13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) ( ) ( ) 0 0 0 0 0 0 ( ) exp exp A A A A A i i B T GSt BT E E t GS R T GSt GS RT BE E E E E RGS R T GSt RT ⎡ ⎤ ⎛ ⎞ + Φ = − − − ⎢ ⎥ ⎜ ⎟ + ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ + − − − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ (4) in which Ei( ) is the exponential integral defined by ' ( ) ' ' x x i e E x dx x −∞ = ∫ (5) 2.2. An equation of state (EOS) for gas To account for the compressibility of gas under subsurface conditions, we adopt an EOS for methane developed by Duan et al. [16], since natural gas consists primarily of methane. The EOS, which is capable to describe the behavior of methane with high accuracy over wide temperature and pressure range (0 - 1000 oC and 0 - 800 MPa, respectively), takes the form 3 5 1 2 4 2 4 5 2 2 2 1 exp r r r r r r r r r r C C PV C C C T V V V V V V V γ γ β ⎛ ⎞ ⎛ ⎞ = + + + + + + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (6) where c rP P P/ = , c rT T T/ = , c rV V V/ = , P and T are the pressure and temperature of the gas, respectively, V = m/ρgas is the molar volume with m denoting the molar mass, Tc is the critical temperature above which methane can not be liquefied regardless of the pressure applied, Pc is the critical pressure required to liquefy methane at the critical temperature Tc, Vc = RTc/Pc, and 3 2 1 1 2 3 r r a a C a T T = + + 5 6 2 4 2 3 r r a a C a T T = + + , 8 9 3 7 2 3 r r a a C a T T = + + , 11 12 4 10 2 3 r r a a C a T T = + + , 5 3 r C T α = (7) The EOS contains 15 material constants: ai (i = 1, 2… 12), α, β and γ，which can be found in [16]. From Eq. (6) we can express gas pressure in terms of gas density and temperature as follows 2 2 4 4 5 5 2 4 5 2 2 2 2 2 2 2 2 2 ( ) 1 exp c r c gas c gas c gas c gas c gas gas c gas c gas c gas PTV BV CV DV EV P P m m m m m FV V V m m m ρ ρ ρ ρ ρ ρ ρ γ ρ γ ρ β ⎡ = = + + + + ⎢ ⎢⎣ ⎤ ⎛ ⎞ ⎛ ⎞ + + − ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎥ ⎝ ⎠ ⎝ ⎠⎦ (8) Figure 1. Subhorizontal periodically spaced collinear microcracks filled by oil and gas 2.3. Fracture mechanics model Consider a row of periodic collinear microcracks initially filled by oil within a source rock under continuous burial as shown in Figure 1, where 2a = 2a (t) denotes the crack length at time t and
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