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.

In our work, we numerically investigated how the roughness of interfaces between different layers in 3D layered models affects the upscaled and downscaled elastic properties. We considered layered models with random rough interfaces and fixed elastic properties in the background and the middle layer, upscaled them to compute the effective stiffness tensor and then downscaled them to obtain new elastic parameters of the middle layer in the statistically equivalent models with smooth interfaces. Afterward, we investigated the dependence of the obtained results on four input varied parameters (the correlation length, standard deviation of the roughness and two physical elastic parameters of the middle layer). Next, we proposed an algorithm for recovering the covariance matrix of such stiffness tensors for arbitrary values from the four-parameter space, applying bilinear regression to the parameters of the interface and the linear component-by-component interpolation by two nearest points to the material parameters. Verification of this algorithm showed that for limited standard deviation values the errors in restoring the covariance matrix do not exceed 0.07. Thus, smooth interface models with random material parameters in the middle layer can be statistically simulated with arbitrary values of the input parameters with high accuracy.

The paper describes an algorithm for constructing relative phase permeability curves based on a numerical simulation of primary drainage and subsequent water re-saturation. These processes are simulated as two-phase fluid flows in three-dimensional models generated from CT-images. We use the Navier-Stokes equations to simulate the flow and the Cahn-Hilliard equation to model phase transport. In the applied phase field method, phases are specified by a concentration function. This function is smooth so that the interphase boundary is replaced by a fairly thin layer in which the values of the concentration function continuously change. This representation makes it possible to robustly calculate surface tension forces and take into account the contact angle when using the finite difference method for numerical modeling of fluid flows and phase transport.

В статье описан новый подход к численной оценке абсолютной проницаемости керна горных пород по их микротомографическим изображениям, позволяющий выполнять эту оценку на персональных компьютерах с использованием ограниченных вычислительных ресурсов. Подход обусловлен тем, что современные микротомографические изображения зачастую имеют размеры 1000 вокселей и выше при разрешении 1–3 мкм на воксель. В работе показано, что даже относительно небольшие подобласти изображений могут использоваться для оценки проницаемости с последующим построением устойчивых корреляционных зависимостей между открытой пористостью и абсолютной проницаемостью. В результате построенные зависимости могут быть как обобщены на весь образец при известной связной пористости (оцененной численно или лабораторно), так и экстраполированы на масштаб полноразмерного керна. The paper describes a new approach to the numerical estimation of the absolute permeability of rocks using their CT-scans. The algorithm is designed to be used on a limited computational resources, such as single GPU or a desktop. Modern microtomographic images often have sizes of 1000 voxels and a resolution of 1–3 µm per voxel. In this case, relatively small sub-regions can be used to estimate permeability, followed by the construction of stable correlations between open porosity and absolute permeability. As a result, the constructed dependences can be either generalized to the entire sample with a known connected porosity (estimated numerically or laboratory), or extrapolated to the scale of a full-sized core.

We present a resource-saving algorithm for numerical evaluation of the absolute permeability of a rock from the sample’s CT-images of huge size, which makes it possible to perform such an assessment using limited computing resources, in particular, personal computers. It is based on a decomposition of the 3D sample image into small representative sub-samples, for which the absolute permeability is estimated based on the numerical solution of the Stokes equation in a static formulation, followed by a creation of the functional dependencies between the permeability and open porosity. After that these dependencies can be extended to the entire original sample or to the full-sized core sample.

