Petroleum Reservoir Simulation Aziz

Petroleum Reservoir Simulation

Modeling Multiphase, Multi-Component Fluid Flow in Complex Geological Rocks

Khalid Aziz Engineering Resources Engineering

Reservoir and Facilities

El Shargi field, Occidental

Deepwater Challenge • BP operated Thunder Horse field lies beneath some 6000m of mud, rock and salt, topped by 1900m of ocean • Reservoir at over 1200 bar and 135°C • Advanced wells are required

Motivation • Development costs for typical oil fields are many billions of dollars • Every field is different • Development and operating actions are irreversible • Models are needed to develop “optimum” strategies • Annually around $10 billion spent on reservoir models, and it is increasing

Growing Energy Demand Total World Energy

Q u a d r illio n B T U

• Energy demand is outpacing new discoveries 800.0 700.0 600.0 • By 2030 energy demand 500.0 400.0 will increase by 50% 300.0 200.0 • Only about 35% 100.0 0.0 of OOINP is recovered • Impact of technology can be huge (~30-70%) • Technology can also reduce environmental footprint

1990 2002 2003 2010 2015 2020 2025 2030 Year

UGS Estimates • About half the reserves of conventional oil have been produced • Unconventional oil much harder to recover

We are not likely to be free of oil soon! • 85 million barrels/day now • 120 million barrels/day by 2030

Outline • What is reservoir simulation? • Underlying equations and solution techniques • Use

Reservoir Simulation

Gringarten, 2002

Integration of data from all sources (wells, cores, seismic, outcrops, well tests, etc.)

Data to Decisions Geosciences

Engineering

Simmodels Geomodels Data Collection, Interpretation and Integration

History Matching and Predictions

Analysis, Optimization and Control (Decisions)

Characteristics of the System • Complex and generally unknown geology • Multicomponent, multiphase flow – Poorly understood fluid mechanics – Thermodynamic complexity

• Complex wells and reservoir well interactions – Multiphase flow

• Strong connections to facilities and surroundings

Stanford VI reservoir model 6 million nodes – Castro et al.

• • • • 10-5

Data

Many sources Many scales (10-5 to 108 cm) Sparse Not always reliable

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

109

1010

Simulation Cells Geological Model Cells

Thin Sections

Well Test Core Data Well Log

Upscaling

Seismic Data

Downscaling Hamdi Tchelepi Pipat 2006

Process • Build one or more geological descriptions on a fine scale • Upscale to a computational grid • Establish boundary conditions and choose development and operating strategies • Solve appropriate equations describing flow • Predict reservoir performance • Maximize or minimize some objective function • Estimate uncertainty

GeoModel and Upscaling • Optimum level of and techniques for upscaling to minimize errors • Gridding and upscaling are interconnected

Gurpinar, 2001

Gridding • Honor geology • Preserve numerical accuracy • Be easy to generate

Gurpinar, 2001

Castellini, 2001

Prevost 2003

Wolfsteiner et al., 2002

Equations • Mass balance for each component in the system in each block (CVFD) • Additional Constraints • Wells and Facilities • Large number of non-linear equations

OGJ

Simulator Equations

∑∑

mcn,,pnl+,,1i

− ∑∑

mcw, ,pni +1

l p w p  

 

Flow Rate into Block i from Connected Blocks l

p - phase c - component

Flow Rate out of Block i through Well w in i

i

=

( ∑ Δt 1

M cn,+p1i − M cn, pi

)

i

p  

Accumulation Rate in Block i

l

Definitions • Flow Rate

mc , p l ,i = ( ϒ c , p )l ,i ⎡⎣ Φ p ,l − Φ p ,i ⎤⎦

M c , p = V (φ S pω c , p )

• Mass Accumulation • Rock

φ = φ [1 + c R ( p − p o

kA ϒ c , p = ωc , p λ pT , T = α Δx kr , p ωc , p = ρ p yc , p , λ p =

μp

o

)]

Methods of Solution ∑∑

mcn,,pnl+,,1i − ∑∑ mcw, ,pni +1 =

   

l

p

w

Flow Rate into Block i from Connected Blocks l

n ,n +1 mc , pl ,i

• • • •

p

Flow Rate out of Block i through Well w in i

= ( ϒ c , p )l ,i

n ,n +1

M ( ∑ Δt 1

n +1 c , pi

− M cn, pi

)

i

 

p

Accumulation Rate in Block i

⎡⎣Φ p ,l − Φ p ,i ⎤⎦

n +1

i

Explicit impractical Fully implicit most robust, but expensive Partially implicit (IMPES, IMPEC) cheaper Adaptive implicit is generally the optimum approach

l

General Formulation G G Non-linear equations set: F ( X ) = 0

G G G Rewrite it as: ⎧ ⎪ Fp ( X p , X s ) = 0 ⎨G G G ⎪⎩ Fs ( X p , X s ) = 0

⎧ p: ⎨ s: ⎩

primary secondary

• Appropriate variables, equations and alignment • All primary variables or a subset treated implicitly

