a family of methods for large scale dfn flow simulations...

Intro Model Discretization Numerical Results Future work

A Family of Methods for Large Scale DFN Flow Simulationswith non-conforming meshes

Stefano Berrone

Dipartimento di Scienze Matematiche“ Giuseppe Luigi Lagrange ”Politecnico di [email protected]

joint work withMatias Benedetto, Sandra Pieraccini, Stefano Scialo

July 18, 2014Politecnico di Milano

Stefano Berrone Simulations of fluid flows in DFNs with non-conforming grids 1


DFN

Figure : Example of DFN

Discrete fracture network models:

3D network of intersectingfractures

Fractures are represented asplanar polygons

Rock matrix is consideredimpervious

Hydraulic head evaluated withDarcy law

Flux balance and hydraulic headcontinuity imposed across traceintersections

Challenges:

Complex domain: difficulties in good quality mesh generation



Nomenclature

Fi ⊂ R3, i ∈ I is a generic fracture of the system.

The computational domain is denoted Ω =⋃

i∈I Fi, ∂Ω =⋃

i∈I ∂Fi.

Fracture intersections are called traces and denoted by S, S ∈ S.

Each trace is shared by exactly two fractures: S = Fi ∩ Fj, andIS = i, j.Si is the set of traces on fracture Fi while S denotes the set of all thetraces.


Intro Model Discretization Numerical Results Future work Fracture model PDE Optimization

Let ∂Fi = ΓiN ∪ ΓiD with ΓiN ∩ ΓiD = ∅ and ΓiD 6= ∅.Find Hi ∈ H1

D(Fi) such that:

(Ki∇Hi,∇v) = (qi, v) + 〈GiN, v|ΓiN〉

H−12 (ΓiN),H

12 (ΓiN)

, ∀v ∈ H10,D(Fi)

provides the hydraulic head Hi ∈ H1D(Fi).

Hi is the hydraulic head on the fracture Fi, H is the hydraulic head on Ω;

Ki is the fracture transmissivity tensor: a symmetric and uniformlypositive definite tensor;∂Hi∂νi

= nti Ki∇Hi = GiN is the outward co-normal derivative of the

hydraulic head and ni the unit outward vector normal to the boundaryΓiN;[[∂Hi∂ν i

S

]]S

denotes the jump of the co-normal derivative along the unique

normal niS fixed for the trace S on the fracture Fi, this jump is

independent of the orientation of niS and is the net (incoming) flux

provided by the trace S to the fracture Fi.



For each trace S ∈ S on the fracture Fi, let use denote by

USi :=

[[∂Hi

∂ν iS

]]S

USi ∈ US⊆H−

12 (S)

the flux entering in the fracture through the trace, and Ui ∈ USi the tuple offluxes US

i ∀S ∈ Si.Let further be ∂Fi = ΓiN ∪ ΓiD with ΓiN ∩ ΓiD = ∅ and ΓiD 6= ∅ (for ease ofexposition).Solving ∀i ∈ I the problem: find Hi ∈ H1

D(Fi) and Ui ∈ Si such that:

(Ki∇Hi,∇v) = (qi, v) + 〈Ui, v|Si〉USi ,USi ′

J(U)

+〈GiN, v|ΓiN〉

H−12 (ΓiN),H

12 (ΓiN)

, ∀v ∈ Vi = H10,D(Fi)

with additional conditions

Hi|S − Hj|S = 0, for i, j ∈ IS, ∀S ∈ S,

USi + US

j = 0, for i, j ∈ IS, ∀S ∈ S,

provides the hydraulic head H ∈ V = H1D(Ω).



PDE constrained optimization approach

Instead of solving the differential problems on the fractures coupled by thecorresponding matching conditions we look for the solution as the minimumof a PDE constrained quadratic optimal control problem, the variable Ubeing the control variable.Let us define the “observation” spaces

H12 (S) ⊆ HS(= US′), ∀S∈S, HSi =

∏S∈Si

HS, H =∏i∈I

HSi .

Let us define the differentiable functional J : U → R as

J(U) =∑S∈S

(||Hi(Ui)|S − Hj(Uj)|S ||

2HS + ||US

i + USj ||2US

)H(U)

We look for the control variable U providing the minimum of the functionalJ(U) constrained by the equation for Hi on each fracture.



