Boundary value problem with a displacement for a hyperbolic equation of third order with a derivative under boundary conditions

В работе исследована краевая задача со смещением для гиперболического уравнения третьего порядка, которая содержит производную в граничных условиях. Доказана теорема единственности и существования регулярного решения исследуемой задачи. The paper investigates a boundary value problem with a shift for a third-order hyperbolic equation, which contains a derivative in the boundary conditions. A uniqueness and existence theorem for a regular solution of the problem under study is proved.

Cauchy problem for a substantially loaded hyperbolic equation

В работе проводится исследование задачи Коши для существенно нагруженного уравнения колебания одномерной струны. Приводятся примеры характеристических многообразий, для которых задача Коши поставлена корректно, а также нехарактеристических многообразий, для которых задача Коши поставлено некорректно. In this work, we study the Cauchy problem for a substantially loaded vibration equation of a one-dimensional string. Examples are given of characteristic manifolds for which the Cauchy problem is posed correctly, as well as non-characteristic manifolds for which the Cauchy problem is posed incorrectly.

On the stability of recovering two sources and initial status in a stochastic hyperbolic-parabolic system

Consider an inverse problem of determining two stochastic source functions and the initial status simultaneously in a stochastic thermoelastic system, which is constituted of two stochastic equations of different types, namely a parabolic equation and a hyperbolic equation. To establish the conditional stability for such a coupling system in terms of some suitable norms revealing the stochastic property of the governed system, we first establish two Carleman estimates with regular weight function and two large parameters for stochastic parabolic equation and stochastic hyperbolic equation, respectively. By means of these two Carleman estimates, we finally prove the conditional stability for our inverse problem, provided the source in the elastic equation be known near the boundary and the solution be in a prior bound set. Due to the lack of information about the time derivative of wave field at final moment, the stability index with respect to the wave field at final time is found to be halved, which reveals the special characteristic of our inverse problem for the coupling system.

Numerical simulation of thermal processes occurring in materials under the action of femtosecond laser pulses

In this work, a numerical study of the solutions of the parabolic and hyperbolic equations of heat conduction with the same physical parameters is carried out and a comparative analysis of the results obtained is carried out. The mathematical formulation of the problem is discussed. The action of the laser is taken into account through the source function, which was chosen as a double femtosecond laser pulse. In the hyperbolic equation, in contrast to the parabolic one, there is an additional parameter that characterizes the relaxation time of the heat flux. In addition, the source of the hyperbolic equation contains an additional term - the derivative of the power density of the source of the parabolic equation. This means that the temperature of the sample is influenced not only by the power density of the source, but also by the rate of its change. The profiles of the sample temperature at different times and its dynamics at different target depths are shown. The calculations were carried out for different time delays between pulses and for different relaxation parameters.

Impediments to Refracturing Success in Shale Reservoirs

Stimulations in early horizontal wells in most shale plays are characterized by few and widely spaced perforation clusters, and low amounts of injected fracturing fluid and proppant. Low recovery from these wells has motivated refracturing although outcomes have been interpreted to range from successful to minimal impact based on operator specific evaluations. To tailor available technologies and improve quantification of upsides, there is need for mapping the spatial distribution of remaining resources and developing simpler but reliable analytical techniques. In this study, hydraulic fractures were assumed to be planar in a matrix with low porosity and ultra-low permeability. Consideration of natural fractures and their interaction with stimulation fluids led to addition of distributed fracture networks adjacent to the planar hydraulic fractures to define the composite fracture corridors. A sector model with the aforementioned architecture was used in reservoir simulation to investigate induced temporal and spatial drainage. These findings were used to explain the efficacy of widely used refracturing techniques and how post-refracturing reservoir response can be analyzed. Results from reservoir simulation showed remaining reserves were in the matrix between earlier placed hydraulic fractures aligned along initial perforation clusters, and beyond tips of hydraulic fractures. Upside from refracs could come from creation of new fractures in the matrix between earlier placed fractures and extension of tips of early fractures into virgin matrix. Assessment of these scenarios found the former to be optimal although depletion and existing perforations would limit the stimulation efficiency of new perforations. The second scenario would require large volumes of fracturing fluid to re-initiate fracture propagation. Yet this could trigger interference with offsets or affect drilling and stimulation of planned wells in adjacent acreage. For treatment efficiency, re-casing horizontal wells with competent liners and use of coiled tubing with straddle packers appears a better solution for bypassing old perforations. For the near wellbore and far field, re-stimulating new perforations at low injection rates could allow extension of fractures in virgin matrix surrounded by depleted strata. Real-time surveillance would be essential for mapping flow paths of refracturing fluid. For assessment of refracturing, actual and simulated flow exhibited persistent linear flow (PLF) that could be matched by Arps hyperbolic equation with a b value of 2. Incorporation of a novel fracture geometry factor (FGF) yielded an Arps-based equation that was tested on North American shale refracturing cases that often use post-treatment peak rate and wellhead pressure as measures of success. This study identified factors hindering the success of refracturing and proposed a modified Arps hyperbolic equation to analyze refracturing production data.