**Scientific results:**

- 2017: We assembled an experimental installation to research flow of foam in a Hele–Shaw cell. The device was launched into operation. We conducted a number of experiments to study flow of foam around an obstacle with variation of geometric parameters. It was found that the shape of an obstacle affects the dynamics of flow of foam around it.
- The dependence of the velocity of foam flow downstream from the object depending on the liquid content in foam.
- The Laboratory conducted preliminary experiments in which we simultaneously recorded a video of destruction of a material and logging acoustic emission data during the propagation of cracks in synthetic materials (in particular, plexiglass). The experiments allowed to refine methodologies for conducting experiments, as well as comparing destruction of material and acoustic emission.
- We obtained theoretical estimates of the electric field induced near a hydraulic fracturing crack by leaking contrasting fluid from the crack into the bed under the influence of the initial high pressure in the crack and the elastic forces in the rock that make the crack close after the pumping is complete.
- A mathematical model was built that describes hydraulic fracturing in a poroelastic medium on suggestions that the medium is inhomogeneous and its physical properties change over space, the crack is flat, a well crossing the plane of the crack is used for pumping. The main feature of the model is that it accounts for pore pressure and full conjugation of the stress-strain state of the rock mass and the filtration of liquid.
- Using a device that creates static and pulsatile pressure in grease, we conducted an experimental research of hydraulic fracturing of thin-walled cylinders with a hole. The cylinders were made of the GF–177 cement material as well as of a mixture of aluminous cement and sand. We refined the methodology for producing samples and working with the experimental bench.
- Our researchers obtained and processed results of intraoperative monitoring: of hemodynamic parameters (velocity and pressure) in brain vessels conducted in 2017 during 13 neurosurgeries with the use of the Volcano ComboMap/ComboWire measurement complex.
- 2018: We conducted a series of experiments to study flow of foam around an obstacle in a Hele–Shaw cell. Due to the presence of an obstacle in a foam flow, deformation of bubbles occurs. When foam flows around an obstacle, surface energy first accumulates due to an increase in the surface area of bubbles and is then released, causing the so-called «negative» footprint, in which bubbles accelerate to velocities exceeding the velocity of the oncoming flow. It was demonstrated that as the height of an obstacle increases, the maximum value of the longitudinal velocity defect in the negative footprint behind the body first increases linearly, then reaches the maximum at some value of the height of the obstacle, after which it decreases to the value corresponding to the case of complete blockage of the gap of the cell by the obstacle. The existence of the maximum of the effect at some height of the obstacle can be explained by the balance between the growth of the surface energy and the increase in viscous loss as the height of the obstacle grows. We researched the nature of changes of the velocity field depending on the share of the liquid fraction in it, demonstrated the smooth transition from flow of foam having pronounced non -Newtonian properties to flow of bubbles of liquid that is close to Newtonian in its properties. The obtained results allow to quantitatively assess the range of applicability of the classical Durian model for computing flows of «dry» and «wet» foam.
- In collaboration with a contractor, we developed an embolizing compound. The embolizing agent was used in a study of a model setting of a problem of the embolization of a microchannel corresponding to the racemose part of an arteriovenous malformation. As a result of the visualization, we determined two modes of the polymerization process that lead to significantly different pictures of microchannel embolization. In the first mode the polymerization reaction runs in a thin layer on the interface between the embolizing agent and water, while the sedimentation of the polymerized embolizing agent occurs upstream in the form of a thread-like structure. The second mode is characterized by a higher degree of mixing on the interface between the embolizing agent and water, which leads to a complete embolization of the microchannel. To research the capabilities of the proposed mathematical model, reviewing flat flow in a channel under the influence of a pressure gradient for liquids with rod-like particles. At the initial moment the particles are oriented transversely to the flow. It was found that for low Reynolds numbers that correspond to the velocity of rotation of the particles, the model is reduced to a stationary system of equations for the velocity of the flow and micro-rotations. while time is a parameter. These equations allow to determine the organization of the particles and their micro-inertia depending on the transverse spatial coordinate and time.
- We introduced a model of computation of the general influx of fluid to a horizontal well with multi-stage hydraulic fracturing accounting for the mutual influence of fracturing cracks, the inhomogeneity of the properties of the rock formation, factoring in the finite permeability of the cracks and the hydraulic resistance of the well. A numerical model has been developed that describes the development of a hydraulic fracturing crack in a poroelastic medium when a liquid with a power-law rheology is pumped. We obtained proof of the fact that at a lower value of the fluid flow index, the crack becomes narrow and long.
