Optimal Reconstruction of Material Properties in Complex Multiphysics Phenomena

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1 Annual Meeting of the Canadian Society of Applied and Industrial Mathematics CAIMS 23 Optimal Reconstruction of Material Properties in Complex Multiphysics Phenomena Vladislav Bukshtynov Department of Energy Resources Engineering Stanford University CAIMS SCMAI Quebec City June 7, 23

2 he Cecil Graham Doctoral Dissertation Award: Acknowledgments Prof. Bartosz Protas (McMaster University) Words of appreciation to: Dr. Oleg Volkov (postdoc at McMaster, now at Stanford) Prof. Dmitry Pelinovsky (McMaster) Profs. David Novog, Nikolas Provatas and James Wadsley (McMaster) Profs. Khalid Aziz and Lou Durlofsky (Stanford)

3 Generous Funding Provided by: School of Computational Science & Engineering and Department of Mathematics & Statistics, McMaster University Natural Sciences and Engineering Research Council of Canada (Collaborative Research and Development Program) Ontario Centres of Excellence Centre for Materials and Manufacturing General Motors of Canada Energy Resources Engineering Department and Smart Fields Consortium, Stanford University

4 Acknowledgments Motivation Simple Model Multiphysics Model Reduced-Order Models Special hanks to My Family My wife Katerina and daughter Ksenia Open Problem & Outlook

5 Agenda Motivation Generalized Framework (Simple Model Problem) Multiphysics Problem Reduced-Order Models Open Problem & Outlook

6 Motivation Generalized Framework (Simple Model Problem) Multiphysics Problem Reduced-Order Models Open Problem & Outlook

Reconstruction of Constitutive Relations: Motivation

Canada) Problem: Unknown material properties of alloys in

surface tension σ( ) all functions of temperature.

7 Reconstruction of Constitutive Relations: Motivation Application: Welding in automotive industry (with GM of Canada) Problem: Unknown material properties of alloys in liquid phase viscosity µ( ) thermal conductivity k( ) surface tension σ( ) all functions of temperature. Goal: How to reconstruct µ( ), k( ), σ( ), etc., based only on available measurements?

8 wo distinct formulations: Parameter Estimation as Inverse Problem a) material properties dependent on the space variable (i.e., the independent variable), and b) material properties dependent on the state variable (i.e., the dependent variable), less attention in the literature

9 Generalized Problem Setup o propose and validate a suite of computational techniques to reconstruct certain material properties based on partial (pointwise) measurements available for a particular process and a particular material Strategy: Formulate an inverse problem of the second type (b) as partial differential equation (PDE) constrained optimization problem which can be solved using an adjoint based gradient method. Propose an optimize then discretize approach where the optimality conditions and the optimization algorithm are both formulated based on the continuous (PDE) form of the problem.

10 Problem Extensions he approach is easily extendable to many problems arising in nonequilibrium thermodynamics and continuum mechanics and is independent of the domain geometry and discretization. simple model multiphysics problem: complications of mathematical character only, e.g. in derivation of the cost functional gradient adding time dependence and/or changing the model geometry affects only the computational part of the solution Examples: D model problem (heat conductivity) 2D multiphysics problem (viscosity in coupled Navier-Stokes and energy equations) reduced-order models of hydrodynamic instabilities (inertial manifolds) open problem in petroleum engineering

11 Motivation Generalized Framework (Simple Model Problem) Multiphysics Problem Reduced-Order Models Open Problem & Outlook

12 Statement of the Problem Assuming the constitutive relation in thermodynamics in a general form [ ] [ ] thermodynamic thermodynamic = k (state variables), flux force in heat conduction problem this relation takes the form of the BVP for steady state heat equation: [k( ) ] = g(x) in Ω, = on Ω spatial domain x Ω, where Ω R n, n =, 2, 3 state (temperature) (x) : Ω R distribution of heat sources g(x) : Ω R control (thermal diffusivity) k( ) : R R well-posedness of the problem: g(x) >, k( ) > measurements { i } M i= at a number of points {x i } M i= in the domain Ω

