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Transient flow in Radial Composite Reservoir



Equations

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q^{\downarrow}_{AQ}(t)= J \cdot \frac{\partial p_a}{\partial r}



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\frac{d Q^{\downarrow}_{AQ}}{dt} = q^{\downarrow}_{AQ}(t)



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p_a(t, r)= p(0) + \int_0^t p_1(t-\tau) \dot p(\tau) d\tau


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\frac{\partial p_1}{\partial t_D} =  \frac{\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_D
} LaTeX Math Block
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\frac{d Q^{\downarrow}_{AQ}}{dt
}
= q^{\downarrow}_{AQ}(t)

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p_1(t
 = 0,
 
r)
= 0
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p_a(t
, r)= 
p(
0
) + \int_0^t p_1(t-\tau) \dot p(\tau) d\tau


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p_1(t, r=r_e) = 1



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\frac{\partial p_1}{\partial r_D}(t, r=r_a) = 0



 Expand
titleDerivation

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\frac{\partial p_a}{\partial t} = \chi \cdot \left[ \frac{\partial^2 p_a}{\partial r^2} + \frac{1}{r}\cdot \frac{\partial p_a}{\partial r} \right]

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