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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi_a}{r_e^2}, \frac{r_a}{r_e} \right) \dot p(\tau) d\tau |
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| q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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| W_{eD}(t, r)= \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r_D = 1} dt_D |
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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} |
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| p_1(t_D = 0, r_D)= 0 |
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| p_1(t_D, r_D=1) = 1 |
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| \frac{\partial p_1}{\partial r_D}
\bigg|_{(t_D, r_D=r_a/r_e)} = 0 |
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| Q^{\downarrow}_{AQ}(t)= B \cdot \sum_\alpha W_{eD}
\left( \frac{ (t-\tau_\alpha) \chi_a}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_\alpha |
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This computational scheme is called superposition in time domain.
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The benefit of VEH approach is that net pay formation pressure history
is usually
known known and the aquifer drive model parameters
can be calibrated using
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iterations.
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