Interwell network models have been proposed by many authors as a good physics-based alternative to machine-learning methods for building data-driven flow models in subsurface applications. Herein, we suggest an alternative approach, in which a conventional simulator, formulated on top of a very coarse volumetric 3D grid, is used as a data-driven proxy model. What distinguishes this conceptually from standard history matching is that the standard tunable parameters in the simulator (pore volumes, transmissibilities, well-connection factors, initial saturations, relative permeability exponents and scalings, etc.) are calibrated freely without regard to the physical interpretation of the resulting parameter values.

In its most basic form, our CGNet models are formulated as a minimal Cartesian grid that covers the assumed map outline and base and top surface of the reservoir. The parameters of the resulting 3D simulation model are then calibrated to match observed well responses. Using simulation cases built on top of public data sets (the Egg model, the Norne field model, etc.), we show that surprisingly accurate proxy models can be developed using grids with a few tens or hundreds of cells, depending upon the geological complexity of the model. For the Norne case, we show that it is important that the proxy model has several vertical layers because of the poor vertical connection inside the true reservoir volume. We also show that starting with a good ballpark estimate of the reservoir volume is a precursor to a good calibration.

The resulting CGNet models fit immediately in any standard simulator and are very fast to evaluate because of the low cell count. Compared with an interwell network model like GPSNet (Ren et al., 10.2118/193855-MS), a typical CGNet model has fewer computational cells but a richer connection graph and more tunable parameters. In our experience, CGNet models calibrate better and are simpler to set up to reflect known (or pre-modelled) fluid contacts or geobodies.

Altering the state of the stress of the subsurface reservoirs can lead fractures to slip and extend their lengths (i.e., to propagate). This process can even be engineered, in many applications, e.g., enhanced geothermal systems. As such, accurate and efficient simulation of the mechanical deformation of the subsurface geological reservoirs, allowing for fracture propagation, is at the core of many geoscientific operational designs .
Subsurface reservoirs entail many fractures at multiple scales. Implementation of 3D complex grids on these complex fractured systems, for mechanical deformation analyses, is extremely challenging. An alternative approach can be developed by using extended finite element methods (XFEM). XFEM allows for capturing the fractures effects on a conveniently-generated structured matrix mesh. The cracks are introduced by extra degrees of freedom (DOFs) on the nodes of the matrix rock mesh. For geoscientific applications, however, XFEM results in too many DOFs which are beyond the scope of simulators. Additionally, for propagating fractures, these DOFs need to be updated in response to the dynamic extension of the fractures in the domain. The propagation process not only adds to the sensitivity of the outputs to the accuracy of the estimated stress field, but also increases the size of the linear systems. In addition to these, matrix rocks are often highly heterogeneous, at high resolutions.
In this work, we present a novel multiscale procedure for propagating fractures in heterogeneous geological reservoirs. For the first time in the community, we present the highly fractured systems at coarser resolutions via XFEM-based basis functions, which also account for the propagating effects. Fractures are allowed to extend their scale and the enriched basis functions are locally updated. Using these bases, the coarse scale system is obtained in which no extra DOFs due to fractures exist. This significantly reduces the computational complexity.
As a significant step forward compared with our recently-published journal paper [Xu, Hajibeygi, Sluys, Journal of Computational Physics, 2021], in this conference contribution we allow the fractures to propagate. Specially, we introduce a local-global-based approach, in which fracture propagation is treated only at local stage; while the stress and deformation are modelled at global scale. In the search of convenient implementation, the procedure is presented algebraically.
Through several test cases, we demonstrate the applicability of the method for complex fractured media. Specially we demonstrate that propagation can be modeled at local scale, while accurate stress and deformation fields are obtained at global scale.

Control volume finite element (CVFE) methods are commonly used for modeling flow and transport in geometrically complex porous media domains with unstructured meshes. The CVFE approach is based on the finite element method to approximate the pressure and velocity fields, and uses the finite volume method to model saturation ensuring mass conservation. Control volumes are constructed by spanning element boundaries, leading to an artificial smearing of the numerical solution in the presence of sharp material interfaces. Recently, a CVFE method based on discontinuous pressure was introduced that enabled the construction of discontinuous control volumes, thus preventing control volumes from spanning element boundaries. This modification of the method provides accurate solutions but is computationally very expensive due to the discontinuous approximation which incorporates additional degrees of freedom per element.

In this work, we propose using hybrid finite element pressure approximations to capture flow and transport in highly heterogeneous porous media. The CVFE element pair $P_{0,DG}-P_{1,H}$ denotes a constant, element-wise, discontinuous Galerkin velocity vector approximation and a hybrid (continuous/discontinuous) Galerkin first-order pressure scalar approximation of the flow model. The method exploits the efficient continuous CVFE method in most of the model domain while the discontinuous CVFE approach is applied exclusively along material discontinuities. We demonstrate that this hybrid scheme outperforms the classical CVFE continuous approach as well as the discontinuous Galerkin modification by incorporating the best of both approaches. The presented hybrid approach computational requirements are comparable to the continuous approach while the accuracy of the transport solution corresponds to that of the discontinuous pressure method. We validate the presented hybrid approach and discuss the convergence of the method. The effectiveness of the new scheme is demonstrated with several numerical experiments for highly heterogeneous subdomains.