Motivation

Inputs & Outputs

Inputs

Outputs

field-average formation pressure at time moment

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. Radial VEH aquifer drive schematic

Mathematical Model

Q^{\downarrow}_{AQ}= B \cdot \int_0^t W_{eD} \left( \frac{(t-\tau)\chi}{r_e^2}\right) \dot p(\tau) d\tau

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

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

p_1 = p_1(t_D, r_D)

\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

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

Derivation of Radial VEH Aquifer Drive @model

Computational Model

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 .

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.

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