13 Parameter Estimation as Optimization Problem Reformulation as minimization problem: defining the cost functional J : X R as J (k) 2 M [ (x i ; k) ] 2 i i= optimal reconstruction ˆk to be obtained as an unconstrained minimizer of cost functional J ˆk = argmin J (k) k X minimizer ˆk can be computed with the gradient descent algorithm as ˆk = lim n k (n), where { k (n+) = k (n) τ (n) k J (k (n) ), n =,... k () = k minimizer ˆk can be found by using a set of adjoint based techniques

14 Extraction of Cost Functional Gradients Gâteaux (directional) differential: J (k; k ) = M [ (x i ; k) i ] (x i ; k, k ), i= where perturbation variable (x i ; k, k ) satisfies the perturbation system obtained from the governing equation. Riesz representation theorem (for some Hilbert space X ) J (k; k ) = k J, k X, e.g. setting X = L 2 gives J (k; k ) = k J k dx. Inconsistency: k is hidden in the system defining (x i ; k, k ) Solution: J (k; k ) requires a form consistent with a Riesz representation. Ω

15 Gradient via Adjoint Analysis Adjoint variable (Lagrange multiplier) satisfies the adjoint system k( ) = = M [ i (x)]δ(x x i ) in Ω, i= where δ( ) is the Dirac delta function. on Ω, Gâteaux differential (after integration by parts and Kirchhoff transformation) β [ ] J (k; k ) = χ [α, (x)](s) dx k (s) ds α Ω β [ χ [α, (x)](s) ] α Ω n dσ k (s) ds = k J, k X, {, s [α, β ], χ [α,β ](s) =, s / [ α, β ]. [Proved in Bukshtynov, Volkov & Protas (2)]

16 Cost Functional Gradient: Solution Commonly used structure adapted to steady-state problem after Chavent & Lemonnier (974, in French) L 2 k J (s) = δ( (x) s) dx, Ω where Dirac delta function δ( (x) s) is used to define a level set of the temperature field (x) Γ s {x Ω, (x) = s} Alternative (computationally more efficient) structure of gradient L 2 k J (s) = M i= χ [α, (x i )](s) i (x i ) k( (x i )) χ [α, (x)](s) Ω n dσ [Equivalence proved in Bukshtynov, Volkov & Protas (2)]

17 Differentiability of Map k Solving parameter estimation problem for [k( ) ] = g in Ω, = on Ω or in a variational formulation in the space = { H (Ω); = on Ω} k( ) u dx = g u dx,, u (2) Ω is equivalent to finding a solution to the operator equation F(k) =, where F : K L 2(Ω) and K is the set of constitutive relations Ω K = {k( ) piecewise C on L; < m k < k( ) < M k, L} () heorem [Bukshtynov, Volkov & Protas (2)] Assume that m k is sufficiently large and solutions of () satisfy L (Ω) <. hen the map k ( ; k) from K to L 2(Ω) defined by (2) is Fréchet differentiable in the norm H (I).

18 Main Complications. L 2 gradient L 2 k J (s) is in general a discontinuous function necessary regularity (smoothness) achieved by preconditioning

19 Main Complications. L 2 gradient L 2 k J (s) is in general a discontinuous function necessary regularity (smoothness) achieved by preconditioning 2. Min problem is in general nonconvex condition J (k; k ) = characterizes only a local, rather than global, minimizer choice of initial guess k

20 Main Complications. L 2 gradient L 2 k J (s) is in general a discontinuous function necessary regularity (smoothness) achieved by preconditioning 2. Min problem is in general nonconvex condition J (k; k ) = characterizes only a local, rather than global, minimizer choice of initial guess k 3. Gradient L 2 k J (s) is nontrivial only for s I, and sensitivity of the functional J (k) to perturbations of k can only be determined on the identifiability region I algorithm for shifting (extending) identifiability region I

