new numerical approaches for discrete fracture network...

Intro Model Discretization Numerical Results Large scale DFNs

New numerical approaches for Discrete Fracture Network flowsimulations

Stefano Berrone

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

joint work withMatias Benedetto, Andrea Borio, Sandra Pieraccini, Stefano Scialo,

Fabio Vicini

February 24, 2015Politecnico di Milano

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


DFN

Figure : Example of DFN

Discrete fracture network models forflows simulations:

3D network of intersectingfractures

Fractures are represented asplanar polygons

Rock matrix is consideredimpervious - flow confined in theDFN

Flow driven by hydraulic headgradientss modeled by Darcylaw in the fractures

Flux balance and hydraulic headcontinuity imposed across traceintersections

Challenges:

Complex domain: difficulties in good quality mesh generation



Notation

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 Large scale DFNs Fracture model PDE Optimization

Single fracture model

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.

In the following[[∂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.



Full fracture formulation

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 us set ∂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).



Totally/Partially conforming meshes

Figure : Totally conforming mesh

Figure : Partially conforming mesh



non-conforming meshes: 36 fractures, 65 traces DFN

Figure : Non-conforming mesh on a 36 fracture DFN



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 Large scale DFNs 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 =12

uTGu + uT q.

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) = Gu + q = 0.



Back to local problems

Proposition

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

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.



Main advantages of the proposed method

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

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 triangulation for the dis-crete solution is fully indepen-dent on each fracture and oneach trace.

Extended Finite Ele-ments: catch the ir-regular behaviour ofthe solution across thetraces improving accu-racy;

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 Large scale DFNs 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 20


36 fractures, 65 traces DFN

Figure : Solution with XFEM on a 36 fracture DFN



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



Numerical results

LargeDFNs

Meshes of uSi and uS

j are given by the intersection of the independenttriangulations on the twin-fractures Fi and Fj with the trace S, respectively(different independent coarse meshes for uS

i and uSj ).

Φ =12

∑i∈Iin

∑Sm∈Si

∫Sm

(umi − αhi|Sm)−

∑i∈Iout

∑Sm∈Si

∫Sm

(umi − αhi|Sm)

,∆cons =

∣∣∣∑i∈Iin ∪Iout

∑Sm∈Si

∫Sm

(umi − αhi|Sm)

∣∣∣Φ

,

∆cont =

√∑Mm=1 ‖hi|Sm

− hj|Sm‖2

hmax∑M

m=1 |Sm|,

∆flux =

√∑Mm=1 ‖um

i + umj − α(hi|Sm

+ hj|Sm)‖2

Φ∑M

m=1 |Sm|,



Results

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

XFEM FEMGrid u-dofs ∆cont ∆flux ∆cont ∆flux

36 fractures120 744 0.1363E-4 0.1596E-4 0.2536E-4 0.1740E-430 1292 0.1344E-4 0.1109E-4 0.1560E-4 0.1239E-47 2474 0.7623E-5 0.5183E-5 0.9475E-5 0.5347E-5

120 fractures120 2676 0.5610E-5 0.5570E-5 0.6866E-5 0.4544E-530 4616 0.3636E-5 0.2812E-5 0.4421E-5 0.2824E-57 8793 0.1875E-5 0.1517E-5 0.2496E-5 0.1703E-5



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



Figure : 3D view of DFN709 (left) and DFN1425 (right)



0 10 20 30 40 50 60 70 80 90

100

101

102

103

104

Angle[DEG]

# O

ccu

rren

ces

7091425

0 1 2 3 4 5

100

101

102

Angle[DEG]

# O

cc

urr

en

ce

s

7091425

Figure : DFN709 and DFN1425. Left: distribution of angles between pairs ofintersecting traces in the same fracture; right: zoom of lower values.



−2 −1 0 1 2 3

100

101

102

103

log10

(Length)

# O

cc

urr

en

ce

s

7091425

−4 −3 −2 −1 0 1 2

100

101

102

103

104

105

log10

(Length)

# O

ccu

rren

ces

7091425

Figure : DFN709 and DFN1425: distribution of trace lengths (left) and of distancesbetween couples of non-intersecting traces in the same fracture (right)



Table : Number of degrees of freedom for the hydraulic head h and the control variableu

NF NT

DFN δ = 0.5 δ = 2 δ = 7 δ = 0.5 δ = 2 δ = 7

709 1378796 356527 107467 363007 187645 105817878 1753469 452856 136549 329584 170733 96759915 1808368 467474 140799 351899 182739 103701

1425 2838981 733588 220980 1157310 598596 338489



Delta

Table : Relative residuals and flux conservation error ∆cons through iterations

relative residual ∆cons# iter δ = 0.5 δ = 2 δ = 7 δ = 0.5 δ = 2 δ = 7

DFN7091000 3.749e-4 4.826e-4 5.767e-4 5.864e-6 8.381e-4 1.056e-310000 1.014e-5 1.244e-5 1.845e-5 1.018e-5 1.031e-6 4.497e-520000 3.615e-6 4.715e-6 8.539e-6 4.814e-6 2.275e-6 2.177e-5






