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Hydraulic Modellingfor DrillingAutomationCASA Day

Harshit BansalApril 19, 2017

2 Supervisors and Collaborators

Team:

I Supervisors at TU/e :W.H.A. Schilders, N. van de Wouw, B. Koren, L. Iapichino

I Collaborators:1. Norway : G. O. Kaasa (Kelda Drilling Controls, NTNU)2. France : F. de Meglio (Ecole de Mines)

I PhD Group : M.H. Abbasi (affiliated to CASA and Kelda DrillingControls)

Project SponsorsThe project HYMODRA (HYdraulic MOdelling for DRilling Automation) issponsored by Shell and NWO-I under the aegis of Shell NWO/FOM PhDProgramme in Computational Sciences for Energy Research.

3 Outline

I Application Perspective

I Mathematical Notion of Drift Flux Model

I Physical and Numerical Boundary Conditions

I Sound Speed Model

I Numerical Schemes

I Numerical results

I Conclusions and perspectives

4 Objective

Figure: Drilling Schematic

Main Goal : Develop hydraulic modelsand supporting model reductiontechniques that are

1. accurate enough

2. simple enough

to be employed in the context of drillingscenario simulations and real timeestimation and control in case of gasinflux.

Managed Pressure Drilling!!

5 Characteristics/Features

Figure: Managed PressureDrilling

Characteristics/Features:I The downhole pressure must be kept

within allowable limitsI Delays in transmission of informationI No downhole measurements during

certain phases of drilling operationsI Top-side measurements are available

Hydraulic Model:I Serve as a model for controller/

estimatorI Aid to design operations before hand

6 Complexities from physical perspective

Complexities from physical perspective

Timescales in the drilling processI Slow transient corresponding to mass transportI Fast transient corresponding to the propagation of the acoustic

waves

Nonlinearities:I Acoustic velocity changes very rapidly in the one-phase to

two-phase transition regions and vice versa.I Disappearance and Appearance of Phases.I Various flow regimes across different sections of the well.I Distributed non-linearities due to source terms.

7 Governing Equations of Drift Flux Model

Governing Equations of Drift Flux Model

∂t (ρlαl ) + ∂x(ρlαlvl ) = Γl

∂t (ρgαg) + ∂x(ρgαgvg) = Γg

∂t (ρlαlvl + ρgαgvg) + ∂x(ρlαlvl 2 + ρgαgvg2 + P ) = Qg + Qv

Qg = −g(ρlαl + ρgαg)sin(θ);Qv = −32µvd2

αl = Liquid Void Fraction ; αg = Gas Void Fractionρl = Liquid Density ; ρg = Gas Densityvl = Liquid Velocity ; vg = Gas Velocity ; v = Mixture Flow VelocityΓl and Γg are the phase change termsP = Pressure ; d = hydraulic diameter ; µ = mixture viscosityθ is the well inclination ; g = acceleration due to gravity

8 Closure Laws Modelling

Closure Laws αg + αl = 1

ρg = P/ag2

ρl = ρl0 + (P − Pl0)/al 2

vg = (Kvlαl + S )/(1 − Kαg)

K and S are flow dependent parameters. There is singularity in the sliplaw when we approach pure gas region.al is the speed of sound in the liquid phaseag is the speed of sound in the liquid phasePl0 : standard atmospheric pressureρl0 : density of liquid at standard atmospheric pressure

9 System of Conservation Laws

The 1-D non-linear conservation law:

wt + (f (w))x = s

is hyperbolic if the Jacobian matrix ∂f∂w is diagonalizable with real

eigenvalues for each physically relevant w .

w =

ρlαl

ρgαg

ρlαlvl + ρgαgvg

, f (w) =

ρlαlvlρgαgvg

ρlαlvl 2 + ρgαgvg2 + P

s =

Γl

Γg

Qg + Qv

For further discussion, Γl = 0, Γg = 0

10 Eigenvalues of the Jacobian Matrix

Eigenvalues of the Jacobian MatrixI The corresponding eigenvalues are given by:

λ1 = vl − ω ,λ2 = vl + ω ,λ3 = vgwhere, ω is the speed of sound in two-phase mixture

I Two eigenvalues are linked to the compressibility effectsI Third eigenvalue is coincident to the gas velocityI One pressure pulse propagates downstream and the other pressure

pulse propagates upstream.I Gas Volume wave travels downstream.