21 Main Complications. L 2 gradient L 2 k J (s) is in general a discontinuous function necessary regularity (smoothness) achieved by preconditioning 2. Min problem is in general nonconvex condition J (k; k ) = characterizes only a local, rather than global, minimizer choice of initial guess k 3. Gradient L 2 k J (s) is nontrivial only for s I, and sensitivity of the functional J (k) to perturbations of k can only be determined on the identifiability region I algorithm for shifting (extending) identifiability region I 4. Positivity condition k( ) > (well-posedness) is required for all L augmenting functional J (k) with the barrier function straightforward approach using slack variables

22 Regularization via Preconditioning Riesz representation theorem J (k; k ) = L 2 k J, k L 2 = H k J, k H and Sobolev space H is endowed with the inner product β [ ] z, z 2 H (L) = z z 2 + l 2 z z 2 ds, z, z 2 H (L), s s α where l R is a length scale (smoothening) parameter. Sobolev gradient H k J can be determined as a solution of the elliptic BVP H k J l 2 d 2 ds 2 H k J = L 2 d ds H k J on (a, b), k J = for s = a, b.

23 Single Identifiability Interval Reconstruction k() L 2 J vs. H J (l =.5 and l =.2) ˆk( ) using H J

24 Single Identifiability Interval Reconstruction k() L 2 J vs. H J (l =.5 and l =.2) ˆk( ) using H J

25 Single Identifiability Interval Reconstruction k() L 2 J vs. H J (l =.5 and l =.2) ˆk( ) using H J

26 Single Identifiability Interval Reconstruction k() L 2 J vs. H J (l =.5 and l =.2) ˆk( ) using H J

27 Reconstruction with Extended Gradients k() H k J (s) = H k J ( α) for s [ a, α], H k J (s) = H k J ( β ) for s [ β, b ]

28 Reconstruction with Shifting Reconstruction of k( ) over the entire interval of interest L by determining new heat source distribution boundary conditions g (j+) (x) g (j) (x) h [k (j) ( (j) (x))],(j+) =,(j) + h to perform shifting to right (h > ) or/and left (h < ) with step h << k() 2 k()

29 Reconstruction with Shifting Reconstruction of k( ) over the entire interval of interest L by determining new heat source distribution boundary conditions g (j+) (x) g (j) (x) h [k (j) ( (j) (x))],(j+) =,(j) + h to perform shifting to right (h > ) or/and left (h < ) with step h << k() 2 k()

30 Reconstruction with Shifting Reconstruction of k( ) over the entire interval of interest L by determining new heat source distribution boundary conditions g (j+) (x) g (j) (x) h [k (j) ( (j) (x))],(j+) =,(j) + h to perform shifting to right (h > ) or/and left (h < ) with step h << k() 2 k()

31 Motivation Generalized Framework (Simple Model Problem) Multiphysics Problem Reduced-Order Models Open Problem & Outlook

32 General Statement Goal: o reconstruct the viscosity coefficient µ( ) in the momentum equation (Navier-Stokes equation) coupled with energy equation (in 2D or 3D): [ ] tu + u u + p µ( )[ u + ( u) ] = in Ω, subject to boundary and initial conditions u = in Ω, t + u [k ] = in Ω, u = u b = b on Ω, on Ω, u(, ) = u, (, ) = in Ω. spatial domain x Ω, where Ω R d, d = 2, 3 states: u(x) : Ω R n, (x) : Ω R control (viscosity) µ( ) : R R + measurements { i } M i= at a number of points {x i } M i= in the domain Ω

33 Computations for Iterative Reconstruction. Direct problem (forward in time) 2. Adjoint problem (backward in time) tu u u σ + u ( u) + = in Ω, t u [k ] + dµ d ( )[ u + ( u) ] : u = u = in Ω, M [ (x i ; µ) i ]δ(x x i ) in Ω, where σ p I + (θ 2 ( ) + m µ) [ u + ( u ) ] and subject to 3. Cost functional gradient tf L 2 µ J (s) = i= u =, = on Ω, u ( ; t f ) =, ( ; t f ) = in Ω. [Derived in Bukshtynov & Protas (23)] Ω δ( (x) s)[ u + ( u) ] : u dx dτ

34 Positivity Constraint µ( ) > Positivity condition µ( ) > required for all L (well-posedness) ˆµ = argmin µ X, J (µ) µ( )>, L Augmenting cost functional by adding penalization term (barrier function).

