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

 

Combining Aquifer Models with Material Balance Equation (MBE)

One of the advantages of Carter-Tracy’s model over Van Everdingen-Hurst model is

that; it does not require superposition and can be easily combined with MBE. Thus,

Carter-Tracy’s model is combined with undersaturated MBE as follows:

 

Combination Drive Reservoir

Oil Saturation Adjustment Due to Combination Drive

For the case of combination drive, both water and gas invasion zone is incorporated

in the saturation equation given as: 

Determination of Present GOC and OWC from

Material Balance Equation

Step 1: Determine the bulk volume of the reservoir rock at each depth interval

Step 2: Make a plot of depth versus the bulk volume

Step 3: Calculate the cumulative water influx from the general material balance

equation (We)

Step 4: Calculate the volume of oil displaced by water (Net water influx into the

reservoir) (OW ¼ We Wp)

Step 5: Calculate the reservoir volume liberated gas (GL)

GL¼ NRsi N Np

 Rs

 Bg

Step 6: Calculate the expansion of the primary gas cap (Ge)

Ge ¼ mNBoi

Bg

Bgi

 1

 

Step 7: Calculate the gas drive (GD)

GD ¼ GL þ Ge

Step 8: Calculate the produced excess gas (Gpe)

Gpe ¼ Np Rp Rs

    Bg

Step 9: Calculate the volume of oil displaced by the gas (Og)

Water Drive Reservoir 5.8.3.1 Undersaturated Reservoir with Water Drive

Undersaturated Reservoir with Water Drive

Saturated Water Drive Reservoir

Oil Saturation Adjustment Due to Water Influx

Representation of Material Balance Equation under Different Reservoir Type

Depletion Drive Reservoir

5.8.1.1 For Undersaturated Reservoir (P > Pb) with No Water Influx

That is, above the bubble point; the assumptions made are:

Material Balance Time Concept for Pseudo Steady State

for Undersaturated Reservoir

From the expression of the isothermal compressibility in terms of effective com￾pressibility, we can express it in terms of 

total compressibility, Ct

 

Saturated Reservoir (P < Pb) Without Water Influx

Calculation of Oil Saturation

As hydrocarbon is produced from the porous rock, water moves to replace the

corresponding space or void left by the produced hydrocarbon because nature avoids

vacuum. In some cases, the effects of the reservoir drive mechanisms need to be

accounted for; which are presented subsequently in this chapter. Mathematically, oil

saturation is given as:

Gas Drive Reservoir

 Oil Saturation Adjustment Due to Gas Cap Expansion

The volume of oil in the gas-invaded zone is represented as:

Water Drive Mechanism

Water drive is the mechanism wherein the displacement of the oil is accomplished by

the net encroachment of water into the oil zone from an underlined water body called

aquifer (Fig. 5.10a).

Production of oil or gas will often change the water saturation which in turn

affects the oil and gas saturation, but the amount of change varies with the reservoir

drive mechanism. In an aquifer driven reservoir on an efficient water flood, as the oil

is produced to the surface facilities via the production tubing, the water saturation

increases accordingly to fill the space previously occupied by the withdrawn oil

(Fig. 5.10b).

This mechanism is represented mathematically as

Water Drive Index ¼ Net water influx

Hydrocarbon Voidage

Production Characteristics (Prof Onyekonwu MO, Lecture Note

on Reservoir Engineering)

• Pressure

– Pressure is maintained (remains high) when water influx is active.

Pressure declines slowly at first but then stabilizes due to increasing influx

with increasing pressure differential, but not when water influx is moderate.

• Oil Rate

– Rate remains constant or gradually declines prior to water breakthrough

– Rate decreases as water rate increases

• Producing GOR

– GOR remains constant as long as P > PBP

– Gradually increases if P is below the saturation pressure

• Water Production

– Dry oil until water breakthrough

– Increasing water production to an appreciable amount from the flank wells; a

sharp increase due to water coning in individual wells.

• Ultimate Recovery

– The expected oil range is 35–75%

Rock Compressibility and Connate Water Expansion

Drive

As the reservoir pressure declines, the rock and fluid expand due to the expansion of

the individual rock grains and formation compaction (individual compressibility).

The compressibility of oil, rock and water is generally relatively small which makes

the pressures in the undersaturated oil reservoirs to drop rapidly to the bubble point if

there is no aquifer support. Sometimes, this drive mechanism is not considered or it

is neglected when performing material balance calculation, especially for saturated

reservoirs.

