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

Water influx Fetkovich Aquifer Model

 Fetkovich (1971) proposed a model to simplify water influx calculations further.

This method uses a pseudo-steady-state aquifer productivity index (PI) and an

aquifer material balance to represent the system compressibility. Like the Carter￾Tracy method, Fetkovich’s model eliminates the use of superposition and there￾fore, it is much simpler than van Everdingen-Hurst method. However, because

Fetkovich neglects the early transient time period in these calculations, the calcu￾lated water influx will always be less than the values predicted by the previous two

models.

The Fetkovich model applies to finite-acting aquifers; the model is applicable to

both radial and linear aquifers. The Fetkovich aquifer model applies to edge-water

and bottom-water drive reservoirs, while the Carter-Tracy aquifer model applies to

edge-water drive reservoirs. In edge-water drive, water influx occurs around the

flanks of the reservoir. In bottom-water drive, the reservoir is underlain by the

aquifer which encroaches vertically into the reservoir. These are represented in the

Fig. 4.4.

Fetkovich used an inflow equation similar to fluid flow from a reservoir to a well,

to model the water influx to the reservoir. Assuming constant pressure at the original

reservoir/aquifer boundary, the rate of water influx is derived as follow:

The inflow equation is given as:

q ¼

Where qw ¼ water influx rate, j ¼ aquifer productivity index, P ¼ Pressure at the

reservoir fluid contact i.e. inner aquifer boundary pressure, Pa ¼ average pressure in

the aquifer & We ¼ cumulative water influx.

The total aquifer influx due to the total pressure drop is:

The results are plotted in the figure below. This shows that there is a closeness in

value between the Van Everdingen and Fetkovich model with little deviation

from the Carter-Tracy model but that does not mean that Carter-Tracy model cannot

estimate water influx well. In some reservoir, Carter-Tracy model fits the aquifer

model used in matching historical data. Thus, these aquifer models are tested on the

reservoir to see which matches the past field performance with a minimum tolerance

of error.

 

Water influx Carter-Tracy Model

 This method is an approximate solution to the diffusivity equation. It can be

combined conveniently with a suitable material balance equation to predict the

performance of water-drive reservoirs. The Carter-Tracy aquifer models can be

applied to both finite and infinite-acting aquifers. It can be applied to both radial

and linear aquifers and also applies to edge-water drive reservoirs only. Mathemat￾ically, 

it is calculated as

Steps in Calculating Carter-Tracy’s Aquifer Model

Step 1: Calculate the total pressure drop at each time step

Step 2: Calculate the dimensionless time at each time step

Step 3: Calculate the dimensionless pressure and pressure derivative at each time

step

Step 4: Calculate the water influx at each time step (Table 4.8)

Table 4.8 Carter-Tracy aquifer model calculation

Example 4.5

Repeat Example 4.3 using the Carter Tracy’s aquifer model to calculate the cumu￾lative water influx at each time step.

Water Influx

Introduction

Water influx can also be referred to as water encroachment or aquifer influx. It can be

defined as an underground layer of water-bearing porous rock which flows out into

any available space in the reservoir rock. In this context, an aquifer is referred to as a

large pool of water body underlying a hydrocarbon accumulation in the reservoir

structure that is made up of more than one fluid arranged according to density

differences. Prior to hydrocarbon accumulation, the original system was occupied

or filled with water and during the drainage process; the migrated hydrocarbons from

the source rock displaced some of the water out of the pore space in the reservoir.

This means that majority of hydrocarbon pools discovered globally have an associ￾ated aquifer which could be a key source of energy (primary recovery) for the

hydrocarbon production once a well is drilled.

4.1.1 Classification of Aquifer Influx

Aquifer influx can be classified based on pressure maintenance, outer boundary

conditions, flow regime, flow geometry as shown in Fig. 4.1.

The classification of aquifer system as shown in Fig. 4.1, is key to understanding

and evaluation of hydrocarbon reservoirs performance. As hydrocarbon is produced

from the reservoir, the pressure of the reservoir declines (changes) and the aquifer

responds to offset the pressure decline due to fluids production, which is dependent

on the strength of the aquifer. Besides, if there is a strong support from the aquifer,

there will be a gradual decline in the reservoir pressure leading to a good hydrocar￾bon recovery. Also, there will be fairly steady gas-oil ratio during the life of the

reservoir with excessive water production in shallow wells.

