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The mobility of a foam depends heavily on its texture, which is the distribution of bubble sizes in the dispersion. To incorporate this variable in a mechanistic simulator, the usual conservation equations are coupled with balances on the densities of flowing and stationary bubbles in the foam. This approach to modeling foam flow is illustrated with a simulation of a displacement in which foam is generated in situ by capillary snap-off.

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...

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Relative permeability and capillary pressure functions define how much oil can be recovered and at what rate. These functions, in turn, depend critically on the geometry and topology of the pore space, on the physical characteristics of the rock grains and the fluids, and on the conditions imposed by the recovery process. Therefore, imaging and characterizing the rock samples and the fluids can add crucial insight into the mechanisms that control field-scale oil recovery. The fundamental equations of immiscible flow in the imaged samples are solved, and one can elucidate how relative permeability and capillary pressure functions depend on wett ability, interfacial tension and the interplay among viscous, capillary and gravitational forces. This knowledge enables one to answer questions such as: Can a change of injected brine salinity increase oil recovery and by how much? How much more oil would be recovered if advancing contact angles could be modified? Does water injection help to recover sufficiently more oil or is it just for pressure maintenance? How can water imbibition be enhanced and oil trapping limited? Can relative permeabilities be modified with a polymer or with a chemical agent, such as an electrolyte or surfactant? Can one rely on gravity drainage of oil films to increase recovery? These and many other questions may be answered through a combination of imaging and calculations presented here. This paper summarizes the development of a complete quasi-static pore network simulator of two-phase flow, "ANetSim," and its validation against Statoil's state-of-the-art proprietary simulator. ANetSim has been implemented in MATLAB® and it can run on any platform. Three-dimensional, disordered networks with complex pore geometry have been used to calculate primary drainage and secondary imbibition capillary pressures and relative permeabilities. The results presented here agree well with the Statoil simulations and experiments. Introduction The world works differently at different scales [1], and earth sciences must therefore rely on different methods of modeling the diverse earth systems that range in size from atoms and molecules to the whole planet. Fig. 1 shows the characteristic volume scales encountered in computational earth sciences, and Fig. 2 shows the corresponding time scales. The respective scales of interest in this paper are highlighted in yellow. Both figures underscore the need for an understanding of the interactions among the numerous characteristic scales in earth sciences and for the appropriate computational tools. In particular, it is apparent that the molecular level approach, such as the Lattice-Gas or Boltzmann methods [2], cannot be extended to describe rock cores, oil-reservoirs, contaminant plumes and the earth's crust. Conversely, a continuum description of a gas condensate reservoir will fail to describe the motion of individual gas molecules that condense into thin films covering the rock surfaces.

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