Energy Resources Engineering Department Stanford University, CA, USA A multiscale method for large-scale inverse modeling example of inverting single-phase flow data Jianlin Fu Jef Caers Hamdi Tchelepi Energy Resources Engineering Department Stanford University, CA, USA

Multiscale modeling Motivation Engineering of subsurface is dependent on large- and small-scale heterogeneity Different practical problems are solved at different representation scales Different data have different scale of information High-resolution reservoir modeling entails multiscale data integration Can we integrate all well data, seismic data, and production data into a model at any appropriate scale, including the fine scale?

State-of-the-art Methods & problems High-resolution modeling to integrate production data is CPU demanding: flow simulation is expensive Most approaches ignore the important fine-scale details Upscaling may filter out important small-scale variations Statistical downscaling ignores physics Physical methods ignore geology

Existing philosophies (upscaling) High-resolution inverse modeling Model updating (Stochastic) Optimization: GDM PPM upscaling ? Flow simulation problem solution 4

Existing philosophies (downscaling) High-resolution inverse modeling Model updating Upscaling Flow simulation Statistical downscaling … History matching may not be preserved ! 5

Summary of challenges Cannot run fully high-resolution flow simulations at the fine scale Reconcile two types of data Production data (ill-posed inverse problem) Spatial continuity model (interpreted from geological knowledge, e.g., variogram, TI)

A new philosophy High-resolution inverse modeling s s p p m Coupled PDEs m Gradient Gradient m Model updating Coupled PDEs 7

What are the difficulties ? How to run multiscale flow simulations? How to compute fine-scale gradient for optimization (history matching)? How to maintain geological consistency? 8

The multiscale solution (1) p p m Coupled PDEs m Model updating Coupled PDEs 9

Multiscale flow simulation Fine-scale flow simulation? Too expensive or impossible! Anxnpnx1=qnx1 Solution: reduce n Upscaling vs multiscale

Multiscale flow simulation Multiscale flow simulation? How …? p=? x Solution: Fine-scale reconstruction 11

Multiscale flow simulation Fine-scale pressure Multiscale pressure Fine-scale saturation Multiscale saturation Courtesy of Zhou

Multiscale flow simulation Error and speedup Courtesy of Wang 13

The multiscale solution (2) Coupled PDEs Gradient Gradient Model updating Coupled PDEs 14

Gradient  Define an objective function:  Minimization of J requires:

Basic adjoint method = 0  Forward flow PDE:  The Lagrangian: Lagrange multiplier  Forward flow PDE: Forward simulation  The Lagrangian: = 0  Adjoint PDE: Adjoint simulation  Gradient: 16

Multiscale adjoint simulation Run forward simulation Compute objective function Solve coarse-scale adjoint equation Solve fine-scale adjoint equation Assemble fine-scale gradient 17

The multiscale solution (3) Coupled PDEs Gradient m Model updating Coupled PDEs 18

Model updating + . Model updating: Model updating: Gradient is used: Gradient-based gradual deformation (Hu, 2004) Z1 Z2 Z3 Model updating: Model updating: + Z(α) Optimize α Gradient is used: 1) Optimization of α 2) Selection of Z2, Z3, … . 19

An illustrative example Experimental configuration (A small 2D case) Horizontal injector Horizontal producer lnK field Initial pressure field Constant Constant p=15 p=0

An illustrative example History matching to pressure data Reference field Initial field

An illustrative example Proposed method Computationally efficient Multiscale matched Fine-scale matched Almost identical convergence rate as fine-scale 22

An illustrative example Physically accurate Proposed method Multiscale simulation Fine-scale simulation Absolute error decreases as the iteration proceeds

An illustrative example Physically accurate Observations Before matching Multiscale matching Fine-scale simulation

An illustrative example Multiscale matched models Realization 1 Realization 2 Realization 3 25

An illustrative example Experiment: what if we do not enforce geological Consistency (pure deterministic optimization)? Matched lnK field Need for enforcing geological consistency 26

Conclusions Computationally efficient Physically accurate Similar iterations but more efficient Capable of handling large-scale cases Physically accurate Small absolute error btw multi- and fine-scale Geologically consistent Spatial structure preserved Let’s just give a very simple summary on the observations.

Thank you ! Questions ?