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# Class Note for C&PE 940 at KU

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Date Created: 02/06/15

STOCHASTIC SIMULATION And RESERVOIR MODELING WORKFLOW CampPE 940 21 October 2005 Geoff Bohling Assistant Scientist Kansas Geological Survey geoffkgskuedu 8642093 Overheads and other resources available at httppeoplekuedugbohlingcpe940 Stochastic Simulation Stochastic simulation is a means for generating multiple equiprobable realizations of the property in question rather than simply estimating the mean Essentially we are adding back in some noise to undo the smoothing effect of kriging This possibly gives a better representation of the natural variability of the property in question and gives us a means for quantifying our uncertainty regarding what s really down there The two most commonly used forms of simulation for reservoir modeling applications are sequential Gaussian simulation for continuous variables like porosity and sequential indicator simulation for categorical variables like facies The basic idea of sequential Gaussian simulation SGS is very simple Recall that kriging gives us an estimate of both the mean and standard deviation of the variable at each grid node meaning we can represent the variable at each grid node as a random variable following a normal Gaussian distribution Rather than chooses the mean as the estimate at each node SGS chooses a random deviate from this normal distribution selected according to a uniform random number representing the probability level For our sixpoint example in the porosity data ordinary kriging gave a mean estimate of 1293 with a standard deviation of 049 In this case if we happened to generate a uniform random number ofp 0665 for this grid node then the assigned porosity would be 1314 the corresponding value of the cumulative normal probability function 10 M cm for N1293049 e a 1 Probability 6 a WWII 4 02 11 12 13 14 15 Porosity 00 So the basic steps in the SGS process are 0 Generate a random path through the grid nodes 0 Visit the first node along the path and use kriging to estimate a mean and standard deviation for the variable at that node based on surrounding data values 0 Select a value at random from the corresponding normal distribution and set the variable value at that node to that number 0 Visit each successive node in the random path and repeat the process including previously simulated nodes as data values in the kriging process We use a random path to avoid artifacts induced by walking through the grid in a regular fashion We include previously simulated grid nodes as data in order to preserve the proper covariance structure between the simulated values Sometimes SGS is implemented in a multigrid fashion first simulating on a coarse grid a subset of the fine grid maybe every 10th grid node and then on the finer grid maybe with an intermediate step or two in order to reproduce largescale semivariogram structures Without this the screening effect of kriging quickly takes over as the simulation progresses and nodes get filled in so that most nodes are conditioned only on nearby values so that small scale structure is reproduced better than large scale structure For SGS it is important that the data actually follow a Gaussian distribution If they do not we can use a normal score transform Empirical CDF Normal CDF 10 10 3 08 03 o E z 4 06 a E n E 2 0 04 E E E E u E U 02 02 00 I 00 v 9 10 11 12 3 2 1 0 1 2 3 Porosity Normal Score Empirical Density Normal Density 04 04 03 03 lt gt 02 02 01 01 00 00 a 10 12 Porosity quotoi 14 3 0 3 Normal Score Here are six sequential Gaussian simulations of our porosity data using the spherical semiVaIiogram model and a 16 nearest neighbor search 365 Porosity 5 500010000 18 17 is E 15 U7 E g 14 U z 13 12 11 5000 10000 Easting m Sequential indicator simulation SIS is very similar to sequential Gaussian simulation expect that indicator kriging is used to build up a discrete cumulative density function for the individual categories at each case and the node is assigned a category selected at random from this discrete CDF Very brie y an indicator representation for a categorical variable such as facies would be formulated as 1 if facies kis resent at u zuk P a 0 otherwise where you would have one indicator variable for each of the K different facies We can then use kriging based on indicator semivariograms to produce a set of facies membership probabilities at each grid point build up a CDF from the probabilities and select a facies at random from the CDF Cherty dolomitic ls 002 Dolomitic ls 006 Cherty dolomite 006 10 o as l o 01 l Dolomite 030 Cumulative Probability E l 02 Dolomitic shale 006 00 l l l l l 1 2 3 4 5 Lithology Number For a continuous variable such as permeability indicator variables are built by comparing data values to a set of thresholds Zki iuak 1 if 2uaS Zk 0 otherw1se We might define thresholds for example at the 10th 25th 50th 75th and 90th percentiles of the data distribution for example In this case kriging the indicator values for the klh threshold Zk gives estimates of PZ u S 2k at each estimation point Since this already is a cumulative probability we don t need to go through the process of summing to get a CDF although we will need to correct any violations of the eXpected order relationships PZu S zkS PZ u S 261 that happen to occur In this case SIS assigns each node to a corresponding range eg upper 10 by random selection from the CDF and the resulting indicator vector gets added to the conditioning data for the remaining nodes Typical Reservoir Modeling Work ow Basically work from largescale structure to smallscale structure and generally