Significance: Ten years ago, US natural gas cost 50% more than that from Russia. Now, it is threefold less. US gas prices plummeted because of the shale gas revolution. However, a key question remains: At what rate will the new hydrofractured horizontal wells in shales continue to produce gas? We analyze the simplest model of gas production consistent with basic physics of the extraction process. Its exact solution produces a nearly universal scaling law for gas wells in each shale play, where production first declines as 1 over the square root of time and then exponentially. The result is a surprisingly accurate description of gas extraction from thousands of wells in the United States’ oldest shale play, the Barnett Shale.

Assessing the production potential of shale gas can be assisted by constructing a simple, physics-based model for the productivity of individual wells. We adopt the simplest plausible physical model: one-dimensional pressure diffusion from a cuboid region with the effective area of hydrofractures as base and the length of horizontal well as height. We formulate a nonlinear initial boundary value problem for transient flow of real gas that may sorb on the rock and solve it numerically. In principle, solutions of this problem depend on several parameters, but in practice within a given gas field, all but two can be fixed at typical values, providing a nearly universal curve for which only the appropriate scales of time in production and cumulative production need to be determined for each well. The scaling curve has the property that production rate declines as one over the square root of time until the well starts to be pressure depleted, and later it declines exponentially. We show that this simple model provides a surprisingly accurate description of gas extraction from 8305 horizontal wells in the United States’ oldest shale play, the Barnett Shale. Good agreement exists with the scaling theory for 2133 horizontal wells in which production started to decline exponentially in less than 10 yr. We provide upper and lower bounds on the time in production and original gas in place. NOMENCLATURE Symbols and dimensions of key quantities Symbol SI dimensions Field dimensions c –compressibility Pa − 1 μ cip d –half-distance between hydrofractures m ft D –production decline coefficient k –permeability m 2 darcy K –partitioning coefficient m –gas pseudopressure Pa s − 1 psi 2 / cp m –cumulative produced mass kg ton M –Original gas in place kg ton M –molecular weight kmol lbmol H –formation thickness m ft p –pressure Pa psi q –volumetric flow rate m 3 s − 1 bbl/d Q –volumetric cumulative production m 3 bbl R —universal gas constant J/kmol-K psi-ft3/lb-mol RF –recovery factor S –saturation t –time in production s month, y T –temperature K °F, °C v ^ –specific volume m 3 / kg ft 3 / lbm y –molar composition Z –compressibility factor α –hydraulic diffusivity m 2 s − 1 ft 2 / y κ –dimensionless constant for gas production in square root phase K –dimensional constant for gas production in square root phase kg / s ton / month μ –gas viscosity Pa s cp ρ –density kg m − 3 lbm / ft 3 τ –time to interference s y ϕ –porosity Subscripts and Superscripts Symbol Meaning a adsorbed f (hydro)fracture g gas i initial L Langmuir S T stock tank conditions w water w c connate water ∼ dimensionless ∧ specific 0 reference or standard conditions

A proper First Law energy balance of the soybean biodiesel cycle shows that the overall efficiency of biodiesel production is 0.18, i.e., only 1 in 5 parts of the solar energy sequestered as soya beans, plus the fossil energy inputs, becomes biodiesel. Soybean meal is produced with an overall energetic efficiency of 0.38, but it is not a fossil fuel. If both biodiesel and soybean meal were treated as finished fossil fuels, the overall energy efficiency of their production would be 0.56, ∼ 40% higher than the average overall efficiency, 0.4, of producing ethanol and DDGS from corn patzek 2006z, patzek 2007a. This difference is caused by the high energy efficiency of producing soybean meal, which is 57% of soya beans by mass. Production of corn ethanol alone, with the average overall efficiency of 0.25, is more energy-efficient than production of biodiesel from soybeans with the overall efficiency of 0.18. The results of the current analysis are compared with those by NREL 1998 and Hill et al. 2006.

The switchgrass-driven process for producing ethanol has received much popular attention. However, a realistic analysis of this process indicates three serious limitations: (a) If switchgrass planted on 140 million hectares (the entire area of active U.S. cropland) were used as feedstock and energy source for ethanol production, the net ethanol yield would replace on average about 20% of today’s gasoline consumption in the U.S. (b) Because nonrenewable resources are required to produce ethanol from switchgrass, the incremental gas emissions would be on average 55 million tons of equivalent carbon dioxide per year to replace just 10% of U.S. automotive gasoline. (c) In terms of delivering electrical or mechanical power, ethanol from 1 hectare (10,000 m2) of switchgrass is equivalent, on average, to 30 m2 of low-efficiency photovoltaic cells. This analysis suggests that investing toward more efficient and durable solar cells, and batteries, may be more promising than investing in a process to convert switchgrass to ethanol.