This document describes the Van Everdingen-Hurst (VEH) model for simulating water influx into oil reservoirs from surrounding aquifers. The VEH model considers two geometries: radial flow and linear flow. For the radial model, flow is assumed to be strictly radial between a cylindrical reservoir and surrounding aquifer. For the linear model, flow is strictly linear between adjacent rectangular reservoir and aquifer volumes. The document provides detailed equations for calculating water influx over time using the VEH model for both radial and linear geometries and both infinite and finite aquifer extents.

1. WATER INFLUX-2: VEH Model
Prof. K. V. Rao
Academic Advisor
Petroleum Courses
JNTUK

2. Van Everdingen-Hurst (VEH) model
van Everdingen and Hurst considered two geometries:
radial- and linear-flow systems. The radial model assumes
that the reservoir is a right cylinder and that the aquifer
surrounds the reservoir. Fig. 1 illustrates the radial aquifer
model, where:
• ro = reservoir radius
• ra = aquifer radius

3. Fig. 1 – Radial aquifer model.

4. Flow between the aquifer and reservoir is strictly radial. This model is
especially effective in simulating peripheral and edgewater drives but also
has been successful in simulating bottomwater drives.
In contrast, the linear model assumes the reservoir and aquifer are
juxtaposed rectangular parallelepipeds. Fig. 2 shows examples. Flow
between the aquifer and reservoir is strictly linear.
This model is intended to simulate edgewater and bottomwater drives. The
model definition depends on the application. For edgewater drives, the
thicknesses of the reservoir and aquifer are identical; the widths of the
reservoir and aquifer are also the same, and the aquifer and reservoir
lengths are La and Lr, respectively (Fig. 2a).
For bottomwater drives, the width of the reservoir and aquifer are identical;
the length (L) of the reservoir and aquifer are also the same; the aquifer
depth is La, and the reservoir thickness is h (Fig. 2b).

5. Fig. 2 – Linear aquifer model for (a) an edge water drive
(b) a bottom water drive.

6. van Everdingen and Hurst solved the applicable differential equations
analytically to determine the water influx history for the case of a
constant pressure differential at the aquifer/reservoir boundary.
This case assumes the reservoir pressure is constant. They called this
case the "constant terminal pressure" and reported their results in
terms of tables and charts.
This solution is not immediately applicable to actual reservoirs because
it does not consider a declining reservoir pressure. To address this
limitation, van Everdingen and Hurst applied the superposition theorem
to a specific reservoir pressure history.
This adaptation usually requires that the reservoir’s pressure history be
known. The first step in applying their model is to discretize the time
and pressure domains.

7. Discretization
The time domain is discretized into (n+1) points (t0, t1, t2, …., tn), where
t0 < t1 < t2 < …..> tn and t0 corresponds to t = 0. The average reservoir
pressure domain also is discretized into (n+1) points (𝑝0, 𝑝1, 𝑝2 …., 𝑝𝑛)
where 𝑝0 is the initial pressure pi. The time-averaged pressure between
levels j and j-1 is
The time-averaged pressure at level j = 0 is defined as the initial
pressure pi. Table 1 shows discretization of t, 𝑝, and 𝑝. The time-
averaged pressure decrement between levels j and j-1 is
[1]
[2]

8. No value is defined for j = 0. Table 1 shows the complete discretization
of t, 𝑝, 𝑝, and p.
Table 1- Time and pressure domain discretization for
van Everdingen-Hurst model.

9. Cumulative water influx
The cumulative water influx at kth level is
where U is the aquifer constant and WD is the dimensionless
cumulative water influx. This equation is based on the superposition
theorem. The term WD (tDk – tDj) is not a product but refers to the
evaluation of WD at a dimensionless time difference of (tDk – tDj). If we
apply Eq. 3 for k = 1, 2, and 3, we obtain
[3]

10. The length of the equation grows with the time. The aquifer
constant, U, and the dimensionless cumulative water influx, WD(tD),
depend on whether the radial or linear model is applied.

