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. Water flowrate at  sector interface with oil reservoir will be: q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h_a \cdot u(t,r_e) |
where  is flow velocity at aquifer contact boundary, which is: u(t,r_e) = M_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
where  is aquifer mobility. Water flowrate becomes: q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
Cumulative water flux: Q^{\downarrow}_{AQ}(t) = \int_0^t q^{\downarrow}_{AQ}(t) dt = C_a \cdot \int_0^t \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} dt |
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