This mechanism is represented mathematically as:

formation Drive Index ¼ rock and connate water expansion

Hydrocarbon Voidage

 Gravity Drainage Reservoirs (Prof Onyekonwu MO,

Lecture Note on Reservoir Engineering)

• The mechanism of gravity drainage is operative in an oil reservoir as a result of

difference in densities of the reservoir fluids.

• Gas coming out of solution moves updip to the crestal areas while oil moves

downdip to the wells located low on the structure (Fig. 5.11).

• Reservoir must have:

– High Dip

– High Permeability

– High Kv/Kh ratio

– Homogeneity

– Low Oil Viscosity

• Production Characteristics:

– Formation of a secondary gas cap

– Low GOR from structurally low wells

– Increasing GOR from high structure wells

– Rapid pressure decline to near dead conditions (stripper wells)

– Little or no water production

• While rates are low, RE will be high (70–80% of the initial oil in place)

eventually.

• Gravity drainage is most significant in fractured tight

Combination Drive Reservoirs

Most oil reservoirs produce under the influence of two or more reservoir drive

mechanisms, referred to collectively as a combination drive. A common example

is an oil reservoir with an initial gas cap and an active water drive as shown in the

Fig. 5.12.

5.7.7.1 Production Trends

The production trends of a combination drive reservoir reflect the characteristics of

the dominant drive mechanism. A reservoir with a small initial gas cap and a weak

water drive will behave in a way similar to a solution gas drive reservoir, with rapidly

decreasing reservoir pressure and rising GORs. Likewise, a reservoir with a large gas

cap and a strong water drive may show very little decline in reservoir pressure while

exhibiting steadily increasing GORs and WORs. Evaluation of these production

trends is the primary method a reservoir engineer has for determining the drive

mechanisms that are active in a reservoir.

Recovery

The ultimate recovery obtained from a combination drive reservoir is a function of

the drive mechanisms active in the reservoir. The recovery may be high or low

depending on whether displacement or depletion drive mechanisms dominate. Water

drive and gas cap expansion are both displacement type drive mechanisms and have

relatively high recoveries. Solution gas drive is a depletion type drive and is

relatively inefficient.

Recovery from a combination drive reservoir can often be improved by minimiz￾ing the effect of depletion drive mechanisms by substituting or augmenting more

efficient ones through production rate management or fluid injection. To do this, the

drive mechanisms active in a reservoir must be identified early in its life

5.7.7.3 Characteristics of Combination Drive Reservoirs (Prof

Onyekonwu MO, Lecture Note on Reservoir Engineering)

• Gradually increasing water-cut in structurally low wells

• Pressure decline may be rapid if no strong water influx and no gas cap expansion.

• Continuously increasing GOR in structurally high wells if the gas cap is

expanding

• Recovery > depletion Drive but may be less than in water drive or gas-cap drive.

• When an oil reservoir is associated with a gas cap above and an aquifer below, all

drive mechanisms may be operative.

• Development strategy and well rate control are very important in the economic

recovery process.

A. If oil production rate is faster than the encroachment rates of gas cap and

water advance, pressure depletion occurs in the oil zone.

B. If oil production rate is controlled to equal voidage, it is better to have water

displace oil than gas displacing oil.

– Danger: Oil migration into gas cap due to shrinkage of gas cap volume;

some oil will be left trapped as residual.

• RE is usually greater than recovery from depletion drive but less than water drive

or gas-cap drive. The expected recovery is between 25 and 40% OOIP

 

Reservoir Drive Mechanisms

 The production of hydrocarbon from a reservoir into the wellbore involves several

stages of recovery. The available drive mechanisms determine the performance of

the hydrocarbon reservoir. When the hydrocarbon fluids are produced by the natural

energy of the reservoir, it is termed primary recovery; which is further classified

based on the dominant energy responsible for primary production. There are six

primary drive mechanisms, they are:

• Solution Gas (Depletion) Drive

• Water Drive

• Gas Cap Expansion (segregation) Drive

• Rock Compressibility and Connate Water Expansion Drive

• Gravity Drainage

• Combination Drive

5.7.1 Basic Data Required to Determine Reservoir Drive

Mechanism

• Reservoir pressure and rate of decline of reservoir pressure over a period of time.

• The character of the reservoir fluids.

• The production rate.

• Gas-Oil ratio.

• Water-oil ratio.

• The cumulative production of oil, gas and water.

5.7.2 Solution Gas (Depletion) Drive

A solution gas or depletion drive reservoir is a recovery mechanism where the gas

liberating out of the solution (oil) provides the major source of energy. We simply

define it as the oil recovery mechanism that occurs when the original quantity of oil

plus all its original dissolved gas expansion as a result of fluid production from its

reservoir rock (Fig. 5.7).

