Effective Solvers for Reservoir Simulation - Ricam · PDF fileE ective Solvers for Reservoir Simulation Xiaozhe Hu The Pennsylvania State University Numerical Analysis of Multiscale

Effective Solvers for Reservoir Simulation

Xiaozhe Hu

The Pennsylvania State University

Numerical Analysis of Multiscale Problems & Stochastic Modelling,

RICAM, Linz, Dec. 12-16, 2011

X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 1 / 47

Collaboration with:I ExxonMobil Upstream Research CompanyI China National Offshore Oil CooperationI Monix Energy Solutions

Main Collaborators:I James Brannick (PSU), Yao Chen (PSU), Chunsheng Feng (XTU),

Panayot Vassilevski (LLNL), Jinchao Xu (PSU),Chensong Zhang (CAS), Shiquan Zhang (Fraunhofer ITWM),and Ludmil Zikatanov (PSU)


Outline

Outline

1 Reservoir Simulation

2 Solvers for Reservoir Simulation

3 Applications & Numerical Tests

4 Ongoing Work and Conclusions


Reservoir Simulation







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Oil Recovery

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List of Oil Products


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!  Primary Recovery !  Secondary Recovery !  Enhanced Oil Recovery

Primary Recovery

Secondary Recovery

Enhanced Oil Recovery



Enhanced Oil Recovery

Enhanced Oil Recovery (EOR) isa generic term for techniques forincreasing the oil recovery ratefrom an oil field:

Thermal Method

Microbial Injection

Gas Injection

Chemical FloodingI AlkalineI PolymerI SurfactantI GelI · · ·· · ·

Recovery Rate:

Primary and Secondaryrecovery: 20-40%

Enhanced oil recovery:30-60%



Black Oil Model

Mass Conservation:

∂

∂t

[φ

(So

Bo+

Rv Sg

Bg

)]= −∇ ·

(1

Bouo +

Rv

Bgug

)+ qO

∂

∂t

[φ

(Sg

Bg+

RsSo

Bo

)]= −∇ ·

(1

Bgug +

Rs

Bouo

)+ qG

∂

∂t

(φ

Sw

Bw

)= −∇ ·

(1

Bwuw

)+ qW

Darcy’s Law:

uα = −kkrα

µα(∇Pα − ραg∇z), α = o, g ,w

Constitutive Laws:

So + Sg + Sw = 1, Pcow = Po − Pw , Pcog = Pg − Po ,



Modified Black Oil Model

Add polymer and sodium chloride:

∂

∂t

(φS∗

w CP

Br Bw

)+

∂

∂t(ρr (1− φ)Ca) = −∇

(CP

BwuP

)+ qW CP

∂

∂t

(φSw CN

Br Bw

)= −∇

(CN

BwuN

)+ qW CN

Effects on Viscosity:

uw = − kkrw

µw ,ef f Rk(∇Pw − ρwg∇z)

Add more components (brine, gel, surfactant, · · · )



Numerical Method

Discretization in time:I Fully Implicit Method (FIM)

(Douglas, Peaceman & Rachford 1959)I Implicit Pressure Explicit Saturation (IMPES)

(Sheldon, Zondek & Cardwell 1959; Stone & Garder 1961)I Sequential Solution Method (SSM)

(MacDonald & Coats 1970)I Adaptive Implicit Method (AIM)

(Thomas & Thurnau 1983)

Discretization in space: finite difference, finite volume, finiteelement, etc.

Linearization: Newton-Raphson Method.

Grid: Structured grids are still used by the main-stream; butunstructured grids are catching up.



Fully Implicit Method

1

∆t

[φ

(Sg

Bg+

RsSo

Bo

)]n+1

−[φ

(Sg

Bg+

RsSo

Bo

)]n

=

∇ · (T n+1g ∇Φn+1

α + Rn+1s T n+1

o ∇Φn+1o )

1

∆t

[(φSw

Bw

)n+1

−(φSw

Bw

)n]

= ∇ · (T n+1w ∇Φn+1

w )

1

∆t

[(φSo

Bo

)n+1

−(φSo

Bo

)n]

= ∇ · (T n+1o ∇Φn+1

o )

Potential:Φα := Pα − ραgz , α = w , o, g ,

Transmissibility:

Tα :=krαk

µαBα, α = w , o, g .



