# Steam-assisted gravity drainage: a posteriori estimates with simplified eval- uation and application of adaptive mesh refinement

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## Balancing and stopping criteria, adaptive algorithm, and efficiency

The individual error component estimators of Corollary 1.3.5 are used in this section to define adaptive criteria to stop the iterative linearizations, to select the value of the regularization parameter ǫ, to locally adapt the quadrature rule, to adjust the time step, and to select the mesh elements to refine/derefine. These criteria are incorporated in a fully adaptive algorithm detailed in Section 1.4.2. Finally, in Section 1.4.3 we show the efficiency of our estimators when the adaptive balancing and stopping criteria are used.

### Balancing and stopping criteria

Following [26, 15, 19], this section introduces stopping criteria for the iterative algorithms based on the estimators of Corollary 1.3.5. The goal is to stop the iterations as soon as the corresponding error component no longer affects significantly the overall error. We assume in what follows that we are marching in time from time tn−1 to time tn. Let three user-given parameters 􀀀lin, 􀀀reg, 􀀀qd ∈ (0, 1) be given. The criteria are: (i) Linearization. The linearization iteration is pursued until step kn such that ηn,ǫ,kn lin ≤ 􀀀lin ηn,ǫ,kn sp + ηn,ǫ,kn tm + ηn,ǫ,kn qd + ηn,ǫ,kn reg .
(ii) Regularization. The regularization parameter ǫ is reduced until the value ǫn such that ηn,ǫn,kn reg ≤ 􀀀reg ηn,ǫn,kn sp + ηn,ǫn,kn tm + ηn,ǫn,kn qd .
(iii) Quadrature. The quadrature rule is improved until ηn,ǫn,kn qd ≤ 􀀀qd ηn,ǫn,kn sp + ηn,ǫn,kn tm .

#### Application to a vertex-centered finite volume discretization

In this section, we consider the vertex-centered finite volume spatial and backward Euler temporal discretization of the Stefan problem (1.1.1). The regularization of Section 1.2.1.2 is considered and the Newton linearization is used. We show how to construct the equilibrated flux tn,ǫ,k h , the linearized flux ln,ǫ,k h , and the interpolation operator n of Assumption 1.3.4 (in generalization of Assumptions 1.3.1 and 1.3.2) and verify Assumptions 1.4.2 and 1.4.3. Thus, all the results of Sections 1.3–1.5 will apply.

Dual and tertial space meshes

The vertex-centered finite volume method is defined using a sequence of dual meshes {Dn}0≤n≤N of the space domain . For a given family of matching simplicial primal meshes {T n}0≤n≤N, we construct {Dn}0≤n≤N as follows: for any 1 ≤ n ≤ N and with every vertex a of the mesh T n, we associate one dual volume D, constructed by connecting the barycenters of the simplices sharing a through edge (and face for d = 3) barycenters, see Figure 1.2, left,for d = 2. We split every set Dn into interior dual volumes Dn,i and boundary dual volumes Dn,b. The simplicial mesh Kn appearing in Sections 1.2–1.5 is constructed by dividing each D ∈ Dn into a mesh KD as indicated in Figure 1.2, right, if d = 2 and similarly for d = 3.

Introduction
General context
General properties for an a posteriori error estimate
A posteriori error analysis based on equilibrated fluxes
Motivations
Moving boundary problems
Reservoir simulation
Outline
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Bibliography
1 Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem
1.1 Introduction
1.2 Continuous and discrete settings
1.2.1 Continuous setting
1.2.2 Discrete setting
1.3 An a posteriori error estimate for the dual norm of the residual
1.3.1 Dual norm of the residual
1.3.2 General assumptions
1.3.3 A basic a posteriori error estimate
1.3.4 An a posteriori error estimate distinguishing the space, time, regularization, linearization, and quadrature errors
1.4 Balancing and stopping criteria, adaptive algorithm, and efficiency
1.4.1 Balancing and stopping criteria
1.4.3 Efficiency of the a posteriori error estimate
1.5 An a posteriori error estimate for the error in temperature and enthalpy
1.5.1 Bounding the error of the temperature and enthalpy by the dual norm of the residual
1.5.2 The a posteriori error estimate
1.6 Application to a vertex-centered finite volume discretization
1.6.1 Dual and tertial space meshes
1.6.2 The vertex-centered finite volume scheme
1.6.3 Newton linearization
1.6.4 Flux reconstruction
1.7 Numerical experiments
1.7.1 Setting
1.7.2 Computing approximately the negative norms
1.7.3 Stopping criteria
1.7.4 Balancing criteria
1.7.5 Overall performance
1.A Proofs
1.A.1 Proof of Theorem 1.4.4
1.A.2 Proof of Theorem 1.5.2
Bibliography
2 A posteriori error estimates, stopping criteria, and adaptivity for multi- phase compositional flows in porous media
2.1 Introduction
2.2 Setting
2.2.1 The multiphase compositional model
2.2.2 An implicit finite volume scheme with phase-upwind and two-point discretization of diffusive fluxes
2.3 A basic a posteriori error estimate
2.3.1 Weak solution
2.3.2 A generic approximate solution
2.3.3 Error measure
2.3.4 Flux and pressure reconstructions
2.3.5 A posteriori error estimate
2.4 Application to finite volume method and adaptivity based on distinguishing the different error components
2.4.1 Linearization and algebraic resolution
2.4.2 Approximate solution
2.4.3 Phase pressure reconstructions
2.4.4 Component flux reconstructions
2.4.5 Distinguishing the space, time, linearization, and algebraic errors
2.5 Numerical results
2.5.1 Common setting
2.5.2 Compressible flow in a homogeneous porous medium
2.5.3 Compressible flow in a heterogeneous porous medium
2.5.4 Five-spots pattern
Bibliography
3 A posteriori error estimates for thermal multiphase compositional flows in porous media
3.1 Introduction
3.2 Setting
3.3 Discretization of the energy equation
3.3.1 Two-point finite volume discretization
3.3.2 Linearization and algebraic resolution
3.4 Approximate solution and reconstructions
3.4.1 Postprocessing of the temperature
3.4.2 Saturations, molar fractions, and molar energy
3.4.3 H1 0 -conforming temperature reconstruction
3.4.4 H(div; )-conforming energy flux reconstructions
3.5 A posteriori error estimate
3.5.1 Weak solution
3.5.2 Error measure
3.5.3 An a posteriori error estimate distinguishing the space, time, linearization, and algebraic errors
3.5.4 Balancing and stopping criteria
3.A Application to the thermal dead oil model
3.A.2 A posteriori error estimate for the thermal dead oil model
Bibliography
4 Steam-assisted gravity drainage: a posteriori estimates with simplified eval- uation and application of adaptive mesh refinement
4.1 Introduction
4.2 SAGD characterization and modeling
4.2.1 Common characteristics
4.2.2 Types of SAGD
4.2.3 SAGD modeling
4.2.4 Mathematical model of the thermal dead oil system
4.3 Evaluation of the estimators using a practical simplified formula
4.3.1 A general simplification formula
4.3.2 Evaluation of the estimators
4.4 SAGD test case
4.4.1 Model description
4.4.2 Initialization and production scheme
4.4.3 Model simulation
4.4.4 Approximate solution and a posteriori estimate