Equations • Number of equations per block varies from 3 to around 10 (nc) • Number of blocks hundred thousand to several million (nb) • Optimum time step is selected automatically • Number of nonlinear equations to be solved every timestep: nc x nb • Equations are linearized using Newton’s method • Typical problems take about 3 iterations per timestep, difficult problems may not converge

Linearized Equations 0

200

400

600

800

1000

1200

1400 0

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

200

400

600

From Jiang 2006

800 nz = 74960

1000

1200

1400

⎞ ⎛ ⎞ ⎛ ⎤ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ → → ⎥ ⎥ • ⎜ X ⎟ = −⎜ R ⎟ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎥ ⎟ ⎜ ⎟ ⎜ ⎦ ⎠ ⎝ ⎠ ⎝

Multi-level Sparse Block Matrix 0

• R ~ reservoir • F ~ facilities and wells

200

400

600

800

1000

1200

1400 0

200

400

600

800 nz = 74960

1000

1200

RR

RF

FR

FF

RW1 RW2

1400

From Jiang 2006

Process 1. Create one or more images of the reservoir based on available data 2. Set objectives 3. Create a grid 4. Select time step 5. Iteratively solve equations to advance solution 6. Go to 3 and continue until • •

Desired time is reached, or Some constraint is violated

7. Go to 2

Block-Based Linear Solvers • Block Solvers – GMRES & BiCGstab

(from IML)

• Multi-Level Block Preconditioners – CPR – BILU(0) – BILU(k) From Jiang 2006

Performance of Block Solvers 9 components, 100x100x5 grid (FIM), solver time 18

14.9

Speedup Factor

15 12

8.3

9 6 3

1

1.6

0 PGMRES+ ILU

From Jiang 2006

BGMRES+ BILU

PGMRES+ CPR(ILU)

BGMRES+ CPR(BILU)

Other Complications • • • • •

Fractured Systems Mutiphase flow in wells and facilities Complex recovery processes Unconventional resources Geomechanics

Modeling Fractures Image source: http://210.42.35.8/ybs/images/jcz/lar7.jpg

From: Bin Gong 06

• Most reservoirs are fractured • Modeling individual fractures is neither possible or desirable • Usually dual media approach is used

Dual Porosity Model Real fractured system

Idealized sugar-cube model

matrix

matrix block

fracture

fracture

(Aziz & Settari, 1979, after Warren & Root, 1963) From: Bin Gong 06

ƒ Main transport through fractures ƒ Flow between matrix and fracture is modeled by transfer functions ƒ Number of equations doubles

Enhancements to Dual Porosity Models

Matrix

Fracture From: Bin Gong 06

• May allow flow between matrix blocks • Subgrid matrix blocks

Modeling Fine Scale Features

• Explicitly model major faults and fractures • Near well modeling Karimi Fard 2006

Discrete Feature Model Features are represented as interfaces between matrix control volumes Matrix

Fracture

Karimi Fard 2006

Treatment of Intersections Fractures

Grid domain

Computational domain

Intermediate control-volume

Karimi Fard 2006

Connectivity list

Modified connectivity list

Star-Delta transformation

Well Model in Reservoir Simulator

• Predicting pressure drop in wellbores is an important component • Wellbore flow model needs to be simple, continuous, and differentiable

Gas-Liquid Flow in Pipes Horizontal Flow Stratified Smooth Flow

Stratified Wavy Flow Elongated Bubble Flow Slug Flow

Annular Flow

Bubble Flow

Slug Flow

Figures from Shoham (1982)

Churn Flow

Annular Flow

Dispersed Bubble Flow

Modeling of Complex Processes • Limited ability to model processes involving – Fast phase changes – Chemical reactions (in situ upgrading) – Unstable fronts

• Unconventional Resources

Rock Deformation Coupled Geomechanics and Fluid Flow Geomechanics Simulator Δ δ K L F ⎡ ⎤⎡ t ⎤ ⎡ ⎤ = ⎢ LT E⎥ ⎢Δ P ⎥ ⎢ R⎥ Flow Simulator ⎣ ⎦⎣ t ⎦ ⎣ ⎦

Subsidence in North Adriatic

From ENI

General Purpose Research Simulator (GPRS) Design belonging inheritance

field SimMaster

core concepts facilities

other surfac

wells

solvers

wellgroup

grid …

stdwell mswell

reservoir

smart wells

fluid rock

From Jiang 2006

Object Oriented Design field

SimMaster

facilities surfac

solvers

SimMaster

reservoir

wells

grid rock

stdwells mswells

……

smart wells

reservoir 1 From Jiang 2006

facilities surfac

solvers

SimMaster

reservoir

wells

grid rock

stdwells mswells smart wells

reservoir 2

……

facilities surfac

solvers

reservoir

wells

grid rock

stdwells mswells smart wells

reservoir 3

……

Concluding Remarks • There have been continuous improvements in simulation techniques over the past 50 or so years • Many challenges remain to make reservoir simulators more accurate, efficient and robust • Benefits can be huge

Acknowledgements • Based on the work of many students and colleagues • Supported by SUPRI-B and SUPRI-HW consortia, and the new Smart Fields Consortium (SUPRI-SFC) • Additional support from DOE and several oil companies