Newton-Raphson Method

In each Newton Iteration, we need to solve the following Jacobian system:

JgP JgSw JgSo

JwP JwSw JwSo

JoP JoSw JoSo

δPδSw

δSo

=

Rg

Rw

Ro

Different blocks have different properties, for example

JgP =1

∆tcgP −∇ · DgP∇− CgP · ∇ −RgP ,

usually the diffusion term in JgP is dominating, which makes JgP like an ellipticproblem. However,

JoSo =1

∆tcoSo − CoSo · ∇ −RoSo ,

here convection term is dominating, which makes JoSo like a transport problem.



Including Wells

Peaceman Model

One-dimentional radial flow

Single-phase flow in thewellbore

qi,α = −NperfX

j

2πhperf Kkrα

ln( rerw +s

)µα(Pj−Pbhp−Hwj)

Jacobian system with wells:

[JRR JRW

JWR JWW

] [δRδW

]=

[RR

RW

]


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∂

∂t

φS∗

wCP

Bw

+

∂

∂t(ρr(1− φ)Ca) = −∇

CP

BwuP

+ qW CP

uw = − kkrw

µw,effRk(∇P − ρwg∇z)

Add polymer(s) to the drive

Effects on viscosity

Add more components (polymer, brine, gel, surfactant, etc)


.&))'!"#&)/

•  One-dimensional radial flow into the wellbore; •  Single-phase flow; •  Pseudo steady-state pressure behavior.

Peaceman Model

o  Multi-segment o  Multi-lateral o  Complex wells o  Couple wells with reservoir



Numerical ChallengesStrongly coupled and complicated PDE system

High nonlinearity

Complicated geometry: irregular domain with faults, pinch-out and erosion

Heterogeneous porous media; large permeability jumps

Complex wells

Unstructured grids: corner point grid, anisotropic mesh

Highly non-symmetric and indefinite large-scale Jacobian system

Different properties of the physical quantities (the equation that describesthe pressure is mainly elliptic, the equation that describes saturation ismainly hyperbolic)

Commercial Simulators set a high bar: ECL2009 (Schlumberger), STARS(CMG), VIP (Halliburton), · · ·

Goal: develop fast solvers, deliver a solver package


Solvers for Reservoir Simulation







Solvers

“For a reservoir simulation with a number of gridblocks of order 100,000,about 80%− 90% of the total simulation time is spent on the solution oflinear systems with the Jacobian.”—- Chen, Huan, & Ma, Computational Methods for Multiphase Flows in Porous Media, 2005

Direct solvers: (Price & Coats 1974)

Incomplete factorization preconditioner: (Watts 1981, Behie & Vinsome 1982; Meyerink 1983;

Appleyard & Cheshire 1983)

Constrained Pressure Residual (CPR) preconditioner: (Wallis 1983; Wallis,Kendall, Little &

Nolen 1985; Aksoylu & Klie 2009)

Algebraic Multigrid (AMG) Method :

I Used in CPR preconditioner (pressure equation): (Klie 1997; Lacroix, Vassilevski &

Wheeler 2001, 2003; Scheichl, Masson & Wendebourg 2003; Cao, Tchelepi, Wallis & Yardumian 2005;

Hammersley & Ponting 2008; Dubois, Mishev & Zikatanov 2009; Al-Shaalan, Klie, Dogru & Wheeler 2009;

Jiang & Tchelepi 2009)

I For whole Jacobian system: (Papadopoulos & Tchelepi 2003; Stuben 2004; Clees & Ganzer

2007)



Solvers

“For a reservoir simulation with a number of gridblocks of order 100,000,about 80%− 90% of the total simulation time is spent on the solution oflinear systems with the Jacobian.”—- Chen, Huan, & Ma, Computational Methods for Multiphase Flows in Porous Media, 2005

Direct solvers: (Price & Coats 1974)

Incomplete factorization preconditioner: (Watts 1981, Behie & Vinsome 1982; Meyerink 1983;

Appleyard & Cheshire 1983)

Constrained Pressure Residual (CPR) preconditioner: (Wallis 1983; Wallis,Kendall, Little &

Nolen 1985; Aksoylu & Klie 2009)

Algebraic Multigrid (AMG) Method :