This drive mechanism is represented mathematically as:

Production Characteristics (Prof Onyekonwu MO, Lecture Note

on Reservoir Engineering)

• Pressure

– declines rapidly and steadily

– decline rate is dependent on production rate

• Oil Rate

– declines rapidly at first as oil mobility decreases

– steady decline thereafter

• Producing GOR

– Increases rapidly as free gas saturation increases.

– Thereafter, decreases rapidly as the remaining oil contains less solution gas.

• Water Production

– Mostly negligible as depletion type reservoirs are volumetric (closed) systems.

• Ultimate Oil Recovery

– It may vary from less than 5% to about 30%. Thus, according to Cole (1969)

these characteristics can be use to identify a depletion drive reservoir.

5.7.3 Gas Cap Expansion (Segregation) Drive

Segregation drive (gas-cap drive) is the mechanism wherein the displacement of oil

from the formation is accomplished by the expansion of the original free gas cap as

shown in Fig. 5.8.

The following are some of the points to note in a gas cap expansion drive

mechanism:

• A gas cap, existing above an oil zone in the structurally higher parts of a reservoir,

provides a major source of energy. The pressure at the original GOC (Fig. 5.8) is

the bubble point pressure since the underlain oil is saturated.

• As pressure declines in the oil column, two things happen:

– Some dissolved gas comes out of oil

– Gas cap expands to replace the voidage

Formation of free gas in the oil column should be minimized as much as possible.

This is achieved if:

– Gas is re-injected in the gas cap, and

– Gas is allowed to migrate upstructure (Gravitational Segregation) (Fig. 5.9).

 Production Characteristics (Prof Onyekonwu MO, Lecture Note

on Reservoir Engineering). The characteristics trend for gas cap

reservoir listed below were comprehensively summarized by

Clark (1969)

• Pressure

– The reservoir pressure falls slowly and continuously

• Oil Rate

Increase in gas saturation leading to increase in the flow of gas and a drop in

the effective permeability of oil.

• Producing GOR

– The gas-oil ratio rises continuously in up-structure wells. As the expanding gas

cap reaches the producing intervals of upstructure wells, the gas-oil ratio from

the affected wells will increase to high values.

• Water Production

– Absent or negligible water production

• Ultimate Oil Recovery

– The expected oil recovery ranges from 20% to 40%

Oil Material Balance Equation

 Figure 5.6 shows an initial condition of a reservoir with original gas cap and the

setting when the reservoir pressure as dropped due to fluid expansion. The material

balance equation uses the principle of conservation of mass. It states that the total

amount of hydrocarbon withdrawn is equal to the sum of the expansion of the oil

plus the original dissolved gas plus the primary gas plus the expansion of the connate

water & decrease in pore volume plus the amount of water the encroached into the

reservoir.

From the diagram, we have that

The derivation of the general material balance is presented below

Quantity of Oil Initially in the Reservoir

NBoi

5.6.2.2 Quantity of Oil Remaining in the Reservoir

 The Free/Liberated Gas in the Reservoir

Expansion of Oil Zone

In the oil zone, will have the original volume of oil plus the original dissolved gas in

the oil

Expansion of Connate Water and Decrease in Pore Volume

The rock compressibility is expressed as

Total Underground Withdrawal

The total underground withdrawal (TUW) due to the pressure drop is the sum of the

oil + gas + water production. Mathematically, it is

Quantity of Injection Gas and Water

Derivation of Material Balance Equations

Gas Reservoir Material Balance Equation

5.6.1.1 Dry Gas Reservoir Without Water Influx

Applying the law of conservation of mass on Fig. 5.1, it states that the mass of the

gas initially in place in the reservoir is equal to the amount of gas produced plus the

amount of gas remaining in the reservoir. Recall that gas expands to fill the shape of

its container. Hence, in terms of volume balance, it simply states that the volume of

gas originally in place at the reservoir conditions is equal to the volume of gas

remaining in the reservoir at the new pressure-temperature conditions after some

amount of gas has been produced. Since the pressure of the reservoir has dropped

with a corresponding decrease in the volume of gas due to the amount that have been

produced, therefore the remaining amount of gas in the reservoir would have

expanded to occupy the same volume as that initially in place. Mathematically, we

have that;

A plot of P=z versus Gp gives the x-intercept as the initial gas in place and the

y-intercept as Pi =zi (Fig. 5.2)

Adjustment to Gas Saturation in Water Invaded Zone

The initial gas in place in reservoir volume expressed in terms of pore volume

(PV) is:

GBgi ¼ PVð

 

Material Balance

 Introduction

Globally, there are different techniques applied in the oil and gas industry to estimate

hydrocarbon reserves. These techniques include the analogy, volumetric, decline

curve, material balance and reservoir simulation. The application of these techniques

is dependent on the volume and quality of data available with some level of

uncertainties. In Chap. 2, we have established that the analogy method is applied

by comparing factors for the current field or wells while the volumetric or geologic

method combined the extent of the reservoir (area), the pore volume of the reservoir

rock, the content of fluid within the reservoir pore volume and PVT properties.