Численная оценка абсолютной проницаемости горной породы на основе корреляционной зависимости от геометрии микротомографического изображения1

В. В. Лисица 1 *, Т. С. Хачкова 2 , Е. А. Гондюл 2 , В. В. Крутько 3 , А. С. Авдонин 3 Моделирование флюидопотоков в образцах с микропористыми включениями 1 Институт математики СО РАН, г.Новосибирск, Российская Федерация 2 Институт нефтегазовой геологии и геофизики СО РАН, г.Новосибирск, Российская Федерация 3 Газпромнефть НТЦ, г.Санкт-Петербург, Российская Федерация * e-mail:

The paper presents an original numerical algorithm to simulate coupled two-phase fluid flow in domains containing open pores and microporous material. To simulate the coupled flow we use the time-dependent Navier-Stokes-Brinkman equation. The transport of the phases is governed by the Cahn-Hilliard equation in the open pores and by the Buckley-Leverett equation in the porous material. We suggested a unified finite-difference approximation of the two transport equations, that satisfy the natural conjugation conditions. However, Cahn-Hilliard requires an additional boundary condition, that must be satisfied at the interface to ensure the wetting angle.

In this paper, we study the effect of the interface roughness on the elastic parameters of layered media. We consider the three-dimensional models of a layered medium with two different elastic materials inside and outside the layer. We generate the first class of models, where the interfaces between the layers are rough, and the elastic parameters of the inner layers are fixed. Then, the numerical upscaling technique is applied to estimate the effective stiffness tensor. Next, we downscale the stiffness tensor to reconstruct the new elastic parameters of the inner layer for the model of second class with flat interfaces; that is the uncertainty of the model geometry is mapped to the uncertainty of the stiffness tensor component for a fixed geometry of the model. After that, we propose an algorithm for extending the results of restoring the elastic tensors for arbitrary parameters of uncertainty applying the bilinear regression with respect to interface rough parameters and bilinear interpolation using two nearest points with respect to the physical parameters of the inner layers. Verification of the algorithm shows that the errors in the recovering covariance matrix do not exceed 7%; that is, it can be used to statistically simulate models of the second class with a flat interface by arbitrary values of the interface roughness and the physical parameters of the layers in the first class of models.

The paper presents a numerical algorithm for conjugated reactive transport at two spatial scales, in application to $$C0_2$$ chemosorption. Transport at the macroporous scale (intergranular space) is supported by both the fluid flow and diffusion. At the microporous scale diffusion is assumed as the only transport mechanism. The mathematical model used in this research operates with two parameters that are unavailable from laboratory measurements, they are reaction rate and diffusion coefficient in the microporous space. We present a series of numerical experiments to calibrate these parameters to match the laboratory-measured rates of the concentration changes rates.

In the paper, the effect of interface roughness on the elastic parameters of a layered medium is studied. Three-dimensional models of a layered medium with different material inside and outside the layer are considered. Initially, the first class of models is statistically generated with rough interfaces between the layers and fixed elastic parameters of the inner layers. Then the effective stiffness tensor is estimated for the equivalent homogeneous model. After that, new elasticity parameters of the inner layer are reconstructed for second class of models with flat interfacies. Thus, the roughness of the model interfacies is mapped into a statistical distribution of the stiffness tensor components for a fixed model geometry. In addition, an algorithm for extending the results of reconstructing the elastic tensors for arbitrary interface roughness parameters using bilinear regression of covariance matrices is proposed. The algorithm verification shows that the error in restoring the covariance matrix does not exceed 7%; i.e., one can use it in statistical modeling the second class of models with flat interfacies for the arbitrary interface roughness values and given physical parameters of layers in the first class of models.

This paper presents a numerical algorithm to simulate reactive transport at the pore scale. The aim of the research is the direct study the changes in the pore space geometry. Thus, the fluids flow and transport of chemically active components are simulated in the pore space. After that the heterogeneous reactions are used to compute the fluid-solid interaction. Evolution of the interface is implemented by the level-set methods which allows handling the changes in the pore space topology. The algorithm is based on the finite-difference method and implemented on the GP-GPU.

