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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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| \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_a}{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 within sector angle at 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 \int_0^t \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} dt |
Substituting | LaTeX Math Block Reference |
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into leads to:| LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot r_e \cdot h_a \cdot M_a \cdot \int_0^t d\xi \ \frac{1\partial }{\partial r_e} \frac{\partialleft[
\int_0^\xi p_a(t1 \cdotleft( \frac{r_e^2}{\chi_a},R_E \cdot r_D)}{\partial r_D} \bigg|(\xi-\tau)\chi_a}{r_e^2}, \frac{r}{r_e} \right) \, \dot p(\tau) d\tau
\right]_{r=r_e} \frac{r_e^2}{\chi_a} dt_D |
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| Q^{\downarrow}_{AQ}(t) = \theta r_e^2 \cdot h_a \cdot cM_ta \phi \cdot \int_0^t d\xi \ \frac{\partial }{\partial p_a(t \cdot \frac{r_e^2}{r_D} \left[
\int_0^\xi p_1 \left( \frac{(\xi-\tau)\chi_a},{r_e \cdote^2}, r_D)}{\partial r_D} \bigg| \right) \, \dot p(\tau) d\tau
\right]_{r_D=1} dt_D = B |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \int_0^t d\xi \
\int_0^\xi \frac{\partial p_a(t \cdot_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1} \, \dot p(\tau) d\tau
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The above integral represents the integration over the area in plane (see Fig. 1):| LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \iint_D d\xi \ d\tau \, \dot p(\tau)
\frac{\partial p_1\chi_a},r_e \cdot r_D)}{\partial r_D} \bigg\left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1} dt_D
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| Fig. 1. Illustration of the integration area in plane |
Changing the integration order from to 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 rh_e^2a \cdot hM_a \cdot c\int_t0^t d\tau \int_\cdottau^t d\phixi \cdot \int_0^t \left[dot p(\tau)
\frac{\partial p_1}{\partial r_D} \left( \int_0^{t} p_a(t \cdot \frac{r_e^2}{\chi_a},r_e \cdot r_D)frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1}
=
\theta \cdot h_a \cdot M_a \cdot \int_0^t \dot p(\tau) d\tau \int_\tau^t d\xi \
\frac{\partial p_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right]) \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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