We study CO2 injection into a saline aquifer intersected by a tectonic fault using a coupled modeling approach to evaluate potential geomechanical risks. The simulation approach integrates the reservoir and mechanical simulators through a data transfer algorithm. MUFITS simulates non-isothermal multiphase flow in the reservoir, while FLAC3D calculates its mechanical equilibrium state. We accurately describe the tectonic fault, which consists of damage and core zones, and derive novel analytical closure relations governing the permeability alteration in the fault zone. We estimate the permeability of the activated fracture network in the damage zone and calculate the permeability of the main crack in the fault core, which opens on asperities due to slip. The coupled model is applied to simulate CO2 injection into synthetic and realistic reservoirs. In the synthetic reservoir model, we examine the impact of formation depth and initial tectonic stresses on geomechanical risks. Pronounced tectonic stresses lead to inelastic deformations in the fault zone. Regardless of the magnitude of tectonic stress, slip along the fault plane occurs, and the main crack in the fault core opens on asperities, causing CO2 leakage out of the storage aquifer. In the realistic reservoir model, we demonstrate that sufficiently high bottomhole pressure induces plastic deformations in the near-wellbore zone, interpreted as rock fracturing, without slippage along the fault plane. We perform a sensitivity analysis of the coupled model, varying the mechanical and flow properties of the storage layers and fault zone to assess fault stability and associated geomechanical risks.

We propose a new method for construction of the absolute permeability map consistent with the interpreted results of well logging and well test measurements in oil reservoirs. Nadaraya-Watson kernel regression is used to approximate two-dimensional spatial distribution of the rock permeability. Parameters of the kernel regression are tuned by solving the optimization problem in which, for each well placed in an oil reservoir, we minimize the difference between the actual and predicted values of (i) absolute permeability at the well location (from well logging); (ii) absolute integral permeability of the domain around the well and (iii) skin factor (from well tests). Inverse problem is solved via multiple solutions to forward problems, in which we estimate the integral permeability of reservoir surrounding a well and the skin factor by the surrogate model. The last one is developed using an artificial neural network trained on the physics-based synthetic dataset generated using the procedure comprising the numerical simulation of bottomhole pressure decline curve in reservoir simulator followed by its interpretation using a semi-analytical reservoir model. The developed method for reservoir permeability map construction is applied to the available reservoir model (Egg Model) with highly heterogeneous permeability distribution due to the presence of highly-permeable channels. We showed that the constructed permeability map is hydrodynamically similar to the original one. Numerical simulations of production in the reservoir with constructed and original permeability maps are quantitatively similar in terms of the pore pressure and fluid saturations distribution at the end of the simulation period. Moreover, we obtained an good match between the obtained results of numerical simulations in terms of the flow rates and total volumes of produced oil, water and injected water.

Obtaining reliable permeability maps of oil reservoirs is crucial for building a robust and accurate reservoir simulation model and, therefore, designing effective recovery strategies. This problem, however, remains challenging, as it requires the integration of various data sources by experts from different disciplines. Moreover, there are no sources to provide direct information about the inter-well space. In this work, a new method based on the data-fusion approach is proposed for predicting two-dimensional permeability maps on the whole reservoir area. This method utilizes non-parametric regression with a custom kernel shape accounting for different data sources: well logs, well tests, and seismics. A convolutional neural network is developed to process seismic data and then incorporate it with other sources. A multi-stage data fusion procedure helps to artificially increase the training dataset for the seismic interpretation model and finally to construct an adequate permeability map. The proposed methodology of permeability map construction from different sources was tested on a real oil reservoir located in Western Siberia. The results demonstrate that the developed map perfectly corresponds to the permeability estimations in the wells, and the inter-well space permeability predictions are considerably improved through the incorporation of the seismic data.

