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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi_a}{r_e^2}, \frac{r_a}{r_e} \right) \dot p(\tau) d\tau |
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| q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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| W_{eD}(t, r)= \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(t_D, r_D)}{\partial r_D}
\Bigg|_{r_D=r_{aD}} = 0 |
or | LaTeX Math Block |
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| p_1(t_D, r_D = \infty) = 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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| \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 | 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 \cdot h_a \cdot M_a \cdot \int_0^t d\xi \ \frac{\partial }{\partial r} \left[
\int_0^\xi p_1 \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, \frac{r}{r_e} \right) \, \dot p(\tau) d\tau
\right]_{r=r_e} |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \int_0^t d\xi \ \frac{\partial }{\partial r_D} \left[
\int_0^\xi p_1 \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \, \dot p(\tau) d\tau
\right]_{r_D=1} |
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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_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}{\partial r_D} \left( \frac{(\xi-\tau)\chi_a}{r_e^2}, r_D \right) \Bigg|_{r_D=1}
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| Fig. 1. Illustration of the integration area in plane |
Changing the integration order from to leads to:| LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \int_0^t d\tau \int_\tau^t d\xi \ \dot p(\tau)
\frac{\partial p_1}{\partial r_D} \left( \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} |
Replacing the variable: | LaTeX Math Block |
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| \xi = \tau + \frac{r_e^2}{\chi_a} \cdot t_D \rightarrow t_D = \frac{(\xi-\tau)\chi_a}{r_e^2} \rightarrow d\xi = \frac{r_e^2}{\chi_a} \cdot dt_D |
and flux becomes: | LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot h_a \cdot M_a \cdot \frac{r_e^2}{\chi_a} \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi_a/r_e^2}
\frac{\partial p_1( t_D, r_D)}{\partial r_D} \Bigg|_{r_D=1} dt_D = B \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi_a/r_e^2}
\frac{\partial p_1( t_D, r_D)}{\partial r_D} \Bigg|_{r_D=1} dt_D |
where is water influx constant and which leads to | LaTeX Math Block Reference |
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and | LaTeX Math Block Reference |
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| Q^{\downarrow}_{AQ}(t)= B \cdot \sum_\alpha W_{eD}
\left( \frac{ (t-\tau_\alpha) \chi_a}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_\alpha
= B \cdot W_{eD}
\left( \frac{ (t-\tau_1) \chi_a}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_1 +
B \cdot W_{eD}
\left( \frac{ (t-\tau_2) \chi_a}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_2
+ ... + B \cdot W_{eD}
\left( \frac{ (t-\tau_N) \chi_a}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_N |
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This computational scheme is model is using a discrete convolution (also called superposition in time domain.some publications) with time-grid
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| body | \{ \tau_1, \, \tau_2, \ ... \ , \tau_N \} |
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.In practical exercises with manual or spreadsheet-assisted calculations the time-grid is usually uniform:
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| body | \{ \tau_1 =\Delta \tau, \, \tau_2 = 2 \cdot \Delta \tau, \ ... \ , \tau_N = N \cdot \Delta \tau\} |
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with the time step of a month to ensure the formation pressure does In practical exercises with manual or spreadsheet-assisted calculations the time step is usually a month to ensure the formation pressure did not change much since the previous time step.
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The benefit of VEH approach is that net pay formation pressure history
is usually known
and the aquifer drive model parameters so that water influx calculation based on aquifer properties is rather straightforward.
In Material Balance calculations the aquifer model is used to
can be calibrated iteratively by using
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in
Material Balance calculations .
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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