Parallel Simulations of Underground Flow in Porous and

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Parallel Simulations of Underground Flow in Porous and
Sparse linear solvers applied to
parallel simulations of underground flow
in porous and fractured media
A. Beaudoin1, J.R. De Dreuzy2, J. Erhel1 and H. Mustapha1
1 - IRISA / INRIA, Rennes, France
2 - Department of Geosciences, University of Rennes, France
Matrix Computations and Scientific Computing Seminar
Berkeley, 26 October 2005
Parallel Simulations of Underground Flow in Porous and Fractured
Media
2D heterogeneous porous medium
Heterogeneous permeability field
Y = ln(K)
with correlation function
 r
2
CY (r)   Y exp   
 Y 

1  Y  9
Parallel Simulations of Underground Flow in Porous and Fractured
Media
3D fracture network with impervious matrix
length distribution has a great impact : power law n(l) = l-a
3 types of networks based on the moments of length distribution
 mean
 variation
2<a<3
 mean
 variation
 third moment
3<a<4
 mean
 variation
 third moment
a>4
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Flow model

Equations
Q = - K*grad (h)
div (Q) = 0
Boundary conditions
2D porous medium
3D fracture network
Nul flux
Fixed head
Fixed head

Fixed
head
Nul flux
Nul flux
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Numerical method for 2D heterogeneous porous medium
Finite Volume Method with a regular mesh
Large sparse structured matrix with 5 entries per row
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Sparse matrix for 2D heterogeneous porous medium
n=32
zoom
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Numerical method for 3D fracture network
Mixed Hybrid Finite Element Method with unstructured mesh
Conforming
triangular mesh
Large sparse unstructured matrix with about 5 entries per row
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Sparse matrix for 3D fracture network
zoom
N = 8181
Intersections and 7 fractures
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Complexity analysis with PSPASES
Memory requirements for matrices A and L
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Complexity analysis with PSPASES
CPU time of matrix generation, linear solving and flow computation
obtained with two processors
Parallel Simulations of Underground Flow in Porous and Fractured
Media
2D porous medium : memory size and CPU time with PSPASES
Theory : NZ(L) = O(N logN)
Slope about 1
Theory : Time = O(N1.5)
Slope about 1.5
Parallel Simulations of Underground Flow in Porous and Fractured
Media
3D fracture network : memory size and CPU time with PSPASES
Theory to be done
NZ(L) = O(N) ?
Time = O(N) ?
Parallel Simulations of Underground Flow in Porous and Fractured
Media
2D porous medium : condition number estimated by MUMPS
To be ckecked : scaling or not
Parallel Simulations of Underground Flow in Porous and Fractured
Media
2D porous medium : residuals with PSPASES
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Parallel architecture
Parallel architecture
distributed memory
2 nodes of 32 bi – processors
(Proc AMD Opteron 2Ghz with 2Go of RAM)
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Scalability analysis with PSPASES : speed-up
S  2T2
Tp
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Scalability analysis with PSPASES : isoefficiency
N
R 
PTp
TS
E 
PTp
2D medium
  1.5
3D fracture network
P
N
Tp
R
P
N
Tp
2
0.26 106
5.60
1.20 106
2
0.26 106
13.10
8
1.05 106
11.33
1.18 106
8
1.05 106
22.06
32
4.19 106
25.70
1,04 106
32
4.19 106
38.41
4
0.26 106
2.92
1.15 106
4
0.26 106
7.94
16
1.05 106
6.06
1.11 106
16
1.05 106
16.05
64
4.19 106
13.08
1,05 106
64
4.19 106
 ?
R
No value No value
Parallel Simulations of Underground Flow in Porous and Fractured
Media
2D porous medium : number of V cycles with HYPRE/SMG
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Comparison between PSPASES and HYPRE/SMG : CPU time
HYPRE
PSPASES
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Comparison between PSPASES and HYPRE/SMG : speed-up
HYPRE
PSPASES
Parallel Simulations of Underground Flow in Porous and Fractured
Media
Perspectives
• porous medium : large sigma, up to 9 and large N, up to 108
• porous medium : 3D problems, N up to 1012
• porous medium : scaling, iterative refinement,
multigrid adapted to heterogeneous permeability field
• 3D fracture networks : large N, up to 109
• model for complexity and scalability issues
• 2-level nested dissection
• subdomain method
• parallel architectures : up to 128 processors
• Monte-Carlo simulations
• grid computing with clusters for each random simulation
• parallel advection-diffusion numerical models

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