Rheological experiments and measurements of particle settling velocity under static conditions were conducted for two types of polyacrylamide (PAA-based) fluids. The flow behavior of the viscoelastic fluids is described by a simplified, three-parameter model that integrates a constant viscosity at low shear rates with a power-law viscosity dependency at higher shear rates. The estimated dependencies of the rheological parameters and particle settling velocity on PAA concentration align with the classical power-law scaling for semi-dilute polymer solutions. The identified scaling offers valuable insight for optimizing the chemical composition of fracturing fluids, thereby improving the efficiency of hydraulic fracturing technology. By applying the lubrication approximation to the Navier-Stokes equations, a set of equations for the suspension flow in a vertical plane channel, characterized by the three-parameter rheology model, was derived. In typical fracturing conditions, the conventional power-law viscosity model used in current fracturing simulators significantly overestimates the gap-averaged apparent fluid viscosity compared to the proposed model. The novel model of the suspension flow aims to enhance the accuracy of proppant transport models applied in hydraulic fracturing simulators. Based on the three-parameter rheology model, a new model of particle settling in the studied PAA-based fluids is developed. It is based on smart Stokes' law, where viscosity is calculated according to either the plateau or power-law region, depending on the self-induced shear rate. The new model more accurately approximates experimentally determined settling velocity of proppant under static conditions across a broad range of PAA concentrations than existing models.

The key goal of the work is to optimize the process of flowback in wells after multi-stage hydraulic fracturing in fields with unconventional oil reserves. The procedure for selecting the optimal flowback operation is to tune the model based on field data, adapt the fracture cleanup model using sensitivity analysis on input parameters, and select the most effective scenario of bottomhole pressure decreasing. This article can be separated into three parts. The first one consists cleanup model description. The second one shows sensitivity analysis of the results of flowback modeling with respect to key input parameters. The last one includes optimization and history matching methods to fit the modeling and field data. Model adaptation to field data is carried out using iterative Newton's method (MSE < 10

Continuum models of media with zero pressure are widely used in various branches of physics and mechanics, including studies of a dilute dispersed phase in multiphase flows. In zero-pressure media, the particle trajectories may intersect, “folds” and “puckers” of the phase volume may arise, and “caustics” (the envelopes of particle trajectories) may appear, near which the density of the medium sharply increases. In recent decades, the phenomena of clustering and aerodynamic focusing of inertial admixture in gas and liquid flows have attracted increasing attention of researchers. This is due to the importance of taking into account the inhomogeneities in the impurity concentration when describing the transport of aerosol pollutants in the environment, the mechanisms of droplet growth in rain clouds, scattering of radiation by dispersed inclusions, initiation of detonation in two-phase mixtures, as well as when solving problems of two-phase aerodynamics, interpretation of measurements obtained by LDV or PIV methods, and in many other applications. These problems gave an impetus to a significant increase in the number of publications devoted to the processes of accumulation and clustering of inertial particles in gas and liquid flows. Within the framework of classical two-fluid models and standard Eulerian approaches assuming single-valuedness of continuum parameters of the media, it turns out impossible to describe zones of multi-valued velocity fields and density singularities in flows with crossing particle trajectories. One of the alternatives is the full Lagrangian approach proposed by the author earlier. In recent years, this approach has been further developed in combination with averaged Eulerian and Lagrangian (vortex-blob method) methods for describing the dynamics of the carrier phase. Such combined approaches made it possible to study the structure of local zones of accumulation of inertial particles in vortex, transient, and turbulent flows. This article describes the basic ideas of the full Lagrangian approach, provides examples of the most significant results which illustrate the unique capabilities of the method, and gives an overview of the main directions of further development of the method as applied to transient, vortex, and turbulent flows of “gas-particle” media. Some of the ideas discussed and the results presented below are of a more general interest, since they are also applicable to other models of zero-pressure media.

