E. V. Dontsov

An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity and leak-off

This paper develops a closed-form approximate solution for a penny-shaped hydraulic fracture whose behaviour is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness and fluid leak-off. The primary assumption that permits one to construct the solution is that the fracture behaviour is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared with the existing solutions for all limiting regimes of propagation. Then, a solution map, which indicates applicability regions of the limiting solutions, is constructed. It is also shown that the constructed approximation accurately captures the scaling that is associated with the transition from any one limiting solution to another. The developed approximation is tested against a reference numerical solution, showing that accuracy of the fracture width and radius predictions lie within a fraction of a per cent for a wide range of parameters. As a result, the constructed approximation provides a rapid solution for a penny-shaped hydraulic fracture, which can be used for quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.

An approximate solution for a plane strain hydraulic fracture that accounts for fracture toughness, fluid viscosity, and leak-off

The goal of this paper is to develop an approximate solution for a propagating plane strain hydraulic fracture, whose behavior is determined by a combined interplay of fluid viscosity, fracture toughness, and fluid leak-off. The approximation is constructed by assuming that the fracture behavior is primarily determined by the three-process (viscosity, toughness, and leak-off) multiscale tip asymptotics and the global fluid volume balance. First, the limiting regimes of propagation of the solution are considered, that can be reduced to an explicit form. Thereafter, applicability regions of the limiting solutions are investigated and transitions from one limiting solution to another are analyzed. To quantify the error of the constructed approximate solution, its predictions are compared to a reference numerical solution. Results indicate that the approximation is able to predict hydraulic fracture parameters for all limiting and transition regimes with an error of under one percent. Consequently, this development can be used to obtain a rapid solution for a plane strain hydraulic fracture with leak-off, which can be used for quick estimations of fracture geometry or as a reference solution to evaluate accuracy of more advanced hydraulic fracture simulators.

A Lagrangian Approach to Modelling Proppant Transport with Tip Screen-Out in KGD Hydraulic Fractures

AbstractThis study introduces a continuum approach to model proppant transport in hydraulic fractures in a Lagrangian frame of reference. The model for the proppant transport is based on the recently obtained slurry flow solution inside a channel, where the latter utilizes a phenomenological constitutive relationship for a slurry. This approach allows us to describe the transition from Poiseuille flow with an effective viscosity to Darcy flow as the particle concentration increases towards the maximum value. The algorithm is presented for the one-dimensional case, for which propagation of a symmetric Kristinovich–Zheltov–Geertsma–De Klerk fracture is considered. To examine the effectiveness of the Lagrangian approach for proppant transport modelling, a set of parameters, for which proppant particles reach the fracture tip and cause the development of a proppant plug is selected. In this situation, the coupling between the hydraulic fracture propagation and proppant transport is the most significant. To estimate the accuracy of the Lagrangian proppant transport model, the results are compared to the predictions of an Eulerian proppant transport model, which utilizes the same algorithm for hydraulic fracture propagation. It is shown that, although both approaches have the same convergence rate, the error of the Lagrangian approach is three to five times smaller, which depends on the number of proppant elements used in the Lagrangian approach. This permits us to use a coarser mesh for hydraulic fracture propagation, and to obtain results with similar accuracy up to a hundred times faster.

Effect of low-frequency modulation on the acoustic radiation force in newtonian fluids

This study investigates the nature of the acoustic radiation force (ARF) in Newtonian fluids generated by a high-intensity sound field in situations when the latter is modulated using frequencies that are, relative to the frequency of sound, on the order of the Mach number. In the context of nonlinear acoustics, problems of this class turn out to be unresponsive to the usual asymptotic reduction via the concept of the ARF owing to the fact that the mean acoustic fields, computed as averages over the period of sound vibrations, retain rapid oscillation features of the latter. To meet the challenge, an asymptotic treatment is pursued within the framework of plane waves via a scaling paradigm that splits the temporal variable into “fast” and “slow” time, permitting one to track the contribution of (time-harmonic) sound and its modulation separately in the solution. In this setting the second-order asymptotic solution, written in terms of the “fast” time averages of acoustic fields, is shown to (i) be free of...