Proposition

Let us define the spaces US andHS as

US = H−12 (S), HS = H

12 (S),

then the hydraulic head H ∈ H1D(Ω) giving the unique exact solution of DFN

problem satisfies the differential problems on Fi, ∀i ∈ I and corresponds toJ(U) = 0.



PDE constrained optimization approach

In order to remove the requirement of having a non-empty portion of Dirichletboundary on each fracture it is necessary to modify the definition of thecontrol variables on each trace as follows:

USi =

[[∂Hi

∂ν iS

]]S+αHi|S US

i ∈ US =H−12 (S), ∀S∈S

where α is a strictly positive scalar parameter. The definition of the functionalis modified accordingly as:

J(U) =∑S∈S

(||Hi(Ui)|S−Hj(Uj)|S ||

2HS

+F(Ki,Kj)||USi + US

j − α(Hi(Ui)|S − Hj(Uj)|S)||2US

)J(u)

The constraint equation on each fracture ∀i ∈ I becomes:find Hi ∈ H1

D(Fi) such that:

(Ki∇Hi,∇v) + α(

Hi|Si, v|Si

)Si

= (qi, v) + 〈Ui, v|Si〉USi ,USi ′

+〈GiN, v|ΓiN〉

H−12 (ΓiN),H

12 (ΓiN)

, ∀v ∈ Vi = H10,D(Fi) h(u)


Intro Model Discretization Numerical Results Future work The discrete problem Numerical resolution

Definition of the discrete problem

Introduce a finite element triangulation on each fracture, completelyindependent of the triangulation on the intersecting fractures. Letus further define on this triangulation a finite element-like discretization hfor H.

Introduce also a discretization u for the control variable U, on the tracesof each fracture independently.

Let us choose US = HS = L2(S) for the discrete norms.

With arbitrary triangulations the minimum of the functional is not null.



The problem under consideration is therefore the equality constrainedquadratic programming problem

minu,h

J(h, u) := 12 hTGhh− αhTBu + 1

2 uTGuu J(U)

s.t. Ah− B u = q H(U)

Gh ∈ RNF×NF, Gu ∈ RNT×NT

are symmetric positive semidefinite sparsematrices;

B ∈ RNF×NTis a sparse matrix;

vectors h ∈ RNFand u ∈ RNT

collect all DOFs for the hydraulic head onfractures and for the control variable on traces;

matrix A ∈ RNF×NFis the block diagonal stiffness matrix;

B ∈ RNF×NTis a “block diagonal” sparse matrix;

q ∈ RNFis a vector which accounts for possible source terms and

boundary conditions.



The unconstrained problem

The first order optimality conditions for the constrained minimization problemare: Gh −αB AT

−αBT Gu −BT

A −B 0

hu−p

=

00q

.Use the linear constraint to remove h = A−1(B u + q) from J:

J(u) = J(u, h(u)) =12

uT(BT A−TGhA−1 B+Gu − αBT A−TB− αBTA−1 B)u

+qTA−T(GhA−1 B−αB)u

such that the minimum of the constrained problem is the same as the solutionto the unconstrained problem: min J(u)

This is equivalent to solve the global linear system

∇J(u)=BTA−TGhA−1(B u+q)+Guu−αBTA−TBu−αBTA−1(B u+q) = Gu+q = 0.

Given u, compute h and the Lagrange multiplier p defined by

Ah = B u + qATp = Ghh− αBu,

∇J(u) = BT p + Guu− αBTh = 0.



Back to local problems

Proposition

Matrix G = BT A−TGhA−1 B+Gu − αBT A−TB− αBTA−1 B is symmetricpositive definite. pcg

Instead of solving the KKT system we solve the system Gu = −q with a PCGmethod performing the same operations.

Let us observe that, given a value to the control variables ui, ∀i ∈ I onlyLOCAL small independent s.p.d. linear systems on each fracture are solvedin order to evaluate the gradient.

Ah = B u + q,ATp = Ghh− αBu,

∇J(u) = BT p + Guu− αBTh.



Resolution of the discrete problem

Algorithm

Preconditioned conjugate gradient method

Set k=0

Choose an initial guess uk

Compute hk, pk and gk = BT pk + Guuk − αBThk

Solve Pyk = gk

Set dk = −yk

While gk 6= 01 Compute λk with a line search along dk2 Compute uk+1 = uk + λkdk3 Update gk+1 = gk + λkGdk4 Solve Pyk+1 = gk+1

5 Compute βk+1 =gT

k+1yk+1

gTk yk

6 Update dk+1 = −yk+1 + βk+1dk7 k = k + 1



We remark that the computation of gk+1 is clearly performed without formingmatrix G, but rather computing vector δgk

= Gdk through the following steps:1 Solve Aδh = B dk

2 Solve ATδp = Ghδh − αBdk

3 Compute δgk= BT δp + Gudk − αBTδh

Furthermore, since J is quadratic, the stepsize λk in Step 1 can be computedvia an exact line search:

λk =rT

k yk

dTk Gdk

.

