Calculating the effective permeability entails considerable computation, even with local upscaling techniques. This study proposes a convolutional neural network architecture that estimates effective permeability. The network's input is the permeability maps of those layers of the fine-scale model that are intended to be upscaled into a single layer with horizontally coarsened cells. It treats each layer of the fine-scale permeability maps as a channel of a high-resolution image. The output is a 3-channel lower-resolution image where each channel presents the upscaled permeability map in one major direction. The proposed architecture is simple and robust. It consists of two 2D convolutional hidden layers with small kernels and a 2D MaxPooling layer.
The network was tested using two different datasets, (1)- a binary-facie fluvial model and (2)- a continuous Gaussian model. For each dataset, 500 geological realisations were created and upscaled using a pressure-solver with periodic boundary conditions. 50% of the realisations were used for training and the remaining for testing. The network captured the nonlinear behaviour very promisingly with no overfitting signs. The results were not only visually acceptable, but also the mean of the L2 norms was negligible (below 0.05 mD in the first dataset and below 0.01 mD in the second dataset). Two-phase simulation results also verified the accuracy of the estimated permeabilities. The proposed method can reduce the upscaling computation significantly. The training time was considerably less than the computation time needed for upscaling using the pressure-solver method. The proposed method can be a step towards more computationally efficient extended-local and quasi-global permeability upscaling methods.

Capillary imbibition is a major process that controls many transport phenomena in porous media for many applications. In the counter-current case, the saturation S of the wetting fluid that imbibes the porous medium may be represented as the solution of a strongly non-linear diffusion equation ∂S(x,t)/∂t=∇.[D(S(x,t))∇S(x,t)] which involves a saturation-dependent diffusion function D(S). This function D(S) depends non-linearly on S through an expression involving relative permeabilities and capillary pressure. Also, it vanishes as a power law near the extreme saturations S = 0 and 1, leading to a singular boundary problem that was investigated by many authors. Considering a finite block, two regimes can be observed: an early-time regime involving the Boltzmann variable x/sqrt(t), and an asymptotic late-time regime that remains to be elucidated.
In this study, after analytically deriving a new asymptotic late-time regime solution, we compare it with numerical simulations while recommending good practices related to the numerical simulation of such a process. We then show the corresponding flux at the boundary of the block exhibits two dynamic regimes using analytical solutions of two extreme saturation cases. At short times, a self-similar behavior is recovered and leads to a matrix to fracture flux varying as t^(-1⁄2). The asymptotic solution of the late time regime proves that the flux is proportional to a power of time that is a function of the diffusion function exponent. The numerical calculations for the flux of the full problem (non-linear D(S)) accord with the analytical predictions. Also, the flux calculation may be represented as a non-linear exchange term involving the average saturation on the block, weighted by a shape factor. To finish with, we set up a spatially averaged macroscopic dual-porosity representation of the imbibition process and demonstrate its accuracy in reproducing the two dynamical imbibition regimes.

An increasing number of geo-energy applications require the quantitative prediction of hydromechanical response in subsurface. Integration of mass, momentum, and energy conservation laws becomes essential for performance and risk analysis of enhanced geothermal systems, stability assessment of CO2 sequestration and hydrogen storage, resolving the issue of induced seismicity. The latter problem is of particular interest because it exposes safety risks to people and surface infrastructure.

Implicit coupling of conservation laws is computationally demanding and the solution procedure often uses different numerical methods for different laws that complicates simulation. Recently developed Finite Volume (FV) schemes for poromechanics present a unified approach for the modeling of conservation laws in geo-energy applications. Contact mechanics at faults requires special attention due to the inequality constraints it imposes and nonlinear friction laws that strongly affect the occurrence of seismicity.

We develop a cell-centered FV scheme for the purpose of integrated simulation in Delft Advanced Research Terra Simulator (DARTS) platform. The scheme proposes a unified numerical framework capable to resolve conservation laws in a fully implicit manner using a single collocated grid. Coupled multi-point flux and multi-point stress approximations provide mass, momentum, and heat fluxes at the faces of the computational grid. We use a conformal discrete fracture model to incorporate faults, where the multi-point approximations of fluxes respect the discontinuity in displacements. The block-partitioned preconditioner that takes the advantage of linear structure of the coupled problem is developed to facilitate the performance of the simulation.

The proposed numerical scheme are validated against analytical and numerical solutions in a number of test cases. The convergence and stability of the schemes are investigated. It is found that the developed scheme is indeed accurate, stable, and efficient. Thereafter, we demonstrate the applicability of the approach to model fault reactivation at the laboratory scale. In a core injection test, we validate the results of simulation against experimental measurements. Next, we investigate the performance of the different preconditioning strategies. The proposed block-partitioned preconditioning strategy demonstrates the scalability and efficiency of the numerical framework.