Aquifer Models

Overview

Water influx ($W_e$) from an adjacent aquifer is often the most uncertain term in the material balance equation. The aquifer is usually unmapped, undrilled, and its properties inferred indirectly from reservoir pressure behavior.

Four aquifer models of increasing complexity are available, each suitable for different situations:

---

Pot Aquifer Model

Concept

The simplest model treats the aquifer as a finite tank that responds instantaneously to pressure changes:

$$W_e = c_t W_{ei} \Delta p$$

Where:

• $c_t$ = total aquifer compressibility ($c_w + c_f$)
• $W_{ei}$ = initial encroachable water volume
• $\Delta p = p_i - p$ = pressure decline

Aquifer Volume

$$W_{ei} = \frac{\pi (r_a^2 - r_e^2) h \phi}{5.615}$$

Where:

• $r_a$ = aquifer outer radius
• $r_e$ = reservoir outer radius
• $h$ = aquifer thickness
• $\phi$ = aquifer porosity

Limitations

• Assumes instant pressure communication — valid only for small aquifers
• Linear response — no time delay or transient behavior
• Underestimates influx for large aquifers where pressure waves travel slowly

---

Schilthuis Steady-State Model

Concept

Assumes a constant influx rate per unit pressure difference. The aquifer is recharged from an external source (or is very large), maintaining a constant pressure at its outer boundary:

$$\frac{dW_e}{dt} = J_s (p_i - p)$$

Cumulative Influx

For discrete time steps:

$$W_e = J_s \sum_{j=0}^{n} (p_i - \bar{p}_j) \Delta t_j$$

Where $J_s$ is the aquifer productivity index (bbl/d/psi) and $\bar{p}_j$ is the average reservoir pressure during interval $j$.

When to Use

• Active aquifer with strong pressure support
• Reservoir pressure decline is moderate (< 30% of initial)
• Aquifer-to-reservoir size ratio > 10:1

---

Fetkovich Pseudo-Steady-State Model

Concept

The Fetkovich model treats the aquifer as a finite reservoir that depletes as water flows into the oil/gas zone. This is the most practical model for most field applications:

$$W_e = W_{ei} c_t (p_i - \bar{p}_a)$$

Where $\bar{p}_a$ is the average aquifer pressure, which declines over time.

Key Parameters

Maximum encroachable water:

$$W_{ei} = c_t f \pi (r_a^2 - r_e^2) h \phi / 5.615$$

Where $f$ = fraction of circumference open to flow (1.0 for full encirclement).

Aquifer productivity index:

For radial flow: $$J = \frac{0.00708 k h f}{\mu_w [\ln(r_a/r_e) - 0.5]}$$

For linear flow: $$J = \frac{0.003381 k h w}{\mu_w L}$$

Step-by-Step Calculation

For each time step $n$:

1. Calculate average aquifer pressure at start of step: $$\bar{p}_{a,n} = p_i - \frac{W_{e,n-1}}{c_t W_{ei}}$$

2. Calculate influx rate: $$q_{w,n} = J(\bar{p}_{a,n} - p_n)$$

3. Calculate cumulative influx: $$W_{e,n} = W_{e,n-1} + \frac{W_{ei} c_t(\bar{p}_{a,n} - p_n)}{1} \left[1 - \exp\left(-\frac{J \Delta t}{c_t W_{ei}}\right)\right]$$

---

Van Everdingen-Hurst (VEH) Unsteady-State Model

Concept

The most rigorous model solves the radial diffusivity equation for the aquifer. It uses dimensionless variables and superposition to handle arbitrary pressure-time histories.

Dimensionless Variables

Dimensionless time: $$t_D = \frac{k t}{\phi \mu_w c_t r_e^2}$$

Dimensionless radius: $$r_{eD} = \frac{r_a}{r_e}$$

Aquifer influx constant: $$B = 1.119 \phi c_t r_e^2 h f$$

Cumulative Influx

Using superposition principle:

$$W_e(t_n) = B \sum_{j=0}^{n-1} \Delta p_j \cdot Q_D(t_{D,n} - t_{D,j})$$

Where:

• $\Delta p_j = p_{j-1} - p_j$ (pressure increment at step $j$)
• $Q_D$ = dimensionless cumulative influx function

Dimensionless Influx Functions

For infinite aquifers at large $t_D$: $$Q_D \approx \frac{2 t_D}{\ln t_D}$$

When to Use VEH

• Large aquifer relative to reservoir ($r_{eD} > 5$)
• Transient pressure behavior is important
• High-quality pressure data available for superposition
• Maximum accuracy required

---

Model Selection Guide

---

Related Topics

• MBE Overview — Where aquifer models fit in material balance
• Oil Reservoirs — Campbell plot with water influx
• Gas Reservoirs — Detecting water drive in gas reservoirs
• Underground Withdrawal — Production terms in MBE

---

References

1. van Everdingen, A.F. and Hurst, W. (1949). "The Application of the Laplace Transformation to Flow Problems in Reservoirs." Transactions of AIME, 186, 305-324.

2. Fetkovich, M.J. (1971). "A Simplified Approach to Water Influx Calculations — Finite Aquifer Systems." Journal of Petroleum Technology, 23(7), 814-828. SPE-2603-PA.

3. Schilthuis, R.J. (1936). "Active Oil and Reservoir Energy." Transactions of AIME, 118, 33-52.

4. Dake, L.P. (1978). Fundamentals of Reservoir Engineering. Elsevier, Chapter 9.

5. Ahmed, T. (2019). Reservoir Engineering Handbook, 5th Edition. Gulf Professional Publishing, Chapter 10.