Motivation




Inputs & Outputs



InputsOutputs

Cumulative subsurface water influx from aquifer

initial formation pressure

Subsurface water flowrate from aquifer

water influx constant





aquifer diffusivity

net pay area



Detailing Inputs

water influx constant

central angle of net pay area ↔ aquifer contact

aquifer effective thickness

aquifer porosity

aquifer total compressibility

aquifer pore compressibility 

aquifer water compressibility



Physical Model



Edge-water Drive Aquifer

Radial Composite Reservoir
Transient flow









Fig. 1. VEH aquifer drive schematic



Mathematical Model



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



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




W_{eD}(t, r)= \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg|_{r_D = 1} dt_D 





p_1 = p_1(t_D, r_D)


Radial DriveLinear drive


\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}



p_1(t_D = 0, r_D)= 0





\frac{\partial p_1}{\partial t_D} =  \frac{\partial^2 p_1}{\partial x_D^2}



p_1(t_D = 0, x_D)= 0



p_1(t_D, r_D=1) = 1



\frac{\partial p_1(t_D, r_D)}{\partial r_D} 
\Bigg|_{r_D=r_{aD}} = 0

or

 p_1(t_D, r_D = \infty) = 0



p_1(t_D, x_D=1) = 1



\frac{\partial p_1(t_D, x_D)}{\partial r_D} 
\Bigg|_{x_D=x_{aD}} = 0

or

 p_1(t_D, x_D = \infty) = 0






Transient flow in Radial Composite Reservoir:


\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]



p_a(t = 0, r)= p(0)



p_a(t, r=r_e) = p(t)



\frac{\partial p_a}{\partial r} 
\bigg|_{(t, r=r_a)} = 0



Consider a pressure convolution:


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



\dot p(\tau) = \frac{d p}{d \tau}



One can easily check that honors the whole set of equations and as such defines a unique solution of the above problem.

Water flowrate within sector angle at interface with oil reservoir will be:

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:

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:

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:

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 into leads to:

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}  


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}   


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 area in plane (see Fig. 1):

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} 

   


Fig. 1. Illustration of the integration area in plane



Changing the integration order from to leads to:

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:

\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:

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 and .



Computational Model



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


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

In practical exercises with manual or spreadsheet-assisted calculations the time-grid is usually uniform:  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.  


Polynomial approximation of   are available (http://dx.doi.org/10.2118/15433-PA).


Table 1. Polynomial approximation of  for infinite aquifer



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  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