In this article, we propose an original algorithm for numerical simulation of the conjugated reactive transport. The algorithm is essentially oriented toward the use of GPUs. We simulate the $$CO_2$$ chemosorption by soda-lime sorbent, assuming a multi-scale process. Macropore space supports both advective and diffusive transport of the reactive species, whereas impermeable microporous sorbent granules admit diffusion as the only transport mechanism. We simulated $$CO_2$$ pass-through an evenly distributed granules illustrating agreement with lab measurements. We also studied the $$CO_2$$ break-through rates in dependence of sorbent granules compaction in the reactor.

The paper presents a numerical algorithm for simulation of the reactive transport at the pore scale. The algorithm allows simulating pore space evolution, porosity, absolute permeability, and form factor changes due to core matrix dissolution or precipitation. We also, introduce the topological measure; the persistence diagrams of independent cycles in pore space to classify different dissolution scenarios. Using derived classification, we constructed the statistically reliable porosity-permeability relations for different dissolution scenarios.

We present an algorithm for the pore-scale simulation of the reactive transport in a 3D case. The algorithm is designed to facilitate the observation of pore space changes caused by chemical fluid-solid interaction. Additionally, the algorithm allows estimation of the main macroscopic properties evolution of the porous material, such as permeability, hydraulic tortuosity, and formation factor. Also, we develop an algorithm to compute the persistence diagrams for the independent cycles in the pore space, which quantitatively characterizes the changes in the pore space topology. Moreover, we speed up this algorithm by using the original digital image reduction approach. Applying the clustering technique to the persistence diagrams, we show that different matrix dissolution scenarios can be distinguished based on the persistence homology. These scenarios depend on the flow rate, reaction rate, and species concentration at the inlet. At the same time, the samples from the different clusters illustrate utterly different behavior of the cross-property (porosity-permeability) relations. This is extended version of our conference paper [1].

The article presents a numerical algorithm for modeling the chemically reactive transport in a porous medium at a pore scale. The aim of the study is to research the change in the geometry of the pore space during the chemical interaction of the fluid with the rock. First, fluid flow and transport of chemically active components are simulated in the pore space. Heterogeneous reactions are then used to calculate their interactions with the rock. After that, the change in the interface between the liquid and the solid is determined using the level-set method, which allows to handle changes in the topology of the pore space. The algorithm is based on the finite-difference method and is implemented on the GP-GPU.

We present a numerical algorithm of seismic wave propagation in anisotropic fractured fluid-saturated porous media and estimation of seismic attenuation. The algorithm is based on numerical solution of anisotropic Biot equations of poroelasticity. We use finite-difference approximation of Biot equations on the staggered grid. We perform a set of numerical experiments of wave propagation in fractured media. Fractures in the media are connected and filled with anisotropic material providing wave induced fluid flow within connected fractures. Numerical estimations of inverse quality factor demonstrate the effect of fracture-filling material anisotropy on seismic wave attenuation.

We present a numerical study of the effect of the interfaces' roughness on layered media's upscaled elastic parameters. First, we consider a layered model with two types of elastic materials, assuming that the interfaces are not flat but rough, and apply the numerical upscaling technique to estimate the effective elastic properties of such models. After that, we apply a downscaling technique to reconstruct a layered media with flat interfaces but with uncertainties in elastic moduli of the layers. Next, we compute the covariance of the elements of the reconstructed stiffness matrix and prove that the logarithm of this matrix is linearly related to the logarithms of the standard deviation and the correlation length of the interfaces of the original problem. Finally, we use generated dataset to estimate covariance matrices of the stiffness matrix for arbitrary interface roughnesses.

In this paper, we present an iterative solver for Poisson equation. The approach is essentially oriented on upscaling the physical parameters of porous materials, such as electrical resistivity, thermal conductivity, pressure field computation. The algorithm allows solving Poisson equation for strongly heterogeneous media with small-scale high-contrast heterogeneities. The solver is based on the Krylov-type iterative method with a pseudo-spectral preconditionner, thus the convergence rate is independent on the sample size, but sensitive to the contrast of physical properties in the model. GPU-based implementation of the algorithm allows performing simulations for the samples of the size \(400^3\) using a single GPU.