The paper is devoted to the development of a surrogate model describing Hydraulic Fracture (HF) flowback, which is a short time period after the start of an oil or gas well, during which it is cleaned out of fracturing fluid and set for a long-term production. The model is based on the combination of the in-house mechanistic model and Machine Learning (ML) algorithms. The mechanistic flowback model describes the dynamics of fracture conductivity during the cleanup, considering several hydro- and geomechanical effects. We conducted a series of computations to assemble the synthetic dataset consisting of 10 000 runs covering the range of input parameters typical for Western Siberia’s conventional terrigenous oil reservoirs. We obtained that the constructed surrogate model is sufficiently accurate at solving the forward problem in terms of the Mean Absolute Percentage Error (MAPE) metric showing the difference between the results acquired by the metamodel and the mechanistic simulator. In particular, we obtained MAPE of 5.12

We investigate the impact of fluid drainage on the stress–strain state of a fluid–saturated reservoir. Our focus is on the transition from an elastic to an elastoplastic state of the rock mass and the appearance of constitutive instability during plastic yield. We determine the onset of inelastic deformations using the Drucker–Prager yield criterion and Eaton’s solution for an elastic medium. Our findings illustrate that the transition to an elastoplastic state occurs with increasing depth and decreasing pore fluid pressure at a fixed depth. When dealing with inelastic rock deformation, we analytically solve the Prandtl–Reuss equations under uniaxial strain conditions to obtain the distribution of minimum horizontal stress within the reservoir characterized by both hydrostatic and abnormally high pore fluid pressure. Furthermore, for a formation undergoing inelastic deformations, we identify the critical value of the plastic hardening modulus at which material instability emerges. The applied analytical approach relies on the Prandtl–Reuss equations, Darcy’s law, and continuity equation for an incompressible fluid.

The aim of this study is to create a fast and stable iterative technique for numerical solution of a quasi-linear elliptic pressure equation. We developed a modified version of the Anderson acceleration (AA) algorithm to fixed-point (FP) iteration method. It computes the approximation to the solutions at each iteration based on the history of vectors in extended space, which includes the vector of unknowns, the discrete form of the operator, and the equation's right-hand side. Several constraints are applied to AA algorithm, including a limitation of the time step variation during the iteration process, which allows switching to the base FP iterations to maintain convergence. Compared to the base FP algorithm, the improved version of the AA algorithm enables a reliable and rapid convergence of the iterative solution for the quasi-linear elliptic pressure equation describing the flow of particle-laden yield-stress fluids in a narrow channel during hydraulic fracturing, a key technology for stimulating hydrocarbon-bearing reservoirs. In particular, the proposed AA algorithm allows for faster computations and resolution of unyielding zones in hydraulic fractures that cannot be calculated using the FP algorithm. The quasi-linear elliptic pressure equation under consideration describes various physical processes, such as the displacement of fluids with viscoplastic rheology in a narrow cylindrical annulus during well cementing, the displacement of cross-linked gel in a proppant pack filling hydraulic fractures during the early stage of well production (fracture flowback), and multiphase filtration in a rock formation. We estimate computational complexity of the developed algorithm as compared to Jacobian-based algorithms and show that the performance of the former one is higher in modelling of flows of viscoplastic fluids. We believe that the developed algorithm is a useful numerical tool that can be implemented in commercial simulators to obtain fast and converged solutions to the non-linear problems described above.

The purpose of this study is to develop a model for the flow of suspensions consisting of Herschel-Bulkley fluid mixed with spherical particles. In particular, the focus is to investigate the effect of non- Newtonian rheology of the carrying fluid on the flow behavior of a suspension. Two-dimensional steady flow problem in a vertical channel is considered, in which both the pressure gradient and gravity drive the suspension flow. Dependence of the velocity profile and particle concentration across the channel on the fluid rheology parameters and orientation of the pressure gradient is investigated. It is found that the non-uniform particle distribution in the flow across the channel leads to the non-uniform density of the suspension, which causes sinkage of the denser regions and promotes downward migration of the particles even without slip velocity. Particle and suspension fluxes are calculated for various fluid rheologies and pressure gradient orientations. The effect of slip velocity between the phases is added via filtration term that captures fluid flow once particles reach the maximum concentration and stall, and via the settling term that describes gravitational particle settling.

