Seismic monitoring of the accumulation and burial of greenhouse gases in a reservoir is attracting increasing attention due to its relevance to mitigating the effects of climate change. Changes in properties in the reservoir during gas injection demonstrate how seismic data changes. Among the seismic surveys, specialists are modeling the propagation of seismic wave fields for seismic monitoring. However, such methods are too resource-intensive, especially when solving for a large number of known velocity models and the number of sources in the acquisition system. Therefore, new effective algorithms for tracking changes in seismic data are needed. The work uses a neural network, which is used in two stages: to refine and obtain seismograms at the next step of seismic monitoring using a coarse computational grid and to suppress numerical dispersion. The developed algorithm makes it possible to accelerate classical seismic monitoring up to 3 times, taking into account the physics of multiphase flows.

We present an original algorithm to simulate seismic wave propagation in anisotropic elastic media. The algorithm is essentially oriented towards solving real-life applications. It is designed to simulate multi-shot multi-receiver seismic data for anisotropic seismic models. Implementation of the algorithm is based on three parallelization levels: direct parallelism to the right-hand sides, domain decomposition-based parallelism governed under MPI functionality, and stencil computations which are easy to perform with multi-core GPU architecture. We illustrate its applicability and high performance in solving real-life problems by simulating two full-coverage datasets consisting of 6600 sources each.

The paper presents a parallel algorithm for numerical simulation of the two-phase flow in the digital rock samples. The algorithm is based on a finite-difference approach, including the solution of the Poisson equation with spectral preconditioning. The phase-field method is used as the interface-capturing method. Since the algorithm is oriented on multi-GPU systems, the domain decomposition approach is implemented. To simplify parallelization, we investigate the CUDA unified memory model, which allows the transfer of halo data from different domains between GPUs automatically. The implementation performance with unified memory is examined in the case of the Cahn-Hilliard equation, which is the basis of the phase-field method, and compared with another implementation based on MPI data transfers. Using the developed algorithm, we perform drainage simulation, which is the underlying process in many lab measurements, such as an unsteady-state method for determining relative permeability curves and the centrifuge method for capillary pressure estimation.

The present study investigates the influence of wettability on the hysteresis of electrical parameters through numerical simulation of drainage and imbibition processes. Using a three-dimensional digital sandstone sample and numerical algorithms for simulating two-phase filtration and electric current flow, a series of numerical experiments are conducted over a full range of contact angles. The dependence of the resistivity index on the water saturation is then established for each value of angle, and the parameters of Archie’s law – namely, the saturation exponent and the saturation distribution factor – are estimated. The principle objective is to derive functional relationships between these parameters and the contact angle. The resulting dependencies are constructed as piecewise linear functions, exhibiting different trends for drainage and imbibition. Validation at intermediate contact angles revealed that the relative errors of the estimates remain below 3

One of the important stages in training neural networks is the construction of a training dataset, especially when it is necessary to minimise its size in order to achieve efficiency. Using the example of a neural network that approximates the fine-grid solution from a coarse-grid solution of a seismic modelling problem, we use a clustering algorithm to form a representative training dataset. In this case, the metric used is the distance between sources. This achieves a solution accuracy that is 10

We present numerical algorithm to estimate the formation factor of porous materials using the micro-tomographic images. The key part of the algorithm is the numerical solution of the 3D Poisson equation with rapidly varying high-contrast coefficients. The suggested algorithm is based on the preconditioned Conjugate Gradient method. The preconditioner is constructed as the inverse Laplace operator corresponding to a homogeneous model. It can be inverted using the spectral decomposition of two tridiagonal matrices corresponding to the approximation of 1D derivatives. The resulting series of 1D problems is solved by Thomas algorithm. We prove analytically and illustrate numerically that the condition number, and thus the convergence rate of the preconditioned problem depends on the contrast of the equation coefficients, but it is independent on the problems size. We illustrate that the preconditioner can be efficiently applied to the original problem with rapidly varying high-contrast coefficients and to the statement where the solution is computed only in the pore space. The algorithm is implemented using Graphic Processor Units. The use of modern GPUs allows us to solve problems of up to size 109 with a single unit.

