Transient flow in Radial Composite Reservoir:
\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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\frac{\partial p_a}{\partial r}
\bigg|_{(t, r=r_a)} = 0 |
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Consider a pressure convolution:
p_a(t, r) = p(0) + \int_0^t p_1 \left(\frac{(t-\tau) \cdot \chi}{r_e^2}, \frac{r}{r_e} \right) \dot p(\tau) d\tau |
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\dot p(\tau) = \frac{d p}{d \tau} |
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One can easily check that honors the whole set of equations – and as such defines a unique solution of the above problem. The the water flowrate at interface with oil reservoir will be: q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
and cumulative flux: \frac{d Q^{\downarrow}_{AQ}}{dt} = q^{\downarrow}_{AQ}(t) |
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