When production and pressure data from the field become available, decline

curve analysis and material balance calculations become the predominant methods

of calculating reserves since the hydrocarbon reserve estimation is a continuous

process for a field that is producing. These methods greatly reduce the uncertainties

in reserves estimation; however, during early depletion, caution should be exercised

in using them.

Material balance equation (MBE) makes use of the basic concept of conservation

of mass which states that the cumulative observed production, expressed as an

underground withdrawal, must be equal to the expansion of the fluids in the reservoir

resulting from a finite pressure drop or expressed as the mass of fluids originally in

place equal to mass of fluid remaining plus the mass of fluid produced. MBE is seen

by the Reservoir Engineers as the basic tool for interpreting and predicting the

performance of oil and gas reservoirs. It helps engineers to get a feel of the

reservoir. To better understand this subject, several textbooks and materials were

consultated. these are: Craft & Hawkins (1991), Dake (1978, 1994), Mattar &

Aderson (2005), Numbere (1998), Pletcher (2002), Steffensen (1992), Matter &

McNeil (1998), Tracy (1955) & Tarek (2010).

5.1.1 Assumptions of Material Balance Equation

To apply the material balance equation, there are several assumptions made by the

engineers to successfully carry out an evaluation on oil and gas reservoirs. These are:

• The reservoir is considered to be a tank

• Pressure, temperature, and rock and fluid properties are not space dependent

• Uniform hydrocarbon saturation and pressure distribution (homogenous

reservoir)

• Thermodynamic equilibrium always attained.

• Isothermal condition apply

• Production data is reliable

Limitations of Material Balance Equation

The implication/limitation of the above stated assumptions in evaluating reservoir

performance is that, material balance uses a model that is existing as an imagination

of the reservoir to actually tell or forecast the behaviour of the reservoir. This is

established as a result of the production of hydrocarbon from the reservoir with

natural energy or by gas or water injection. These implications are given below:

• It is considered to be a tank model with a zero dimension which implies that it

does not reflect the area drained

• the shape or geometry of the reservoir

• the manner in which the wells drilled into the reservoirs are positioned and

orientation are not considered

• the dynamic effects of fluid are not considered

• the heterogeneous nature of the reservoir and no time parameters

These implications lead to the statement made by Warner et al. (1979) that the

material balance method has some limitations, though it can be used as a

pre-processing tool to infer fluid in place, drive mechanisms and identify aquifer

for a more sophisticated tool “reservoir simulation”. This sophisticated tool gives an

insight into dynamic rock and fluid properties for evaluation of past reservoir

performance, prediction of future reservoir performance, and reserves estimation.

5.2 Data Requirement in Performing Material Balance

Equation

5.2.1 Production Data

• Cumulative oil, gas and water volume produced

• cumulative gas-oil ratio

5.2.2 PVT Properties

• Oil, gas and water formation volume factor

• Compressibility of water

• Solution Gas-Oil Ratio

5.2.3 Reservoir Properties

• Rock Compressibility

• Connate water saturation

Other Terms

• Initial volume of oil in reservoir

• Initial gas cap

• Water and gas injection if any

5.3 Sources of Data Use for the MBE

Uses of Material Balance Equation

However, despite the assumptions and limitations of the material balance approach,

there some basic uses which could guide reservoir engineers prior to full field

reservoir study. These are:

• Determination of the hydrocarbon in place, gas cap size etc.

• Reservoir pressure estimation from historical production and/or injection

schedule.

• Predict the future performance of the reservoir and the average production of the

wells sunk into the reservoir for a given pressure schedule

• Determine the presence, type and size of an aquifer.

• Estimation of fluid contacts (Gas/Oil, Water/Oil, Gas/Water).

• Material balance equation can be used to calculate fluid saturation as production

increases

5.5 PVT Input Calculation

The PVT properties can either be obtained from the laboratory analysis or generated

from existing correlations. Some of these developed correlations are given below.

Standing Correlations

Glaso Correlations

 Al-Marhouns