The stepsize λk is therefore computed without much effort, as quantity Gdk isthe same needed in Step 3.

The preconditioning matrix P is block diagonal. Each block of P can be builton a fracture independently from other fractures and is a tri-diagonalsymmetric positive definite matrix. As a consequence preconditioning can beperformed locally on each fracture and is computationally inexpensive. G



Main advantages of the proposed method

the gradient method makes the optimization approach to DFNsimulations nearly inherently parallel ;

on a multicore or GPU architecture we can associate each fracture to adifferent core;

exchange of very small amount of data between processes;

each process only exchanges data with a limited number of other knownprocesses;

resolution of small linear systems independently performed;

the optimization approach is independent of the discretization approachused on the fractures.



Circumventing mesh generation problems, different discretizations

The numerical triangulation forthe discrete solution is fullyindependent on each fractureand on each trace.

Extended Finite Ele-ments: used to catchthe irregular behaviourof the solution acrossthe traces improvingaccuracy adding suit-able basis functions;

Figure : Enrichment functionaway from trace tip

−1

−0.5

0

0.5

1

−1−0.8−0.6−0.4−0.200.20.40.60.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure : Tip enrichmentfunction



Circumventing mesh generation problems

Figure : Example of non-conforming triangulation for the FEM-XFEM

Standard Finite Elements: less accurate around the traces, where thesolution has a sudden change of gradient. Nonetheless Standard FiniteElements can be used, maybe in conjunction with a mesh adaptivealgorithm, to refine the triangulation near the traces;eXtended Finite Elements: more accurate around the traces, where thesolution has a sudden change of gradient. A bit more complex;Virtual Element Method: starting from an arbitrary mesh requires a meshmodification. As accurate as XFEM.


Intro Model Discretization Numerical Results Future work DFN Problems DFNs with Virtual Element Method

36 fractures, 65 traces DFN

Fractures size spans from 100m× 100m to 50m× 50m, different aspect ratios;

Figure : Solution with XFEM on a 36 fracture DFNStefano Berrone Simulations of fluid flows in DFNs with non-conforming grids 18



Figure : Solution with XFEM on a 36 fracture DFN




Figure : Mesh with XFEM on a 36 fracture DFN



36 fractures DFN

Figure : 36F: Solution on fracture F3Figure : 36F: Functional trend inminimization process



120 fractures - 256 traces DFN

Size of fractures spans from 100m× 100m to 50m× 50m,different aspect ratios;



Figure : 120F, Contour plot of the solution: detail



Figure : 120F, Solution on the sourcefracture

Figure : 120F, Solution on the fracture 40



Results

Table : Results for DFNs 36F and 120F with Induced Node for u. XFEM and standardFEM compared.

XFEM FEMGrid Node ∆cont ∆flux ∆cont ∆flux

36 fractures120 744 0.001363 0.001596 0.002536 0.00174030 1292 0.001344 0.001109 0.001560 0.0012397 2474 0.000762 0.000518 0.000947 0.000534

120 fractures120 2676 0.0005610 0.0005570 0.0006866 0.000454430 4616 0.0003636 0.0002812 0.0004421 0.00028247 8793 0.0001875 0.0001517 0.0002496 0.0001703



DFNs with first order Virtual Element Method

Virtual Element Methods [1] can be easily ap-plied in conjuction with our optimization ap-proach to the DFNs models.Let Th be a decomposition of Fi in non overlap-ping polygons E with straight faces (differentnumber of edges are allowed). The discreteVirtual Finite Element space Vi,δ ⊂ H1

D(Fi) willbe defined element-wise, by introducing a localspace Vi,δ|E for each element E.

s ss

sss s

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppFigure : General P1 VEM elementE

For all E ∈ Th:

Vi,δ|E =

v ∈ H1(E) : ∆v = 0, v|e ∈ P1(e) ∀e ∈ ∂E.

the functions v ∈ Vi,δ|E are continuous (and known) on ∂E;

the functions v ∈ Vi,δ|E are unknown inside the element E;

P1(E) ⊆ Vi,δ|E.