- Our researchers determined the limit strains in concrete both in homogeneous and inhomogeneous strain states. Destruction in an inhomogeneous strain field was performed during clean shift on the surface of the hole under conditions of hydraulic fracturing. Using non-local criteria of the limit state, we modeled destruction in an inhomogeneous stain field. The results of the modeling were compared to experimental data. It was demonstrated that a satisfactory description of experimental data on destruction in an inhomogeneous strain field is achieved only with the help of non-local limit state criteria.
- In the course of a comparative analysis of clinical data of numerical 3D modeling, it turned out that for the problem of determining a potential location of an aneurysm rupture, a significant role is played by the patient-specific profile of blood flow that is provided as a boundary condition, while data on the strength of a cerebral aneurysm can be taken from literature. For the same problem we demonstrated the advantage of hydroelastic computation before rigid computations for the purpose of using such computations for preliminary modeling.
- 2019: we built a model that allows to predict changes in the velocity of foam on an arbitrary obstacle bounded by two plates that is a monolayer of bubbles. Initially, the model is based on potential flow of a Newtonian fluid around a cylinder in a Hele–Shaw cells: a cylindrical obstacle partially filling the gap between the plates is considered. Further, we introduced the notion of limiting pressure exerted on a foam bubble during its passage over an obstacle. The obtained result is used to set conditions on the boundary of an obstacle when transitioning to a more generalized model describing the motion of a monolayer of bubbles in a cell.
- Our researchers have obtained results of an experimental modeling of hydraulic fracturing under conditions of an inhomogeneous (stepwise) distribution of horizontal compression strains in rock masses. The experiments were conducted at various values of the velocity of the pumping of hydraulic fracturing fluid and the mean compression strain between PMMA blocks.
- We found dependencies of the motion of a hydraulic fracture and the size of the boundaries of the propagation of fluid and the fluid lag on the volume of the pumped fluid. It was demonstrated that in experiments with a profiled block, the fluid lag had significantly larger dimensions in comparison with experiments on flat polished blocks. The maximum pressure of the hydraulic fracturing fluid equals to the sum of the application pressure in the middle zone of the blocks and the characteristic net pressure depending solely on the rate of pumping of the hydraulic fracturing fluid and its viscosity. The value of the characteristic net pressure grows linearly as the rate of pumping of the fluid increases.
- The Laboratory conducted experiments to study the propagation of a crack in synthetic materials. The fast and slow modes of the propagation of a crack were reviewed, as well as the transitional mode. A simplified qualitative model was proposed that is based on statistical physics and thermal dissipation. The model predicts an increase in temperature up to several thousand degrees.
- For a flow in a channel, we have built correlations between the pressure gradient and the flow rate of phases, such as the carrier liquid and the granulation phase (particles). A criterion has been found for the stationarity of the granulated phase.
- It was determined that, depending on the limit shear strain such a phase does not flow if either the pressure gradient or the thickness of the channel is small. A numerical analysis of liquid flow rate with various input parameters was performed.
- We presented and used an experimental device for researching the morphology and dynamics of cracks slowly put into motion in disordered media. We used a transparent sample made of two PMMA sheets, in which the propagation of cracking front can be observed in a videotape of the experiments. Various conditions of loading were used, allowing to have a wide range of mean velocities of the cracking front. Thanks to the high resolution of recording and the large volume of data, it is possible to review intermittent crackling in a sample in detail. The changes of the velocity of the local front demonstrates an avalanche dynamic, where every local avalanche is characterized by its size S, its normalized duration D and the maximum velocity V
_{max}. It was shown that each possibility density function of these values are a power-law with exponential cutoff. An experimental assessment of these power-law exponents confirmed that the critical damping of a long elastic string propagating in a two-dimensional disordered medium quantitatively defines the large-scale behavior of experimental observations.
- It was shown that when modeling a hydraulic fracturing crack with the use of a purely elastic medium, there is a risk of not capturing a breakthrough into undesirable layers if the permeability of the productive layer is sufficiently high. Controlling the realization of one or another scenario of the behavior of a crack is possible by changing the parameters pumping: the flow rate and viscosity of the hydraulic fracturing fluid. If the flow rate is high, the possibility of a breakthrough into the non-permeable neighboring layer increases. When the viscosity of the hydraulic fracturing fluid decreases, the probability of a breakthrough also diminishes.
- We developed the «PREVELCOM» program allowing to synchronously view and visually analyze data from recording equipment presented in the form of a text file and data of imaging equipment recorded in the standard medical format DICOM. An application for the state registration of a computer program was submitted. Using the developed software, we processed intraoperative monitoring data, built a model of optimal embolization.
