Imaging of flow in porous media - from optimal transport to prediction

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1 Imaging of flow in porous media - from optimal transport to prediction Eldad Haber Dept of EOS and Mathematics, UBC October 15, 2013

2 With Rowan Lars Jenn Cocket Ruthotto Fohring

3 Outline Prediction is very difficult, especially about the future. Niels Bohr

4 Outline Multiphysics imaging The mathematical problem Discretization Solution through Variable Projection Summary and future work

5 Flow in porous media Flow in porous media is used for Enhanced Oil Recovery CO 2 sequestration monitoring Salt water intrusion monitoring

6 Enhanced Oil Recovery Inject CO 2 to push oil out Goal: image and control the flow

7 CO 2 Sequestration monitoring Is the CO 2 staying in the ground? Where does it flow to?

8 Salt water intrusion monitoring Is salt water polluting fresh water aquifer?

9 Flow in porous media Governing equations (IMPES formulation) u = q u = λ s (s)κ p s t + ( u λ(s)) = 0 IMP ES Given s 0 and parameters possible to solve for p and s(t) In realistic situations κ, λ and s 0 are known to very low accuracy (or not at all) Difficult to predict the flow

10 Flow in porous media Prediction is very difficult Long term prediction impossible Improving prediction Drill History match well data

11 Flow in porous media Prediction is very difficult Long term prediction impossible Improving prediction Drill History match well data. Use imaging to see the fluids

12 Imaging flow In general, consider the dynamical system ṡ = f(s, u) s(0) = s 0 Dynamical system with uncertain inputs Let the dynamics run for a short time and use data to update parameters Improve flow model Data assimilation

13 Imaging flow Use time laps imaging for fluid flow Fluids change the physical properties Goal: Combine imaging and dynamics to better predict the flow

14 Imaging fluids and flow Electromagnetic methods µ 1 e + iωσ(s) e = iω q d = Q e = QF(σ) e - electric field σ - conductivity Seismic methods u + ω 2 γ(s)u = q d = Qu = QF(γ) u - pressure field γ - seismic velocity In general: F(m) + ɛ = d

15 Model Flow Problem - Tracer flow Flow equations u = q u = κ(x) p s t + ( u s) = 0 s - saturation p - pressure κ - hydraulic conductivity tensor

16 Model Imaging - Borehole tomography Place sources and receivers in boreholes/surface and measure seismic/electric fields

17 Assumptions Flow s t + ( u(κ, p)s) = 0 s(0, x) = s 0 The imaging problem is linear w.r.t s Tomography d(t) = As(t) + ɛ

18 Goals Prediction and control s t + ( u(κ, p)s) = 0 s(0, x) = s 0 As(t) + ɛ = d No need for the pressure! Recover the velocity u and the saturation s

19 4 Numerical examples Similarity to super resolution In this section we demonstrate that the coupled algorithms can be superior to uncoupled approaches for the super-resolution problem. For the numerical Super Resolution tests reported in this section, - Use we use aa number magnetic resonance of(mr) low-res image, which is available in Matlab. The original high resolution image with 128 images to obtain a single high-res image 2 pixels, together with three low resolution images of 32 2 pixels, is shown in Figure 2. (a) (b) (c) (d) Figure 2: The high resolution image is shown in (a), and three selected low resolution images are shown in (b-d). A I(u)s + ɛ = d We assume that we have 32 low resolution images which are generated by a sequence of rotations and translations of the original image. For the

20 Similarity to super resolution Super Resolution - Use a number of low-res images to obtain a single high-res image Solve for s ans u Similar to the problem of super resolution [Elad & Furer, 90, Chung, H & Nagy 06, Borzi & Kunisch 07] Main differences - More complex dynamics and observation operators Similar mathematical structure

21 Solution through optimization min s, u J (s 0, u) s.t. s t + ( us) = 0 s(0, x) = s 0 Similar to the optimal control approach to OMT of Benamou & Brenier But there are major differences

22 Solution through optimization min s, u J (s 0, u) = j As(t j ) d j 2 + α s R s (s) + α u R u ( u) s.t. s t + ( us) = 0 s(0, x) = s 0

23 Solution through optimization min s, u J (s 0, u) = j As(t j ) d j 2 + α s R s (s) + α u R u ( u) s.t. s t + ( us) = 0 s(0, x) = s 0 Optimal mass transport - optimality criteria based on data OMT does not have a unique solution and require regularization Choice of regularization- motivated by the physics of the problem

24 Solution through optimization Continuous problem min J (u) Discretize then optimize Optimize then discretize Discretize u and J Compute g(u) = u J (u) = 0 compute g(u) = u J(u) Discretize g h (u) = 0 Solve the discrete problem Solve the discrete PDE

25 Solution through optimization Continuous problem min J (u) Discretize then optimize Optimize then discretize Discretize u and J Compute g(u) = u J (u) = 0 compute g(u) = u J(u) Discretize g h (u) = 0 Solve the discrete problem Solve the discrete PDE In general g h (u) g(u) g h (u) is not a gradient of any discrete function No guaranteed descent Convergence only when h is small enough

26 Solution through optimization Our framework: Discretize and optimize Gradient of the discrete function can be calculated exactly (linear algebra vs calculus) Best optimization algorithms can be used Gradient flow = steepest decent (ssssslllllooooowwww) Variations of Newton s method Multilevel Newton methods How to discretize the hyperbolic PDE?

