Simulation of Three-Component Two-Phase Flow in Porous Media Using Method of Lines

Goudarzi, Salim; Mathias, Simon A.; Gluyas, Jon

doi:10.1007/s11242-016-0639-5

Cited by 10 publications

(8 citation statements)

References 39 publications

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“…The SciPy ODE solvers ode and solve_ivp have the ability to define a banded Jacobian pattern: setting uband and lband arguments to 1 tells the ODE solver that the Jacobian is a tridiagonal matrix. The SciPy ODE solver solve_ivp can also handle a general n×n Jacobian pattern, which is more adaptable for multi-dependent variable coupled problems (e.g., Goudarzi et al, 2016). The MATLAB ODE solvers can read the Jacobian sparsity pattern matrix from the JPattern argument.…”

Section: Defining the Jacobian Patternmentioning

confidence: 99%

A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0)

Ireson

Spiteri²,

Clark³

et al. 2023

Geosci. Model Dev.

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Abstract. A simple numerical solution procedure – namely the method of lines combined with an off-the-shelf ordinary differential equation (ODE) solver – was shown in previous work to provide efficient, mass-conservative solutions to the pressure-head form of Richards' equation. We implement such a solution in our model openRE. We developed a novel method to quantify the boundary fluxes that reduce water balance errors without negative impacts on model runtimes – the solver flux output method (SFOM). We compare this solution with alternatives, including the classic modified Picard iteration method and the Hydrus 1D model. We reproduce a set of benchmark solutions with all models. We find that Celia's solution has the best water balance, but it can incur significant truncation errors in the simulated boundary fluxes, depending on the time steps used. Our solution has comparable runtimes to Hydrus and better water balance performance (though both models have excellent water balance closure for all the problems we considered). Our solution can be implemented in an interpreted language, such as MATLAB or Python, making use of off-the-shelf ODE solvers. We evaluated alternative SciPy ODE solvers that are available in Python and make practical recommendations about the best way to implement them for Richards' equation. There are two advantages of our approach: (i) the code is concise, making it ideal for teaching purposes; and (ii) the method can be easily extended to represent alternative properties (e.g., novel ways to parameterize the K(ψ) relationship) and processes (e.g., it is straightforward to couple heat or solute transport), making it ideal for testing alternative hypotheses.

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Section: Defining the Jacobian Patternmentioning

confidence: 99%

A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0)

Ireson

Spiteri²,

Clark³

et al. 2023

Geosci. Model Dev.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The scipy ODE solvers ode and solve_ivp have the ability to define a banded Jacobian pattern: setting uband and lband arguments to 1 tells the ODE solver that the Jacobian is a tri-diagonal matrix. The scipy ODE solver solve_ivp can also handle a general 𝑛 × 𝑛 Jacobian pattern, which is more adaptable for multi-dependent variable coupled problems (e.g., Goudarzi et al, 2016). The MATLAB ODE solvers can read the Jacobian sparsity pattern matrix from the JPattern argument.…”

Section: Defining the Jacobian Patternmentioning

confidence: 99%

A simple, efficient, mass conservative approach to solving Richards’ Equation (openRE, v1.0)

Ireson

Spiteri²,

Clark³

et al. 2022

Preprint

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Abstract. We show that a simple numerical solution procedure – namely the method of lines combined with an off-the-shelf ODE solver – can provide efficient, mass conservative solutions to the pressure-head form of Richards’ equation. We present a novel method to quantify the boundary fluxes that reduces water balance errors without negative impacts on model runtimes. We compare this solution with alternatives, including the classic Modified Picard Iteration method of Celia et al. (1990) and the Hydrus 1D model (Šimůnek et al., 2005, 2016). We reproduce a set of benchmark solutions with all models. We find that Celia’s solution has the best water balance, but it can incur significant truncation errors in the simulated boundary fluxes, depending on the time steps used. Our solution has comparable run-times to Hydrus and better water balance performance (though both models have excellent water balance closure for all the problems we considered). Our solution can be implemented in an interpreted language, such as MATLAB or Python, making use of off-the-shelf ODE solvers. We investigated alternative scipy ODE solvers in Python and make practical recommendations about the best way to implement them for Richards’ equation. There are two advantages of our approach: i) the code is short and simple, making it ideal for teaching purposes; and ii) the method can be easily extended to represent alternative properties (e.g., novel ways to parameterize the K(ψ) relationship) and processes (e.g., it is straightforward to couple heat or solute transport), making it ideal for testing alternative hypotheses.

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“…following Goudarzi et al (2016), it is better to solve for fluid pressure, P, temperature, T , and the mass fractions of gaseous methane and hydrate in the pore-space, z g [-] and z h [-], respectively, found from…”

Section: Recasting In Terms Of Primary Dependent Variablesmentioning

confidence: 99%

“…Following Mathias et al (2014) and Goudarzi et al (2016), the above set of equations are solved using a method of lines approach. The spatial domain is discretized into N x equally-spaced points in the x direction and N r equally-spaced points in the y direction using Godunov's method (LeVeque, 1992).…”

Section: Numerical Solutionmentioning

confidence: 99%

Masuda’s sandstone core hydrate dissociation experiment revisited

Hardwick

Mathias

2018

Chemical Engineering Science

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. (1999) represents an important experimental data set that can be used for benchmarking numerical simulators for this purpose. Data collected includes gas production, water production, boundary pressure and temperature from three internal observation points. At least six modeling studies exist within the literature seeking to simulate the gas production data and the temperature data. However, the pressure data and water production data are generally overlooked. In this article we present a set of numerical simulations capable of reconciling the Masuda et al. (1999) data set in its entirety. Improvements on existing modeling studies are achieved by: (1) using improved estimates of the initial hydrate saturation; (2) obtaining relative permeability parameters, a hydrate stability depression temperature and a convective heat transfer coefficient by calibration with the observed data; (3) applying a new critical threshold permeability model, specifically to reconcile a relatively fast gas production with a relatively slow far-field boundary pressure response. A subsidiary finding is that permeability is significantly reduced in the presence of very low hydrate saturations. But more importantly, the multi-faceted effectiveness of the data set from Masuda's experiment is clearly demonstrated for numerical simulation benchmarking in the future.

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Simulation of Three-Component Two-Phase Flow in Porous Media Using Method of Lines

Cited by 10 publications

References 39 publications

A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0)

A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0)

A simple, efficient, mass conservative approach to solving Richards’ Equation (openRE, v1.0)

Masuda’s sandstone core hydrate dissociation experiment revisited

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