Processes of Dissolution and Hydrate Formation behind a Shock Wave in a Liquid with Carbon Dioxide Bubbles

The processes of dissolution and hydrate formation behind the front of a shock wave of moderate amplitude in water with carbon dioxide bubbles are studied experimentally at various initial static pressures. The influence of a surface-active substance (SAS) in the medium on the processes of dissolution and hydrate formation behind the shock wave is investigated. It is demonstrated that behind a shock wave of moderate amplitude in a liquid with carbon dioxide bubbles an intensive process of dissolution and hydrate formation takes place, resulting in complete disappearance of the gas phase in a matter of a few milliseconds. The presence of an SAS in the medium does not significantly influence the processes of dissolution and hydrate formation within the investigated periods of time.

The 'sixth sense' of ultrasound: probing nonlinear elasticity with acoustic radiation force.

Prompted by a recent finding that the magnitude of the acoustic radiation force (ARF) in isotropic tissue-like solids depends linearly on a particular third-order modulus of elasticity-hereon denoted by C, this study investigates the possibility of estimating C from the amplitude of the ARF-generated shear waves. The featured coefficient of nonlinear elasticity, which captures the incipient nonlinear interaction between the volumetric and deviatoric modes of deformation, has so far received only a limited attention in the context of soft tissues due to the fact that the latter are often approximated as (i) fluid-like when considering ultrasound waves, and (ii) incompressible under static deformations. On establishing the analytical and computational platform for the proposed sensing methodology, the study proceeds with applying the prototype technique toward estimating via ARF the third-order modulus C in a series of tissue-mimicking phantoms. To help validate the concept and its implementation, the germane third-order modulus is independently estimated in each phantom via an established technique known as acoustoelasticity. The C-estimates obtained respectively via acoustoelasticity and the new theory of ARF show a significant degree of consistency. The key features of the new sensing methodology are that: (a) it requires no external deformation of a material other than that produced by the ARF, and (b) it estimates the nonlinear C-modulus locally, over the focal region of an ultrasound beam-where the shear waves are being generated.

Fluid Flow in Unconventional Gas Reservoirs

State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou 221116, China Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77004, USA School of Mechanical and Mining Engineering, The University of Queensland, St. Lucia, QLD 4072, Australia Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China Department of Earth Sciences Institute of Geophysics, ETH Zürich, 8092 Zurich, Switzerland

Direction-independent algorithm for simulating nonlinear pressure waves

AbstractThis study formulates a frequency-domain computational scheme for simulating nonlinear wave propagation in a homogeneous medium governed by the Westervelt equation. The need for such numeri...

A new paradigm for proppant schedule design

This study introduces a novel methodology for the design of the proppant pumping schedule for a hydraulic fracture, in which the final proppant distribution along the crack is prescribed. The method is based on the assumption that the particles have relatively small impact on the fracture propagation, unless they reach the tip region. This makes it possible to relate the proppant velocity to the clear fluid velocity inside the fracture, which is calculated assuming no proppant. Having the history of the clear fluid velocity distribution, the prospective proppant motion can be computed. Then, volume balance is used to relate the final concentration at some point inside the fracture to the corresponding input concentration at a specific time instant, which helps to avoid solving an inverse problem. One exceptional feature of the approach lies in the fact that it is applicable to multiple fracture geometries and can be implemented using various hydraulic fracturing simulators. To verify the technique, two fracture geometries are considered Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D (P3D). It is shown that the developed approach is capable of properly estimating the pumping schedule for both geometries. In particular, the proppant placement along the fracture at the end of the pumping period, calculated according to the adopted proppant transport model, shows close agreement with the design distribution. The comparison with Nolte’s scheduling scheme shows that the latter is not always accurate, and cannot capture the essential differences between the schedules for the fracture geometries considered.

Dual-time-scale method for modeling of the nonlinear amplitude-modulated ultrasound fields

This study focuses on modeling of the nonlinear acoustic wave propagation in situations when the amplitude of the focused ultrasound field is modulated by a low-frequency signal. This problem is relevant to both ultrasound imaging applications entailing the use of the acoustic radiation force, and treatment applications such as histotripsy. The difficulty of predicting the pressure wavefield lies in a fact that the excessive length of the low-frequency modulated signal may significantly increase the computational effort. To tackle the problem, this study utilizes the dual-time-scale approach, where two temporal variables are introduced to distinguish between ultrasound-scale and modulation-scale variations. In this case, the Westervelt-type equation can be effectively solved using hybrid time-frequency algorithm for any transient (sufficiently smooth) modulation envelope. To validate the proposed approach, the Khokhlov-Zabolotskaya-Kuznetsov equation was solved in the time domain for an example pressure p...