Motivation

The most accurate way to simulate Aquifer Expansion (or shrinkage) is full-field 3D Dynamic Flow Model where Aquifer Expansion is treated as one of the fluid phases and accounts of geological heterogeneities, gas fluid properties, relperm properties and heat exchange with surrounding rocks.

Unfortunately, in many practical cases the detailed information on the aquifer is not available which does not allow a proper modelling of aquifer expansion using a geological framework.

Besides many practical applications require only knowledge of cumulative water influx from aquifer under pressure depletion.

This allows building an Aquifer Drive Models using analytical methods.

Inputs & Outputs

Inputs

Outputs

$\begin{array}{l}p(t)\end{array}$

field-average formation pressure at time moment $\begin{array}{l}t\end{array}$

$\begin{array}{l}Q^{\downarrow}_{AQ}(t)\end{array}$

Cumulative subsurface water influx from aquifer

$\begin{array}{l}p_i\end{array}$

initial formation pressure

$\begin{array}{l}q^{\downarrow}_{AQ}(t) = \frac{dQ^{\downarrow}_{AQ}}{dt}\end{array}$

Subsurface water flowrate from aquifer

$\begin{array}{l}B\end{array}$

water influx constant

$\begin{array}{l}\chi\end{array}$

aquifer diffusivity

$\begin{array}{l}A_e\end{array}$

net pay area

Detailing

Detailing Inputs
$\begin{array}{l}B = \frac{\theta}{2\pi} \cdot A_e \cdot h \cdot \phi \cdot c_t\end{array}$	water influx constant
$\begin{array}{l}\theta\end{array}$	central angle of net pay area ↔ aquifer contact
$\begin{array}{l}h\end{array}$	aquifer effective thickness
$\begin{array}{l}\phi\end{array}$	aquifer porosity
$\begin{array}{l}c_t=c_r +c_w\end{array}$	aquifer total compressibility
$\begin{array}{l}c_r\end{array}$	aquifer pore compressibility
$\begin{array}{l}c_w\end{array}$	aquifer water compressibility

Physical Model

Edge-water Drive Aquifer
Radial Composite Reservoir
Transient flow

	Fig. 1. Radial VEH aquifer drive schematic

Mathematical Model

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

(2)	$\begin{array}{l}\displaystyle W_{eD}(t) = \int_0^{t} \frac{\partial p_1}{\partial r_D} \bigg\|_{r_D = 1} dt_D\end{array}$

(3)	$\begin{array}{l}\displaystyle q^{\downarrow}_{AQ}(t)= \frac{dQ^{\downarrow}_{AQ}}{dt}\end{array}$

(4)	$\begin{array}{l}\displaystyle p_1 = p_1(t_D, r_D)\end{array}$

(5)	$\begin{array}{l}\displaystyle \frac{\partial p_1}{\partial t_D} = \frac{\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_D}\end{array}$

(6)	$\begin{array}{l}\displaystyle p_1(t_D = 0, r_D)= 0\end{array}$

(7)	$\begin{array}{l}\displaystyle p_1(t_D, r_D=1) = 1\end{array}$

(8)	$\begin{array}{l}\displaystyle \frac{\partial p_1(t_D, r_D)}{\partial r_D} \Bigg\|_{r_D=r_{aD}} = 0\end{array}$

or

(9)	$\begin{array}{l}\displaystyle p_1(t_D, r_D = \infty) = 0\end{array}$

Derivation

Derivation of Radial VEH Aquifer Drive @model

Computational Model

(10)

$\begin{array}{l}\displaystyle Q^{\downarrow}_{AQ}(t)= B \cdot \sum_\alpha W_{eD} \left( \frac{ (t-\tau_\alpha) \chi}{r_e^2} \right)\Delta p_\alpha = B \cdot W_{eD} \left( \frac{ (t-\tau_1) \chi}{r_e^2} \right)\Delta p_1 + B \cdot W_{eD} \left( \frac{ (t-\tau_2) \chi}{r_e^2} \right)\Delta p_2 + ... + B \cdot W_{eD} \left( \frac{ (t-\tau_N) \chi}{r_e^2} \right)\Delta p_N\end{array}$

This computational model is using a discrete convolution (also called superposition in some publications) with time-grid $\begin{array}{l}\{ \tau_1, \, \tau_2, \ ... \ , \tau_N \}\end{array}$ .

In practical exercises with manual or spreadsheet-assisted calculations the time-grid is usually uniform: $\begin{array}{l}\{ \tau_1 =\Delta \tau, \, \tau_2 = 2 \cdot \Delta \tau, \ ... \ , \tau_N = N \cdot \Delta \tau\}\end{array}$ with the time step $\begin{array}{l}\Delta \tau\end{array}$ 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 approximations for WeD

Polynomial approximation of $\begin{array}{l}W_{eD}(t_D)\end{array}$ are available (http://dx.doi.org/10.2118/15433-PA).

Table 1. Polynomial approximation of $\begin{array}{l}W_{eD}(t_D)\end{array}$ for infinite aquifer

$\begin{array}{l}t_D< 0.01\end{array}$	$\begin{array}{l}W_{eD}=\sqrt{\frac{t_D}{\pi}}\end{array}$
$\begin{array}{l}0.01 < t_D< 200\end{array}$	$\begin{array}{l}\displaystyle W_{eD}=\frac {1.2838 \cdot t_D^{1/2} + 1.19328 \cdot t_D +0.269872 \cdot t_D^{3/2} +0.00855294 \cdot t_D^2} {1+0.616599 \cdot t_D^{1/2}+0.0413008 \cdot t_D}\end{array}$
$\begin{array}{l}t_D > 200\end{array}$	$\begin{array}{l}\displaystyle W_{eD}=\frac{-4.29881+2.02566 \cdot t_D}{\ln t_D}\end{array}$

Scope of Applicability

The benefit of VEH approach is that net pay formation pressure history $\begin{array}{l}p(t)\end{array}$ is usually known so that water influx calculation based on aquifer properties $\begin{array}{l}\{ B, \, r_a, \, \chi \}\end{array}$ 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

Page tree

Motivation

Inputs & Outputs

Physical Model

Mathematical Model

Computational Model

Scope of Applicability

See Also

Reference

Page tree

Radial VEH Aquifer Drive @model (van Everdingen-Hurst)

Motivation

Inputs & Outputs

Physical Model

Mathematical Model

Computational Model

Scope of Applicability

See Also

Reference