27 Discretization of the PDE s t + ( u s) = 0 s(0, x) = s 0 Some things to consider u unknown - CFL condition unknown Unconditionally stable methods Upwinding - non-differentiable! Most high resolution methods (ENO, WENO, ) are highly nonlinear and non-differentiable Keeping discontinuities not relevant(?)

28 Discretization of the PDE Explicit methods Careful control over time stepping Differentiability - no flux limiters Implicit methods No stability issues Invert linear systems Semi-Lagrangian methods No stability issues Can be designed to be differentiable

29 Example for difficulty - Explicit Methods Test Equation: s t us x = 0 Upwind s k+1 = s k + t x non differentiable {}}{ max(u, 0)D + + min(u, 0)D s k Lax - Friedrichs s k+1 = A v s k + t 2 x diag(u)dc s k

30 Discretization - Particle in Cell

31 PIC Discretization Can be written as s k+1 I(u)s k = 0 s(0, x) = s 0 Exact conservation Unconditionally stable Can be made differentiable [H. Modersitzki, 06] Low accuracy Low diffusion

32 The discrete optimization problem min s, u 1 2 A s(t j ) d(t j ) 2 + α s R s (s 0 ) + α u R u (u) j s.t. s k+1 I(u)s k = 0 s(0, x) = s 0 To complete need to choose regularization scheme

33 Choosing regularization for s 0 Problem highly ill-posed, L 1 & TV not appropriate choice [Schwarzbach & H 12, Ascher, van Den Doel & H. 12] Choice of regularization for s 0 Smoothness R s (s 0 ) = 1 s dv Ω Weighted smoothness R s (s 0 ) = 1 2 w - weighted support Ω w(x) s 0 2 dv

34 Choosing regularization for u u - vector quantity Recall that u = 0 AE u can have discontinuous tangential components u jumpy Set R( u) = Ω α 1 2 u 2 + α 2 u 1 dv

35 The discrete optimization problem min s,u 1 2 As(t j ) d(t j ) 2 + α s R s (s 0 ) + α u R u (u) j s.t. s k+1 I(u)s k = 0 s(0, x) = s 0 The problem is linear in s nonlinear in u Use Variable Projection (VarPro) [Golub Pereyra (73,02)]

36 Solution through Variable Projection Eliminate Constraint s = F (u) 1 I 0 s 0 where I I(u) F (u) = I(u) I... I 0 = Unconstrained problem... I(u) I min s 0,u 1 2 AF (u) 1 I 0 s 0 d 2 + α s R s (s 0 ) + α u R u (u)

37 Solution through Variable Projection Two step iteration [Chung, Nagy & H (06), Chung Thesis (08)] Minimize wrt s 0 ŝ (k) 0 = ( I 0 F A AF 1 I 0 + α s 2 R s ) 1 F A d Fix s 0 = ŝ (k) 0 and minimize over u 1 min u 2 AF (u) 1 I 0 ŝ (k) 0 d 2 + α u R u (u) Advantages Decoupling the inverse problems Easy to choose regularization parameters

38 Solution through Variable Projection No need to form matrices Use GCV for regularization parameter for s 0 Lagged diffusivity for the u 1 regularization [Vogel (96)] Solution of the problem for u need not be exact

39 Example - Imaging CO2 Flow Experimental Setting: Borehole Experiment

40 Example - Imaging CO 2 Flow Assume 625 rays (data points) 20 times observed Prediction after Velocity field obtained by solving the pressure equation with highly discontinuous coefficients

41 Example - Imaging CO 2 Flow Flow simulation

42 Example - Imaging CO 2 Flow Observed Data

43 Recovered and predicted flow Flow simulation

44 Comments Reconstruction Excellent reconstruction of initial saturation Reasonable recovery of flow field Prediction Short term predictions - excellent Long term prediction - fail No information on the velocity in regions where there is no flow

45 Summary and prediction Summary Combine flow in porous media and imaging Basic framework - super resolution Requires special regularization VarPro for the solution Prediction Algorithm speedup Use joint inversion criteria for unknown petrology Experimental design