Hydraulic Modelling for Drilling Automation

Transcription

1 Hydraulic Modelling for Drilling Automation CASA Day Harshit Bansal April 19, 2017 Where innovation starts

2 Team: Supervisors at TU/e : W.H.A. Schilders, N. van de Wouw, B. Koren, L. Iapichino Collaborators: 1. Norway : G. O. Kaasa (Kelda Drilling Controls, NTNU) 2. France : F. de Meglio (Ecole de Mines) PhD Group : M.H. Abbasi (affiliated to CASA and Kelda Drilling Controls) Project Sponsors The project HYMODRA (HYdraulic MOdelling for DRilling Automation) is sponsored by Shell and NWO-I under the aegis of Shell NWO/FOM PhD Programme in Computational Sciences for Energy Research. 2 Supervisors and Collaborators

3 Application Perspective Mathematical Notion of Drift Flux Model Physical and Numerical Boundary Conditions Sound Speed Model Numerical Schemes Numerical results Conclusions and perspectives 3 Outline

4 Main Goal : Develop hydraulic models and supporting model reduction techniques that are 1. accurate enough 2. simple enough to be employed in the context of drilling scenario simulations and real time estimation and control in case of gas influx. Managed Pressure Drilling!! Figure: Drilling Schematic 4 Objective

5 Characteristics/Features: The downhole pressure must be kept within allowable limits Delays in transmission of information No downhole measurements during certain phases of drilling operations Top-side measurements are available Figure: Managed Pressure Drilling Hydraulic Model: Serve as a model for controller/ estimator Aid to design operations before hand 5 Characteristics/Features

6 Complexities from physical perspective Timescales in the drilling process Slow transient corresponding to mass transport Fast transient corresponding to the propagation of the acoustic waves Nonlinearities: Acoustic velocity changes very rapidly in the one-phase to two-phase transition regions and vice versa. Disappearance and Appearance of Phases. Various flow regimes across different sections of the well. Distributed non-linearities due to source terms. 6 Complexities from physical perspective

7 Governing Equations of Drift Flux Model t (ρ l α l ) + x (ρ l α l v l ) = Γ l t (ρ g α g ) + x (ρ g α g v g ) = Γ g t (ρ l α l v l + ρ g α g v g ) + x (ρ l α l v 2 l + ρ g α g v 2 g + P) = Q g + Q v Q g = g(ρ l α l + ρ g α g )sin(θ); Q v = 32µv d 2 α l = Liquid Void Fraction ; α g = Gas Void Fraction ρ l = Liquid Density ; ρ g = Gas Density v l = Liquid Velocity ; v g = Gas Velocity ; v = Mixture Flow Velocity Γ l and Γ g are the phase change terms P = Pressure ; d = hydraulic diameter ; µ = mixture viscosity θ is the well inclination ; g = acceleration due to gravity 7 Governing Equations of Drift Flux Model

8 Closure Laws α g + α l = 1 ρ g = P /a 2 g ρ l = ρ l0 + (P P l0 )/a 2 l v g = (Kv l α l + S)/(1 K α g ) K and S are flow dependent parameters. There is singularity in the slip law when we approach pure gas region. a l is the speed of sound in the liquid phase a g is the speed of sound in the liquid phase P l0 : standard atmospheric pressure ρ l0 : density of liquid at standard atmospheric pressure 8 Closure Laws Modelling

9 The 1-D non-linear conservation law: 9 System of Conservation Laws w t + (f (w)) x = s is hyperbolic if the Jacobian matrix f w is diagonalizable with real eigenvalues for each physically relevant w. ρ l α l ρ l α l v l w = ρ g α g ρ l α l v l + ρ g α g v g, f (w) = ρ g α g v g ρ l α l v 2 l + ρ g α g v 2 g + P Γ l s = Γ g Q g + Q v For further discussion, Γ l = 0, Γ g = 0

10 Eigenvalues of the Jacobian Matrix The corresponding eigenvalues are given by: λ 1 = v l ω, λ 2 = v l + ω, λ 3 = v g where, ω is the speed of sound in two-phase mixture Two eigenvalues are linked to the compressibility effects Third eigenvalue is coincident to the gas velocity One pressure pulse propagates downstream and the other pressure pulse propagates upstream. Gas Volume wave travels downstream. 10 Eigenvalues of the Jacobian Matrix