I Used in CPR preconditioner (pressure equation): (Klie 1997; Lacroix, Vassilevski &

Wheeler 2001, 2003; Scheichl, Masson & Wendebourg 2003; Cao, Tchelepi, Wallis & Yardumian 2005;

Hammersley & Ponting 2008; Dubois, Mishev & Zikatanov 2009; Al-Shaalan, Klie, Dogru & Wheeler 2009;

Jiang & Tchelepi 2009)

I For whole Jacobian system: (Papadopoulos & Tchelepi 2003; Stuben 2004; Clees & Ganzer

2007)



Auxiliary Space Preconditioning

1 construct preconditioner via auxiliary (simpler) problems

2 solve the auxiliary problems by efficient solvers (such as multigrid)

3 apply preconditioned Krylov subspace methods

Examples:

Fictitious Domain Methods (Nepomnyaschikh 1992)

Method of Subspace Correction (Xu 1992)

Auxiliary Space Method (Xu 1996)

Nodal Auxiliary Space Preconditioning in H(curl) and H(div) spaces(Hiptmair & Xu 2007)

· · · · · ·



Auxiliary Space Method (Xu 1996)

Abstract problem: A : V → V is SPD

Au = f

Auxiliary spaces:V = V ×W1 × · · · ×WJ

Additive preconditioner:

B = S +J∑

j=1

Πj A−1j Π∗

j

With Πj : Wj 7→ V (transfer) and S : V 7→ V (smoother). It can beshown that under some assumptions

κ(BA) ≤ C



Auxiliary Space Method for Reservoir SimulationFast Auxiliary Space Preconditioning

Auxiliary Spaces:

V = V ×WP ×WS ×Wwell

FASP Algorithm: Given u0, Bu0 := u4 where

u1 = u0 + ΠwellA−1wellΠ

∗well(f − Au0)

u2 = u1 + ΠSA−1S Π∗

S(f − Au1)

u3 = u2 + ΠPA−1P Π∗

P(f − Au2)

u4 = u3 + S(f − Au3)

Auxiliary problems Awell , AS and AP are solved approximately.


Applications & Numerical Tests






Applications & Numerical Tests Single Phase Flow

Single Phase Flow (pressure equation)Intergrated Reservoir Performance PredictionExxonMobil Upstream Research Company

Possion-like Model Problem (After linearization):

−∇ · (a∇p) + cp = f , in Ω

p = gD , on ∂ΩD ,

(a∇p) · nnn = gN on ∂ΩN ,

Discretization: Hybrid-Mixed FEM (Kuznetsov & Repin 2003)

Linear system: Symmetric and Positive definite



Difficulties for the solver

Models are from real problems,the geometry is complicated

Mesh (hexahedral grid): Highlyunstructured; Degenerated,nonmatching, and distortedgridblock.

Permeability: Large jumps: varyform 10−4 to 104; Anisotropic.

Only two types of pinchout hexahedrons satisfy these assumptions. Dueto the terminology in [6], we call them the “Element H1” (the vertical edgeconnecting node 0 with node 4 vanishes) and “horizontal triangular prism”(two neighboring vertical edges (0, 4) and (1, 5) disappear), see Figure 4.

5

6

70, 4

1

23

6

70, 4

1, 5

3 2

Figure 4: Element H1 (left) and horizontal triangular prism (right)

The element H1 has seven vertexes, two triangular and four quadrilateralfaces, and 11 edges. The horizontal prism has six vertexes, two triangularand three quadrilateral faces.

13

Also, due to the presence of the so called fault surfaces, nonmatchingmeshes are possible. For instance, consider a case when we have a domaindivided into geological layers which allows us to construct a hexahedral meshfitting to the interfaces between layers. Suddenly, something happens, forinstance, an earthquake or any other natural disaster. In this case, certainhexahedral cells may change their positions, and some of the hexahedral cellsmay have two or more neighbors from the side of the fault. Therefore, weobtain a nonmatching mesh. Several examples of nonmatching hexahedralmeshes were considered in the previous report [2]. In this report, the side of ahexahedral cell which is adjacent to a fault interface was divided into severalparts. The number of this parts is equal to the number of the neighboring cellsfrom the other side of the fault interface. For instance, the green hexahedralcell shown in Figure 3 has three neighbors adjacent to its right side.

