Lower Monte Carlo Sw limits?


•(0.356 – 0.076) < Sw < (0.356 + 0.076)

•0.280 < Sw < 0.432

•Be aware of how Excel ‘bins’ data

•RollingTheDice.pdf


0

100

200

300

400

500

0.00 0.10 0.20 0.30 0.40 0.50

Freq

uency

Sw

Sw


“S ” MC


•There is a 95% likelihood that Sw is contained within + / - 2 σ


•What is the Upper deterministic Sw limit?•What is the Lower deterministic Sw limit?


There is a 95% likelihood that Sw is contained within + / - 2 σ

•0.280 < Sw < 0.432

•The Best / Worst case would yield considerably more uncertainty

•0.239 < Sw < 0.50

•In practice, it’s unlikely (but not impossible) that Best / Worst of all attributes would occur simultaneously, so that Sw(Monte Carlo) provides a better representation


7/28/2009

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•Supplemental material•Excel can handle common probability distributions, and can thus serve as a Monte Carlo simulator. •Quantitative estimation of the

ncertaint allo s one to d t i


RollingTheDice.PDF

uncertainty allows one to determine where time / money is most effectively spent, and to further avoid the trap of being misled as a result of a previous bad experience with a poorly defined parameter•The importance of the various input parameters will change according toparameters will change, according to the various magnitudes. There may be a linkage in that one parameter becomes more or less important as another parameter value is change. •One size does not fit all feet.

•Supplemental material•An alternative, deterministicapproach to error analysis is accomplished by taking the derivative of Sw(Archie) with respect to each

Monte Carlo Evaluation of Sw(Archie)RiskyBusiness.PDF

f ( ) pattribute

Swn = a Rw / (Φ m Rt)

•The same approach will suffice for a shaly sand equation•The various terms in the derivative expression quantify the individual impact of uncertainty in each termimpact of uncertainty in each term, upon the result•The relative magnitude of each then allow one to recognize where the biggest bang for the buck, in terms of a core analyses program or suite of potential logs, is to be found (Figure 1).


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Monte Carlo Evaluation of Sw(Archie) Summary

•Sw uncertainty is a dynamic issue •Should be evaluated for each set of circumstances

•Attributes are ‘linked’•A change in one can cause another to become more, or less, g , ,important

•The evaluation can be done with either a (deterministic) derivative, or (probabilistic) Monte Carlo simulation approach•In general, it’s unlikely that Best / Worst of all attributes would occur simultaneously

•A deterministic or probabilistic evaluation will be more•A deterministic or probabilistic evaluation will be more representative


montecarlo illustrations

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