11 Physical Boundary Conditions: (ρ l α l v l )(0, t) = f (t) (ρ g α g v g )(0, t) = h (t) P(L, t) = r(t) or (α l v l )(0, t) = f (t) (α g v g )(0, t) = h (t) P(L, t) = r(t) 11 Physical Boundary Conditions

12 Compatibility relations for multi phase system, which are: Characteristic 1 and 2: Compatibility relation corresponding to the pressure wave propagating in the upstream direction and downstream direction of the flow: d dt p + ρ l ω(v g v l ) d dt α g ρ l α l (v g v l + ω) d dt v l = q(v g v l + ω) where, d dt = t + (v l + ω) x is the directional derivative Characteristic 3: Compatibility relation corresponding to the gas volume wave: d dt p + p d α g (1 K α g ) dt α g = 0 where, d dt = t + (v g) x is the directional derivative 12 Numerical Boundary Conditions

13 Sound Speed Model Approximate Sound Speed Model is written as: a l if α g < ɛ ω = c(p, α g, ρ l, K) if ɛ α g 1 ɛ if α g > 1 ɛ a g where ɛ is a small parameter P c(p, α g, ρ l, K) = α g ρ l (1 K α g ) a l and a g are the sound speeds in liquid and gas medium respectively 13 Sound Speed Model

14 Assumptions for Sound Speed in the two phase mixture Liquid is incompressible αg ρ g << α l ρ l Why is sound speed in the two phase mixture important? Numerical flux computations are heavily dependent on the mixture sound speed Numerical dissipation depends on the sound speed of the two phase mixture Enable correct determination of locations and speeds of the wave fronts 14 Sound Speed Model

15 Reasons for Model Improvement Drilling fluids are highly compressible Existing models for sound speed in two phase mixture are singular at low and high void fractions Existing models become singular before rendering the Drift Flux Model non-hyperbolic Existing models also fail in modelling the realistic effects at high operating pressures Need of a unified model for single phase flow and two phase flow modelling 15 Sound Speed Model

16 ρ l ρ C = g (ρ g ( Kv l (1 αg )+S 1 K αg ) ρ l v l ) ( (1 αg )v l a 2 l (1 αg )v ρ l v l l a l 2 D = ρ g ( Kv l (1 αg )+S αg ( Kv l (1 αg )+S 1 K ) αg 1 K ) αg ag 2 ( v l 2ρ l + ( Kv l (1 αg )+S 1 K ) 2 ρ αg g ) (1 + ( Kv l (1 αg )+S 2 1 K ) αg αg ag 2 16 Modified Sound Speed Model 1 αg a l 2 0 αg ag αg ( Kv l (1 αg )+S 1 K ) αg ag 2 ) ((1 α g )ρ l + α g ρ g ( K(1 αg ) 1 K αg )) ρ l (1 α g ) α g ρ g ( K(1 αg ) 1 K αg ) + v2 l (1 αg ) a l 2 ) (2ρ l (1 α g )v l + 2ρ g α g v g ( K(1 αg ) 1 K αg )) Eigenvalues of the Jacobian matrix C 1 D can be computed numerically. In particular for, K=1 and S=0 i.e. assuming zero slip between the liquid and gaseous phase. Modified sound speed comes out to be: 1/2 ω new (ρ g ρ = a g a l ( l ) ((ρ l + α g ρ g α g ρ l )(ag 2 ρ g α g ag 2 ρ g + α g al 2ρ l )) )

17 Figure: Comparative plot Figure: Zooming the comparative plot 17

18 W n+1 i = W n i w t + (f (w)) x = s t { } F n (W x i + 1 L, W R ) F n (W 2 i 2 1 L, W R ) + ts n i W L and W R are estimated value of variables at left and right cell interface respectively 18 Full Discretization Figure: Stencil for discretization in space and time

19 Features of Hyperbolic PDE: Information propagates with finite speed and has preferred direction Discontinuities or shock waves develop in a finite time and propagate even if initial and boundary data are smooth. Requirements from the Numerical Method: Sharp Resolution of discontinuities No spurious oscillations Minimal smearing effect Consistent, Stable and Convergent Conservation property in discrete sense 19 Numerical Methods