11 Physical Boundary Conditions

Physical Boundary Conditions:(ρlαlvl )(0, t) = f (t)

(ρgαgvg)(0, t) = h (t)

P (L , t) = r(t)

or

(αlvl )(0, t) = f (t)

(αgvg)(0, t) = h (t)

P (L , t) = r(t)

12 Numerical Boundary Conditions

Compatibility relations for multi phase system, which are:Characteristic 1 and 2: Compatibility relation corresponding to thepressure wave propagating in the upstream direction and downstreamdirection of the flow:

ddt

p −+ ρlω(vg − vl )ddt

αg − ρlαl (vg − vl +− ω)ddt

vl = q(vg − vl +− ω)

where, ddt =

∂∂t + (vl −+ ω) ∂

∂x is the directional derivative

Characteristic 3: Compatibility relation corresponding to the gas volumewave:

ddt

p +p

αg(1 − Kαg)

ddt

αg = 0

where, ddt =

∂∂t + (vg) ∂

∂x is the directional derivative

13 Sound Speed Model

Sound Speed ModelApproximate Sound Speed Model is written as:

ω =

al if αg < ε

c(P , αg , ρl ,K ) if ε ≤ αg ≤ 1 − ε

ag if αg > 1 − ε

where ε is a small parameter

c(P , αg , ρl ,K ) =

√P

αgρl (1 − Kαg)

al and ag are the sound speeds in liquid and gas medium respectively

14 Sound Speed Model

Assumptions for Sound Speed in the two phase mixture

I Liquid is incompressibleI αgρg << αlρl

Why is sound speed in the two phase mixture important?

I Numerical flux computations are heavily dependent on themixture sound speed

I Numerical dissipation depends on the sound speed of thetwo phase mixture

I Enable correct determination of locations and speeds of thewave fronts

15 Sound Speed Model

Reasons for Model Improvement

I Drilling fluids are highly compressibleI Existing models for sound speed in two phase mixture are

singular at low and high void fractionsI Existing models become singular before rendering the Drift

Flux Model non-hyperbolicI Existing models also fail in modelling the realistic effects at

high operating pressuresI Need of a unified model for single phase flow and two phase

flow modelling

16 Modified Sound Speed Model

C =

−ρl

1−αga2l

0

ρgαga2g

0

(ρg (Kvl (1−αg )+S

1−Kαg)− ρl vl ) (

(1−αg )vla2l

+αg (

Kvl (1−αg )+S1−Kαg

)

a2g

) ((1 − αg )ρl + αg ρg (K (1−αg )1−Kαg

))

D =

−ρl vl(1−αg )vl

a2l

ρl (1 − αg )

ρg (Kvl (1−αg )+S

1−Kαg)

αg (Kvl (1−αg )+S

1−Kαg)

a2g

αg ρg (K (1−αg )1−Kαg

)

(−v2l ρl + (

Kvl (1−αg )+S1−Kαg

)2

ρg ) (1 +(Kvl (1−αg )+S

1−Kαg)2

αg

a2g

+v2l (1−αg )

a2l

) (2ρl (1 − αg )vl + 2ρg αg vg (K (1−αg )1−Kαg

))

Eigenvalues of the Jacobian matrix C−1D can be computed numerically.In particular for, K=1 and S=0 i.e. assuming zero slip between the liquidand gaseous phase. Modified sound speed comes out to be:

ωnew = agal ((ρgρl )

((ρl + αgρg − αgρl )(a2g ρg − αga2

g ρg + αga2l ρl ))

)1/2

17

Figure: Comparative plot

Figure: Zooming the comparative plot

18 Full Discretization

wt + (f (w))x = s

Wn+1i = Wn

i − ∆t∆x

{Fni+ 1

2(WL ,WR )− Fni− 1

2(WL ,WR )

}+ ∆tSn

i

WL and WR are estimated value of variables at left and right cell interfacerespectively

Figure: Stencil for discretization in space and time

19 Numerical Methods

Features of Hyperbolic PDE:I Information propagates with finite speed and has preferred

directionI Discontinuities or shock waves develop in a finite time and

propagate even if initial and boundary data are smooth.

