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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 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.Water flowrate at sector interface with oil reservoir will be: LaTeX Math Block |
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| 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: LaTeX Math Block |
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| u(t,r_e) = M_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
where LaTeX Math Inline |
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body | M_a = \frac{k_a}{\mu_w} |
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| is aquifer mobility.Water flowrate becomes: LaTeX Math Block |
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| q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h_a \cdot M_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
Cumulative water flux: LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \int_0^t q^{\downarrow}_{AQ}(t) dt = \theta \cdot r_e \cdot h_a \cdot M_a \cdot \cdot \int_0^t \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} dt = \theta \cdot r_e \cdot h_a \cdot M_a \cdot \cdot \int_0^t \frac{1}{r_e} \frac{\partial p_a(t \cdot \frac{r_e^2}{\chi_a},R_E \cdot r_D)}{\partial r_D} \bigg|_{r=r_e} \frac{r_e^2}{\chi_a} dt_D = \theta r_e^2 \cdot h_a \cdot c_t \phi \cdot \int_0^t \frac{\partial p_a(t \cdot \frac{r_e^2}{\chi_a},r_e \cdot r_D)}{\partial r_D} \bigg|_{r_D=1} dt_D |
Substituting LaTeX Math Block Reference |
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| into LaTeX Math Block Reference |
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| leads to: LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot r_e^2 \cdot h_a \cdot c_t \cdot \phi \cdot \int_0^t \frac{\partial p_a(t \cdot \frac{r_e^2}{\chi_a},r_e \cdot r_D)}{\partial r_D} \bigg|_{r_D=1} dt_D |
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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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