Uncertainty quantification in Discrete Fracture Network models: stochastic geometry

We consider the problem of uncertainty quantification analysis of the output of underground flow simulations. We consider in particular fractured media described via the discrete fracture network model; within this framework, we address the relevant case of networks in which the geometry of the fractures is described by stochastic parameters. In this context, due to a possible lack of smoothness in the quantity of interest with respect to the stochastic parameters, well assessed techniques such as stochastic collocation may fail in providing reliable estimates of first-order moments of the quantity of interest. In this paper, we overcome this issue by applying the Multilevel Monte Carlo method, using as underlying solver an extremely robust method

Towards effective flow simulations in realistic Discrete Fracture Networks

We focus on the simulation of underground flow in fractured media, modeled by means of Discrete Fracture Networks. Focusing on a new recent numerical approach proposed by the authors for tackling the problem avoiding mesh generation problems, we further improve the new family of methods making a step further towards effective simulations of large, multi-scale, heterogeneous networks. Namely, we tackle the imposition of Dirichlet boundary conditions in weak form, in such a way that geometrical complexity of the DFN is not an issue; we effectively solve DFN problems with fracture transmissivities spanning many orders of magnitude and approaching zero; furthermore, we address several numerical issues for improving the numerical solution also in quite challenging networks

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

In this paper we propose a modified construction for the polynomial basis on polygons used in the Virtual Element Method (VEM). This construction is alternative to the usual monomial basis used in the classical construction of the VEM and is designed in order to improve numerical stability. For badly shaped elements the construction of the projection matrices required for assembling the local coefficients of the linear system within the VEM discretization of Partial Differential Equations can result very ill conditioned. The proposed approach can be easily implemented within an existing VEM code in order to reduce the possible ill conditioning of the elemental projection matrices. Numerical results applied to an hydro-geological flow simulation that often produces very badly shaped elements show a clear improvement of the quality of the numerical solution, confirming the viability of the approach. The method can be conveniently combined with a classical implementation of the VEM and applied element-wise, thus requiring a rather moderate additional numerical cost

Hybrid numerical methods for multiphysics simulation of flow in coal

Coalbeds are important resources for natural gas, referred to as coalbed methane. They can also be used for storage of carbon dioxide or hydrogen. Pore/cleat-scale modelling is a valuable tool for capturing flow physics in these multi-scale fractured media. Direct numerical simulations (DNS) and pore-network modelling (PNM) have been used for predicting multi-scale multi-physics flow on images of the porous media obtained from micro-computed tomography (micro-CT) scanning. Both approaches have advantages and disadvantages, while the development of hybrid methods can produce more efficient simulation tools. A fracture pore-network (PN) model is coupled with a diffusion flow solver to capture the transient gas flow physics in coal. The Finite Volume Method (FVM) is used for the discretisation of Fick’s second law and results are used to update fluxes in the coal matrix and the adjacent fractures. The Langmuir equation is solved to account for sorption and to correlate between the gas pressure and concentration when coupling PN convection and FVM diffusion models. To simulate two-phase flow in coal fractures, a hybrid model using PNM and the volume of fluid (VOF) advection scheme is proposed. The model is validated against the conventional VOF realisation by conducting a comprehensive sensitivity analysis. The developed VOF-PNM model allows for dynamic tracking of fluid interfaces and for predicting relative permeability and capillary pressure curves. The developed solver can be one to two orders of magnitude faster than the conventional VOF implementation while its accuracy is within 5%. A two-phase Darcy-Brinkman-Stokes (DBS) framework is employed for multi-scale multiphysics fluid flow in coal. Diffusion, sorption, gas and rock compressibility are embedded into the numerical scheme. Variation of permeability due to coal deformation is accounted for via the Palmer-Mansoori analytical model, while surface diffusion is introduced by Fick’s second law. The Langmuir equation is used to introduce sorption via its implicit coupling with pressure equations within a DBS framework. A hybrid of VOF, PN, and the continuum Darcy model is applied for computationally efficient multi-scale multiphysics flow simulations in coal. The model applies the Hagen–Poiseuille analytical solution and VOF advection scheme for two-phase flow in fractures coupled with the continuum Darcy biphasic flow in the coal matrix. The model has similar functionality to the previously developed DBS framework while the computational cost is more than 90 times lower. Each stage of the conducted research introduces a novel hybrid approach for multi-scale multiphysics flow simulations in coal seams by an efficient numerical combination of PNM, DNS, and continuum models. The developed hybrid models capture several complex physical mechanisms occurring in coalbed methane reservoirs at different stages of gas production and storage and can be used as a predictive tool for optimisation of gas recovery and monitoring possible environmental impacts