We present an original preconditioner to solve the discretized Poisson equation with the conjugate gradient method. The preconditioner is based on the approximation of the inverse Poisson operator. In particular, we construct the inverse operator corresponding to a layered model which is close to the original one. The action of the preconditioner at each iteration requires the system of equations to be solved corresponding to the layered media. It is done by applying the spectral decomposition of the matrix corresponding to the 1D problem (spectral decomposition along one spatial direction) and further direct solution of a series of 1D problems along the other direction. We provide the numerical and theoretical study of the convergence rate depending on the way the layered model is constructed to approximate the original. We consider four cases: maximal value over the layer, minimal value, arithmetic averaging, and harmonic averaging. For the first two cases, we prove analytically that the convergence rate of the preconditioned conjugate gradient method is independent of the size of the problem but depends on the coefficient contrast. For the other two cases, the weak dependence on the size of the problem is illustrated numerically.

We present an analysis of the applicability of previously reported approach (NDM-net) to the construction of depth seismic images. The NDM-net allows us to simulate seismic data by combining finite differences with deep learning method. Coarse computational grids are used for the fast generation of inaccurate datasets. After that, the training dataset is simulated for a small number of shot points on fine grids. Then, the NDM-net is trained to map inaccurate data to accurate common shot gathers. Previously, we illustrated the high quality of the generated shot gathers. In this research, we applied the prestack depth migration to estimate the effect of the NDM-net numerical errors on the resulting seismic images.

Accurate monitoring of CO2 migration in subsurface reservoirs is critical for understanding the behavior of injected greenhouse gases. This study proposes a neural network-based approach to improve the accuracy of seismograms used in time-lapse seismic monitoring. The method consists of two stages: first, a neural network predicts changes in seismograms corresponding to velocity model variations between consecutive monitoring steps, allowing for the approximation of spatio-temporal dependencies and facilitating wavefield extrapolation. The seismograms at this stage are generated using a coarse computational grid to reduce computational costs. In the second stage, a neural network is employed to mitigate numerical dispersion in the predicted seismogram differences generated via classical modeling under the assumption of an unchanged velocity model. The trained network is then applied to all seismograms obtained in the first stage. This approach enables a more precise estimation of CO2 migration patterns, providing valuable insights into subsurface dynamics. The proposed approach significantly accelerates seismic modeling and its application to monitoring greenhouse gases in reservoir rocks.

In this paper, we present an attempt to construct a preconditioner based on the machine learning to solve Poisson equation. We use the Conjugate Gradient method. To precondition the algorithm we suggest approximating the inverse Laplace operator with using the U-Net. We consider the supervised learning where the vector of unknowns and right-hand sides are known; thus, we use the relative L-2 error as the loss function of the network training. We illustrate that U-Net with five convolutional layers provide insufficient accuracy of inverse Laplace operator approximation, so that the constructed conjugate gradient method stabilizes and possesses irreducible residual.

In this paper we consider the problem of numerical solution of the boundary value problem of the theory of elasticity in static formulation in a rectangle with arbitrary boundary conditions. For this purpose, we use the approach of splitting in the direction of the Laplace operator based on its spectral decomposition, which is similar to the discrete Fourier transform but does not require periodicity of the boundary conditions. A fast matrix-to-vector multiplication algorithm is proposed using an efficiently software-implemented matrix multiplication algorithm. Numerical experiments are performed to show the effectiveness of the proposed method with the ability to solve the elasticity problem on a mesh of 10(9) nodes on systems with 128G RAM.