- The Laboratory studied the mechanics of tissues of cerebral aneurysms. In the course of the research of the applicability of hyperelastic models to the modeling of such tissues, we built their hierarchy depending on the value of deformation and the status of the aneurysm.
- 2020: We conducted an experimental research of the propagation of a hydraulic fracturing region between blocks of PMMA and concrete based on the GTs–50 aluminous cement that are pressed against each other, as well as between a concrete block and a 12-mm thick silicate glass plate.
- We conducted experiments using a device for hydraulic fracturing modeling to determine the development of instability on the interface when fluids with a contrast in rheological properties are pumped alternately into a crack with elastic walls. We reproduced the picture of the Saffman–Taylor instability, mixing of fluids, the heterogeneous placement of the proppant in the crack. The conducted experiments are necessary for the validation of mathematical models and the testing of technological approaches aimed at increasing the efficiency and reducing the costs of hydraulic fracturing.
- The problem of convective sedimentation of particles in a vessel with vertical walls at various angles of inclination of the vessel was reviewed. It is known that in an inclined vessel sedimentation runs faster (A. E. Boycott, Nature 1920). A mathematical model was proposed that is in agreement with such an effect. It turned out that the main role is played not by rotational diffusion but by gravitational mobility in the Fick’s law of diffusion for a flow for concentration of particles.
- We reviewed the problem of flow of suspensions in a channel at a specified pressure gradient for the case of rod-like particles, we developed a mathematical model accounting for impulse momenta within the Kosser continuum, when each particle of the material continuum is considered as a solid body that has not only linear velocity but also angular velocity and the micro-inertia tensor. It was established that the phenomenon of hysteresis on the gradient-flow rate plane, which corresponds to real flows in oil pipes under conditions of introduction of optimizing additives.
- The Laboratory developed software based on machine learning for the batch processing of experimental data on flows of fluid foam in a Hele–Shaw cell consisting of two glass plates divided by a millimeter-wide gap in the presence of a local permeable obstacle.
- We conducted a large-scale processing of experimental data that allowed to reconstruct every element of fluid foam. Relying on the obtained data, we conducted a statistical analysis of topological reconfigurations (Т1 events) of foam cells as well as built fields of foam deformation in a Hele–Shaw cell with local obstacles. We demonstrated the dependence of the deformation on the height of the obstacle and on the share of the liquid fraction in foam.
- During mechanical testing of bacterial cellulose, we found a statistically significant influence of such a factor as the use of the chitosol-vancomycin in the process of growing bacterial nanocellulose on the strength characteristics of the material (limit strain). An application for the state registration of a useful model of a retaining device for testing was submitted.
- We performed a statistical analysis of leaps in stress-strain graphs of diagrams of single-axis testing of tearing of tissues of cerebral aneurysms. We isolated statistically significant differences for cohorts between burst and intact anerysms in the nature of such leaps (evaluation).
- 2021: We built a device for studying the propagation of a crack using optical methods as well as load sensors and acoustic emission sensors. An algorithm was created for processing visual data, we obtained the distribution of the front of the avalanche and compared it to acoustic data. Two automated experimental devices have been created for studying the effect of bridging in the two-dimensional and three-dimensional case.
- The Laboratory has performed a statistical and optical analysis of obtained experimental data in air or water with variation in the size of the hole and the sizes of the used granules.
- We obtained the results of an experimental research of the interaction of two cracks propagating towards each other in a sheet material. As the research material we used matte tracing paper with a thickness of 80 µm. The experiments were conducted using the Zwick&Roell Z100 testing bench installed inside a climatic chamber.
- Our researchers built a combined model of embolization, in which, along with the flow of blood and the embolizing agent in arteriovenous malformations, we also account for blood flowing to neighboring healthy vessels. Moreover, using a one-dimensional model of two-phase filtration that relies on real clinical data, modeled joint flow of blood and the embolizing compound inside an arteriovenous malformation. We conducted an experimental research of the rheological properties of embolizing compounds and performed a numerical 3D modeling on the basis of experimental data.
- We developed a user interface for a computer program that allows to rapidly input a patient’s parameters in one or another configuration of the circle of Willis and to compute the optimal parameters of anastomosis.
- 2022: Our researchers conducted a study of the uncertainty of the boundary conditions during the numerical modeling of the hemodynamics of a cerebral aneurysm.
- We experimentally researched the rheology of embolizing agents in a wide range of shear models and temperatures.

