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.

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.

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 ?

desired 40  20 flow-scale model Case example 200  100 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

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

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

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

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

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

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

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

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

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

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

Comparison

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