1. Brent Lowry & Jef Caers Stanford University, USA
Direct conditioning of upscaled reservoir models to fine scale well data using direct sampling
Brent Lowry & Jef Caers
Stanford University, USA
Good afternoon, we’ve talked about faster ways to model uncertainty, such as in Orhun’s talk, and today I’d like to talk about ways to directly include fine-scale data into coarse-scale models.

2. Scale Issue Account for fine-scale geological heterogeneity
With limited fine-scale simulation
Addressing the “missing scale” issue
How to account for fine-scale data in coarser grid blocks
Direct generation of upscaled realizations
Limiting amount of upscaling
Conditioning to fine scale data
The scale problem has been historically important for several reasons. For one, flow simulation usually necessitates a coarse-scale grid, but the support volume of well data is much smaller. How can this fine-scale data be used in the modeling of coarse-scale realizations? In particular, when looking at coarse grid blocks intersecting a well, this has been known as the missing scale problem. In this talk I will investigate with a small example some initial ideas on how we can directly integrate fine-scale well data into coarser scale flow models generated with MPS.

3. Information content of fine-scale data
One or multiple fine scale information v : zv
Reservoir simulation grid block V with property ZV
The coarse support volume “V” may contain multiple fine-scale well data as shown here by circles. The target therefore is to derive the conditional density of the coarse grid permeability given some fine-scale permeability from well data.
The question that we will address here is therefore twofold. Namely, how can we derive these conditional pdf. Evidently this will require some calibration and this calibration will require some limited amount of modeling of fine-scale models and upscaling those. This will be our training set for deriving the conditional pdfs. Then secondly, in MPS, we will replace the hard well data now with a conditional pdfs. So the question I will need to address is on how to integrate those pdfs into MPS algorithms.
Fine scale informs coarse scale :
Two questions:
How to derive these conditional distributions?
How to integrate them with MPS methods ?

4. desired 40  20 flow-scale model
Case example
200  fine-scale model
desired 40  20 flow-scale model
1,000 mD
1,000 mD
Aquifer storage and recovery project: Inject fresh water, recover
Purpose: uncertainty on recovery efficiency (RE)
Data: fine-scale data along a single well path
Questions: How to condition coarse scale models directly?
Does it affect uncertainty on RE?
Recover when needed

5. Generating coarse scale realizations
Direct Sampling
Randomly search a TI for the matching data event
Advantages: MPS with continuous, cosimulated variables
Mariethoz et. al. 2010

6. Training Image fine-scale permeability training image 1200  600
coarse-scale permeability training image
240  120
1,000 mD
1,000 mD

7. Probability Functions of Permeability in the X-Direction
Calibrating
Generate fine-scale models at the well
Upscale to desired coarseness
Each upscaled block has an associated distribution
unconditional
conditional
Probability Functions of Permeability in the X-Direction 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
1,000 mD

8. Multiple sets of block values
Search for replicates in training data
Create a PDF from accepted values
Multiply conditional and unconditional PDFs
Draw a sample
Continue until all nodes are populated ?
data event

9. Multiple coarse-scale realizations
Create many sets of coarse block values
Generate a coarse-scale realization from each

10. Multiple coarse-scale realizations
e-type
variance
1,000 mD
40,000

11. Comparison Fine-scale simulation reference
Multiple fine-scale realizations simulated
Each realization is upscaled
Ideal, time-consuming
Fixed block values
Hard data upscaled before simulation
Common, fast

12. Multiple realizations
40  20
realization
fixed block values
fine-simulation reference
e-type
1,000 mD
1,000 mD
variance
100,000
100,000

13. Flow modeling Aquifer storage and recovery system
Injecting and recovering potable water
Recovery Efficiency (RE)
Total recovered fresh water / total injected
Objective-centered evaluation
Way to compare each method

14. Flow simulation Water injected into water Three-step process
Injection for 87.5 days
Storage for 200 days
Production for 87.5 days

15. Comparison

16. Conclusions
Accounting for fine-scale data in coarse grid cells by calibrated conditional PDFs
Integrating these PDFs into MPS methods by direct sampling
Future:
Discover when this is needed
Integration with pattern techniques