We present a boundary integral formulation for steadily propagating semi-infinite plane strain tensile and shear fractures in poroelastic media. By combining fundamental solutions of plane strain poroelasticity for an instantaneous fluid source and instantaneous edge dislocations (normal and slip modes) with temporal and spatial superposition principles, we derive boundary integral equations for steadily moving fractures under the adopted hydraulic boundary conditions. These equations relate the tractions (normal and shear stresses) and the pore fluid pressure on the fracture surfaces to the fracture opening, slip, and the fluid displacement function. Assuming prescribed traction and pore fluid pressure profiles, we develop a numerical methodology to solve the governing equations for fracture opening, slip, and the fluid displacement function. The formulation is systematically verified on several relevant problems, including a tensile fracture with exponential normal loading, a stress-free tensile fracture with an imposed exponential pore fluid pressure, and a shear fracture under uniform shear loading over a finite region, demonstrating excellent agreement with analytical and semi-analytical solutions. The resulting boundary integral framework provides an accurate and efficient tool for analyzing semi-infinite steadily propagating cracks in permeable poroelastic media. By supplementing the formulation with appropriate closure relations and additional physics, such as lubrication flow in hydraulic fractures or frictional strength evolution in shear fractures, it can be used to investigate a broad range of coupled fracture-fluid problems. The approach may also be adapted to other classes of elasto-diffusive problems by modifying the underlying physical parameters.

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.

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.

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.

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.

We propose a novel approach to data-driven modeling of a transient production of oil wells. We apply the transformer-based neural networks trained on the multivariate time series composed of various parameters of oil wells measured during their exploitation. By tuning the machine learning models for a single well (ignoring the effect of neighboring wells) on the open-source field datasets, we demonstrate that transformer outperforms recurrent neural networks with LSTM/GRU cells in the forecasting of the bottomhole pressure dynamics. We apply the transfer learning procedure to the transformer-based surrogate model, which includes the initial training on the dataset from a certain well and additional tuning of the model's weights on the dataset from a target well. Transfer learning approach helps to improve the prediction capability of the model. Next, we generalize the single-well model based on the transformer architecture for multiple wells to simulate complex transient oilfield-level patterns. In other words, we create the global model which deals with the dataset, comprised of the production history from multiple wells, and allows for capturing the well interference resulting in more accurate prediction of the bottomhole pressure or flow rate evolutions for each well under consideration. The developed instruments for a single-well and oilfield-scale modelling can be used to optimize the production process by selecting the operating regime and submersible equipment to increase the hydrocarbon recovery. In addition, the models can be helpful to perform well-testing avoiding costly shut-in operations.

This chapter considers a model for a radial hydraulic fracture propagation in a permeable, linear elastic rock formation driven by a point source fluid injection. The linear elastic fracture mechanics theory controls the quasi-static propagation. The hydraulic fracturing fluid is slickwater -- pure water solution with polymeric additives which allow reducing the fluid flow friction in the wellbore and fracture in reservoir field applications. We focus on the possible transformation of the fluid flow regime inside the fracture channel from laminar to turbulent with distance from the fracture front. We assume that the turbulent friction of slickwater is described by the maximum drag reduction asymptote, while Carter's law governs the leak-off into the permeable rock. The solution is obtained numerically using the algorithm based on the Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We compute solution examples for typical field cases and demonstrate a significant impact of the turbulent flow regime during the initial few minutes of propagation, namely, shorter radius and wider maximum aperture than the laminar model provides. Moreover, we observe higher fluid pressure values at the wellbore within tens of minutes of the start of the injection. This leads to a larger hydraulic pumping power requirement than the laminar model predicts. We also find that the fluid leak-off into the permeable rock enhances the turbulent flow effect in the fracture when compared to the impermeable rock case. In order to analyze the parametric dependence of the general solution, we convert the governing equations into the dimensionless form. We perform an extensive exploration of the normalized solution in space of two non-dimensional parameters, leak-off and characteristic Reynolds numbers, and normalized time. Specifically, we determine the applicability domains of the limiting propagation regimes to frame the general solution, investigate the alterations of the crack characteristics depending on the governing parameters, and identify zones where the turbulent flow is important.