Consequently, in evaluating the performance of hydrocarbon reservoirs, we need

to accurately determine the amount of water encroaching into the reservoir whose

value is dependent on the water viscosity, the permeability of the rock in the aquifer

and the cross-sectional area between the water zone and the region where the

hydrocarbon is accumulated

Aquifer Models

There are several analytical aquifer models presented in the past to estimate the

amount of water encroaching into hydrocarbon reservoirs and some of these models

are briefly presented below. The aquifer analytical models make use of simplified

assumptions that do not consider the heterogeneous nature of the reservoir but a

relatively homogeneous reservoir which has deterred the ideal comparison that is

adopted in the analytical solutions. But when the equations are accurately

discretization, they are relatively easy to program in computer spreadsheets with

the exception of the Van Everdingen & Hurst, whose model does not demand much

computer power.

4.2.1 Pot Aquifer Model

This method is one the simplest model for estimating the amount of water

encroaching into hydrocarbon reservoirs. Mathematically, it is given as

Schilthuis Model

Schilthuis (1936) was the first to develop useful expressions for calculating water

influx in a hydrocarbon reservoir. His steady-state expression is given by:

Hurst Modified Steady-State Model

Analysis of water expansion into a hydrocarbon reservoir indicates that water influx

should often be an unsteady-state process. Hence, the Hurst modified steady-state

eq. (1958) should give better results. The equation is:

Van Everdingen & Hurst Model

Van Everdingen & Hurst method of calculating water influx requires the principle of

superposition which is a tedious exercise, but it provides an exact solution to the

radial diffusivity equation and can be applied at the early stage. To abate the

intricacy of water influx calculations, Carter and Tracy (1960) proposed a direct

water influx calculation technique that does not require superposition. The primary

difference between Carter-Tracy and Van Everdingen & Hurst techniques is that the

former assumes constant water influx rates over each finite time interval. Hence, the

cumulative water influx at any time “tn” can be calculated directly from previous

values obtained at tn-1.

 

Condensate Reservoir Calculation

Condensate Reservoir Calculation

The example applies here for calculating condensate in place was written by Engr.

Ogbarode Napoleon Ogbon in his Lecture note on Natural Gas Engineering II.

3.5.1 Applications of Gas and Condensate Inplace Value

• Determination of economic value of gas and condensate in place to make a

decision on project economic viability

3.5.2 Major Points for Consideration

• As the gas-condensate reservoir fluid pressure drops below the dew point, liquid

hydrocarbon (condensate) will begin to drop.

• It is necessary to recombine the condensate with the gas in a proper ratio to

calculate the original volume of gas-in-place 

in the reservoir 

Data Required to Allow Estimates of the Gas-in-Place

Volume Are

• The geologic data

• The reservoir data

• The production data

• The geologic and reservoir data are used to provide plots of gas

compressibility, etc.

• This method uses standard charts and simple equations to calculate hydrocarbon￾in-place volumes in gas-condensate reservoirs.

3.5.4 Method Basic Requirements

• It is based on correlations established by Rzasa and Katz (2011) and provides a

means to calculate the gas-in-place volume in a gas-condensate reservoir Based

on

– The amount of produced gas

– The amount of produced associated condensate.

• Plots of correlations based on this method are readily available for use.

• However, it requires a clear understanding of the behaviour of oil and gas under

various reservoir and surface operating conditions including:

– Reservoir pressure and temperature, or depth to calculate the required

parameters,

– Compositions of oil and gas or their gravities and molecular weights,

– Gravities and production rates of separator condensate and gas,

– Rock porosity,

– Gas or interstitial water saturation

– Area-thickness, in the absence of which calculations are based on one acre of

reservoir volume.

Deterministic Versus Probabilistic Volumetric estimation reserve

The aspect of uncertainty in hydrocarbon reserves estimation cannot be

overemphasized since the estimation of reserves is done under conditions of uncer￾tainties. There are basically two methods of returning the results of reserves estima￾tion for any of the techniques such as volumetric, material balance, decline curve etc.

employed for reserves estimation. These methods are the deterministic and proba￾bilistic methods. Thus, if a single best estimate of reserves is made based on known

geological, engineering and economic data, the method is called deterministic whose

procedure is to select a single value for each parameter to input into an appropriate

equation (volumetric, material balance, decline curve etc.), to obtain a single answer.

In volumetric method, all input parameters are exactly known and variability is

sometimes ignored.

On the other hand, when the known geological, engineering, and economic data

are used to generate a range of estimates and their associated probabilities; the

method of estimation is called probabilistic. This method is more rigorous and less

commonly used; it utilizes a distribution curve for each input parameter and through

the use of Monte Carlo Simulation. In this method, all input parameters are not

exactly known and variability cannot be ignored.