35 Positivity Constraint µ( ) > Positivity condition µ( ) > required for all L (well-posedness) ˆµ = argmin µ X, J (µ) µ( )>, L Augmenting cost functional by adding penalization term (barrier function). Straightforward approach: slack variable θ( ) : R R µ( ) = θ 2 ( ) + m µ, m µ R +, m µ.

36 Positivity Constraint µ( ) > Positivity condition µ( ) > required for all L (well-posedness) ˆµ = argmin µ X, J (µ) µ( )>, L Augmenting cost functional by adding penalization term (barrier function). Straightforward approach: slack variable θ( ) : R R µ( ) = θ 2 ( ) + m µ, m µ R +, m µ. New unconstrained (except governing PDE) optimization problem where µ( ) > is satisfied automatically with L 2 gradient ˆθ = argmin J (θ), J (θ) tf θ X 2 L 2 θ J (s) = 2 tf Ω M i= [ (x i ; θ) i ] 2 dτ δ( (x) s) θ(s) [ u + ( u) ] : u dx dτ

37 Setup for Numerical Simulations algebraic expression (Andrade Law) to represent viscosity µ( ) µ( ) = C e C 2/ geometry: simple 2D lid driven (shear driven) cavity flow thermal conductivity k =.2 = const local Re = 25 BCs: top boundary (moving lid) u x = u top = cos(αt), u y =, = 5 all other: u x = u y =, = 3 synthetic temperature measurements: 9 thermal sensors capture the temperature correspondent to µ( ) μ()

38 Setup for Numerical Simulations algebraic expression (Andrade Law) to represent viscosity µ( ) µ( ) = C e C 2/ geometry: simple 2D lid driven (shear driven) cavity flow thermal conductivity k =.2 = const local Re = 25 BCs: top boundary (moving lid) u x = u top = cos(αt), u y =, = 5 all other: u x = u y =, = 3 synthetic temperature measurements: 9 thermal sensors capture the temperature correspondent to µ( )

39 Vorticity ω Simulations for Direct and Adjoint Fields: Vorticity Adjoint vorticity ω x t =.5

40 Vorticity ω Simulations for Direct and Adjoint Fields: Vorticity Adjoint vorticity ω x 4 x t =.5 t =.5

41 Vorticity ω Simulations for Direct and Adjoint Fields: Vorticity Adjoint vorticity ω x 4 x 4 x t =.5 t =.5 t =.95

42 Simulations for Direct and Adjoint Fields: emperature emperature Adjoint temperature t =

43 Simulations for Direct and Adjoint Fields: emperature emperature Adjoint temperature t =.5 t =.5

44 Simulations for Direct and Adjoint Fields: emperature emperature Adjoint temperature t =.5 t =.5 t =.95

45 Integration over Level Sets (emperature Isolines) Cost functional gradient L 2 µ J (s) = tf Ω δ( (x) s)[ u + ( u ) ] : u dx dτ Integration over codimension manifolds defined by a level set function φ(x) Γ: φ(x)= h(x)dσ = δ(φ(x)) φ(x) h(x) dx Ω

46 Integration over Level Sets (emperature Isolines) Cost functional gradient L 2 µ J (s) = tf Ω δ( (x) s)[ u + ( u ) ] : u dx dτ Integration over codimension manifolds defined by a level set function φ(x) Γ: φ(x)= h(x)dσ = Ω δ(φ(x)) φ(x) h(x) dx Points to consider: regularity of isolines φ(x, s) = (x) s discrete definition of field (x) computational efforts