Summary We investigate the influence of fluid yield stress on propagation of a radial hydraulic fracture in a permeable reservoir. The hydraulic fracturing fluid rheology is governed by Herschel-Bulkley model including yield stress and non-linearity of the shear stress. The rock is linear elastic, and the fracture is formed due to fluid injection at a constant volumetric rate. The crack propagation criterion follows the theory of linear elastic fracture mechanics, and Carter’s leak-off law describes the fluid leak-off into formation. We developed two numerical approaches to compute the problem solution: fully numerical (Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques) and approximate (the global fluid balance equation combined with fracture tip asymptote). The presented simulations representing typical field cases demonstrate that the yield stress can lead to a fracture with a shorter radius and larger aperture compared to the radial fracture model with simpler power-law fluid. We derived limiting propagation regimes characterised by dominance of certain physical phenomena and built parametric maps showing their applicability domains. Such analysis enables one to identify whether the yield stress provides a substantial impact for any given problem parameters.

We analyse the influence of fluid yield stress on propagation of a radial (penny-shaped) hydraulic fracture in a permeable reservoir. In particular, the Herschel-Bulkley rheological model is adopted that includes yield stress and non-linearity of the shear stress. The rock is assumed to be linear elastic, and the fracture is driven by the point source fluid injection with a constant volumetric rate. The fracture propagation condition follows the theory of linear elastic fracture mechanics, and Carter’s leak-off law is selected to govern the fluid exchange process between the fracture and formation. The numerical solution for the problem is found using the algorithm based on Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We also construct an approximate solution with the help of the global fluid balance equation and the near-tip region asymptote. The latter approximation is computationally efficient, and we estimate its accuracy by comparing the primary crack characteristics such as opening, pressure, and radius with that provided by the full numerical solution. We present examples corresponding to typical field cases and demonstrate that the addition of yield stress can lead to shorter radius and wider opening compared to the corresponding case with a simpler power-law fluid rheology. Further, we quantify the limiting propagation regimes (or vertex solutions) characterised by dominance of a particular physical phenomenon. Relative to the power-law results, there are two new vertices that are associated with domination of yield stress: storage-yield-stress and leak-off-yield-stress. To understand the influence of various problem parameters, we utilise the constructed approximate solution to investigate the dimensionless parametric space for the problem, in which the applicability domains of the limiting solutions are quantified. This enables one to quickly determine whether the yield stress provides a strong influence for a given problem parameters.

This paper investigates the problem of a radial (or a penny-shaped) hydraulic fracture propagating in a permeable reservoir. In particular, we consider the fluid exchange between the crack and ambient porous media as a pressure-dependent process. In most of the existing models, the fluid exchange process is represented as one-dimensional pressure-independent leak-off described by Carter’s law. We modify this mechanism by including the dependence of the fluid-exchange rate on the fluid pressure inside the fracture. The proposed approach allows the liquid not only to flow out of the fracture but also to leak-in along the region adjacent to the fracture front. A complete model for hydraulic fracturing involves several physical processes that determine the crack propagation, namely, brittle rock failure, elastic equilibrium of rock, viscous fluid flow inside the fracture channel, and fluid exchange. In order to resolve these phenomena near the fracture front in the numerical scheme, we utilise a special asymptotic multi-scale model for the near-tip region, which is used as a propagation condition for the finite fracture. The main aspect of the analysis is a comparison of fracture characteristics such as aperture profile, radius and others calculated via the proposed model with the results of the radial fracture model that assumes standard pressure-independent fluid exchange mechanism governed by Carter’s law. Based on the comparison, we determine parameter ranges, for which the effect of the pressure-dependent fluid exchange mechanism is essential, and, on the other hand, we outline the zones for which Carter’s leak-off model provides accurate results. Finally, by using an approximate solution for the radial fracture with Carter’s leak-off, we obtain estimates of the effect in the entire parametric space, which allows one to evaluate the influence of the pressure-dependent leak-off for any problem parameters.

