A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media

Radu, Florin A.; Nordbotten, Jan Martin; Pop, Sorin; Kumar, Kundan

doi:10.1016/j.cam.2015.02.051

Cited by 65 publications

(79 citation statements)

References 29 publications

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“…An efficient combination of the modified Picard and the Newton method, the Picard/Newton method is proposed in [20]. For the sake of completeness we mention also the accelerated Picard method [21] for Richards' equation and the semi-smooth Newton method [19] and L−method [30] for two-phase flow in porous media, as valuable linearization methods.…”

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

A study on iterative methods for solving Richards’ equation

2016

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This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards' equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards' equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L−scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L−scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L−scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.

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Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

A study on iterative methods for solving Richards’ equation

2016

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“…(a) FOU (b) FVBJ We note that [14] employed a similar linearisation with constant derivative in the iteration applied to a global pressure formulation and convergence of the iterative process was shown. We now consider a variety of tests on structured and unstructured grids.…”

Section: Predictor-corrector Fixed-point Iteration Methods An Iterativmentioning

confidence: 99%

Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids

Xie

Edwards

2017

Comput Geosci

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Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting.

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“…For solving the nonlinear problem, we propose an iteration scheme that builds on the ideas in (the ‘L’‐scheme). The idea is to construct a sequence of triplets

false(s_{w}^{n + 1, i}, p_{n}^{n + 1, i}, p_{c}^{n + 1, i} false)

converging as

i \to \infty

to the solution

false(s_{w}^{n + 1}, p_{n}^{n + 1}, p_{c}^{n + 1} false)

of Problem .…”

Section: Numerical Approximationmentioning

confidence: 99%

“…The previous texts has motivated the linearization schemes proposed in for the finite element, finite volume and the mixed finite element discretization of porous media flow models. The idea of the linearization scheme is to add an additional term in the form of

L \cdot (Solution_Current_Iteration - Solution_Old_Iteration),

with L being a parameter that has to be chosen sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase flow in porous media with dynamic capillarity effects

Karpinski

Pop

Radu

2017

Numerical Meth Engineering

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SUMMARYAbstract We present a linearization scheme for an interior penalty discontinuous Galerkin method for two-phase porous media flow model which includes dynamic effects in the capillary pressure. The fluids are assumed immiscible and incompressible, and the solid matrix non-deformable. The physical laws are approximated in their original form, without using the global or complementary pressures. The linearization scheme does not require any regularization step. Furthermore, in contrast with Newton or Picard methods, there is no computation of derivatives involved. We prove rigorously that the scheme is robust and linearly convergent. Numerical results, including examples in realistic, heterogeneous media are presented to sustain the applicability of the scheme.

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A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media

Cited by 65 publications

References 29 publications

A study on iterative methods for solving Richards’ equation

A study on iterative methods for solving Richards’ equation

Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids

Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase flow in porous media with dynamic capillarity effects

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