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Natural gas from tight shale formations will provide the United States with a major source of energy over the next several decades. Estimates of gas production from these formations have mainly relied on formulas designed for wells with a different geometry. We consider the simplest model of gas production consistent with the basic physics and geometry of the extraction process. In principle, solutions of the model depend upon many parameters, but in practice and within a given gas field, all but two can be fixed at typical values, leading to a nonlinear diffusion problem we solve exactly with a scaling curve. The scaling curve production rate declines as 1 over the square root of time early on, and it later declines exponentially. This simple model provides a surprisingly accurate description of gas extraction from 8,294 wells in the United States' oldest shale play, the Barnett Shale. There is good agreement with the scaling theory for 2,057 horizontal wells in which production started to decline exponentially in less than 10 y. The remaining 6,237 horizontal wells in our analysis are too young for us to predict when exponential decline will set in, but the model can nevertheless be used to establish lower and upper bounds on well lifetime. Finally, we obtain upper and lower bounds on the gas that will be produced by the wells in our sample, individually and in total. The estimated ultimate recovery from our sample of 8,294 wells is between 10 and 20 trillion standard cubic feet.hydrofracturing | shale gas | scaling laws | energy resources | fracking T he fast progress of hydraulic fracturing technology (SI Text, Figs. S1 and S2) has led to the extraction of natural gas and oil from tens of thousands of wells drilled into mudrock (commonly called shale) formations. The wells are mainly in the United States, although there is significant potential on all continents (1). The "fracking" technology has generated considerable concern about environmental consequences (2, 3) and about whether hydrocarbon extraction from mudrocks will ultimately be profitable (4). The cumulative gas obtained from the hydrofractured horizontal wells and the profits to be made depend upon production rate. Because large-scale use of hydraulic fracturing in mudrocks is relatively new, data on the behavior of hydrofractured wells on the scale of 10 y or more are only now becoming available.There is more than a century of experience describing how petroleum and gas production declines over time for vertical wells. The vocabulary used to discuss this problem comes from a seminal paper by Arps (5), who discussed exponential, hyperbolic, harmonic, and geometric declines. Initially, these types of decline emerged as simple functions providing good fits to empirical data. Thirty-six years later, Fetkovich (6) showed how they arise from physical reasoning when liquid or gas flows radially inward from a large region to a vertical perforated tubing, where it is collected. For specialists in this area, the simplicity and familiarity of hyperbolic de...

This paper demonstrates that rapid fracture of ideal brittle lattices naturally involves phenomena long seen in experiment, but which have been hard to understand from a continuum point of view. These idealized models do not mimic realistic microstructure, but can be solved exactly and understood completely. First it is shown that constant velocity crack solutions do not exist at all for a range of velocities starting at zero and ranging up to about one quarter of the shear wave speed. Next it is shown that above this speed cracks are by and large linearly stable, but that at sufficiently high velocity they become unstable with respect to a nonlinear micro-cracking instability. The way this instability works itself out is related to the scenario known as intermittency, and the basic time scale which governs it is the inverse of the amount of dissipation in the model. Finally, we compare the theoretical framework with some new experiments in Plexiglas, and show that all qualitative features of the theory are mirrored in our experimental results.

Basic equations of nonequilibrium thermo field dynamics of dense quantum systems are presented. A formulation of nonequilibrium thermo field dynamics has been performed using the nonequilibrium statistical operator method by D.N.Zubarev. Hydrodynamic equations have been obtained in thermo field representation. Two levels of the description of kinetics and hydrodynamics of a dense nuclear matter are considered. The first one is a quantum system with strongly coupled states, the second one is a quark-gluon plasma. Generalized transfer equations of a consistent description of kinetics and hydrodynamics have been obtained, as well as limiting cases are considered.

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