Dynamic mesh optimisation for efficient numerical simulation of density-driven flows: Application to the 2- and 3-D Elder problem

Accurate modelling of the fate of injected CO 2 is necessary if geological storage is to be used at a large scale. In one form of geological storage, CO2 is injected into an aquifer that has a sealing caprock, forming a CO2 cap beneath the caprock. The diffusion of CO2 into underlying formation waters increases the density of water near the top of the aquifer, bringing the system to a hydro-dynamically unstable state. Instabilities can arise from the combination of an unstable density profile and inherent perturbations within the system, e.g., formation heterogeneity. If created, this instability causes convective mixing and greatly accelerates the dissolution of CO2 into the aquifer. Accurate estimation of the rate of dissolution is important for risk assessments because the timescale for dissolution is the timescale over which the CO2 has a chance to leak through the caprock or any imperfectly sealed wells. A new 2D numerical model which has been developed to study the diffusive and convective mixing in geological storage of CO2 is described. Effects of different formation parameters are investigated in this paper. Results reveal that there are two different timescales involved. The first timescale is the time to onset the instability and the second one is the time to achieve ultimate dissolution. Depending on system Rayleigh number and the formation heterogeneity, convective mixing can greatly accelerate the dissolution of CO2 in an aquifer. Two field scale problems were studied. In the first, based on the Nisku aquifer, more than 60% of the ultimate dissolution was achieved after 800 years, while the computed timescale for dissolution in the same aquifer in the absence of convection was orders of magnitude larger. In the case of the Glauconitic sandstone aquifer, there was no convective instability. Results suggest that the presence and strength of convective instability should play an important role in choosing aquifers for CO2 storage. risk of leakage of CO2 from a storage formation may need to analyze leakage mechanisms and their likelihood of occurrence during the full-time period over which mobile free-phase CO 2 is expected to remain in the reservoir. Once dissolved, risk assessments may well ignore the leakage pathways resulting from the very slow movement of CO2-saturated brines. An accurate assessment of the timescales for dissolution are therefore of the first order of importance. The CO2 injected into a saline reservoir is typically 40 – 60% less dense than the resident brines (4) . Driven by density contrasts, CO 2 will flow horizontally (in a horizontal aquifer) spreading under the caprock, and flow upwards, potentially leaking through any high permeability zones or artificial penetrations, such as abandoned wells. The free-phase CO2 (usually supercritical fluid) slowly dissolves in the brines. The resulting CO2-rich brines are slightly denser than undersaturated brines, making them negatively buoyant, and thus greatly reducing or eliminating the possibility of leakage. The rate of dissolution depends on the rate at which diffusion or convection brings undersaturated brine in contact with CO2. Convective mixing enhances the dissolution rate as compared to diffusion by distributing the CO2 into the aquifer (5) . Therefore, the role of convective mixing in CO2 sequestration and the timescales involved in the process are important. The dissolution time of the injected CO2 into brine is important because during this time the injected CO2 has a chance to leak into the atmosphere through the caprock and wellbores. Accurate modelling of the convective mixing in heterogeneous porous media plays a central role in predicting the fate of CO2 injected into aquifers. In this paper, geological CO2 storage is modelled by solving the convection-diffusion equation while considering the CO2-brine interface as a boundary condition. Geochemical reactions that can reduce the timescale of sequestration of CO2 are not included, since they generally occur on longer timescales (6) . The paper is organized as follows. First, the mathematical model for simulating density-driven flow through porous media is briefly presented. The model is validated with a benchmark problem for density-driven flow in porous media. Then, the geological CO2 sequestrations both in small and field scale are simulated using the model. Two important timescales, the effect of formation properties, as well as sensitivity to temporal and spatial discretisations, are discussed. Finally, the results are summarized and their relevance to geological storage of CO 2 in aquifers is discussed.

A three-dimensional non-hydrostatic coupled model for free surface – Subsurface variable – Density flows

Modeling density-driven flow in porous media by physics-informed neural networks for CO2 sequestration

The multiscale finite-volume (MSFV) method has been developed to solve multiphase flow problems on large and highly heterogeneous domains efficiently. It employs an auxiliary coarse grid, together with its dual, to define and solve a coarse-scale pressure problem. A set of basis functions, which are local solutions on dual cells, is used to interpolate the coarse-grid pressure and obtain an approximate fine-scale pressure distribution. However, if flow takes place in presence of gravity (or capillarity), the basis functions are not good interpolators. To treat this case correctly, a correction function is added to the basis function interpolated pressure. This function, which is similar to a supplementary basis function independent of the coarse-scale pressure, allows for a very accurate fine-scale approximation. In the coarse-scale pressure equation, it appears as an additional source term and can be regarded as a local correction to the coarse-scale operator: It modifies the fluxes across the coarse-cell interfaces defined by the basis functions. Given the closure assumption that localizes the pressure problem in a dual cell, the derivation of the local problem that defines the correction function is exact, and no additional hypothesis is needed. Therefore, as in the original MSFV method, the only closure approximation is the localization assumption. The numerical experiments performed for density-driven flow problems (counter-current flow and lock exchange) demonstrate excellent agreement between the MSFV solutions and the corresponding fine-scale reference solutions.

