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to participate as Lecturers.
And special thanks to The American Institute of Mining, Metallurgical,
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2. Managing Uncertainty in the Reservoir Model
Ashley Francis
Managing Director
Earthworks Environment & Resources Ltd
16 January 2019
© Earthworks Environment & Resources Ltd. All rights reserved

3. What is Uncertainty? There is that which we know we know
There is that which we know we don’t know
There is that which we don’t know we don’t know
Who said this?
Plato, 380 BC, Meno
see

4. Uncertainty Precision or Accuracy?
Small Uncertainty
Precise
Inaccurate
Most technical procedures and uncertainty modelling are designed to reduce uncertainty by improving precision
Improving accuracy of predictions is a far more important objective
Accuracy
Actual Reserves
Probability
Large Uncertainty
Imprecise
Accurate
Reserves

5. Typical Stages of Reservoir Modelling
Structural framework modelling and faults
Grid and layer definitions
Scale-up of well logs into grid cells
Lateral prediction of facies and porosity
Permeability prediction

6. Horizons, Zones and Layers
Well #1
Well #2
Layers
Seismic Pick
Horizons
Cells
Zone 1
Zone 2
Formation Tops

7. Geostatistical Reservoir Model Weaknesses and Uncertainty
Top structure uncertainty
Scale-up of core and well log data into cells
Lateral prediction of properties away from wells
Relationships between porosity and permeability
Incorporating seismic property information

8. 1. Top Structure Uncertainty
Top structure surface is often a smooth, best case depth map
Smooth depth maps result in biased volumetric estimates
Top structure uncertainty is often ignored in reservoir modelling
What about gross rock volume uncertainty?
GRV may be largest uncertainty
GRV cannot be measured directly
GRV uncertainty routinely estimated using geostatistical realisations of depth conversion

9. Consequences of Smooth Structure Maps
The volume of the accumulation is generally underestimated (depth/OWC or GWC case)
The area of an accumulation is underestimated
The accumulation appears more contiguous / connected than it really is
These points can easily be illustrated by progressively smoothing a known surface such as digital elevation (topographic) data
Mapping (estimation) will smooth in a similar manner

10. Top Structure Smoothing 3D Seismic Data Quality
Smoothing = 13x13
Volume = 195 Mm3
Smoothing = 9x9
Volume = 227 Mm3
Smoothing = 5x5
Volume = 262 Mm3
Smoothing = 3x3
Volume = 278 Mm3
Smoothing = 1x1
Volume = 300 Mm3

11. Smoothing of the Cumulative Distribution Function (CDF)
Predicted Volume
Cumulative Frequency
Depth
Oil-water contact
CDF of Estimated Depth Surface
True Volume
CDF of Data
Integrate
Samples

12. Kriging and Simulation
Consider an experiment involving tossing a fair coin
Heads is denoted “0” and tails “1”
The Expected Value Ev is (P0.5 * 0 + P0.5 * 1) = 0.5
Ev is the mean, average, deterministic or kriged result
Our best estimate of the result of tossing the coin is always 0.5 because this gives the smallest average square error (least squares)
The only possible realisations (outcomes or simulations) for tossing the coin are “0” or “1”

13. Conditional Simulation Objectives
In order to obtain unbiased estimates of gross rock volume we require geostatistical simulations of the top structure
Each realisations is conditional to (ties) the measured data points at the wells
Each realisation restores the histogram using a Monte Carlo process and so avoids smoothing
Each realisation reproduces the spatial correlation behaviour represented by the variogram model
This is critical for the correct connectivity behaviour

14. 4 Example Realisations 2D Seismic Case
248 Mm3
233 Mm3
213 Mm3
197 Mm3

15. Cutoff Probability Calculations
Realisation #0002
Realisation #0001
Realisation #0003
Realisation #0004
Realisation #0020
Realisation #0010
Realisation #0005
Realisation #0050

16. Isoprobability Cutoff Map 100 Realisations

17. Connectivity Probability 2 Cells in Parallel
50%
100%
100%
50%
If critical nodes are independent connectivity = 75%
If critical nodes are dependent connectivity = 50%

18. Connectivity Calculations
Realisation #0003
Realisation #0002
Realisation #0001
Realisation #0004
Realisation #0005
Realisation #0050
Realisation #0020
Realisation #0010

19. Connectivity Probability Map 100 Realisations
Connected Probability = 72%

20. 2. Scale-up from Log to Cells
After designing the grid and layer definitions we need to perform a scale-up of the well logs into the grid cells penetrated by the wells
Volume scale change from well log sample to model cell is about 500,000 : 1
Typically between 10 and 300 log samples are averaged into a cell depending on the deviated trajectory of the well passing through the cell
Cell never contains the appropriate porosity and uncertainty for its scale

21. Scale of Measurements Log to Cell
Zone C
Zone B
Zone A
50 m
2 m

22. Scale-up or Re-sampling?
Averaging is a smoothing operation and changes the effective scale of measurement
CDF becomes narrower and steeper
Cell size variation and well deviation means CDF is varying from cell to cell across model
Averaging is between 10:1 and 300:1 from log to cell
Resampling by selecting a subset of data does not change the scale of measurement
CDF remains the same

23. Smoothing of the Cumulative Distribution Function (CDF)
Cumulative Frequency
Porosity
Porosity Cutoff
CDF of Averaged Data
Predicted Volume
CDF of Data
True Volume
Integrate
Samples