Performance Analysis of Multi-Task Deep Learning Models for Flux Regression in Discrete Fracture Networks

In this work, we investigate the sensitivity of a family of multi-task Deep Neural Networks (DNN) trained to predict fluxes through given Discrete Fracture Networks (DFNs), stochastically varying the fracture transmissivities. In particular, detailed performance and reliability analyses of more than two hundred Neural Networks (NN) are performed, training the models on sets of an increasing number of numerical simulations made on several DFNs with two fixed geometries (158 fractures and 385 fractures) and different transmissibility configurations. A quantitative evaluation of the trained NN predictions is proposed, and rules fitting the observed behavior are provided to predict the number of training simulations that are required for a given accuracy with respect to the variability in the stochastic distribution of the fracture transmissivities. A rule for estimating the cardinality of the training dataset for different configurations is proposed. From the analysis performed, an interesting regularity of the NN behaviors is observed, despite the stochasticity that imbues the whole training process. The proposed approach can be relevant for the use of deep learning models as model reduction methods in the framework of uncertainty quantification analysis for fracture networks and can be extended to similar geological problems (for example, to the more complex discrete fracture matrix models). The results of this study have the potential to grant concrete advantages to real underground flow characterization problems, making computational costs less expensive through the use of NNs

Tracing back the source of contamination

From the time a contaminant is detected in an observation well, the question of where and when the contaminant was introduced in the aquifer needs an answer. Many techniques have been proposed to answer this question, but virtually all of them assume that the aquifer and its dynamics are perfectly known. This work discusses a new approach for the simultaneous identification of the contaminant source location and the spatial variability of hydraulic conductivity in an aquifer which has been validated on synthetic and laboratory experiments and which is in the process of being validated on a real aquifer

A hybrid mortar virtual element method for discrete fracture network simulations

The most challenging issue in performing underground flow simulations in Discrete Fracture Networks (DFN), is to effectively tackle the geometrical difficulties of the problem. In this work we put forward a new application of the Virtual Element Method combined with the Mortar method for domain decomposition: we exploit the flexibility of the VEM in handling polygonal meshes in order to easily construct meshes conforming to the traces on each fracture, and we resort to the mortar approach in order to ``weakly'' impose continuity of the solution on intersecting fractures. The resulting method replaces the need for matching grids between fractures, so that the meshing process can be performed independently for each fracture. Numerical results show optimal convergence and robustness in handling very complex geometries

Refinement strategies for polygonal meshes applied to adaptive VEM discretization

In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well and states new issues, here tackled, concerning good quality mesh elements and reliability of the simulations. In this paper we propose several new polygonal refinement strategies and numerically investigate the quality of the meshes generated by an adaptive mesh refinement process, as well as optimal rates of convergence with respect to the number of degrees of freedom. Among the several possible problems in which these strategies can be applied, here we have considered a geometrically complex geophysical problem as test problem that naturally yields to a polygonal mesh and tackled it by the Virtual Element Method. All the adaptive strategies here proposed, but the “Trace Direction strategy”, can be applied to any problem for which a polygonal element method can be useful and any numerical method based on polygonal elements and can generate good quality isotropic mesh elements