[1] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo: Basic principles of

Virtual Element Methods, Math. Models Methods Appl. Sci. 23, (2013), 199-214.



The bilinear form ah(·, ·)

For general continuous elements we set one degree of freedom for everyvertex of the mesh that represents the point-wise value of the solution atthe vertex.The global space Vi,δ is built assembling the local spaces Vi,δ|E as usual:

Vi,δ = v ∈ H10(Fi) : v|E ∈ Vi,δ|E ∀E ∈ Th

We have one dof per “non-Dirichlet” vertex.

A discrete bilinear form ai,δ(·, ·) is built element-wise

ai,δ(φ, ϕ) =∑

E∈Th

aEi,δ(φ, ϕ) ∀φ, ϕ ∈ Vi,δ,

whereaE

i,δ(·, ·) : Vi,δ|E × Vi,δ|E −→ R

are symmetric bilinear forms approximating aEi and satisfying a stability and

a consistency condition.



The space discretization: VEM

Use flexibility of VEM in order to catch the behavior of the solution across thetraces, allowing for a partially conforming mesh, but still maintaining anindependent meshing process on each fracture.

1 Start from a given triangularmesh, built without taking intoaccount trace positions orconformity requirements.

2 Whenever a trace intersects oneelement edge, a new node iscreated. New nodes are alsocreated at trace tips. If the tracetip falls in the interior of anelement, the trace is prolongedup to the opposite mesh edge.

3 Intersected elements are thensplit into two new “sub-elements”,which become elements in theirown right. Convex polygons areobtained.



Mesh smoothing

Figure : Left: VEM mesh without modification. Right: Same mesh after modifications.



6 fracture DFN with the VEM

Fracture size from 7m× 7m to 4m× 4m, 6 traces

Figure : DFN: Solution and mesh




Figure : VEM: VEM elements on fracture 2Figure : VEM: Solution on fracture 2




Figure : VEM: VEM elements on fracture 6Figure : VEM: Solution on fracture 6



DFN 36F

VEM XFEMGrid udof hdof ∆flux ∆cont hdof ∆flux ∆cont

50 776 4091 9.515e-04 9.432e-04 5772 1.039e-03 9.521e-0430 942 6048 9.621e-04 8.394e-04 8106 1.147e-03 1.181e-0312 1342 13967 6.736e-04 6.514e-04 16932 7.358e-04 8.189e-045 1885 30782 5.972e-04 6.083e-04 34958 5.930e-04 7.019e-042 2862 74107 4.847e-04 3.949e-04 80403 4.342e-04 4.664e-04



References and future work

ReferencesBenedetto M.F., Berrone S., Pieraccini S., Scialo S. (2014) The VirtualElement Method for discrete fracture network simulations. Accepted forpublication in Computer Methods in Applied Mechanics and Engineering.Berrone S., Pieraccini S., Scialo S. (2014) An optimization approach forlarge scale simulations of discrete fracture network flows. Journal ofComputational Physics, vol. 256 n. 1, pp. 838-853.Berrone S., Fidelibus C., Pieraccini S., Scialo S. (2014) Simulation of thesteady-state flow in Discrete Fracture Networks with non conformingmeshes and extended finite elements. Rock Mechanics and RockEngineering. To appear.Berrone S., Pieraccini S., Scialo S. (2013) On simulations of discretefracture network flows with an optimization-based extended finiteelement method. SIAM J. Sci. Comput., vol. 35 n. 2, A908-A935.Berrone S., Pieraccini S., Scialo S. (2013) A PDE-constrainedoptimization formulation for discrete fracture network flows. SIAM J. Sci.Comput., vol. 35 n. 2, B487-B510.

Ongoing work:Uncertainty Quantification (joint work with Claudio Canuto)Scalability tests, “a posteriori” error estimates, adaptivity, mixedformulation



Scalability results, 120F

Preliminary scalability tests performed on CINECA - Eurora by Fabio Vicini

1 2 4 7 8 16

0

2

4

6

8

10

12

14

16

processes

S

MPI - Master-Slave;MPI+OMP - Master-Slave;MPI - All-to-All;MPI+OMP - All-to-All;

Figure : Time reduction factor vs number MPI-processes or MPI-processes ×OMP-threads - CG algorithm



Monster DFN

Figure : 4937 Fractures, 346196 Traces


a family of methods for large scale dfn flow simulations...

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