Figure 3: An example of a distorted nonmatching hexahedral mesh from theprevious project

11



AMG with ILU smoother?

Preconditioner #iter Setup time Solve timeILU(0) 3458 0.16 96.19AMG 362 0.85 40.32

ILU(0)/1+AMG 2255 1.00 305.17ILU(0)/2+AMG - 1.18 -

ILU or AMG alone does not work well

Using ILU as smoother in AMG may not work

Why AMG with ILU smoother does not work in this case?

Consider (B denotes ILU and S denotes AMG)

u ← u + B(f − Au), u ← u + S(f − Au), u ← u + B(f − Au).

This gives:I − BA = (I − BA)(I − SA)(I − BA).

B might not be positive definite, and cannot be applied to PCG

References about ILU smoother: Wittum 1989; Stevenson 1994





ILU(0)/1+AMG 2255 1.00 305.17ILU(0)/2+AMG - 1.18 -













ILU(0)/1+AMG 2255 1.00 305.17ILU(0)/2+AMG - 1.18 -











AMG + ILU!Consider

u ← u + S(f − Au), u ← u + B(f − Au), u ← u + S(f − Au).

This gives:I − BA = (I − SA)(I − BA)(I − SA).

Theorem (Falgout, H., Wu, Xu, Zhang, Zhang & Zikatanov 2011)

Assume that S : V → V satisfies ‖(I − SA)x‖A ≤ ‖x‖A, ∀x ∈ V andoperator B : V → V is SPD. Then the operator B is SPD.


Assume that ((I − SA)v , v)A ≤ ρ(v , v)A, ρ ∈ [0, 1), B is SPD and thecondition number of BA is κ(BA) = m1/m0. If m1 > 1 ≥ m0 > 0. thenκ(BA) ≤ κ(BA), Furthermore, if ρ ≥ 1− m0

m1−1 , then κ(BA) ≤ κ(SA).



AMG + ILU!Consider

u ← u + S(f − Au), u ← u + B(f − Au), u ← u + S(f − Au).

This gives:I − BA = (I − SA)(I − BA)(I − SA).


Assume that S : V → V satisfies ‖(I − SA)x‖A ≤ ‖x‖A, ∀x ∈ V andoperator B : V → V is SPD. Then the operator B is SPD.


Assume that ((I − SA)v , v)A ≤ ρ(v , v)A, ρ ∈ [0, 1), B is SPD and thecondition number of BA is κ(BA) = m1/m0. If m1 > 1 ≥ m0 > 0. thenκ(BA) ≤ κ(BA), Furthermore, if ρ ≥ 1− m0

m1−1 , then κ(BA) ≤ κ(SA).



ExxonMobil Test Problems


ILU(0)/1+AMG 2255 1.00 305.17ILU(0)/2+AMG - 1.18 -

FASP (AMG+ILU) 41 0.99 5.83

Model 1 Model 2 Model 3 Model 4size 388,675 156,036 287,553 312,851nnz 4,312,175 1,620,356 3,055,173 3,461,087

#iter Setup time (s) Solve time (s) Total time (s)FASP(AMG+ILU) + CG

Model 1 16 2.12 4.63 6.75Model 2 41 0.99 5.83 6.82Model 3 47 1.97 12.81 14.78Model 4 11 1.75 2.68 4.43









ExxonMobil Large Test Problems



ExxonMobil Large Test Problems


Applications & Numerical Tests Enhanced Oil Recovery

(Modified) Black Oil ModelCNOOC and Monix: SOCF Simulator

Decoupling: Alternative Block Factorization (Bank, Chan, Coughran &

Smith 1989)

Auxiliary problems:I WP : pressure (+ bottom hole pressure)I WS : saturation (+ concentrations)I Wwell : wells (+ perforations)

Solvers for each auxiliary problems:I AP : AMG (+ILU)I AS : Block Gauss-Seidel with downwind ordering and crosswind blocks

(Bey & Wittum 1996; Hackbush & Probst 1997; Wang & Xu 1999; Kwok 2007)

I Awell : ILU/Direct solver

Smoother S : Block Gauss-Seidel with downwind ordering andcrosswind blocks.

Krylov iterative method: GMRes.