Summary This work is devoted to an analysis of the near-tip region of a hydraulic fracture driven by slickwater in a permeable saturated rock. We consider a steady-state problem of a semi-infinite fracture propagating with constant velocity. The host rock is elastic and homogeneous, and fracture propagates according to linear elastic fracture mechanics. The fluid exchange between the fracture and reservoir is governed by Carter’s law. The distinguishing feature of the model is an account for the transition of the flow regime inside the crack channel from laminar to turbulent moving away from the fracture front. The main objective is to analyse the influence of the leak-off process on the laminar-to-turbulent transition and, thus, potential prominence of turbulent flow effects. Hydraulic fracturing fluid is water with polymeric additives (slickwater). These additives reduce viscous friction resulting in the decrease of energy consumption required for pumping. Compared to water, the slickwater exhibits significantly delayed transition to the turbulent regime described by the maximum drag reduction asymptote ( Virk 1975 ). The system of governing equations, which consists of elasticity equation, propagation condition, the continuity equation for viscous incompressible Newtonian fluid, and Poiseuille’s law modified for the turbulent flow regime, is solved for the fracture aperture and fluid pressure along the fracture as a function of problem parameters. We find out that the leak-off process enhances the turbulent flow effects by shifting the transition between laminar and turbulent flow regimes closer to the fracture front, as compared to the zero-leak-off case ( Lecampion & Zia, 2019 ), resulting in a broader region of the fracture hosting turbulent flow. Consequently, in the permeable reservoir case, the transition to turbulent flow can be realised at a distance from the front smaller than the typical field hydraulic fracture size (10 – 100 meters). We compare the fracture width profiles with the impermeable rock case and reveal that the fracture volume increases when leak-off occurs. We analyse the problem parametric space where five limiting regimes are identified: toughness, laminar-viscosity and -leak-off, turbulent-viscosity and -leak-off. We derive analytical expressions for the fracture width and pressure profiles in the turbulent-leak-off regime while others have been established previously. By comparing the limiting solutions with the general numerical solution, we can define their applicability domains and corresponding solution regime maps. The toughness and turbulent-viscosity regimes approximate the general solution in the near- and far-fields, while the other three limiting cases can emerge in the intermediate field.

Summary The present work considers the development of a machine learning model trained on synthetic and lab data for the steady-state multiphase pipe flow. We propose a new method for calculating flow characteristics such as liquid holdup, flow regime, and pressure gradient in the pipe segment based on ANNs and transfer learning technique. Besides, the created tool is implemented within the marching algorithm for calculating flow parameters along the whole pipe. The segment module consists of three sub-models, namely, for calculating liquid holdup, defining flow regime, and estimating pressure gradient. For sub-models creation, we use transfer learning methodology: on the first stage, the ANNs are trained on synthetic data, which we generate by using OLGAS mechanistic model; on the second stage, we train meta-models additionally on the real data, which in our case presented by lab measurements. As a result, we create the new multiphase flow correlation, which includes the basics of the physics-based OLGAS model and is tuned for real data that can be in the general case field measurements. At the final stage, we apply marching algorithms with the suggested segment model to the field dataset for testing purposes.

