Abstract
This paper presents a new mathematical representation of multiphase thermodynamic equilibrium using so-called repartition coefficients. Combined with a global mass formulation of multiphase Darcy flow in porous media, it allows the derivation of a computationally efficient family of time schemes. The model accounts for the mass conservation of an arbitrary number of components flowing through an arbitrary number of phases, coupled with thermodynamic equilibrium and pore volume conservation. By separating the thermodynamic equilibrium part from the flow part through the repartition coefficients, the formulation removes the need for any specific handling of phase appearance and disappearance within the flow solver. Any “black box” thermodynamic equilibrium solver can then be used to compute the repartition coefficients, from EOS based solvers to tabulated representation of the thermodynamic equilibrium, each specific choice of thermodynamic solver leading to a new scheme. Three numerical experiments, from a simple beam to a real case, illustrate the good behavior of the approach.
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Appendices
Appendix A: Thermodynamic conversions
By definition, one has
Then
and consequently
and thus \(x_{i}^{\alpha }=\frac {n_{i}^{\alpha }}{n_{\alpha }}\) becomes
In the same way, starting from
we obtain
and thus \(X_{i}^{\alpha }=\frac {m_{i}^{\alpha }}{m_{\alpha }}\) becomes
Thus for the molar masses of phases, depending whether we have molar fractions or mass fractions at our disposal, we can use either:
or
as \({\sum }_{i=0}^{N_{comp}-1}X_{i}^{\alpha }=1\).
In the same way, for phases we have, if mtot denotes the total mass and ntot the total mole number of the system
and \(n_{\alpha }=\frac {m_{\alpha }}{M_{m_{\alpha }}}\) give for \(\theta _{\alpha }=\frac {n_{\alpha }}{n_{tot}}\) and \({\varTheta }_{\alpha }=\frac {m_{\alpha }}{m_{tot}}\):
For the total fractions of components,
and \(m_{i}=n_{i}M_{m_{i}}\) lead for \(z_{i}=\frac {n_{i}}{n_{tot}}\) et \(Z_{i}=\frac {m_{i}}{m_{tot}}\) to:
finally, the mass equilibrium coefficients are defined through:
Indeed, as:
we deduce that we can convert from molar to mass by setting (their is no uniqueness as only the coefficient ratios play a role in the above identity):
Notice that by construction:
which means that mass equilibrium is formally identical to molar equilibrium.
Appendix B: Full discretization using TPFA finite volumes
Meshes and notations
We assume that the computational domain Ω is an open polygonal subset of \(\mathbb {R}^{d}\), d = 2 or 3, such that
where the sets \(({\varOmega }_{i})_{0 \leq i \leq N_{layer}-1}\) are also open polygonal subsets of \(\mathbb {R}^{d}\), in which the geological properties are assumed to evolve continuously (in general, they correspond to geological layers). We recall the usual notations describing a mesh \({\mathscr{M}}=(\mathcal {T},\mathcal {F})\) of Ω. \(\mathcal {T}\) is a finite family of connected open disjoint polygonal subsets of Ω (the cells of the mesh), such that \(\overline {{\varOmega }}=\cup _{K \in \mathcal {T}} \overline {K}\). For any \(K \in \mathcal {T}\), we denote by |K| the measure of |K| and by \(\partial K = \overline {K}\setminus K\) the boundary of K. \(\mathcal {F}\) is a finite family of disjoint subsets of hyperplanes of \(\mathbb {R}^{d}\) included in \(\overline {{\varOmega }}\) (the faces of the mesh) such that, for all \(\sigma \in \mathcal {F}\), its measure is denoted |σ|. For any \(K \in \mathcal {T}\), there exists a subset \(\mathcal {F}_{K}\) of \(\mathcal {F}\) such that \(\partial K =\cup _{\sigma \in \mathcal {F}_{K}}\overline {\sigma }\). Then, for any \(\sigma \in \mathcal {F}\), we denote by \(\mathcal {T}_{\sigma }=\{K \in \mathcal {T} | \sigma \in \mathcal {F}_{K}\}\). Next, for all \(K \in \mathcal {T}\) and all \(\sigma \in \mathcal {F}_{K}\), we denote by nK,σ the unit normal vector to σ outward to K. The set of boundary faces is denoted \(\mathcal {F}_{ext}\), while interior faces are denoted \(\mathcal {F}_{int}\). We complement the mesh by a family of points \(\mathcal {P}=\left ((\boldsymbol {x}_{K})_{K \in \mathcal {T}},(\boldsymbol {x}_{\sigma })_{\sigma \in \mathcal {F}_{ext}}\right )\) indexed by the cells and boundary faces such that xK ∈K̈ for any \(K \in \mathcal {T}\) and xσ ∈ σ and any \(\sigma \in \mathcal {F}_{ext}\). If \(\mathcal {T}_{\sigma } =\{K,L\}\), we assume that xK≠xL. If \(\sigma \in \mathcal {F}_{K}\), we denote dK,σ the distance between xK and σ. Finally, we assume that for any 0 ≤ i ≤ Nlayer − 1, there exists \(\mathcal {F}_{i} \subset \mathcal {F}\) such that:
thus the mesh is assumed adapted to the geological discontinuities.
To any continuous variable p(x,t), we associate a family of discrete variables \(({p_{K}^{n}})_{K \in \mathcal {T}, 0 \leq n \leq N_{T}}\) such that \({p_{K}^{n}}\) is in principle an approximation of p(xK,tn). As we consider the TPFA finite volume approximation, to ensure this approximation property we assume that the mesh is \(({\varLambda }_{K})_{K \in \mathcal {T}}\)- orthogonal, with \(({\varLambda }_{K})_{K \in \mathcal {T}}\) is the discrete permeability tensor. More precisely, there exists a family of straight lines \((\mathcal {D}_{K,\sigma })_{\sigma \in \mathcal {F}_{K}}\), with \(\mathcal {D}_{K,\sigma }\) orthogonal to σ with respect to the scalar product induced by \({\varLambda }_{K}^{-1}\), such that
-
For any \(K \in \mathcal {T}\), \(\bigcap _{\sigma \in \mathcal {F}_{K}} \mathcal {D}_{K,\sigma } = \boldsymbol {x}_{K}\)
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For any \(\sigma \in \mathcal {F}_{int}\) with \(\mathcal {T}_{\sigma }=\{K,L\}\), \(\mathcal {D}_{K,\sigma }\cap \sigma = \mathcal {D}_{L,\sigma } \cap \sigma \neq \emptyset \)
-
For any \(\sigma \in \mathcal {F}_{ext}\) with \(\mathcal {T}_{\sigma }=\{K\}\), \(\mathcal {D}_{K,\sigma }\cap \sigma \neq \emptyset \).
TPFA Finite volume scheme for porous media flow
For simplicity, we assume that the boundary conditions are homogeneous Neumann boundary conditions everywhere, i.e. no flow can leave the computational domain. In this case, the discrete mass balance equations of each component 0 ≤ i ≤ Ncomp − 1 become, for any \(K \in \mathcal {T}\) and any 0 ≤ n ≤ NT − 1:
where we have denoted:
with
and
and also
The upwind relative permeabilities are given by:
In the same way, the upwind mass fractions are defined by:
as are \(\rho _{\alpha ,\sigma }^{n+1}\) and \(\mu _{\alpha ,\sigma }^{n+1}\). Finally \(Q_{i,K}^{n+1}\) is given by the usual Peaceman’s well source term. The discretization of the remaining equations is immediate. We have, for all \(K \in \mathcal {T}\)
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Coatléven, J., Meiller, C. On an efficient time scheme based on a new mathematical representation of thermodynamic equilibrium for multiphase compositional Darcy flows in porous media. Comput Geosci 25, 1063–1082 (2021). https://doi.org/10.1007/s10596-021-10039-0
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DOI: https://doi.org/10.1007/s10596-021-10039-0