Ref: H., Liu, Qin, Xu, Yan and Zhang 2011, SPE 148388



Optimality



East Beverly Oil Field Test

Black Oil Model

Unstructured grid

83,592 grid blocks

169 wells

Faults: 994 non-localconnections

40 years of water flooding



East Beverly Oil Field Test

We use sequential version of FASP in this table. OpenMP version of FASPcosts 46 min (8-core, 2.8GHz).



JZ9-3 Oil Field Test

Polymer flooding(5-compoment)

Number of grids: 157x53x57

Lots of inactive grids

Several faults

37 wells

10 years simulation in total

3 years of polymer flooding

Simulator # Newton Iter Total CPU Time CPU Time/NewtonECL2009 1562 81.0 (min) 3.11 (sec)

SOCF(FASP) 1653 40.0 (min) 1.45 (sec)



JZ9-3 Results Comparison: Oil Production Rate


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!


!"#$%&'()*+,)&-./0123).45&698'

!



JZ9-3 Results Comparison: Gas Production Rate


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!


!"#$%&'()*+,)&-./0123).45&698'

!



JZ9-3 Results Comparison: Water Cut


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!


9:;<$;&=$-./0.4(4,&&8(),

! !SOCF CMG

Field Oil Saturation



ld101-1 7-component Test

Polymer-Gelflooding

5 wells: 1 injectwell, 4 productwells

32 yearssimulation intotal

No wellILU FASP

Size # Iter Time (s) # Iter Time (s)40x40x4 37 2.20 12 0.47

40x40x20 153 19.59 15 2.8640x40x40 296 56.05 13 4.11

5 wellsILU FASP

# Iter Time (s) # Iter Time (s)40x40x4 55 2.70 13 0.50

40x40x20 236 28.21 15 2.9040x40x40 319 65.04 14 5.78



ld101-1 7-component Test

3/20/2011

75

4

149Isoquant contours of water saturation

SOCF CMG

4LD10-1

Polymer flooding Gel profile-control flooding

Isoquant contours of oil saturation150

3/20/2011

75

4

149Isoquant contours of water saturation

SOCF CMG

4LD10-1

Polymer flooding Gel profile-control flooding

Isoquant contours of oil saturation150

3/20/2011

76

4

151

Poly

mer

floo

ding

Gel

pro

file-

cont

rol

flood

ing

Oil profile

LD10-1

4

LD10-1

0 1000 2000 3000 4000 5000 6000 70000

50

100

150

200

250

300

350

400

Fiel

d O

il P

rodu

ctio

n R

ates

(SM

3)

Time(Day)

A=0 A=-3 A=-4

0 1000 2000 3000 4000 5000 6000 70000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time(Day)

Wat

er C

ut

A=0 A=-3 A=-4

Field Oil Production Rate (SM3) Water Cut152

0 2 31 2 3 4 1[1 ( ) exp( )]i

i i i i

spp w i p i p i p sep iA c A c A c c A t

3/20/2011

76

4

151

Poly

mer

floo

ding

Gel

pro

file-

cont

rol

flood

ing

Oil profile

LD10-1

4

LD10-1

0 1000 2000 3000 4000 5000 6000 70000

50

100

150

200

250

300

350

400

Fiel

d O

il P

rodu

ctio

n R

ates

(SM

3)

Time(Day)

A=0 A=-3 A=-4

0 1000 2000 3000 4000 5000 6000 70000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time(Day)

Wat

er C

ut

A=0 A=-3 A=-4

Field Oil Production Rate (SM3) Water Cut152

0 2 31 2 3 4 1[1 ( ) exp( )]i

i i i i

spp w i p i p i p sep iA c A c A c c A t

Oil Saturation (Left: Polymer; Right: Polymer/Gel)


Ongoing Work and Conclusions






Ongoing Work and Conclusions Ongoing Work

Move to GPUNVIDIA

Parallel AMG algorithm suits GPU:

Regular sparsity pattern

Low complexity

Low setup cost

Fine-grain parallelism

Take advantage of hierarchical memory structure

Our choice: Unsmoothed Aggregation AMG (UA-AMG)

Controllable sparsity pattern

Low complexity (less fill-in)

Low setup cost (no triple matrices multiplication)

Large potential of parallelism



UA-AMG with Nonlinear AMLI-cycleProblem: UA-AMG with V-cycle usually dose not converge uniformly!