In this paper, we demonstrate the ability to run an adaptation on field data for a steady-state multiphase pipe flow model for calculating the bottomhole pressure of an inclined well using the deviation survey data, wellhead parameters (pressure, phases flowrates and properties), and the temperature distribution along the wellbore. In the underlying model published earlier in different modifications (Baryshnikov et al., 2020; Kanin et al., 2019), we divide the well into segments and perform the hydraulic calculation of the pressure distribution along the wellbore via the marching algorithm in which the set of PVT correlations and the segment model are utilized. The distinguishing feature of the utilized approach is a data-driven segment model required for calculating the pressure drop along the chosen part of the pipe. This model is based on Artificial Neural Networks (ANNs), which firstly trained on the synthetic data and lab measurements, and after that, additionally tuned on the real field data collected on two wells during the flowback period conducted via the free-flow production method. During the optimization of the ANNs on the field data, we imply the quasi-steady-state approximation of the multiphase flow, i.e. the transient flow is considered as a sequence of the stationary states. In work, we also carry out the comparison of the prediction capability of the constructed pipe segment model with several most common multiphase flow correlations and mechanistic models (within the frame of the marching algorithm) and demonstrate its more accurate outcome. As a result, we develop a method for calculating the flowing bottomhole pressure value adaptable on the required field data. The model adapted on field data allows estimating the flowing bottomhole pressure at each time moment (assuming the quasi-steady-state flow) which is particularly important in the absence of the bottomhole pressure gage at the well.

This paper is concerned with an analysis of the near tip region of a propagating fluid-driven fracture in a saturated permeable rock. The study attempts to accurately resolve the coupling between the physical processes - rock breakage, fluid pressure drop in the viscous fluid flow in the fracture, and fluid exchange between fracture and the rock - that exert influence on the hydraulic fracture propagation, yet occur over length scales often too small to be efficiently captured in existing coarse grid numerical models. We consider three fluid balance mechanisms: storage in the fracture, pore fluid leak-in from the rock into the fracture as the result of dynamic suction at the dilating crack tip, and fluid leak-off from the fracture into the rock as the fluid pressure in the fracture recovers with distance away from the tip. This process leads to the formation of a pore fluid circulation cell adjacent to the propagating fracture tip. We obtain the general numerical solution for the fracture opening and fluid pressure in the semi-infinite steadily propagating fracture model and fully characterize the solution within the problem parametric space. This allows to identify the parametric regimes of fracture propagation, assess the impact of pore fluid leak-in and the associated near-tip circulation cavity on the solution, and explore limitations of the widely-used, pressure-independent Carter’s leak-off model. The obtained solution can be further used as a tip element in a numerical realization of a solution for a transient growth of a finite fracture (e.g., within the Planar3D approach).

Engineering simulators used for steady-state multiphase pipe flows are commonly utilized to predict pressure drop. Such simulators are typically based on either empirical correlations or first-principles mechanistic models. The simulators allow evaluating the pressure drop in multiphase pipe flow with acceptable accuracy. However, the only shortcoming of these correlations and mechanistic models is their applicability. In order to extend the applicability and the accuracy of the existing accessible methods, a method of pressure drop calculation in the pipeline is proposed. The method is based on well segmentation and calculation of the pressure gradient in each segment using three surrogate models based on Machine Learning algorithms trained on a representative lab data set from the open literature. The first model predicts the value of a liquid holdup in the segment, the second one determines the flow pattern, and the third one is used to estimate the pressure gradient. To build these models, several ML algorithms are trained such as Random Forest, Gradient Boosting Decision Trees, Support Vector Machine, and Artificial Neural Network, and their predictive abilities are cross-compared. The proposed method for pressure gradient calculation yields $R^2 = 0.95$ by using the Gradient Boosting algorithm as compared with $R^2 = 0.92$ in case of Mukherjee and Brill correlation and $R^2 = 0.91$ when a combination of Ansari and Xiao mechanistic models is utilized. The method for pressure drop prediction is also validated on three real field cases. Validation indicates that the proposed model yields the following coefficients of determination: $R^2 = 0.806, 0.815$ and 0.99 as compared with the highest values obtained by commonly used techniques: $R^2 = 0.82$ (Beggs and Brill correlation), $R^2 = 0.823$ (Mukherjee and Brill correlation) and $R^2 = 0.98$ (Beggs and Brill correlation).