Abstract
Fast transients in pumps and valves may induce significant variations in pressures and flow rates throughout pipelines that can even cause structural damages. This computational work deals with parameter estimation of a water hammer model, with focus on a transient friction coefficient and an empirical parameter related to the pipeline elasticity. The hyperbolic water hammer model was solved with a total variation diminishing version of the weighted average flux finite volume scheme. Simulated pressure and flow rate measurements taken near the inlet and the outlet of the pipeline were used for the solution of the parameter estimation problem with Markov Chain Monte Carlo methods. This work aimed at the reduction of the computational time of the inverse problem solution with two different strategies: (1) the application of a recent parallel computation version of the Metropolis–Hastings algorithm; and (2) the use of a machine learning metamodel obtained with the evolutionary neural network algorithm and the approximation error model approach. These two approaches are compared in terms of the parameter estimation accuracies and associated computational times.
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Abbreviations
- a :
-
Celerity, m/s
- A :
-
Area of the pipe cross section, m2
- AEM:
-
Approximation error model
- B :
-
Vector of source terms
- c k :
-
Courant number
- D :
-
Pipe diameter, m
- e :
-
Pipe wall thickness, m
- e(P):
-
Vector of modeling errors
- E :
-
Young’s modulus of the pipe, Pa
- EvoNN:
-
Evolutionary neural network
- f :
-
Darcy–Weisbach friction coefficient
- f u :
-
Transient friction coefficient
- F :
-
Vector of fluxes
- G :
-
Pipe shear modulus, Pa
- g :
-
Gravity acceleration, m/s2
- H :
-
Head, m
- K :
-
Elastic modulus of the fluid, Pa
- K p,b :
-
Localized head loss coefficient
- k :
-
Iteration index of each parallel Markov chain
- L :
-
Pipe length, m
- MCMC:
-
Markov Chain Monte Carlo
- I :
-
Number of measurements
- N :
-
Number of waves in the Riemann problem
- N tr :
-
Number of training data for the neural networks
- n :
-
Time step
- P :
-
Vector of parameters
- Pr(Y):
-
Prior distribution
- Pr(P|Y):
-
Posterior distribution
- Pr(Y|P):
-
Likelihood function
- p :
-
Pressure, Pa
- PPGA:
-
Predator–Prey genetic algorithm
- q :
-
Mass flow rate, kg/s
- R :
-
Total number of cores for parallel computation
- R in :
-
Pipe internal radius
- R out :
-
Pipe external radius
- Re:
-
Reynolds number
- r :
-
Processor index
- S LS :
-
Least squares norm
- T(P) :
-
Vector of dependent variables obtained with the high-fidelity model
- T app (P) :
-
Vector of dependent variables obtained with the low-fidelity model
- t :
-
Time, s
- TVD:
-
Total variation diminishing
- u :
-
Velocity in the parallel direction of the flow, m/s
- U :
-
Vector of state variables
- W :
-
Covariance matrix
- WAF:
-
Weighted average flux
- x :
-
Axial position, m
- x b :
-
Position of the localized pressure loss, m
- Y :
-
Vector of measurements
- α :
-
Acceptance rate
- δ(.):
-
Dirac delta function
- ∆p b :
-
Localized pressure loss
- ε :
-
Measurements errors
- θ :
-
Angle between the flow direction and a horizontal reference plane
- ρ :
-
Density, kg/m3
- ϕ :
-
Proposal distribution
- Φ:
-
Limiter function
- Φ( t ) :
-
Diagonal matrix with the standard deviations of the proposal for each parameter
- υ P :
-
Poisson’s ratio
- Ψ:
-
Dimensionless parameter related to the pipe elasticity
- ω :
-
Pipe roughness
- η(P):
-
Total error
- Γ η P :
-
Covariance matrix of η and P
- 0:
-
Steady-state local value
- i :
-
Finite volume position
- k :
-
Index of the core
- max:
-
Maximum
- n :
-
Time step
- –:
-
Mean value
- *:
-
Candidate parameter
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Acknowledgements
This study is mainly supported by Petrobras S.A., through project PT-112.01.13161. The support provided by CNPq, FAPERJ and CAPES (Finance Code 001, PQ1A, Cientista do Estado) is also greatly appreciated.
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Carvalho, R.C., Herzog, I.L., Orlande, H.R.B. et al. Parameter estimation with the Markov Chain Monte Carlo method aided by evolutionary neural networks in a water hammer model. Comp. Appl. Math. 42, 35 (2023). https://doi.org/10.1007/s40314-022-02162-0
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DOI: https://doi.org/10.1007/s40314-022-02162-0
Keywords
- Bayesian statistics
- Metropolis–Hastings algorithm
- Parallel computation
- Machine learning
- EvoNN
- Approximation error model