47 Integration Over Codimension Manifolds Approaches to evaluate h(x)dσ Γ: φ(x)= fall into two main groups: or Ω δ(φ(x)) φ(x) h(x) dx. reduction to a line (contour) integral x ( ) dy 2 h(x, y(x)) + dx x dx t t ( ) dx 2 ( ) dy 2 h(x(t), y(t)) + dt dt dt geometric integration: Min & Gibou (27, 28) 2. evaluation of an area integral regularization δ ɛ of the Dirac delta measure with a suitable choice ɛ: Zahedi & ornberg (2) approximations δ of the Dirac delta function: Smereka (26), owers (27, 29)

48 Approximation of Contour Integrals with Area Integrals

discretization of the space domain and st order accurate with respect to the

49 Approximation of Contour Integrals with Area Integrals heorem [Bukshtynov & Protas (23)] Method (*) using 2-D compound midpoint rule is 2 nd order accurate with respect to the discretization of the space domain and st order accurate with respect to the discretization of the state domain f (s) = Ω δ( (x) s)g(x) dx = h N s,h j= g(x j )hx hy + O(h2 ) + O(h )

50 Approximation of Contour Integrals: Performance 3 2 Ne /2 ξ 2 h.5 time, s N e 3 2 h h Relative error vs. discretization step h = x = y CPU time (sec) vs. number of computational elements N e Ne Line integration ( ) vs. Delta Function Approximation ( ) vs. Area Integration( )

51 Validation of Gradients κ(ε)..99 κ ε ε Directional differential J (µ; µ ) obtained in two ways: finite difference approximation and using adjoint field (different perturbations µ and µ 2) κ(ɛ) ɛ [J (µ + ɛµ ) J (µ)] + µj (s) µ (s) ds Ratio κ as t (- - - and )

52 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

53 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

54 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

55 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

56 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

57 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

58 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

59 Choice of Initial Guess μ().2.5 μ() μ().2.5 μ()

60 Identifiability Region: Initial vs. Extended Cases μ() Reconstruction on I with initial guess µ = µ(α) + µ( β) 2

61 Identifiability Region: Initial vs. Extended Cases μ() Reconstruction on I with initial guess µ = µ(α) + µ( β) 2

62 Identifiability Region: Initial vs. Extended Cases.35 μ() μ() Reconstruction on I with initial guess µ = µ(α) + µ( β) 2 I = L with BCs for b assigned b top = b b else = a

63 Identifiability Region: Initial vs. Extended Cases.35 μ() μ() Reconstruction on I with initial guess µ = µ(α) + µ( β) 2 I = L with BCs for b assigned b top = b b else = a

64 Identifiability Region: Initial vs. Extended Cases.35 μ() μ() Reconstruction on I with initial guess µ = µ(α) + µ( β) 2 I = L with BCs for b assigned b top = b b else = a

65 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.5.4 μ()

66 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.5% α.5.4 μ()

67 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.5.4 μ()

68 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.3% α.5.4 μ()

69 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.5% α.5.4 μ()

70 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.5.4 μ()

71 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.6 λ = μ().3 μ()

72 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.6 λ = μ().3 μ()

73 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.6 λ = μ().3 μ()

74 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.6 λ = μ().3 μ()

75 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.6 λ = μ().3 μ()

76 Noise in Measurements Cost functional J λ (µ) J (µ) + λ 2 tf µ µ 2 Y(I) dτ.6 augmented with ikhonov regularization term, e.g. of H form, where λ 2 µ µ 2 Ḣ (I) = λ β ( dµ 2 ds d µ ) 2 ds ds noise =.% α.6 λ = μ().3 μ()

77 Motivation Generalized Framework (Simple Model Problem) Multiphysics Problem Reduced-Order Models Open Problem & Outlook

78 Optimal Reconstruction of Inertial Manifolds in Reduced-Order Models by Protas, Noack & Morzyński (22) admor et al. (2) Approximation within a low-dimensional space R N by evolution equation d dt a = f(a), a RN, f : R N R N Goal: parameter-free propagator f identification in a system with oscillatory dynamics