A 6M digital twin for modeling and simulation in subsurface reservoirs

Multiphase flow in porous media is involved in various natural and industrial applications, including water infiltration into soils, carbon geosequestration, and underground hydrogen storage. Understanding the invasion morphology at the pore scale is critical for better prediction of flow properties at the continuum scale in partially saturated permeable media. The deep learning method, as a promising technique to estimate the flow transport processes in porous media, has gained significant attention. However, existing works have mainly focused on single-phase flow, whereas the capability of data-driven techniques has yet to be applied to the pore-scale modeling of fluid–fluid displacement in porous media. Here, the conditional generative adversarial network is applied for pore-scale modeling of multiphase flow in two-dimensional porous media. The network is trained based on a data set of porous media generated using a particle-deposition method, with the corresponding invasion morphologies after the displacement processes calculated using a recently developed interface tracking algorithm. The results demonstrate the capability of data-driven techniques in predicting both fluid saturation and spatial distribution. It is also shown that the method can be generalized to estimate fluid distribution under different wetting conditions and particle shapes. This work represents the first effort at the application of the deep learning method for pore-scale modeling of immiscible fluid displacement and highlights the strength of data-driven techniques for surrogate modeling of multiphase flow in porous media.

A sequential fully implicit multi-scale meshless multi-point flux method (MS-MMPFA) for nonlinear hyperbolic partial differential equations of fluid flow in heterogeneous porous media is described in this paper. The method extends the recently proposed the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media [Lukyanov, “Meshless Upscaling Method and its Application to a Fluid Flow in Porous Media”, Proceeding ECMOR XII, 2010] by utilizing advantages of the existing multi-scale finite volume (MSFV) schemes. The MMPFA is based on a gradient approximation commonly used in meshless method and combined with the mixed corrections which ensure linear completeness. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The MMPFA method was successfully tested for a number of problems where it was clearly shown that the MMPFA gives a good agreement with analytical solutions for a given number of particles. However, the level of detail and range of property variability included in reservoir characterization models leads to a large number of particles to be considered in MMPFA method. In this paper this problem is resolved using a sequential fully implicit MS-MMPFA method. The results are presented, discussed.

Flow prediction of heterogeneous nanoporous media based on physical information neural network

A comprehensive modelling investigation of dynamic capillary effect during non-equilibrium flow in tight porous media

Modeling non-Darcian flow and solute transport in porous media with the Caputo–Fabrizio derivative

Simulation of flow in granular porous media using combined Lattice Boltzmann Method and Smoothed Profile Method

Gas/Water Flow in Porous Media in the Presence of Adsorbed Polymer: Experimental Study on Non-Darcy Effects

SummaryNumerical modeling of large deformations in fluid‐infiltrated porous media must accurately describe not only geometrically nonlinear kinematics but also fluid flow in heterogeneously deforming pore structure. Accurate simulation of fluid flow in heterogeneous porous media often requires a numerical method that features the local (elementwise) conservation property. Here, we introduce a new finite element framework for locally mass conservative solution of coupled poromechanical problems at large strains. At the core of our approach is the enriched Galerkin discretization of the fluid mass balance equation, whereby elementwise constant functions are augmented to the standard continuous Galerkin discretization. The resulting numerical method provides local mass conservation by construction with a usually affordable cost added to the continuous Galerkin counterpart. Two equivalent formulations are developed using total Lagrangian and updated Lagrangian approaches. The local mass conservation property of the proposed method is verified through numerical examples involving saturated and unsaturated flow in porous media at finite strains. The numerical examples also demonstrate that local mass conservation can be a critical element of accurate simulation of both fluid flow and large deformation in porous media.

1.1 Motivation This chapter focusses on the description and modeling of mass transfer processes occurring between two fluid phases in a porous medium. The principle underlying physical process comprises a transport of particles from one phase to the other phase. This process takes place across fluid–fluid interfaces (see Fig. 1) and may constitute evaporation, dissolution, or condensation, for example. Such mass transfer processes are crucial in many applications involving flow and transport in porous media. Major examples are found in soil science (where the evaporation from soils is of interest), soil and groundwater remediation (like thermally-enhanced soil vapor extraction where dissolution, evaporation, and condensation play a role), storage of carbon dioxide in the subsurface (where the dissolution of carbon dioxide in the surrounding groundwater is a crucial storage mechanism), CO2-enhanced oil recovery (where after primary and secondary recovery, carbon dioxide is injected into the reservoir in order to mobilize an additional 8-20 per cent of oil), and various industrial porous systems (such as certain types of fuel cells). Let us have a closer look at a few of these applications and identify where interphase mass transfer is relevant. Four specific examples are shown in Fig. 2 and briefly described.

Density-driven flow and heat transport processes in 2-D porous media scenarios are governed by coupled, non-linear, partial differential equations that normally have to be solved numerically. In the present work, a model based on the network method simulation is designed and applied to simulate these processes, providing steady state patterns that demonstrate its computational power and reliability. The design is relatively simple and needs very few rules. Two applications in which heat is transported by natural convection in confined and saturated media are studied: slender boxes heated from below (a kind of Bénard problem) and partially heated horizontal plates in rectangular domains (the Elder problem). The streamfunction and temperature patterns show that the results are coherent with those of other authors: steady state patterns and heat transfer depend both on the Rayleigh number and on the characteristic Darcy velocity derived from the values of the hydrological, thermal and geometrical parameters of the problems.