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| \frac{d Q^{\downarrow}_{AQ}}{dt} = q^{\downarrow}_{AQ}(t)= B \cdot \int_0^t W_{eD}(t - \tau) \dot p d\tau |
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| q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_edQ^{\downarrow}_{AQ}}{dt} |
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| anchor | VEHPCT |
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| alignment | left |
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| pW_a{eD}(t, r)= p(0) + \int_0^t p_1 \left(0^{t} \frac{(t-\tau) \cdot \chi}{r_e^2}, \frac{r}{r_e} \right) \dot p(\tau) d\tau | | LaTeX Math Block |
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| \dot p(\tau) = \frac{d p}{d \tau}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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| Expand |
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Transient flow in Radial Composite Reservoir:
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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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| p_a(t = 0, r)= p(0) |
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| p_a(t, r=r_e) = p(t) |
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| anchor | p1_PSS |
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| alignment | left |
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| \frac{\partial p_a}{\partial r}
\bigg|_{(t, r=r_a)} = 0 |
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Consider a pressure convolution:
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| 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 pressure from | LaTeX Math Block Reference |
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honors the whole set of equations | LaTeX Math Block Reference |
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–| LaTeX Math Block Reference |
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and as such defines a unique solution of the above problem.The the water flowrate at interface with oil reservoir will be: | LaTeX Math Block |
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| q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
and cumulative flux: | LaTeX Math Block |
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| \frac{d Q^{\downarrow}_{AQ}}{dt} = q^{\downarrow}_{AQ}(t) |
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See Also
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Petroleum Industry / Upstream / Subsurface E&P Disciplines / Field Study & Modelling / Aquifer Drive / Aquifer Drive Models
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