The Alternative Time Function Model

 The Alternative Time Function Model

Considering the left hand side of the material balance equation

Where

No Water Drive, a Known Gas Cap

Linear Form of Material Balance Equation

Introduction

The material balance equation is a complex equation for calculating the original oil

in place, cumulative water influx and the original size of the gas cap as compared to

the oil zone size. This complexity prompted Havlena and Odeh to express the MBE

in a straight line form. This involves rearranging the MBE into a linear equation. The

straight lines method requires the plotting of a variable group against another

variable group selected, depending on the reservoir drive mechanism and if linear

relationship does not exist, then this deviation suggests that reservoir is not

performing as anticipated and other mechanisms are involved, which were not

accounted for; but once linearity has been achieved, based on matching pressure

and production data, then a mathematical model has been achieved. This technique

of trying to match historic pressure and production rate is referred to as history

matching. Thus, the application of the model to the future enables predictions of the

future reservoir performance. To successfully develop this chapter, several textbooks

and materials such Craft & Hawkins (1991), Dake (1994), Donnez (2010), Havlena

& Odeh AS (1964), Numbere (1998), Pletcher (2002) and Steffensen (1992) were

consulted.

The straight line method was first recognized by Van Everdigen et al. (2013) but

with some reasons, it was never exploited. The straight line method considered the

underground recoverable F, gas cap expansion function Eg, dissolved gas-oil expan￾sion function Eo, connate water and rock contraction function Ef,w as the variable for

plotting by considering the cumulative production at each pressure.

Havlena and Odeh presented the material balance equation in a straight line form.

These are presented below:

Diagnostic Plot

In evaluating the performance of a reservoir, there is need to adequately identify the

type of reservoir in question based on the signature of pressure history or behaviour

and the production trend. Campbell and Dake plots are the vital diagnostic tools

employed to identify the reservoir type. The plots are established based on the

assumption of a volumetric reservoir, and deviation from this behaviour is used to

indicate the reservoir type.

For volumetric reservoirs whose production is mainly by oil and connate water/

rock expansion, the value of STOIIP, N can be calculated at every pressure where

production data is given. Rearranging the material balance equation as shown below.

If a plot of cumulative oil production versus net withdrawal over the fluid

expansions is created with a volumetric reservoir data, then the calculated values

of STOIIP, N on the horizontal axis should be constant at all pressure points. In

practice, this is often not the case either because there is water influx or because there

may be faulty pressure or production readings.

If a gas cap is present, there will be a gas expansion component in the reservoir’s

production. As production continues and the reservoir pressure decreased, the gas

expansion term increases with an increase in the gas formation volume factor. To

balance this, the withdrawal over oil/water/formation expansion term must also

continue to increase. Thus, in the case of gas cap drive, the Dake plot will show a

continual increasing trend.

Similarly, if water drive is present, the withdrawal over oil/water/formation

expansion term must increase to balance the water influx. With a very strong aquifer,

the water influx may continue to increase with time, while a limited or small aquifer

may have an initial increase in water influx to the extent that it eventually decreases.

The Campbell plot is very similar to Dake’s diagnostic tool, with an exception

that it incorporates a gas cap if required. In the Campbell plot, the withdrawal is

plotted against withdrawal over total expansion, while the water influx term is

neglected. If there is no water influx, the data will plot as a horizontal line. If there

is water influx into the reservoir, the withdrawal over total expansion term will

increase proportionally to the water influx over total expansion. The Campbell plot

can be more sensitive to the strength of the aquifer. In this version of the material

balance, using only ET neglects the water and formation compressibility (compac￾tion) term. The Campbell plot is shown below.

The Linear Form of the Material Balance Equation

According to Tarek (2010), the linear form of MBE is presented in six scenarios to

determine either m, N, G or We as follow:

• Undersaturated reservoir without water influx

• Undersaturated reservoir with water influx

• Saturated reservoir without water drive

• Saturated reservoir with water drive

• Gas cap drive reservoir

• Combination drive reservoir

Scenario 1: Undersaturated Reservoir Without Water

Influx

Applying the above assumption, the equation reduces to

Scenario 2: Undersaturated Reservoir with Water Influx

Applying the above assumption, the equation reduces to

Scenario 3: Saturated Reservoir Without Water Influx

Applying the above assumption, the equation reduces to

Scenario 4: Saturated Reservoir with Water Influx

Applying the above assumption, the equation reduces to

Scenario 5: Gas Cap Drive Reservoir

• Finding the STOIIP, N when the gas cap size, m is known

• Finding the gas cap size, m the STOIIP, N and the GIIP, G

Finding the STOIIP, N when the gas cap size, m is known

Scenario 6: Combination Drive Reservoir

Linear Form of Gas Material Balance Equation

Havlena and Odeh also expressed the material balance equation in terms of gas

production, fluid expansion and water influx as