The structure of a steady 2D gas-droplet flow in the near-wall region behind the point of incidence of a normal or an oblique shock wave on a plane wall is investigated. In this case, the normal wave corresponds to the Mach stem in the Mach reflection mode, and the oblique wave corresponds to the regular reflection mode of the incident oblique wave. The main aim of the study is to evaluate the effect of small liquid droplets present in the free stream on the equilibrium temperature of the adiabatic wall behind the point of wave reflection. The question is investigated: to what extent the presence of an oblique shock wave incident on the wall can enhance the effect of reducing the equilibrium wall temperature by small droplets present in the flow. The flow region is split into the outer region of “effectively inviscid flow” and the region of an asymptotic laminar boundary layer. Flow calculations in each region are based on a two-fluid model of a gas-droplet mixture, taking into account the phase transition (evaporation) on the droplet surface. The most interesting wave configurations from the point of view of heat transfer, corresponding to “fully and partially dispersed waves” with incomplete evaporation of droplets behind the reflected wave, are studied. A simple limiting scheme of the formation of a liquid film by droplets deposited on the wall is adopted, with the effects of film instability and spattering being ignored. Based on numerical calculations, the estimates are obtained for the possible decrease in the equilibrium temperature of the adiabatic wall behind the point of incidence of a shock wave in a steady supersonic gas flow containing a low concentration of liquid droplets.

The present stage of development of artificial intelligence (AI) is based on advanced technologies and modern methods and algorithms of machine learning (ML) including deep machine learning, intellectual data analysis and other fundamental scientific areas, which in turn play an important role in delivering applied solutions in almost all areas of digital economy. However, the corresponding expansion of the AI application range and complication of the class of problems, as well as of the spectrum and the volume of data used for creation of applied AI models and AI-based intelligent system call for a significant expansion of the theoretical and algorithmic basis of AI including development of ML methods that involve mathematical and physical models of objects and phenomena of domains, methods of fusion of multimodal data, methods for constructing geometric and topological components of deep neural networks, methods for modeling 3D objects, etc. In 2021, in order to cope well with these challenges, Skoltech Research Center for Applied Artificial Intelligence was created with support from the Federal Program “Artificial Intelligence.” The aims of the Center are as follows: creation of a scientific and technological basis for solution of a wide range of relevant applied tasks for purposes of sustainable development of the economy of the Russian Federation including optimization problems of managerial decisions aimed at reduction of the carbon footprint and other urgent ESG tasks; environmental monitoring tasks for detection of anomalies and extreme situations forecast; assessment of economic and social risks and their dynamics caused by climatic changes; predictive analytic problems, etc. In the present paper, we describe new AI technologies, models, methods and algorithms developed in the Center, describe the main applied areas of research of the Center, and provide a briefly survey of the already developed scientific and applied results.

CO2 injection into a saline aquifer crossed by a tectonic fault is studied with coupled fluid mechanics - geomechanics modeling. The simulation approach is based on coupling of the MUFITS reservoir simulator and the FLAC3D mechanical simulator via an in-house API (i.e., an algorithm for data transfer between simulators). MUFITS simulates the non-isothermal multiphase flow of CO2 and brine in rock formation accounting for phase transitions and thermal effects. The modeling workflow is sequential, so that hydrodynamical simulations are carried out at a certain time interval, after which pressure, temperature, and density distributions are passed to FLAC3D, which calculates the equilibrium mechanical state. Computed deformations and stresses are utilized to update the porosity and permeability fields for the subsequent hydrodynamic modeling. In particular, we focus on the tectonic fault and its behavior during CO2 injection. We distinguish the damage zone and core inside the fault and derive the closure relations for their permeability alteration analytically. The coupled approach developed here is applied to simulate CO2 injection into synthetic and realistic reservoir models. For the former one, we study the effect of formation depth and presence of the tectonic stresses at the initial mechanical state, while for the latter, we consider different injection modes (bottomhole pressure). In each numerical experiment, we describe the evolution of the fault permeability due to the slip along its plane and the development of plastic deformations leading to the loss of reservoir integrity and CO2 leakage. Sensitivity analysis of the coupled model to realistic values of input parameters to assess the fault stability is carried out.

