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This work investigates the performance of the on-demand machine learning (ODML) algorithm introduced in Leal et al. (Transp. Porous Media133(2), 161–204, 2020) when applied to different reactive transport problems in heterogeneous porous media. This approach was devised to accelerate the computationally expensive geochemical reaction calculations in reactive transport simulations. We demonstrate that even with a strong heterogeneity present, the ODML algorithm speeds up these calculations by one to three orders of magnitude. Such acceleration, in turn, significantly advances the entire reactive transport simulation. The performed numerical experiments are enabled by the novel coupling of two open-source software packages: Reaktoro (Leal 2015) and Firedrake (Rathgeber et al. ACM Trans. Math. Softw.43(3), 2016). The first library provides the most recent version of the ODML approach for the chemical equilibrium calculations, whereas, the second framework includes the newly implemented conservative Discontinuous Galerkin finite element scheme for the Darcy problem, i.e., the Stabilized Dual Hybrid Mixed(SDHM) method Núñez et al. (Int. J. Model. Simul. Petroleum Industry, 6, 2012).

The paper is concerned with guaranteed and fully computable a posteriori error estimates for evolutionary problems associated with the poroelastic media governed by the quasi-static linear Biot equations [17]. It addresses the question of approximation error control, which arises in the iterative and monolithic approaches used for semi-discrete approximations obtained by the implicit Euler time-discretization scheme. The derivation of the error bounds is based on a combination of the Ostrowski-type estimates [58] derived for iterative schemes and a posteriori error estimates of the functional type for elliptic problems originally (also called error majorants and minorants) introduced in [65,66]. The validity of the first estimates is based on the contraction property of the fixed stress splitting scheme [51,49] used for decoupling. The error bounds are applicable for any approximation from the admissible functional space and independent of the discretisation method used. They are fully computable and do not contain mesh dependent constants. Functional estimates provide the reliable global estimates of the error measured in the terms of the energy norm and suggest efficient error indicators for the distribution of local errors that is advantageous for automated mesh adaptation algorithms.

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