The neural network was designed to mitigate numerical dispersion in three-dimensional seismic data. The algorithm operates in two stages. In the first stage, seismic modeling is performed using a coarse computational grid to generate seismograms, followed by the recording of a subset of seismograms with greater accuracy using a fine computational grid. This subset serves as the training data for the neural network. The second stage involves training the neural network using the acquired training dataset. Once trained, the network is applied to rough seismograms to achieve enhanced results. A series of numerical experiments were carried out to assess the efficacy of the proposed methodology.

The paper presents the Numerical Dispersion Mitigation neural network (NDM-net) to speed up seismic modeling. The idea of the NDM-net is to simulate the common-shot gathers for the entire set of source positions using a coarse grid. This solution can be computed fast but inaccurately. In addition, a small number of seismograms are generated using a fine enough grid to get an accurate solution. After that, the NDM-net is trained to map numerically polluted solutions to the accurate one and applied to correct the entire dataset. Previously, it was shown that NDM-net allows to speed up seismic modeling up to six times without noticeable loss of accuracy if the velocity model is fixed. In this paper, we focus on the applicability of NDM-net to the case where both the velocity model discretization and computational grid are corrected. We apply the NDM-net to suppress two types of numerical error: the numerical dispersion and the interface error.

In this paper, we present an original preconditioner to solve Poisson equation for strongly heterogeneous media. We suggest using the Conjugate Gradient method with the preconditioner based on the solution of the Poisson equation for homogeneous media. Corresponding operator is easy to invert by spectral method, where spectral decomposition is applied in two spatial directions and the Gauss elimination method is applied to solve a series of 1D problems. We illustrate that use of such precondtioner strongly decreases the number of iterations to solve the original Poisson equation, moreover the number of iterations weakly depends on the problem size. Implementation of the suggested approach using modern GPUs allows solving problems of up to the size of 1000(3) voxels.

The present study focuses on the numerical evaluation of Archie’s coefficients – specifically, the cementation parameter and saturation exponent – using a digital model of a high-permeability rock sample. Accurate estimation of these parameters is critical for reservoir development planning and optimizing hydrocarbon recovery through formation stimulation techniques. To derive the coefficients, we construct two relationships: the formation factor versus porosity and the resistivity index versus water saturation. The paper presents a methodology and case studies for establishing these dependencies. The proposed approach involves solving two optimization problems to perform a linear regression in a logarithmic scale, based on both the generalized and classical forms of Archie’s law.

In this paper, we present a numerical study of the effect of the bulk sorbent particle packing on the break-through time. In our experiments, we fix the physical and chemical properties of the sorbent particles but vary the packing density (macroporosity) and the CO2 volume fraction at the inlet. After that, the gas flow through the reactor is simulated, and CO2 volume fraction as a function of time at the outlet is measured. We show that, in the case of high macroporosity, the recorded CO2 concentration strongly depends on the macroporosity. On the contrary, if the macroporosity is low, we observe almost no difference in the break-through curves for the models with macroporosity less than 45%.

The paper presents an algorithm for three-dimensional modelling of two-phase flows on the scale of pore size order for numerical evaluation of relative phase permeability curves of porous materials. Such an evaluation is performed based on the results of numerical simulation of primary drainage with subsequent waterflooding. In this case, models of porous materials based on three-dimensional tomographic images of rocks are used. The simulation of the flow considers the Stokes equation and the Cahn-Hilliard equation for modelling phase transfer, which allows us to determine phases using the concentration function. The combination of the phase field method and finite difference method makes it possible to correctly take into account the contact angle and stably calculate surface tension forces in domains with complex topology.

The paper discusses the extension of NDM-net (Numerical Dispersion Mitigation neural network) to pseudo-3D cases and the construction of training datasets. NDM-net is initially designed to reduce numerical dispersion in seismic data, which are generated as a result of simulating elastic waves. Previously, seismograms on a coarse grid with numerical dispersion and a certain number of seismograms on a fine grid are calculated to form a training sample. The paper discusses three approaches for building a representative sample to speed up the learning process in case of pseudo-3D. Additionally, the paper discusses the combination of metrics based on statistical analysis.