from more deterministic methods to more stochastic methods 0 Establish largescale geologic structure for example by deterministic interpolation of formation tops this creates a sete of distinct zones 0 Within each zone use SIS or some other discrete simulation technique such as objectbased simulation to generate realizations of the facies distribution the primary control on the porosity amp permeability distributions 0 Within each facies use SGS or similar to generate porosity distirubtion and then simulate permeability distribution conditional to porosity distribution assuming there is some relationship between the two Porosity and facies simulations could be conditioned to other secondary data such as seismic Methods also eXist for conditioning to well test and production data but these are fairly elaborate and probably not in very common use as yet More typical maybe to run ow simulations after the fact and rank realizations by comparison to historical production amp well tests For more on reservoir modeling work ow and assessment of uncertainty from multiple realizations see C V Deutsch 2002 Geoslalislical Reservoir Modeling Oxford Univeristy Press 376 pp Dealing With Trend Zone A Thickness Data Here are the Zone A thickness Values measured in the same 85 Wells as our example porosity data Northing m 15000 10000 i 5000 Zone A Thickness m O 0 1 1 1 10000 15000 20000 Easting m 1 0 5000 The SWNE trend affecw the omnidirectional semiVariogram Semivariance 44 4o 36 32 28 24 2000 4000 6000 8000 Lag m 10 Approach 1 Fit a global linear trend and krige the residuals Thickness Trend and Residuals m i 0 15000 5 O 10000 Northing m 000 5000 i i i 5000 10000 15000 Easting m Semivariogram of Thickness Residuals m h r o o o o o E ltr 7 o 0 ii 2 m r Exponentiaimodei g sm48m m N 7 Range 4020 m o i i i i 2000 4000 0000 8000 Lag m 11 Northing m Northing m 15000 10000 5000 15000 10000 5000 SimpleKriged Thickness Residuals m 5000 10000 15000 Easting m Thickness Trend Kriged Resid m 5000 10000 15000 Easting m 45 40 35 30 25 Approach 2 Fit Variogram in trendfree direction and use kriging With rstorder trend in X amp Y Here are the directional Variograms directions are azimuths from north computed With a directional tolerance of 225 Directional Variograms for Thickness 2000 4000 6000 8000 i i i i i 90 135 i 20 O 15 o O O 0 710 o O o O o 3 O gXKJLgALfkgi 5 C o E a 0 45 gt 7 O E 20 O O n m o 15 o O o 1039 o o o 7 O O 5 K O O i i i i 2000 4000 6000 8000 Lag m The model shown is that tted to the semiVariogram for N 135 E Which seems to be reasonably trendfree The model is exponential With a sill of61 m2 and arange of5292 m 13 We then krige using a rstorder trend model in X amp Y and the presumably trendfree semivariogram model for N 135 E Thickness m from Kr 45 15000 40 10000 35 30 5000 25 5000 10000 15000 Easting m ng with Trend Northing m 14 Accounting for PorosityPermeability Correlation Permeability data are available from 42 of the 85 Zone A wells Log10Permeability md I 1 16000 i7 o o o A O 0 07 E 10000 k o5 U y 0 3 2 o 01 g f o 01 Z 5000 9 9 e o o o r a o k 0 1 1 1 1 0 5000 10000 15000 20000 Easting m There is a fairly strong correlation between the logperms and porosity at the 42 wells PorosityLogPerm Correlation 9 7 o ltr 7 39 Q in N r 39 E Q g 5 O a o D 4 N 9 Correlation 072 2 r LogPerm 28 020 Porosity 1 1 1 1 1 1 12 13 14 15 16 17 Porosity 15 We Want to account for the observed permeabilityporosity relationship for two reasons 1 to preserve this relationship in our modeled permeability and porosity distributions and 2 to take advantage of the more abundant porosity data in our estimation of the permeability To do this We Will Work With the LogPerm residuals actual LogPerms minus those predicted from the porosity and add the kriged or simulated residuals back into a mean LogPerm grid predicted from the porosity grid The LogPerm residuals at the Wells seem reasonably normally distributed A02 A01 01 02 l l l l Quantiles of Log10Perm Residuals 0 0 l 03 Standard Normal Quantiles 16 and may or may not shoW a bit of trend Log1DPermeability Residuals a c 150007 39 7 o 39 o A 0 0 0 03 100007 39 7 02 ow o 5 0 01 E 0 0 C 15 0 01 o z 50007 0 00 7 0392 0 03 o 0 3 o 0 07 0 7 1 1 1 1 1 0 5000 10000 15000 20000 Easting m Semivariogram of LogPerm Residuals 8 c n 0 2 o o g m7 o o 7 o E g E 7 o m 0 8 o g 1 1 1 1 0 2000 4000 6000 8000 10000 Lag m The semiVaIiogram model is exponential With a sill of0026 and a range of l 1800 m The tted sill matches the global Va1iance quite closely so I am going to treat the residuals as trendfree 17 Here are six sequential Gaussian simulations of the LogPeIm residuals based on simple inging with a 16 nearestneighbor search Six Simulations of LogPerm Residuals 500010000 0 6 0 4 A 0 2 E 07 c 39E 0 0 1 D Z 70 2 70 4 70 6 500010000 Easting m 18 And here are those six simulations ofthe LogPerrn itselfbased on adding the simulated residuals to the LogPerm Values predicted on the basis of the six porosity simulations We developed before using the regression equation developed on the Well data Six LogPenn Simultations 500010000 03 15000 10000 a s 5000 04 g 02 U 5 00 O 2 15000 0 2 10000 04 5000 gt00 5000 10000 Easting m 19 Here are the crossplow of simulated LogPerm Versus simulated porosity along With the regression line developed from the Well data The simulated Values show a somewhat higher correlation overall 079 than seen in the original Well data 072 probably due to the small positive correlation 0 19 between the simulated LogPerm residuals and simulated porosities Simulated LogPermPorosity Crossplots 12 14 16 18 1 1 1 1 1 sim5 sim6 Log Perm 1 12 14 16 18 Porosity 20

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