20 Approximation of Numerical Flux Liquid Gas {}}{{}}{ Fi FVS +1/2 (w L, w R ) = (α l ρ l ) L Ψ + l,l + (α l ρ l ) R Ψ l,r + (α g ρ g ) L Ψ + g,l + (αg ρg ) R Ψ g,r } {{ } Numerical Convective Flux + (F p ) i +1/2 }{{} Numerical Pressure Flux Liquid Contribution Gas Contribution Pressure Contribution Ψ + l,l = Ψ+ (v l l,l, ω i +1/2 ) Ψ l,r = Ψ (v l l,r, ω i +1/2 ) Ψ + (v, ω) = V + (v, ω) 1 0 l v Ψ (v, ω) = V (v, ω) 1 0 l v Ψ + g,l = Ψ+ g (v g,l, ω i +1/2 ) Ψ g,r = Ψ g (v g,r, ω i +1/2 ) Ψ g + (v, ω) = V + (v, ω) 0 1 v Ψg (v, ω) = V (v, ω) 0 1 v (F p ) i +1/2 = ( 0 0 p i +1/2 ) T p i +1/2 = P + (v L, ω i +1/2 )p L + P (v R, ω i +1/2 )p R v = mixture fluid velocity Splitting Functions V ± and P ± are the functions that satisfy the consistency, upwinding, monotonicity, differentiability and positivity property 20 Approximation of Numerical Flux

21 Numerical Test Cases No analytical results exist for Drift Flux Model. We try out numerical benchmark tests for multiphase flow problems. 1. Shock Tube:Shock capturing due to pressure difference 2. Fast Transients: Propagation of pressure pulses. 3. Slow Transients: Propagation of mass transport wave Correct description of fluid transport and pressure waves requires high resolution schemes possessing little numerical diffusion. Both first order and second order schemes were investigated. 21 Numerical Test Cases

22 Figure: Wave Fronts 22 Numerical Benchmarking Figure: Numerical Example

23 23 Shock Tube Figure: Behaviour of Gas Void Fraction using first order FVS

24 24 Shock Tube Figure: Behaviour of Pressure using first order FVS

25 25 Shock Tube Figure: Behaviour of Liquid Velocity using first order FVS

26 Fast Transients : Test Case 1 Figure: Fast Transient Test Case Capturing fast transients allows the modelling of water hammer effects. 26 Fast Transients

27 Figure: Snapshots of fast transients test case using first order FVS at CFL = 0.25; Gas Volume Fraction(left), Liquid Velocity(middle), Pressure(right) 27 Numerical Results

28 Fast Transients : Test Case 2 Figure: Fast Transients Test Case Capturing fast transients allows the modelling of water hammer effects. 28 Fast Transients

29 Figure: Comparison between second order AUSM scheme and second order FVS scheme at CFL = 0.25; Pressure(left) and Liquid Velocity(right) 29 Numerical Results

30 30 Numerical Results Figure: Simulation of Fast Transient using AUSM scheme

31 Slow Transients : Test Case 1 Figure: Slow Transients Test Case Models transient behaviour induced by injecting gas and liquid at the inlet 31 Slow Transients

32 Future Work Non-linear Stability Analysis For hyperbolic conservation laws, the spectrum of the upwind spatial differential operator constitutes eigenvalues that lie in the left half plane near the imaginary axis The absolute stability region of the forward Euler method intersects the imaginary axis only at the origin Forward Euler is typically not a stable choice of time discretization; furthermore it is only first order accurate Nonlinear stability conditions become critical for the convergence in the presence of shocks or sharp gradients Establish order of merit of the numerical scheme 32 Future Work

33 Future Work Model Order Reduction Based on the properties of the fully discretized or semi discretized models, an appropriate model order reduction technique needs to be obtained, which: Handle non-linearities and delays (due to wave propagation) Preserves stability characteristics of the original model Preserves multiple time scales involved in the problem Preserves input-output behaviour of the original system 33 Future Work

34 Future Work Numerical Modelling of Tripping Benchmark Scenario Challenges from Simulation Perspective Cross sectional area changes dynamically as the pipe moves Regridding of the annular region Higher than 1D model would be more accurate Figure: Drilling Process 34 Future Work

35 35 Thank You for your attention!!