Material Balance for Cumulative Water Influx: Havlena and Odeh Approach in Microsoft Excel

Material balance is an interpretation method used to estimate the original fluids-in-place in a reservoir based on production and pressure data. The material balance equation relates the original oil, gas, and water in the reservoir to the production volumes and the current pressure conditions.

The Havlena and Odeh method is a graphical technique that can be applied to any reservoir type, such as gas, oil, or water-drive reservoirs. The method uses a plot of cumulative withdrawal (F) versus a dimensionless time function (Et) to obtain the original fluids-in-place and the aquifer parameters. The cumulative withdrawal is the sum of the produced oil, gas, and water at reservoir conditions, and the dimensionless time function is a function of the average reservoir pressure, the initial reservoir pressure, and the fluid properties.

The Havlena and Odeh method assumes that the water influx is proportional to the pressure drop in the reservoir. The water influx can be computed by using the Van Everdingen-Hurst Q T function values, which depend on the aquifer model and the aquifer parameters3. The aquifer parameters are the effective aquifer radius (reD) and the aquifer transmissibility (kt) for a radial aquifer, or the aquifer length (leD) and the aquifer conductivity (kh) for a linear aquifer.

The Havlena and Odeh method can be used to confirm the producing mechanism, estimate the original fluids-in-place, estimate the water influx history, and estimate the water influx model parameters. The method is based on the law of mass conservation, which states that the net withdrawal from the reservoir must be equal to the expansion of the fluids left in the reservoir plus the water influx.

The material balance equation for cumulative water influx (Havlena and Odeh) is given by:

$N_p = \frac{{W_e - W_{oi}}}{{W_{wi}}}$

where:

$N_p$ is the cumulative oil production,
$W_e$ is the cumulative water influx,
$W_{oi}$ is the initial connate water in the reservoir, and
$W_{wi}$ is the irreducible water saturation.

Procedures

Determine Initial Values:
- $N_p$ (cumulative oil production)
- $W_{oi}$ (initial connate water in the reservoir)
- $W_{wi}$ (irreducible water saturation)
Calculate Cumulative Water Influx ( $W_e$ ):
$W_e = N_p \cdot W_{wi} + W_{oi}$
Apply Havlena and Odeh Equation:
$N_p = \frac{{W_e - W_{oi}}}{{W_{wi}}}$

Explanation

Let’s consider a reservoir with the following properties:

$N_p = 500,000$ STB (cumulative oil production)
$W_{oi} = 50,000$ STB (initial connate water in the reservoir)
$W_{wi} = 20\%$ (irreducible water saturation)

Calculate Cumulative Water Influx ( $W_e$ ):
$W_e = 500,000 \times 0.20 + 50,000 = 150,000$

STB
Apply Havlena and Odeh Equation:
$N_p = \frac{{150,000 - 50,000}}{{0.20}} = 500,000$

STB

Excel Formulas

Cell	Formula	Result
A1	Cumulative Oil Production (Np)	500,000
A2	Initial Connate Water (Woi)	50,000
A3	Irreducible Water Saturation (Wwi)	0.20
A4	Cumulative Water Influx (We)	=A1*A3 + A2 = 150,000
A5	Material Balance – Havlena and Odeh	=(A4 – A2)/A3 = 500,000

MATLAB Comparison

If you wish to perform the same calculations in MATLAB, the script would look like this:

% Given values
Np = 500000;     % Cumulative oil production (STB)
Woi = 50000;     % Initial connate water (STB)
Wwi = 0.20;      % Irreducible water saturation

% Calculate Cumulative Water Influx (We)
We = Np * Wwi + Woi;

% Apply Havlena and Odeh Equation
result_MATLAB = (We - Woi) / Wwi;
disp(['Material Balance - Havlena and Odeh (MATLAB): ', num2str(result_MATLAB), ' STB']);

Result

The material balance calculation using Havlena and Odeh in both Excel and MATLAB yields $N_p = 500,000$ STB, providing consistency and confidence in the reservoir analysis.