Since the oil and gas business is associated with some inherent uncertainties, it

implies that a quality control and assurance should be made before making any

decision to develop the hydrocarbon prospect because a wrong evaluation of the

hydrocarbon initial in place leads to a wrong decision which in turn leads to an entire

failure of the field development. However, a comparison of the deterministic and

probabilistic methods can provide quality assurance for estimating hydrocarbon

reserves. This means that when the values of the reserves calculated deterministi￾cally and probabilistically agree with minimal deviation or tolerance of error, then

confidence on the calculated reserves is increased. On the contrary, when there is a

significant difference in value, then the assumptions made need to be reexamined.

A Monte-Carlo technique is employed to evaluate hydrocarbons in place where

each input parameter required for the reserves estimation are represented by statis￾tical distributions. Monte-Carlo methods are mainly used in three distinct problem

classes, such as optimization, numerical integration and generating draws from a

probability distribution. There are basically five types of statistical distribution used

with this method. 

 

What is a Contour

 Contour is an imaginary line on the ground surface joining points of equal elevation

or a line on which every point is at the same level above or below a chosen reference

surface. In most maps, the reference surface is sea level. This line on the map

represents a contour and is called contour line.

Therefore, a map showing contour lines is known as Contour map. Contour maps

are one of the most effective means of displaying information about the geologic

structure (i.e. the degree of buckling and faulting of the layers) of an area. A contour

map gives an idea of the altitudes of the surface features as well as their relative

positions in the plan. A map showing structure contours for a certain rock layer

throughout an area is called a structure contour map (Fig. 3.1). Such maps are used

to illustrate the size, shape and location of geologic structures.

Contour lines are drawn as fine and smooth freehand curved lines. Sometimes

they are represented by broken lines. They are inked in either in black or brown

colour. A drawing pen gives a better line than a writing pen and French curves

should be used as much as possible. Every fifth contour is made thicker than the rest.

The elevation of contours must be written in a uniform manner, either on the

higher side or in a gap left in the line. When the contour lines are very long, their

elevations are written at two or three places along the contour. In the case of small￾scale maps, it is sufficient to figure every fifth contour. Therefore, the constant

vertical distance between two consecutive contours is called the contour interval.

The contour interval is constant between the consecutive contours

Methods of Contouring

There are basically two main methods of locating contours; these are the Direct

Method and Indirect Method.

Direct Method

This method requires a lot of time to be invested in searching for points of the same

elevation on the ground surface. This implies that it is very slow and tedious but it is

the most accurate method of contouring, thus suitable for small area and where great

accuracy is required. In this method, the contours to be located are directly traced out

in the field by locating and marking a number of points on each contour. These

points are then surveyed and plotted on plan and the contours drawn through them

(Fig. 3.2).

For a radial line, temporary benchmarks are first established at the centre and near

the ends of the radial lines. The contour points are then located and marked on these

lines and their positions are determined by measuring their distances along the radial

lines. They are then plotted on the plan and the contours drawn by joining all the

corresponding points with the help of a plane table instrument (Fig. 3.3).

3.3.1.2 Indirect Method

In this method, the points located and surveyed are not necessarily on the contour

lines but the spot levels are taken along the series of lines laid out over the area. The

spot levels of the several representative points representing hills, depressions, ridge

and valley lines and the changes in the slope all over the area to be contoured are also

observed. Their positions are then plotted on the plan and the contours drawn by

interpolation. This method of contouring is also known as contouring by spot levels.

Conversion from Planimeter Unit to Field Unit

For a map scale of 1:10,000

Volumetric Reserves Estimation

 Overview of Reserve Estimation

The estimation of hydrocarbon reserves for a producing field is a process that

continues throughout the entire life of the field. This process is usually associated

with some level of uncertainties in calculating the reserves. These reserves estima￾tion methods are affected by the reservoir type, sources of reservoir energy (drive

mechanism), quantity and quality of the geologic, engineering and geophysical data,

the assumptions adopted when making the estimation, available technology, the

experience and knowledge of the evaluator(s). The oil and gas reserves estimation

methods can be grouped into the following categories: analogy, volumetric, decline

analysis, material balance calculations for oil and gas reservoirs, and reservoir

simulation.