We aim at the development of a general modelling workflow for design and optimization of the well flowback and startup operation on hydraulically fractured wells. Fracture flowback model developed earlier by the authors is extended to take into account several new fluid mechanics factors accompanying flowback, namely, viscoplastic rheology of unbroken cross-linked gel and coupled "fracture-reservoir" numerical submodel for influx from rock formation. We also developed models and implemented new geomechanical factors, namely, (i) fracture closure in gaps between proppant pillars and in proppant-free cavity in the vicinity of the well taking into account formation creep; (ii) propagation of plastic deformations due to tensile rock failure from the fracture face into the fluid-saturated reservoir. We carried out parametric calculations to study the dynamics of fracture conductivity during flowback and its effect on well production for the set of parameters typical of oil wells in Achimov formation of Western Siberia, Russia. The first set of calculations is carried out using the flowback model in the reservoir linear flow regime. It is obtained that the typical length of hydraulic fracture zone, in which tensile rock failure at the fracture walls occurs, is insignificant. In the range of rock permeability in between 0.01 mD and 1 D, we studied the effect of non-dimensional governing parameters as well as bottomhole pressure drop dynamics on oil production. We obtained a map of pressure drop regimes (fast, moderate or slow) leading to maximum cumulative oil production. The second set of parametric calculations is carried out using integrated well production modelling workflow, in which the flowback model acts as a missing link in between hydraulic fracturing and reservoir commercial simulators. We evaluated quantitatively effects of initial fracture aperture, proppant diameter, yield stress of fracturing fluid, pressure drop rate and proppant material type (ceramic and sand) on long-term well production beyond formation linear regime. The third set of parametric calculations is carried out using the flowback model history-matched to field data related to production of four multistage hydraulically fractured oil wells in Achimov formation of Western Siberia, Russia. On the basis of the matched model we evaluated geomechanics effects on fracture conductivity degradation. We also performed sensitivity analysis in the framework of the history-matched model to study the impact of geomechanics and fluid rheology parameters on flowback efficiency.

A rivulet of a power-law-rheology fluid steadily draining from a point source on an inclined superhydrophobic plane is considered. An equation for the shape of the cross section of the rivulet has been derived in the thin layer approximation with the inhomogeneous slip boundary condition (slip coefficients are power functions of the spatial coordinates). Under the assumption that the rivulet is symmetric with respect to its middle plane, the conditions for the existence of a class of self-similar solutions of one ordinary differential equation of the second order have been determined. For some slip parameters of the superhydrophobic surface and some rheological indices of the draining fluid, analytical and numerical solutions from the found class have been constructed and the shape of the cross section of the rivulet and the geometry of the wetting spot have been analyzed.

We evaluate the CO2 storage potential for the depositional environments of the West Siberian sedimentary basin. Particularly, we investigate the prospect of storing large volumes of CO2 in the alluvial fan, barrier island, deltaic, and fluvial environments. We employ several detailed 3-D models of petroleum reservoirs as proxies for the saline aquifers occurring in the basin under the listed environments. By simulating supercritical CO2 injection into several reservoir sectors, we compare the environments in terms of their prospects for the underground storage of CO2. We estimate the maximum capacity and storage potential for the entire sectors by considering them the resources licensed for an industrial-scale Carbon Capture and Storage (CCS) project. The scenarios with a single and multiple injection wells are assessed in the evaluation, which might be of interest for the elaboration of feasible regulations and policies in the area. We find out that the barrier island deposits are characterized by maximum potential, whereas the fluvial environments are the least suitable for deploying CCS. We demonstrate that for the barrier island environments, the storage potential is most sensitive to the relative permeability end-points. We also test several scaling concepts for extending our estimates for a single-well injection to a multi-well scenarios and generally from a site-specific to regional scale. We conclude that due to the interference between nearby wells, the storage potential for a single well scenario cannot reliably be scaled to the case of CO2 injection through a group of wells.

