Motivation
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| Aquifer Drive |
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| Aquifer Drive |
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Physical Model
Mathematical Model
Radial Drive | Linear drive |
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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi}{r_e^2}, \frac{r_a}{r_e} \right) \dot p(\tau) d\tau |
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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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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi}{r_e^2}, \frac{r_a}{x_e} \right) \dot p(\tau) d\tau |
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| W_{eD}(t, r)= \int_0^{t} \frac{\partial p_1}{\partial x_D} \bigg|_{x_D = 1} dt_D |
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| q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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| p_1 = p_1(t_D, r_D) |
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| q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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| p_1 = p_1(t_D, x_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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| \frac{\partial p_1}{\partial t_D} = \frac{\partial^2 p_1}{\partial x_D^2} |
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| p_1(t_D = 0, x_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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| p_1(t_D, x_D=1) = 1 |
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| \frac{\partial p_1(t_D, x_D)}{\partial r_D}
\Bigg|_{x_D=x_{aD}} = 0 |
or LaTeX Math Block |
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| p_1(t_D, x_D = \infty) = 0 |
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borderColor | wheat |
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borderWidth | 10 |
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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 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 \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 \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e} |
where is aquifer mobility.Water flowrate becomes: LaTeX Math Block |
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| q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h \cdot M \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 \cdot M \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 \cdot M \cdot \int_0^t d\xi \ \frac{\partial }{\partial r} \left[
\int_0^\xi p_1 \left( \frac{(\xi-\tau)\chi}{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 \cdot M \cdot \int_0^t d\xi \ \frac{\partial }{\partial r_D} \left[
\int_0^\xi p_1 \left( \frac{(\xi-\tau)\chi}{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 \cdot M \cdot \int_0^t d\xi \
\int_0^\xi \frac{\partial p_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi}{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 \cdot M \cdot \iint_D d\xi \ d\tau \, \dot p(\tau)
\frac{\partial p_1}{\partial r_D} \left( \frac{(\xi-\tau)\chi}{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 \cdot M \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}{r_e^2}, r_D \right) \Bigg|_{r_D=1}
=
\theta \cdot h \cdot M \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}{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} \cdot t_D \rightarrow t_D = \frac{(\xi-\tau)\chi}{r_e^2} \rightarrow d\xi = \frac{r_e^2}{\chi} \cdot dt_D |
and flux becomes: LaTeX Math Block |
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| Q^{\downarrow}_{AQ}(t) = \theta \cdot h \cdot M \cdot \frac{r_e^2}{\chi} \cdot \int_0^t \dot p(\tau) d\tau \int_0^{(t-\tau)\chi/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/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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Computational Model
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| Q^{\downarrow}_{AQ}(t)= B \cdot \sum_\alpha W_{eD}
\left( \frac{ (t-\tau_\alpha) \chi}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_\alpha
= B \cdot W_{eD}
\left( \frac{ (t-\tau_1) \chi}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_1 +
B \cdot W_{eD}
\left( \frac{ (t-\tau_2) \chi}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_2
+ ... + B \cdot W_{eD}
\left( \frac{ (t-\tau_N) \chi}{r_e^2}, \frac{r_a}{r_e} \right)\Delta p_N |
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This computational model is using a discrete convolution (also called superposition in 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 not change much since the previous time step.
Moving to annual time step may accumulate a substantial mistake if formation pressure has varied substantially in some years.
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title | Polynomial approximations for WeD |
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Polynomial approximation of are available (http://dx.doi.org/10.2118/15433-PA).
Table 1. Polynomial approximation of for infinite aquifer | LaTeX Math Inline |
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body | W_{eD}=\sqrt{\frac{t_D}{\pi}} |
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| | LaTeX Math Inline |
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body | \displaystyle W_{eD}=\frac {1.2838 \cdot t_D^{1/2} + 1.19328 \cdot t_D +0.269872 \cdot t_D^{3/2} +0.00855294 \cdot t_D^2} {1+0.616599 \cdot t_D^{1/2}+0.0413008 \cdot t_D} |
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body | \displaystyle W_{eD}=\frac{-4.29881+2.02566 \cdot t_D}{\ln t_D} |
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Scope of Applicability
The benefit of VEH approach is that net pay formation pressure history
is usually known so that water influx calculation based on aquifer properties
LaTeX Math Inline |
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body | \{ B, \, r_a, \, \chi \} |
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is rather straightforward.
In the past the VEH approach was considered as tedious in calculating superposition during the manual exercises.
In modern computers the convolution is a fast fully-automated procedure and VEH model is considered as a reference in the range of analytical aquifer models.
Although the model is derived for linear and radial flow it also shows a good match for bottom-water drive depletions.
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Field Study & Modelling / Aquifer Drive / Aquifer Drive Models
Reference
1. van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans., AIME 186, 305.
2. Tarek Ahmed, Paul McKinney, Advanced Reservoir Engineering (eBook ISBN: 9780080498836)
3. Klins, M.A., Bouchard, A.J., and Cable, C.L. 1988. A Polynomial Approach to the van Everdingen-Hurst Dimensionless Variables for Water Encroachment. SPE Res Eng 3 (1): 320-326. SPE-15433-PA. http://dx.doi.org/10.2118/15433-PA