**Implemented results of research:**

Number of objects of intellectual property (over the whole course of implementation of the project):

- A program for synchronous viewing and visual analysis of data from «PREVELCOM» recording equipment.
- A module for computing the yield of a horizontal well with multi-stage hydraulic fracturing based on numerical modeling using «FreeFEM++».
- «FEMEngine», a computation package based on the finite-element method with linear finite elements on a triangular lattice.

- A restraining device for single-axis testing of thin biological tissues (membranes).

**Education and career development:**

Participation of members of the research team in schools and advanced training:

- Flowing Matter (CISM School), 2017.
- Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics (CISM School), 2017.
- Mechanics of Fibrous Materials and Application: Physical and Modeling Aspects (CISM School, Italy), 2018.
- Biomechanics, from Protein to Tissue to Organ: Modeling and Computation (Austria), 2018.
- MR image processing - from image data to information (Germany), 2018.
- CISM–AIMETA Advanced School on «Anisotropic Particles in Viscous and Turbulent Flows» (CISM School, Italy), 2019.
- The Art of Modeling in Computational Solid Mechanics (CISM School, Italy), 2019.
- Modern software complexes for problems of the mechanics of continuous media (Novosibirsk, Russia).

Employees of the Laboratory conducted joint international events and workshops:

- Russia–Japan Joint Seminar «Nonequilibrium processing of materials: experiments and modeling, Novosibirsk, Novosibirsk State University–Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, 2018.
- Russia–Japan Workshop «Mathematical analysis of fracture phenomena for elastic structures and its applications, Novosibirsk, Novosibirsk State University–Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, 2019.

**Organizational and structural changes:**

Over the period of work of the Laboratory, we have created a collection of unique equipment for oil and gas engineering and other industrial domains:

- A device for the pumping of a proppant with the capability of visualization of flow in a model of the crack.
- A device for researching bridging both in an air environment and in a fluid.
- A device for monitoring the growth of the crack in the hydraulic fracturing formation.
- A device for monitoring the flow of foam in a Hele–Shaw cell.

We have purchased scientific equipment that allowed to reach a new level of research in the domain of studying properties of materials, including in bioengineering:

- Instron 5944 universal machine with a thermostatic biobath.
- ANTON PAAR rheometer (Austria).

This equipment is used both for obtaining scientific results and for solving hi-tech problems of our commercial partners.

**Collaborations:**

- As a result of the implementation of the project, a new major research direction emerged that studies the dynamics of complex fluids and their flow in elastic media when there is a possibility of destruction and deformation of elastic media, changes in the rheological properties of fluids in the process of their flow as well as the formation of power circuit and effects of clusterization and clogging in granular and fiber materials carried by liquids. The methodologies developed during the implementation of the project allow to design liquid materials that have specified qualities and optimize the parameters of their flow in technical biomedical applications. The work for the implementation of the project at the Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences were conducted in parallel with the work to create the unique Russian simulation of hydraulic fracturing «Kiber–GRP» on order of the Research and Technology Center «Gazprom neft» in collaboration with a consortium of leading Russian universities (Moscow Institute of Physics and Technology, Skolkovo Institute of Technology, Peter the Great Saint Petersburg Polytechnic University, Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences).
- We have published and continue to conduct research in close cooperation with the Meshalkin National Medical Research Center, the Federal Neurosurgical Center (Novosibirsk), the Research Institute of Traumatology and Orthopedics named after Y. L. Tsivyan (Novosibirsk). We conducted joint research with TU Wien (Austria) and communicated with Tohoku University (Japan).