Table : Hydraulic head mismatch ∆head and flux unbalance ∆flux through iterations

∆head ∆flux# iter δ = 0.5 δ = 2 δ = 7 δ = 0.5 δ = 2 δ = 7

DFN7091000 2.116e-6 2.356e-6 2.802e-6 5.683e-10 9.552e-10 1.002e-9

10000 5.138e-7 8.427e-7 1.092e-6 3.278e-10 3.461e-10 4.903e-1020000 5.002e-7 8.278e-7 1.062e-6 3.202e-10 3.409e-10 4.818e-10

DFN8781000 2.892e-6 3.322e-6 3.994e-6 4.863e-10 5.930e-10 7.395e-10

10000 7.318e-7 1.149e-6 1.668e-6 2.382e-10 3.299e-10 4.853e-1020000 7.030e-7 1.127e-6 1.641e-6 2.400e-10 3.319e-10 4.743e-10

DFN9151000 2.774e-6 3.187e-6 3.854e-6 4.397e-10 5.369e-10 6.780e-10

10000 7.595e-7 1.117e-6 1.683e-6 2.196e-10 3.045e-10 4.309e-1020000 7.109e-7 1.095e-6 1.641e-6 2.211e-10 3.004e-10 4.234e-10

DFN14251000 9.025e-7 1.009e-6 1.231e-6 1.311e-10 1.670e-10 2.102e-10

10000 2.168e-7 3.460e-7 5.397e-7 6.186e-11 9.671e-11 1.168e-1020000 2.121e-7 3.424e-7 5.284e-7 6.187e-11 9.553e-11 1.146e-10



Table : Number of communicated h-DOFs and u-DOFs for various numbers of parallelprocesses

DFN Slaves δ = 0.5 [%] δ = 2 [%] δ = 7 [%]

u DOFs comm

709

3 6978 [1.9] 3648 [1.0] 2119 [0.6]

1425

7 17755 [4.9] 9392 [2.6] 5451 [1.5]15 36994 [10.2] 19433 [5.4] 11253 [3.1]31 114061 [31.4] 59340 [16.3] 33779 [9.3]

3 55096 [4.8] 28694 [2.5] 16488 [1.4]7 108229 [9.4] 56773 [4.9] 32788 [2.8]

15 185821 [16.1] 97148 [8.4] 56057 [4.8]31 286723 [24.8] 149710 [12.9] 86405 [7.5]

h DOFs comm

709

3 7216 [0.5] 3885 [1.1] 2355 [2.2]

1425

7 18379 [1.3] 10016 [2.8] 6075 [5.7]15 38240 [2.8] 20682 [5.8] 12497 [11.6]31 117008 [8.5] 62290 [17.5] 36729 [34.2]

3 56671 [2.0] 30269 [4.1] 18066 [8.2]7 111724 [3.9] 60270 [8.2] 36284 [16.4]

15 191661 [6.8] 103000 [14.0] 61908 [28.0]31 295443 [10.4] 158446 [21.6] 95148 [43.1]



Table : Time per iteration of the parallel CG algorithm and efficiency for DFN709

Time [sec] Efficency %Version Slaves δ = 0.5 δ = 2 δ = 7 δ = 0.5 δ = 2 δ = 7

DFN709

Serial 1 0.83754 0.23432 0.08039

MPI

3 0.29631 0.08324 0.02909 94.22 93.83 92.11

MPI*

7 0.16898 0.04758 0.01651 70.81 70.35 69.5515 0.10830 0.03038 0.01057 51.56 51.41 50.6931 0.06239 0.01632 0.00568 43.31 46.33 45.68

15 0.11496 0.04281 0.01624 48.57 36.49 33.0031 0.07350 0.02281 0.00666 36.76 33.13 38.9663 0.03431 0.01194 0.00350 38.74 31.16 36.47

126 0.02243 0.00629 0.00233 29.64 29.56 27.35



Table : Time per iteration of the parallel CG algorithm and efficiency for DFN1425

Time [sec] Efficency %Version Slaves δ = 0.5 δ = 2 δ = 7 δ = 0.5 δ = 2 δ = 7

DFN1425

Serial 1 2.67725 0.70857 0.24254

MPI

3 0.91974 0.24591 0.08328 97.03 96.05 97.08

MPI*

7 0.48278 0.12251 0.04159 79.22 82.63 83.3015 0.28466 0.07176 0.02434 62.70 65.82 66.4431 0.14775 0.03888 0.02072 58.45 58.79 37.76

15 0.28194 0.07920 0.02947 63.31 59.64 54.8731 0.14638 0.04149 0.01434 59.00 55.09 54.5763 0.14838 0.04244 0.01540 28.64 26.50 25.00

127 0.06611 0.01881 0.00676 31.89 29.66 28.26



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P1 P2 P3 P4

P0

Figure : Master-Slave communication scheme



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pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

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P1

P3P2

P0

Figure : Point-to-Point communication scheme


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