Where innovation starts
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
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.
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)
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