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Inputs & Outputs

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Physical Model

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Mathematical Model

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Linear drive

LaTeX Math Block anchor VEH alignment left 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

LaTeX Math Block anchor WeD alignment left W_{eD}(t

,

)

r)

= \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r

_D = 1} dt_D

LaTeX Math Block
anchor VEHL alignment left
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

LaTeX Math Block

anchor	WeDL
alignment	left

W_{eD}(t, r)= \int_0^{t} \frac{\partial p_1}{\partial x_D} \bigg|

_

{x_anchor

D = 1} dt_D	LaTeX Math Block anchor 1 alignment left q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt}	LaTeX Math Block anchor 1 alignment left p_1 = p_1(t_D, r_D)
LaTeX Math Block anchor

1alignmentleft

q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt}

LaTeX Math Block
anchor 1 alignment left
p_1 = p_1(t_D, x_D)

LaTeX Math Block

RC1 alignment left \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}

LaTeX Math Block anchor CT alignment left p_1(t_D = 0, r

_D)= 0

LaTeX Math Block
anchor RC1 alignment left
\frac{\partial p_1}{\partial t_D} = \frac{\partial^2 p_1}{\partial x_D^2}

LaTeX Math Block

anchor	CT
alignment	left

p_1(t_D = 0, x

_D)= 0

LaTeX Math Block anchor CT alignment left p_1(t_D, r_D=1) = 1

LaTeX Math Block anchor gradp alignment left \frac{\partial p_1(t_D, r_D)}{\partial r_D} \Bigg|_{r_D=r_{aD}} = 0 or LaTeX Math Block anchor gradp alignment left p_1(t_D, r_D = \infty) = 0

LaTeX Math BlockanchorCTalignmentleft

p_1(t_D, x_D=1) = 1

LaTeX Math Block
anchor gradp alignment left
\frac{\partial p_1(t_D, x_D)}{\partial r_D} \Bigg|_{x_D=x_{aD}} = 0

or

LaTeX Math Block
anchor gradp alignment left
p_1(t_D, x_D = \infty) = 0

Transient flow in Radial Composite Reservoir:

Expand
title Derivation
Panel
borderColor wheat borderWidth 10

LaTeX Math Block
anchor RC alignment left
\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]

LaTeX Math Block
anchor 1 alignment left
p_a(t = 0, r)= p(0)

LaTeX Math Block
anchor 1 alignment left
p_a(t, r=r_e) = p(t)

LaTeX Math Block
anchor p1_PSS alignment left
\frac{\partial p_a}{\partial r} \bigg|_{(t, r=r_a)} = 0

Consider a pressure convolution:

LaTeX Math Block
anchor VEHP alignment left
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

LaTeX Math Block
anchor 1 alignment left
\dot p(\tau) = \frac{d p}{d \tau}

One can easily check that

LaTeX Math Block Reference
anchor VEHP

honors the whole set of equations

LaTeX Math Block Reference
anchor RC

–

LaTeX Math Block Reference
anchor p1_PSS

and as such defines a unique solution of the above problem.

Water flowrate within

LaTeX Math Inline
body \theta

sector angle at interface with oil reservoir will be:

LaTeX Math Block
anchor 1 alignment left
q^{\downarrow}_{AQ}(t)= \theta \cdot r_e \cdot h \cdot u(t,r_e)

where

LaTeX Math Inline
body u(t,r_e)

is flow velocity at aquifer contact boundary, which is:

LaTeX Math Block
anchor 1 alignment left
u(t,r_e) = M \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e}

where

LaTeX Math Inline
body M = \frac{k}{\mu_w}

is aquifer mobility.

Water flowrate becomes:

LaTeX Math Block
anchor 1 alignment left
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
anchor Qaq1 alignment left
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
anchor VEHP

into

LaTeX Math Block Reference
anchor Qaq1

leads to:

LaTeX Math Block
anchor Qaq1 alignment left
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}

LaTeX Math Block
anchor 1 alignment left
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}

LaTeX Math Block
anchor 1 alignment left
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

The above integral represents the integration over the

LaTeX Math Inline
body D

area in

LaTeX Math Inline
body (\tau, \ \xi)

plane (see Fig. 1):

LaTeX Math Block
anchor 1 alignment left
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}

Image Removed

Fig. 1. Illustration of the integration

LaTeX Math Inline
body D

area in

LaTeX Math Inline
body (\tau, \ \xi)

plane

Changing the integration order from

LaTeX Math Inline
body \tau \rightarrow \xi

to

LaTeX Math Inline
body \xi \rightarrow \tau

leads to:

LaTeX Math Block
anchor 1 alignment left
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
anchor 1 alignment left
\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
anchor 1 alignment left
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

LaTeX Math Inline
body B

is water influx constant and which leads to

LaTeX Math Block Reference
anchor VEH

and

LaTeX Math Block Reference
anchor WeD

.Derivation of Radial VEH Aquifer Drive @model

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}

 \

right)\Delta p_\alpha 


= B \cdot W_{eD} 
\left( \frac{ (t-\tau_1) \chi}{r_e^2

}

  \right)\Delta p_1 +
 B \cdot W_{eD} 
\left( \frac{ (t-\tau_2) \chi}{r_e^2

}  \right)\Delta p_2
+ ... + B \cdot W_{eD} 
\left( \frac{ (t-\tau_N) \chi}{r_e^2}

 \right)\Delta p_N

This computational model is using a discrete convolution (also called superposition in some publications) with time-grid 

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Scope of Applicability

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The benefit of VEH approach is that net pay formation pressure history 

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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.

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