24. 3. Lateral Prediction from Wells
After simulating facies need to populate each facies with appropriate porosity (and permeability) values
Typical procedure is to map porosity in the model, usually by facies type and following the stratigraphic layering
Proper procedure is to geostatistically simulate porosity within the model

25. Radius of Influence Which Map Do You Prefer?
1000
2000
4000
6000
8000
10000

26. Mapping is Estimation
We refer to our reconstruction of the sub-surface as a map but this is misleading
Mapping is the work of a cartographer
A cartographer represents on a sheet of paper that which they already know
Sub-surface “mapping” is actually on estimation procedure
Prediction of the value of a property at an unmeasured location
Estimation obtains accuracy by smoothing the measured values towards the mean or local trend

27. 4. Permeability Transform
Permeability is defined at the core scale
Permeability values are very sparse at each well
Commonly perform a transform to either
Convert well log porosity to a permeability log
Convert mapped porosity to mapped permeability

28. Cutoffs on Predictors are Biased
Equation is best predictor of permeability from porosity (under certain assumptions)
Cutoff 1mD = 9.5% porosity
Porosity cutoff (B+D):
92.5% net
Permeability cutoff (A+B):
82.5% net
Cutoff calculations cannot be applied to the results of estimation A B C D
Actual Perm Range
Predicted Range

29. Poro-Perm Transforms
Transform from porosity to permeability causes smoothing due to estimation
Transform commonly defined at core scale but applied at log or even at cell scale
Typical 300:1 scale change from core to log
Over 100 Million : 1 scale change from core to model cell
Different properties scale-up in different ways
Porosity linear averaging
Permeability harmonic/geometric averaging or tensor upscaling
Estimators and cutoffs should not be used together

30. Scale of Measurements Core Plug to Cell
(x70)
Zone C
Zone B
Zone A
50 m
2 m

31. 5. Seismic Property Constraints
The use of deterministic seismic inversion to generate quantitative seismic impedance estimates is becoming common as an input to reservoir modelling
Deterministic seismic inversion combines data from wells and seismic to create a broad bandwidth impedance model of the earth
Seismic resolution is at a coarser scale than the cell size of most reservoir models
Average properties over a zone

32. Scale of Measurements Core Plug to Seismic
(x70)
Zone C
Zone B
Zone A
50 m
2 m

33. Deterministic Inversion
The impedance information at low frequencies is simply an interpolation of the well data and so deterministic inversions should not be used to condition reservoir models
Frequency
Seismic
Wells

34. CSSI Deterministic Inversion Final Inversion
Sparse Spike
Elastic Impedance
(All Frequencies)
1.5
2.0
2.5
Two-way time (sec)

35. CSSI Deterministic Inversion Low Frequency Model
Sparse Spike
Elastic Impedance
(Low Freq. Model)
1.5
2.0
2.5
Two-way time (sec)

36. Conditioning with Seismic Data
Conditioning a reservoir model to a deterministic inversion is equivalent to conditioning to a map of the wells only
It is ok to condition a reservoir model to
seismic attributes
relative or coloured impedances
deterministic inversion filtered to remove low frequencies
A better strategy is to condition to realisations of impedance generated through stochastic seismic inversion
Incorporates geophysical uncertainty in the reservoir model

37. Constraining Reservoir Models with 3D Seismic Volumes
Ideally our reservoir models should be fully consistent with the large amount of (pre-stack) 3D seismic amplitude information
Probability transforms from seismic to rock properties
Stochastic inversion to give impedance realisations
Conditioning to seismic through full rock physics model

38. Deterministic Inversion
Two-way time (ms)
5,000
10,000
Absolute Impedance (m/s * g/cc)

39. Lithology Discrimination
8,150

40. Deterministic Inversion Sand Prediction
Two-way time (ms)
8,150
12,150
Absolute Impedance (m/s * g/cc)

41. Deterministic Inversion Sand Thickness Prediction
1,000 ft
0.0
50.0
Sand Thickness (ft)

42. Non-Uniqueness Two-way time (ms) 4,500 11,500
Absolute Impedance (m/s * g/cc)

43. Predicted Sand P50 Stochastic Inversion
Two-way time (ms)
40%
100%
Sand Probability
50%

44. Stochastic Inversion Sand Isochron 50% Isoprob
1,000 ft
0.0
50.0
Sand Thickness (ft)

45. Stochastic Inversion Sand Isochron 50% Probability
1,000 ft
0.0
50.0
Sand Thickness (ft)

46. Deterministic Inversion Sand Prediction Failure
Poor Prediction
Of Thin Channel
Thick Sand
Correctly
Predicted
Sand Thickness
Under-Predicted
1,000 ft
0.0
50.0
Sand Thickness (ft)

47. Total Net Sand Volume Distribution Curve
Deterministic Net Sand
Estimate = 8.5%
Stochastic Net Sand
Mean Estimate = 13.5%
Well Net Sand
Average = 13.2%

48. Geostatistical Reservoir Model Coupled Directly to 3D Seismic
3D Reservoir Model
Geology/Petrophysics
Rock/Fluid
Property
Model
Forward Seismic
Seismic Constraint
Update Rock
Properties
Time Lapse
3D Seismic

49. Summary
Significant scale changes are not properly accounted for in reservoir models
Estimation, including linear regression and mapping, involve smoothing and this results in significant volumetric bias and potential misclassification
Deterministic seismic inversion data is not suitable for conditioning reservoir models
Future geostatistical reservoir models should include comprehensive mechanisms for conditioning directly to the seismic response through a coupled rock physics model