Solution: UA-AMG with Nonlinear AMLI-cycle (Variable AMLI-cycle / K-cycle).

Nonlinear AMLI-cycle: Bk [·] : Vk → Vk .

Assume B1[f ] = A−11 f , and Bk−1[·] has been defined, then for f ∈ Vk

Pre-smoothing u1 = Rk f ;

Coarse grid correction u2 = u1 + Bk−1[Qk−1(f − Aku1)]

Post-smoothing Bk [f ] := u2 + Rtk(f − Aku2).

Nonlinear coarse grid correction: Bk−1[·] : Vk−1 → Vk−1

Defined by n iterations of a Krylov subspace iterative methods usingBk−1[·] as the preconditioner.

Ref: Axelsson & Vassilevski 1994; Kraus 2002; Kraus & Margenov 2005; Notay & Vassilevski

2008.



UA-AMG with Nonlinear AMLI-cycleProblem: UA-AMG with V-cycle usually dose not converge uniformly!Solution: UA-AMG with Nonlinear AMLI-cycle (Variable AMLI-cycle / K-cycle).

Nonlinear AMLI-cycle: Bk [·] : Vk → Vk .

Assume B1[f ] = A−11 f , and Bk−1[·] has been defined, then for f ∈ Vk

Pre-smoothing u1 = Rk f ;

Coarse grid correction u2 = u1 + Bk−1[Qk−1(f − Aku1)]

Post-smoothing Bk [f ] := u2 + Rtk(f − Aku2).

Nonlinear coarse grid correction: Bk−1[·] : Vk−1 → Vk−1

Defined by n iterations of a Krylov subspace iterative methods usingBk−1[·] as the preconditioner.

Ref: Axelsson & Vassilevski 1994; Kraus 2002; Kraus & Margenov 2005; Notay & Vassilevski

2008.



Convergence Analysis of Nonlinear AMLI-cycle (SPD)

Theorem (H., Vassilevski & Xu 2011)

Assume the convergence factor of V-cycle MG with fixed level differencek0 is bounded by δk0 ∈ [0, 1), if n is chosen such that (1− δn)δk0 + δn ≤ δhas a solution δ ∈ [0, 1), which is independent of k, then we have

‖v − Bk [Akv ]‖2Ak≤ δ‖v‖2

Akand ‖v − Bk [Akv ]‖2

Ak≤ δn‖v‖2

Ak,

A sufficient condition of n: n > 11−δk0

> 1.

This is a generalization of the result in Notay & Vassilevski 2008.



Primary Results

1024×1024 2048×2048

Setup Solve Total Setup Solve Total

CPU 0.80 7/0.69 1.49 3.28 7/2.75 6.03

GPU (CUSP) 0.63 36/0.35 0.98 2.38 41/1.60 3.98

GPU (FASP) 0.13 19/0.47 0.60 0.62 19/2.01 2.63

Processor:

I CPU: Intel i7-2600 3.4 GHzI GPU: Tesla C2070

Algorithm:

I CPU: Classical AMG + V-cycle;I GPU(CUSP): Smoothed Aggregation AMG + V-cycle;I GPU(FASP): UA-AMG + Nonlinear AMLI-cycle

Ref: Brannick, Chen, H., Xu, Zikatanov, 2011



Other ongoing work

Compositional Model (Equation of State)

Complex wells

Improve robustness of FASP solvers

Improve robustness of nonlinear iteration

Heterogeneous parallel computing


Ongoing Work and Conclusions Conclusions

Conclusions

We developed effective solvers for reservoir simulation, which workswell for reservoirs from real world.

We developed a solver package (FASP). Part of the solver packagehas be used for the chemical flooding simulator (SOCF) by MonixEnergy Solutions and CNOOC.

We are working on parallelization of FASP package, includingOpenMP, MPI and CUDA.

We are working on the robustness of our package.


Ongoing Work and Conclusions Conclusions

Thank you!


Documents

Effective Solvers for Reservoir Simulation - Ricam · PDF fileE ective Solvers for Reservoir Simulation Xiaozhe Hu The Pennsylvania State University Numerical Analysis of Multiscale