Motivation
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| Aquifer Drive |
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Inputs & Outputs
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Physical Model
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Mathematical Model
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| Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi}{r_e^2 |
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}, \frac{r_a}{r_e \right) \dot p(\tau) d\tau |
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q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt},r)= \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r_D = 1} dt_D |
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WeD | alignmentp_1 = p_left | (t_D, r_D)| Radial Drive | Linear drive |
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anchorRC1 | \fracpartial 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}| LaTeX Math Block |
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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^2downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt} |
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CT | alignmentp_left | (t_D = 0, x_D)= 0| LaTeX Math Block |
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anchorCT | (t_D, r_D=1) = 1| LaTeX Math Block |
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\frac{\partial p_1(t_D, r_D)}{\partial r_D}
\Bigg|_{r_D=r_{aD}} 0or
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= \infty = 0CTp_1(t_D, x_D=1) = 1 |
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anchor| alignment | left |
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gradp | (t_D, x_D) r_D}
\Bigg|_{x_D=x_{aD}} =0or
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p_1(, x_D = \infty) = 0...
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Transient flow in Radial Composite Reservoir:
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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 |
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} |
One can easily check that
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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:
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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:
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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
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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 planeChanging the integration order from
to leads to:...
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_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} |
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Replacing the variable:
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Derivation of Radial VEH Aquifer Drive @modelcdot \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_Dwhere 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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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 |
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}, \frac{r_a}{r_e \right)\Delta p_\alpha
= B \cdot W_{eD}
\left( \frac{ (t-\tau_1) \chi}{r_e^2} |
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,\frac{r_a}{r_e} \right)\Delta p_1 +
B \cdot W_{eD}
\left( \frac{ (t-\tau_2) \chi}{r_e^2} |
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,\frac{r_a}{r_e} \right)\Delta p_2
+ ... + B \cdot W_{eD}
\left( \frac{ (t-\tau_N) \chi}{r_e^2} |
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, \frac{r_a}{r_e} 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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.
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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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| 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
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The benefit of VEH approach is that net pay formation pressure history
is usually known so that water influx calculation based on aquifer properties
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| body | \{ B, \, r_a, \, \chi \} |
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is rather straightforward.
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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
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Petroleum Industry / Upstream / Subsurface E&P Disciplines / Field Study & Modelling / Aquifer Drive / Aquifer Drive Models
Reference
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1. van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans., AIME 186, 305.
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