Summary We analyze pressure slopes during buildup when proppant placement into a fracture results in tip screenout (TSO), as observed in real field data on pumping of fracturing jobs. The phenomenon is described by proppant packing and jamming near the fracture tip, leading to TSO. Transient evolution of the zone of packed proppant near the tip and associated additional pressure drop along the packed bank are evaluated. Under the assumption of constant fracture height, the additional pressure drop during TSO grows as t(1+α), where tα is the law of proppant concentration growth (pressure drop grows quadratically in time, if the proppant concentration on surface at the blender grows linearly in time). The model is calibrated on field data from fracturing jobs in Western Siberia, Russia. These approximate models are verified with the two-continua model of proppant transport used to describe the process of proppant packing and jamming near the fracture tip. We also show how predictions of bottomhole pressure (BHP) growth during TSO obtained using proppant transport models implemented into existing hydraulic fracturing simulators can be significantly improved by a reasonable (based on behavior of dense suspensions at the packing limit) modification to parameters, describing suspension shear viscosity.

A two-fluid mathematical model of a supersonic gas-droplet mixture flow with account of droplet growth due to vapor condensation behind a condensation shock in a plane expanding channel is constructed. The flow structure in both the inviscid core and the two-phase boundary layer assumed to be laminar is studied numerically. The range of flow parameters is considered in which the spontaneous condensation zone ("condensation shock") is fairly narrow, and just behind this shock thermal parameters of the phases almost reach thermodynamic equilibrium. Downstream, the droplet radius continues to grow due to the flow expansion and further vapor condensation. Of primary concern is the flow region where the droplets become sufficiently inertial to travel with a velocity slip, and hence to model adequately the velocity and temperature fields in both phases a two-fluid model is required. In addition to the Stokes drag, the Saffman lifting force exerted on the droplets in the boundary layer is taken into account. The Saffman force results in droplet deposition and the formation of a thin liquid film on the channel walls. A parametric numerical analysis of the two-phase flow structure in the inviscid flow region and in the near-wall boundary layer is performed, and the effects of phase transitions (droplet condensation and evaporation, as well as the film formation) on the reduction of the adiabatic channel wall temperature are analyzed.

We describe a stacked model for predicting the cumulative fluid production for an oil well with a multistage-fracture completion based on a combination of Ridge Regression and CatBoost algorithms. The model is developed based on an extended digital field data base of reservoir, well and fracturing design parameters. The database now includes more than 5000 wells from 23 oilfields of Western Siberia (Russia), with 6687 fracturing operations in total. Starting with 387 parameters characterizing each well, including construction, reservoir properties, fracturing design features and production, we end up with 38 key parameters used as input features for each well in the model training process. The model demonstrates physically explainable dependencies plots of the target on the design parameters (number of stages, proppant mass, average and final proppant concentrations and fluid rate). We developed a set of methods including those based on the use of Euclidean distance and clustering techniques to perform similar (offset) wells search, which is useful for a field engineer to analyze earlier fracturing treatments on similar wells. These approaches are also adapted for obtaining the optimization parameters boundaries for the particular pilot well, as part of the field testing campaign of the methodology. An inverse problem (selecting an optimum set of fracturing design parameters to maximize production) is formulated as optimizing a high dimensional black box approximation function constrained by boundaries and solved with four different optimization methods: surrogate-based optimization, sequential least squares programming, particle swarm optimization and differential evolution. A recommendation system containing all the above methods is designed to advise a production stimulation engineer on an optimized fracturing design.

Summary We perform the coupled fluid- and geo-mechanical modeling of CO2 sequestration in a saline aquifer crossed by a tectonic fault. The model is based on hydrodynamical reservoir simulator MUFITS and mechanical simulator FLAC3D linked by the algorithm of the data transfer. The modelling of multiphase filtration of CO2 and brine accounting for phase transitions and thermal phenomena is carried out using MUFITS. At certain time, the hydrodynamical simulation is paused, while current pressure and temperature fields are passed to FLAC3D to calculate the equilibrium mechanical state. Numerically found volumetric strain is used to update the porosity and permeability fields for the subsequent hydrodynamic modelling. The study is focused on the effect of CO2 injection on activation of fault crossing the target aquifer. We derive an analytical expression for the permeability alteration in the rock damage zone due to plastic deformations which is based on the dilatancy evaluated numerically. For the fault core, we apply closure relation available in open literature. Parametric study of fault behavior during CO2 sequestration is carried out by varying the governing parameters including the injection strategy. Safe injection regimes are identified which do not lead to seismic activity and CO2 leakage out of the target aquifer.