ABSTRACT

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 Pa1 μ cip 
d–half-distance between hydrofractures ft 
D–production decline coefficient   
k–permeability m2 darcy 
K–partitioning coefficient   
m–gas pseudopressure Pas1 psi2/cp 
m–cumulative produced mass kg ton 
M–Original gas in place kg ton 
M–molecular weight kmol lbmol 
H–formation thickness ft 
p–pressure Pa psi 
q–volumetric flow rate m3s1 bbl/d 
Q–volumetric cumulative production m3 bbl 
R—universal gas constant J/kmol-K psi-ft3/lb-mol 
RF–recovery factor   
S–saturation   
t–time in production month, y 
T–temperature °F, °C 
v^–specific volume m3/kg ft3/lbm 
y–molar composition   
Z–compressibility factor   
α–hydraulic diffusivity m2s1 ft2/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 kgm3 lbm/ft3 
τ–time to interference 
ϕ–porosity   
Subscripts and Superscripts 
Symbol Meaning  
a adsorbed  
f (hydro)fracture  
g gas  
i initial  
L Langmuir  
ST stock tank conditions  
w water  
wc connate water  
∼ dimensionless  
∧ specific  
reference or standard conditions  
NOMENCLATURE 
Symbols and dimensions of key quantities 
Symbol SI dimensions Field dimensions 
c–compressibility Pa1 μ cip 
d–half-distance between hydrofractures ft 
D–production decline coefficient   
k–permeability m2 darcy 
K–partitioning coefficient   
m–gas pseudopressure Pas1 psi2/cp 
m–cumulative produced mass kg ton 
M–Original gas in place kg ton 
M–molecular weight kmol lbmol 
H–formation thickness ft 
p–pressure Pa psi 
q–volumetric flow rate m3s1 bbl/d 
Q–volumetric cumulative production m3 bbl 
R—universal gas constant J/kmol-K psi-ft3/lb-mol 
RF–recovery factor   
S–saturation   
t–time in production month, y 
T–temperature °F, °C 
v^–specific volume m3/kg ft3/lbm 
y–molar composition   
Z–compressibility factor   
α–hydraulic diffusivity m2s1 ft2/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 kgm3 lbm/ft3 
τ–time to interference 
ϕ–porosity   
Subscripts and Superscripts 
Symbol Meaning  
a adsorbed  
f (hydro)fracture  
g gas  
i initial  
L Langmuir  
ST stock tank conditions  
w water  
wc connate water  
∼ dimensionless  
∧ specific  
reference or standard conditions  

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