Modeling and Simulation for Multiphase Flow in Petroleum Reservoirs

Zhangxing Chen

University of Calgary

Sponsors

Synergia Polygen Ltd

Outline

•  Part I: Modeling and Simulation of Conventional Oil

•  Part II: Investigation of Compositional Grading

•  Part III: Current Research in Heavy Oil Modeling

Outline, Part I

•  My Research Background •  Models •  Current Developments •  Difficulties •  Conclusions and References

Basin Modeling Reservoir Simulation

From Basin Modeling to

Reservoir Filling to

Reservoir Simulation

Problem Description

Idealization

Conceptual Model

Mathematical Model Development of a numerical model

Model verification

Model validation and process identification

Measurements

Lab experiments

Analytical solution Numerical model Simulation

Initial and boundary conditions

Verification

Application

Lab scale

Comparison

Models: History of Numerical Reservoir Simulation

- 1950 – 1970, Study of dynamics of fluid flow and transport through porous media

- 1970 – 1980, Various reservoir simulators (black oil, compositional, thermal, dual porosity) based on the finite difference method

- 1980 – 1990, Commercial reservoir simulators (fully implicit method, fast solvers, EOS, vector computers) - 1990 – 2000, Workstation computer techniques, advanced

GUIs, integration with geo-modeling, geomechanics, parallel computer techniques (PVM, MPI, clusters)

- After 2000, Commercial unstructured grids simulators, large scale simulation on PC (64 bites), new history matching and optimization techniques, new computer hardware (multiple cores, GPUs, OpenMP, hybrid OpenMP-MPI, blue gene)

Models (cont’d): Oil production methods

•  Primary recovery: simple natural decompression

•  Secondary recovery: water injected

•  Enhanced recovery: -Miscible displacement -Chemical processes -Thermal processes

Models (cont’d): Types of fluid flows in porous media

•  Primary recovery: single-phase •  Secondary recovery: two-phase

(above a bubble pressure) or three-phase black oil (water, liquid, and gas)

•  Enhanced recovery: multicomponent, multiphase, isothermal or non-isothermal

Models (cont’d): Major laws

•  Conservation of mass •  Conservation of momentum •  Conservation of energy

Models (cont’d): Single phase flow

- Mass conservation equation:

- Darcy’s law:

Models (cont’d): Two-phase flow

- Mass conservation equation

- Darcy’s law

Pc=Po-Pw

Models (cont’d): Three-phase flow

•  Governing equations

•  Darcy’s Law

Models (cont’d): Three-phase flow

– Constraint equation

– Capillary pressures

Models (cont’d): Compositional flow

Models (cont’d): Thermal flow

•  Mass conservation •  Darcy’s law •  Phase package •  Conservation of energy:

Models (cont’d): Mathematical Issues

•  Existence of a solution •  Uniqueness of the solution •  Solution regularity

Current Developments

Geo-models

Field Scale Models

Gridding Solvers &

Parallelization

Validation & Applications

Software Research

Numerical Models

Journeying to the Reservoir

Current Developments (cont’d): Upscaling

•  Mathematical techniques: homogenization, volume averaging, etc.

•  Numerical upscaling: - purely numerical: renormalization, power law

averaging, harmonic mean, etc.

- multiscale methods

Current Developments (cont’d): Dynamical Gridding

–  Irregular geometric feature presentation • boundaries (and

BCs) •  faults •  fractures • pinch-outs

Current Developments (cont’d): Dynamical Gridding

– Complex features • complicated well

architecture •  local reaction

zones • different spatial

and temporal scales

• geomechanics

Current Developments (cont’d): Numerical Methods

– Finite difference methods – Finite volume (control

volume) methods – Finite element methods

Current Developments (cont’d): Fast Linear Solvers

•  Large scale systems (million unknowns) •  Coupling of different physical variables •  Highly nonsymmetric and indefinite matrices •  Ill conditioned systems •  Matrix structure spoiled by well perforation and

unstructured grids •  80-90% of the total simulation time spent on the

solution of large linear systems •  Limitation of problem size and space resolution on

a single processor

Current Developments (cont’d): Fast Linear Solvers •  Fast and robust solvers: - ORTHOMIN (orthogonal minimum residual) - GMRES (generalized minimum residual) - BiCGSTAB (biconjugate gradient stabilized) •  Efficient preconditioners: - ILU(k) - CPR (constrained pressure residual) - AMG (algebraic multigrid) •  Taking advantage of modern parallel architecture

Strong coupling & nonlinearity

Small diffusion

High resolution Heterogeneity

Irregular geometric features

Complex well

architecture

Difficulties

Instability and

fingering

Large scale systems

RESERVOIR SIMULATION

Surface facilities coupling

Difficulties (cont’d): Upscaling

•  Integration –  Disparate data with different scales –  Coupling of different flow, transport and

chemical processes

•  Upscaling –  Geological models with tens of millions

of cells to reservoir models with over one million cells

•  Speed of computation –  Fast enough for timely decisions

Difficulties (cont’d): Gridding

•  Grid adaptivity in space and time

•  Wells with complex features

•  Easy integration

Difficulties (cont’d): Numerical Methods

–  Multipoint upstream winding

–  Multipoint flux approximation

–  Instability and fingering

–  Small diffusion/dispersion representation

–  Mass and energy conservation

Difficulties (cont’d): Solvers

•  Large scale systems (million unknowns and long time integration )

•  Coupling of different physical variables •  Highly nonsymmetric and indefinite

matrices •  Matrix structure spoiled by well

perforation and unstructured grids •  Ill conditioned systems •  Limitation of problem size and space

resolution on a single processor

Current Research

Oil

Oil & Water Mixture

Water

Wells

Modelling of a Reservoir

Current Research (cont’d)

THAI Model

Modelling Complex Layers & Slanted

Wells

Complex Flow Due to Heterogeneous Geology

Validation of Simulator: n-Component (cont’d)

Rayleigh Number Validation

Reservoir with Baffles for n-Component Mixing (cont’d)

Conclusions

•  Development of simulator integrating geological and reservoir processes

•  Good features: flexibility, speed, accuracy, interface, etc.

•  Incorporation of more physics: fluid flow, heat transfer, chemistry, and geomechanics

•  All these mean significant savings in capital costs

Three Recent Books

•  Finite Element Methods and Their Applications

•  Z. Chen •  Year 2005 •  Over 1,000 copies sold

Three Recent Books (cont’d)

•  Computational Methods for Multiphase Flows in Porous Media

•  Year 2006 •  Z. Chen, G. Huan and Y.

Ma •  1st Edition out

Three Recent Books (cont’d)

•  Reservoir Simulation: Mathematical Techniques in Oil Recovery

•  Year 2007 •  Z. Chen •  NSF Summer School