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Assumptions
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| Transient flow in Radial Composite Reservoir |
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| Fig. 1. VEH aquifer drive schematic |
Equations
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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD}(t - \tau) \dot p d\tau |
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| q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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| W_{eD}(t)= \int_0^{t} \frac{\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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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_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 |
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| Q^{\downarrow}_{AQ}(t) = = \theta \cdot r_e \cdot h_a \cdot M_a \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 = B \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 \left[ int_0^{(t-\tau)r_e^2/\chi_a} \frac{\partial p_a(t \cdot \frac{r_e^2}{\chi_a},r_e \cdot r_D)}{\partial r_D} \bigg|_{r_D=1} \right] 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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