11. Radial model
The radial model is based on the following equations. The effective
reservoir radius is a function of the reservoir PV and is
where:
• ro is expressed in ft
• Vpr is the reservoir PV expressed in RB
• ϕr is the reservoir porosity (fraction)
• h is the pay thickness in ft
[4]

12. The constant f is θ/360, where θ is the angle that defines the portion of
the right cylinder. Fig. 3 illustrates the definition of θ for a radial aquifer
model.
The dimensionless time is
where:
• ka = aquifer permeability (md)
• μw = water viscosity (cp)
• ct = total aquifer compressibility (psi–1)
• ϕa = aquifer porosity (fraction)
• t is expressed in years
where U is in units of RB/psi if h is in ft, ro is in ft, and ct is in psi–1.
[5]
[6]

13. Fig. 3 – Definition of angle, ϕ, for radial aquifer model.

14. The dimensionless aquifer radius is
The dimensionless water influx, WD, is a function of tD and reD and
depends on whether the aquifer is infinite acting or finite.
[7]

15. Infinite radial aquifer
The aquifer is infinite acting if re approaches infinity or if the
pressure disturbance within the aquifer never reaches the aquifer’s
external boundary. If either of these conditions is met, then WD is
where a7 = 4.8534 × 10–12, a6 = –1.8436 × 10–9, a5 = 2.8354 × 10–
7, a4 = –2.2740 × 10–5, a3 = 1.0284 × 10–3, a2 = –2.7455 × 10–2,a1 =
8.5373 × 10–1, a0 = 8.1638 × 10–1, or
[8]
[9]
[10]

16. Finite radial aquifer
For finite aquifers, Eqs. 8 through 10 apply if tD < tD*, where
Where
If tD > tD*, then
Where
[11]
[7]
[12]
[13]

17. Marsal[7] gave Eqs. 11 through 13. These equations are effective in
approximating the charts and tables by van Everdingen and Hurst.
Minor discontinuities exist at some of the equation boundaries. A
slightly more accurate but much more lengthy set of equations has
been offered by Klins et al.[6] Fig. 4 shows WD as a function
of tD for reD = 5, 7.5, 10, 20, and ∞. These equations simplify the
application of the VEH model enormously.

18. Fig. 4 – WD vs. tD for a radial aquifer model.

19. A finite aquifer can be treated effectively as an infinite aquifer if
[14]
[15]
where tDmax is the maximum value of tD. These equations follow
from Eq. 11. For example, if tDmax is 540 and corresponds to a time
of 8 years, then Eq. 15 yields ≥ reD = 38. Therefore, if the aquifer has
a dimensionless radius greater than 38, then the aquifer acts
indistinguishably from and equivalent to an infinite aquifer at all
times less than 8 years.

20. Linear aquifer
The aquifer size in the linear model is given in terms of the
aquifer/reservoir pore-volume ratio, Vpa/Vpr.
The aquifer constant is
For edge water drives, the aquifer length is
[16]
[17]
[18]

21. where La and Lr are defined in Fig. 2a. For bottom water drives, the
aquifer depth is
where La is defined in Fig. 2b. The dimensionless time is
Eqs. 20 and 5 use the same units except La is given in ft. One difference
between the linear and radial models is that tD is a function of the aquifer
size for the linear model, whereas tDis independent of the aquifer size for
the radial model. This difference forces a recalculation of tD in the linear
model if the aquifer size is changed.
[19]
[20]

22. The dimensionless cumulative water influx is an
and
Eq. 21 is by Marsal, and Eq. 22 is by Walsh. Fig. 5 shows WD as a
function of tD. The aquifer can be treated as infinite if the aquifer
length is greater than the critical length
where tmax is the maximum time expressed in years and Lac is in units
of ft.
[21]
[22]
[23]

23. Eqs. 20 and 23 use the same units. Alternatively, the aquifer is
infinite-acting if tD ≤ 0.50. If infinite-acting and an edge water
drive, We can be evaluated directly without computing WD and is
where the units in Eq. 5 apply, and We is in units of RB
and h and w are in units of ft.
[24]

24. Fig. 5 – WD vs. tD for linear aquifer model.

25. Calculation procedure
1. Discretize the time and average reservoir pressure domains and
define tj and 𝑝𝑗 for (j=0, 1, …n) according to table 1.
2. Compute the time-averaged reservoir pressure 𝑝𝑗 for (j = 1, 2, ….,
n) with eq. 1. Note that 𝑝0 = pi.
3. Compute the time-averaged incremental pressure differential pj for
(j=1, 2, …., n) with eq. 2.
4. Compute tDj for (j=0, 1, …, n) with eq. 5 for radial aquifers or with
eq. 20 for linear aquifers.
5. Steps 5 through 9 create a computational loop that is
repeated n times. The loop index is k, where k = 1, ..., n. For
the kth time level, compute (tDk – tDj) for (j = 0, ..., k – 1).

26. 6. For the kth time level, compute WD(tDk – tDj) for (j = 0, ..., k – 1).
7. For the kth time level, compute Δpj + 1 WD(tDk – tDj) for (j = 0,
..., k – 1).
8. For the kth time level, compute Wek with Eq. 3.
9. Increment the time from level k to k + 1, and return to Step 5
until k > n.
This procedure is highly repetitive and well suited for spreadsheet
calculation. The example below illustrates the procedure.