Requirements from the Numerical Method:I Sharp Resolution of discontinuitiesI No spurious oscillationsI Minimal smearing effectI Consistent, Stable and ConvergentI Conservation property in discrete sense

20 Approximation of Numerical Flux

Approximation of Numerical Flux

F FVSi+1/2(wL ,wR ) =

Liquid︷ ︸︸ ︷(αl ρl )L Ψ+

l ,L + (αl ρl )R Ψ−l ,R +

Gas︷ ︸︸ ︷(αg ρg )L Ψ+

g,L + (αg ρg )R Ψ−g,R︸ ︷︷ ︸

Numerical Convective Flux

+ (Fp )i+1/2︸ ︷︷ ︸Numerical Pressure Flux

Liquid Contribution

Ψ+l ,L = Ψ+

l (vl ,L ,ωi+1/2)

Ψ−l ,R = Ψ−

l (vl ,R ,ωi+1/2)

Ψ+l (v,ω) = V+(v,ω)

10v

Ψ−l (v,ω) = V−(v,ω)

10v

Gas Contribution

Ψ+g,L = Ψ+

g (vg,L ,ωi+1/2)

Ψ−g,R = Ψ−

g (vg,R ,ωi+1/2)

Ψ+g (v,ω) = V+(v,ω)

01v

Ψ−g (v,ω) = V−(v,ω)

01v

Pressure Contribution

(Fp )i+1/2 =(

0 0 pi+1/2)T

pi+1/2 = P+(vL ,ωi+1/2)pL +P−(vR ,ωi+1/2)pR

v = mixture fluid velocity

Splitting Functions

V± and P± are the functions that satisfy theconsistency, upwinding, monotonicity, differ-entiability and positivity property

21 Numerical Test Cases

Numerical Test Cases

No analytical results exist for Drift Flux Model. We try out numericalbenchmark 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 highresolution schemes possessing little numerical diffusion. Both firstorder and second order schemes were investigated.

22 Numerical Benchmarking

Figure: Wave Fronts

Figure: Numerical Example

23 Shock Tube

Figure: Behaviour of Gas Void Fraction using first order FVS

24 Shock Tube

Figure: Behaviour of Pressure using first order FVS

25 Shock Tube

Figure: Behaviour of Liquid Velocity using first order FVS

26 Fast Transients

Fast Transients : Test Case 1

Figure: Fast Transient Test Case

Capturing fast transients allows the modelling of water hammer effects.

27 Numerical Results

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

28 Fast Transients

Fast Transients : Test Case 2

Figure: Fast Transients Test Case

Capturing fast transients allows the modelling of water hammer effects.

29 Numerical Results

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

30 Numerical Results

Figure: Simulation of Fast Transient using AUSM scheme

31 Slow Transients

Slow Transients : Test Case 1

Figure: Slow Transients Test Case

Models transient behaviour induced by injecting gas and liquid at theinlet

32 Future Work

Future Work

Non-linear Stability Analysis

I For hyperbolic conservation laws, the spectrum of theupwind spatial differential operator constitutes eigenvaluesthat lie in the left half plane near the imaginary axis

I The absolute stability region of the forward Euler methodintersects the imaginary axis only at the origin

I Forward Euler is typically not a stable choice of timediscretization; furthermore it is only first order accurate

I Nonlinear stability conditions become critical for theconvergence in the presence of shocks or sharp gradients

I Establish order of merit of the numerical scheme

33 Future Work

Future Work

Model Order Reduction

Based on the properties of the fully discretized or semi discretizedmodels, an appropriate model order reduction technique needs tobe obtained, which:

I Handle non-linearities and delays (due to wave propagation)I Preserves stability characteristics of the original modelI Preserves multiple time scales involved in the problemI Preserves input-output behaviour of the original system

34 Future Work

Future Work

Numerical Modelling of Tripping Benchmark Scenario

Challenges from SimulationPerspective

I Cross sectional area changesdynamically as the pipe moves

I Regridding of the annular regionI Higher than 1D model would be

more accurate

Figure: Drilling Process

35

Thank You for your attention!!