Nonlinear Quantum Evolution Equations to Model Irreversible Adiabatic Relaxation with Maximal Entropy Production and Other Nonunitary Processes

We first discuss the geometrical construction and the main mathematical features of the maximum-entropy-production/steepest-entropy-ascent nonlinear evolution equation proposed long ago by this author in the framework of a fully quantum theory of irreversibility and thermodynamics for a single isolated or adiabatic particle, qubit, or qudit, and recently rediscovered by other authors. The nonlinear equation generates a dynamical group, not just a semigroup, providing a deterministic description of irreversible conservative relaxation towards equilibrium from any non-equilibrium density operator. It satisfies a very restrictive stability requirement equivalent to the Hatsopoulos-Keenan statement of the second law of thermodynamics. We then examine the form of the evolution equation we proposed to describe multipartite isolated or adiabatic systems. This hinges on novel nonlinear projections defining local operators that we interpret as ``local perceptions'' of the overall system's energy and entropy. Each component particle contributes an independent local tendency along the direction of steepest increase of the locally perceived entropy at constant locally perceived energy. It conserves both the locally-perceived energies and the overall energy, and meets strong separability and non-signaling conditions, even though the local evolutions are not independent of existing correlations. We finally show how the geometrical construction can readily lead to other thermodynamically relevant models, such as of the nonunitary isoentropic evolution needed for full extraction of a system's adiabatic availability.Comment: To appear in Reports on Mathematical Physics. Presented at the The Jubilee 40th Symposium on Mathematical Physics, "Geometry & Quanta", Torun, Poland, June 25-28, 200

Energy-Efficient Fracturing Based on Stress-Coupled Perforation

Hydraulic fracturing is one of the key technologies for reservoir stimulation in low-permeability/unconventional oil and gas fields. In response to the high energy consumption and greenhouse gas emissions caused by extreme flow-limiting perforation in hydraulic fracturing, stress-coupled perforation technology has been proposed to promote low-energy consumption and high efficiency in artificial fracturing. Based on the stress distribution in perforation hole and fracture propagation mechanism, a mathematical model for fracture propagation was established based on linear elastic fracture mechanics theory. Taking into account rock mechanical parameters, tensile effects at the crack tip, stress on both sides of the main crack and fracturing parameters, the real-time stress distribution and fracturing energy consumption were calculated using Monte Carlo random method and Newton's iterative method. With the unit fracture area energy consumption, total fracture area, and fracture uniformity as objective functions, the number and diameter of perforation holes were fully coupled and optimized. The developed simulator balances the calculation efficiency and accuracy of multiple fracture propagation. The study shows that under the condition of the same reservoir fracture volume, stress-coupled perforation technology will reduce energy consumption by 37%. This technology has been successfully applied to the reservoir stimulation of Fan Ye 1 well horizontal well in Jiyang Depression, improving the energy utilization efficiency of hydraulic fracturing and reducing carbon emissions

Evaluating Data Assimilation Algorithms

Data assimilation leads naturally to a Bayesian formulation in which the posterior probability distribution of the system state, given the observations, plays a central conceptual role. The aim of this paper is to use this Bayesian posterior probability distribution as a gold standard against which to evaluate various commonly used data assimilation algorithms. A key aspect of geophysical data assimilation is the high dimensionality and low predictability of the computational model. With this in mind, yet with the goal of allowing an explicit and accurate computation of the posterior distribution, we study the 2D Navier-Stokes equations in a periodic geometry. We compute the posterior probability distribution by state-of-the-art statistical sampling techniques. The commonly used algorithms that we evaluate against this accurate gold standard, as quantified by comparing the relative error in reproducing its moments, are 4DVAR and a variety of sequential filtering approximations based on 3DVAR and on extended and ensemble Kalman filters. The primary conclusions are that: (i) with appropriate parameter choices, approximate filters can perform well in reproducing the mean of the desired probability distribution; (ii) however they typically perform poorly when attempting to reproduce the covariance; (iii) this poor performance is compounded by the need to modify the covariance, in order to induce stability. Thus, whilst filters can be a useful tool in predicting mean behavior, they should be viewed with caution as predictors of uncertainty. These conclusions are intrinsic to the algorithms and will not change if the model complexity is increased, for example by employing a smaller viscosity, or by using a detailed NWP model