The selection of appropriate method to estimate reserves and resources, and the

accuracy of the estimation, depend largely on the following factors: The type,

quantity, and quality of geoscience, engineering, and economic data available for

technical and commercial analyses, the complexity of the formation geology, the

recovery mechanism, the stage of development, and the maturity or degree of

depletion. More importantly, reserves and resources assessment rely on the integrity,

skill and judgment of the experienced professional evaluators (PRMS

Guideline 2011)

In the early stages of development, reserves estimations are restricted to the

analogy and volumetric calculations. The analogy method is applied to reserves

estimation by comparing factors for the analogous and current fields or wells. This

implies that in analogy method, the reserves are estimated on the basis of a

relationship of resemblance or equivalence between two fields. This method directly

compares a poorly or newly discovered reservoir to a known reservoir that has

similar geologic and petrophysical properties such as lithology of the formation,

depth, porosity to mention a few. Hence, the accuracy with this method is the least

among other methods of reserve estimation.

Furthermore, a close-to-abandonment analogous field is taken as an approxima￾tion to the current field. This method is the most useful technique when running the

economics on the current field; which is supposed to be an exploratory field

(Petrobjects 2003).

3.2 Volumetric Method

The volumetric method is probably the easiest method used by engineers to estimate

reserves. It requires a limited amount of data for the estimation, this implies that

immediately after discovery of the hydrocarbon accumulations, during initial delin￾eation and development of a field, the volumetric method is the key to hydrocarbon

volume estimation. Reserves estimation is often high with this method, because it

does not consider the heterogeneity of the reservoir and it includes the undrained

compartments that do not account to flow and are included in making up the bulk

rock volume of the reservoir or accumulation. At this stage, the level of inherent

error can be reduced if the reservoir is accurately described or characterized.

3.2.1 Errors in Volumetric Method

Volumetric method is subject to considerable error because it is often used to

evaluate reserves when little data are available; it requires the estimation of the

reservoir rock and fluid properties and the reservoir volume from spot measurements

of the properties that are then applied to the entire reservoir. The porosity and

saturation are measured either from core samples or logs that are measured from a

small portion of the reservoir and under best circumstances, it only approximates the

condition in the reservoir. The areal extent of the reservoir is rarely known until

many wells are drilled while the volume is estimated using zone thickness measured

at one or more points in the reservoir. The volumetric method is only seen as a gross

estimate of oil or gas in place.

Application of Volumetric Method

• The volumetric result is useful in reserves estimation of the initial oil and gas in

place.

• The volumetric result is useful in reserves estimation of oil and gas in place at any

time of depletion.

• Volumetric estimation is useful during the development period before reservoirs

limit have been defined.

• Later in the life of the reservoir, when reservoir volume is defined and perfor￾mance data are available, volumetric estimation provide valuable checks on oil

and gas in place estimates obtained from material balance and reservoir simula￾tion methods.

The volumetric method is a straightforward approach which requires determination

of the areal extent of the reservoir or bulk volume (calculated as area times pay

thickness), the rock pore volume, and the fluid content within the pore volume to

calculate the amount of hydrocarbons-in-place. The ultimate recovery can thus be

estimated by applying an appropriate recovery factor. Each of the variables used in

the volumetric reserves calculation above has inherent uncertainties, and when

combined; cause significant uncertainties in the reserves estimate (Petrobjects

2003). Therefore, the following steps consist the volumetric method of reserves

estimation:

Step 1: Determination of hydrocarbon rock bulk volume (hydrocarbon saturated

portion) from area and thickness (isopach map). Explanation of this method is

presented in the next page.

Step 2: Determination of average porosity either from core analysis or well logs.

From core analysis

Calculation of Reservoir Bulk Volume (Table 3.1)

The volumetric method of reserves estimation largely depends on the bulk volume,

calculated as follows:

(a) Prepare a structure map with contours from top to bottom of the reservoir, in

subsea depths

(b) Mark out a small square on the map e.g. (10 cm 10 cm). Use the scale on the

map to determine the area of the square in acres. Planimeter the square and

determine the area in planimeter units. Then determine the planimeter constant in

acres/planimeter unit by dividing the actual area in acres by the area into

planimeter units. Use the planimeter constant to covert the areas of the map

from planimeter units to acres.

Trapezoidal Rule 

Pyramidal Rule

To calculate the bulk volume of the reservoir from Isopach or contour map, there

is need to understand the concept of contouring which can be defined as the process

of tracing contour lines on the surface of the earth. This is not only applicable to

petroleum engineers but contour survey is also carried out at the beginning of any

engineering project such as a road, a railway, a canal, a dam, a building etc.