79 From -dim to dim = 3 Using POD modes as basis consider model reduction } tu + u u + p ν u = in Ω, = u = in Ω, ṙ = g (r) r, θ = g 2(r), a 3 = g 3(r), where r a 2 + a2 2 and θ arctan(a2/a). Measurements: ã(t) = [ã (t), ã 2(t), ã (t)] [ r(t), θ(t), ã (t)], for t [, ] Cost functionals for each optimization problem: J (g ) 2 J 2(g 2) 2 J 3(g 3) 2 [r(t) r(t)] 2 dt [ e ıθ(t) e ı θ(t) ] 2 dt [g 3(r(t)) ã (t)] 2 dt

80 Optimal Reconstructions Model: 2D flow around a circular cylinder g (r) in ṙ = g (r) r.5. g 2(r) in θ = g 2(r) g (r) g 2 (r) r Protas et al. (22) initial guess (mean-field model) [ ] ṙ(t) = σ β α M r 2 (t) r(t), r θ(t) = ω + γ α M r 2 (t) optimal reconstruction measurement data dr (r(t r dt i)), dθ dt (r(t i)), i =,..., N

81 Comparison of Output (state magnitude r = a 2 + a2 2 ) Protas et al. (22) r(t) t initial guess (mean-field model) optimal reconstruction measurement data

82 Comparison of Output (state variable a ) Protas et al. (22) a (t) t initial guess (mean-field model) optimal reconstruction measurement data

83 Motivation Generalized Framework (Simple Model Problem) Multiphysics Problem Reduced-Order Models Open Problem & Outlook

84 Parameter Reconstruction for Large Real Field Models Norne field in the Norwegian Sea operated by Statoil, Statoil (2) Norne field model ( 45, active cells, 3 wells, 9-year production history available)

85 Closed Loop Reservoir Modelling

History Matching as Optimization Problem Goal: to determine geological model parameters m = [m m 2... m Nm ] s.t. simulated results G sim (m) match observed (production) data d obs = [d d 2.

86 History Matching as Optimization Problem Goal: to determine geological model parameters m = [m m 2... m Nm ] s.t. simulated results G sim (m) match observed (production) data d obs = [d d 2... d Nd ] and honor prior information about geological structure of the model. Optimal solution ˆm is sought as Sarma et al. (SPE 676) min m N d i= (G i sim(m) d i obs) 2 + R geology s.t. g(x; m) = m feasibly bounded set where R geology is geology pertaining regularization

87 Open Problem: Reconstruction of Relative Permeability K r (S) Mass balance equation for (O)il, (G)as and (W)ater phases, p = o, g, w (simplified black oil model) where Darcy velocity of phase p (φρpsp) + ρpup =, t u p = k Krp(S) µ p [ p p ρ pg] relative permeability, K r oil water water saturation, S w K r,water and K r,oil as functions of S water setup in tables approximations, e.g. Corey-type (black oil) [ K rw (S w ) = Krw S w S wi S wi S orw ] Nw Enhanced Oil Recovery (EOR) models (multicomponent) require more complicated dependencies K rp(s w ) and K rp(s g )

88 Conclusions Novel computational approach to reconstruct solution dependent constitutive relations based on available measurement data using a gradient based optimization technique Systematic experimental design procedure: reconstruction over a broader range of the state variable required degree of smoothness of the reconstructed material properties superior accuracy and efficiency of the novel numerical approach for evaluation of cost functional gradients

89 References. V. Bukshtynov, O. Volkov, B. Protas, On Optimal Reconstruction of Constitutive Relations, Physica D 24, , (2). 2. V. Bukshtynov, B. Protas, Optimal Reconstruction of Material Properties in Complex Multiphysics Phenomena, Journal of Computational Physics 242, , (23). 3. B. Protas, B. R. Noack and M. Morzyński, An Optimal Model Identification For Oscillatory Dynamics With A Stable Limit Cycle, (submitted), (22).

On Optimal Reconstruction of Constitutive Relations

On Optimal Reconstruction of Constitutive Relations Vladislav Bukshtynov a, Oleg Volkov b, and Bartosz Protas b, a School of Computational Engineering and Science, McMaster University, Hamilton, Ontario,