13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ( ) { } ( ) ( ) 2 2 0 0 0 0 0 1 exp 16 1 exp ln sec 2 th th gas oil M t b p M t a E b ν π ρ ρ π − ⎡−Φ ⎤ − Δ ⎡−Φ ⎤ ⎡ ⎤ ⎛ ⎞ ⎣ ⎦ ⎣ ⎦ + = ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ (18) where th gas ρ is the corresponding gas density at time t0. 2.4. Simulation of subcritical crack propagation and coalescence using finite difference A finite difference formulation is used to study the coupling between gas generation from oil degradation and gas expulsion through microfracture propagation and coalescence. Consider the microcrack propagation from time t0 to current time t. We subdivide the time domain into N intervals to construct a mesh of equally-spaced grids: t0, t1,…, tN-1, tN with tN=t. The following notations are adopted. ai=a(ti) and ( ) i gas gas it ρ ρ = represent the half crack length and the corresponding gas density at time step ti , respectively. Replace the derivative in Eq. (14) by a forward difference approximation at time ti, then we obtain the following expression for the half crack length at ti+1 ( ) /2 /2 1 0 1 [ ( ) ] (2 ) tan ( ) 2 n i n n i i i gas s i i i a a a A P g H St b t t b π ρ ρ + + ⎡ ⎤ ⎛ ⎞ = + − + − ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ (19) Neglecting the mutual solubility of gas and oil, we obtain the oil and gas volumes at ti+1 as follows ( ) 0 1 1 exp i i oil oil M t V ρ + + ⎡⎣ −Φ ⎤⎦ = , ( ) { } 0 1 1 1 1 exp i i gas i gas M t V ρ + + + − ⎡⎣ −Φ ⎤⎦ = (20) where 1 i gas ρ + is the gas density at time step t i+1. The volume of the crack at ti+1 is by Eq. (11) applied at step ti+1. The requirement that crack volume must be equal to the sum of oil and gas volumes yields the excess pressure Δpi+1 at ti+1, ( ) ( ) { } ( ) 2 2 1 1 1 1 0 1 1 exp 16 1 exp ln sec 2 i i i i i oil gas t b t a p M E b ν π ρ ρ π + + + + + ⎡ ⎤ ⎧ ⎫ − ⎡−Φ ⎤ − ⎡−Φ ⎤ ⎡ ⎤ ⎪ ⎪ ⎛ ⎞ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ Δ = + ⎨ ⎬ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦ (21) Note that ( ) 1 1 0 1 ( ) i i gas s i p P g H St ρ ρ + + + Δ = − + , and then we can solve this equation for the unknown gas density 1 i gas ρ + . Consequently the excess pressure Δp i+1 at ti+1 can be obtained using Eq. (21). To drive the crack growth subcritically, Δpi+1 must satisfy the following condition 1 1 / 2 tan 2 i i th a p K b b π + + Ι ⎛ ⎞ Δ ≥ ⎜ ⎟ ⎝ ⎠ (22) Otherwise, we need to adjust time ti+1 by solving Eq. (17) and (18) with a0, t0 and th gas ρ replaced by ai+1, ti+1 and 1+i gas ρ , respectively. 3. Numerical Results and Discussions This section presents numerical examples to illustrate effects of initial burial depth and crack spacing on the excess pressure evolution with time and crack propagation distance for a single crack, and propagation and coalescence of collinear cracks. Since shales are major source rocks for oil and natural gas, in our simulation we study shales for illustration purpose. Typically thermal cracking of oil is initiated at temperatures of 120-160 oC, and ultimate conversion to methane may occur at
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