Petroleum Reserves
Estimation Methods
The
process of estimating oil and gas reserves for a producing field continues
throughout the life of the field. There is always uncertainty in making such
estimates. The level of uncertainty is affected by the following factors:
1. Reservoir type,
2. Source of reservoir energy,
3. Quantity and quality of the
geological, engineering, and geophysical data,
4. Assumptions adopted when making
the estimate,
5. Available technology, and
6.
Experience and knowledge of the evaluator.
The
magnitude of uncertainty, however, decreases with time until the economic limit
is reached and the ultimate recovery is realized, see Figure 1.
Figure
1: Magnitude of
uncertainty in reserves estimates
The
oil and gas reserves estimation methods can be grouped into the following
categories:
1. Analogy,
2. Volumetric,
3. Decline analysis,
4. Material balance calculations for
oil reservoirs,
5. Material balance calculations for
gas reservoirs,
6.
Reservoir simulation.
In
the early stages of development, reserves estimates are restricted to the
analogy and volumetric calculations. The analogy method is applied by comparing
factors for the analogous and current fields or wells. A close-to-abandonment
analogous field is taken as an approximate to the current field. This
method is most useful when running the economics on the current field; which is
supposed to be an exploratory field.
The
volumetric method, on the other hand, entails determining the areal extent of
the reservoir, the rock pore volume, and the fluid content within the pore
volume. This provides an estimate of the amount of hydrocarbons-in-place. The
ultimate recovery, then, can be estimated by using an appropriate recovery
factor. Each of thefactors used in the calculation above have inherent
uncertainties that, when combined, cause significant uncertainties in the
reserves estimate.
As
production and pressure data from a field become available, decline analysis
and material balance calculations, become the predominant methods of
calculating reserves. These methods greatly reduce the uncertainty in reserves
estimates; however, during early depletion, caution should be exercised in
using them. Decline curve relationships are empirical, and rely on uniform,
lengthy production periods. It is more suited to oil wells, which are usually
produced against fixed bottom-hole pressures. In gas wells, however, wellhead
back-pressures usually fluctuate, causing varying production trends and
therefore, not as reliable.
The
most common decline curve relationship is the constant percentage decline (exponential).
With more and more low productivity wells coming on stream, there is currently
a swing toward decline rates proportional to production rates (hyperbolic
and harmonic). Although some wells exhibit these trends, hyperbolic or
harmonic decline extrapolations should only be used for these specific cases.
Over-exuberance in the use of hyperbolic or harmonic relationships can result
in excessive reserves estimates.
Material
balance calculation is an excellent tool for estimating gas reserves. If a
reservoir comprises a closed system and contains single-phase gas, the pressure
in the reservoir will decline proportionately to the amount of gas produced.
Unfortunately, sometimes bottom water drive in gas reservoirs contributes to
the depletion mechanism, altering the performance of the non-ideal gas law in
the reservoir. Under these conditions, optimistic reserves estimates can
result.
When
calculating reserves using any of the above methods, two calculation procedures
may be used: deterministic and/or probabilistic. The
deterministic method is by far the most common. The procedure is to select a
single value for each parameter to input into an appropriate equation, to
obtain a single answer. The probabilistic method, on the other hand, is more
rigorous and less commonly used. This method utilizes a distribution curve for
each parameter and, through the use of Monte Carlo Simulation; a
distribution curve for the answer can be developed. Assuming good data, a lot
of qualifying information can be derived from the resulting statistical
calculations, such as the minimum and maximum values, the mean (average value),
the median (middle value), the mode (most likely value), the standard deviation
and the percentiles, see Figures 2 and 3.
Figure
2: Measures of
central tendency
Figure
3: Percentiles
The
probabilistic methods have several inherent problems. They are affected by all
input parameters, including the most likely and maximum values for the
parameters. In such methods, one cannot back calculate the input parameters
associated with reserves. Only the end result is known but not the exact value
of any input parameter. On the other hand, deterministic methods calculate
reserve values that are more tangible and explainable. In these methods, all
input parameters are exactly known; however, they may sometimes ignore the
variability and uncertainty in the input data compared to the probabilistic
methods which allow the incorporation of more variance in the data.
A
comparison of the deterministic and probabilistic methods, however, can provide
quality assurance for estimating hydrocarbon reserves; i.e. reserves are
calculated both deterministically and probabilistically and the two values are
compared. If the two values agree, then confidence on the calculated reserves
is increased. If the two values are away different, the assumptions need to be
reexamined.
Volumetric
The
volumetric method entails determining the physical size of the reservoir, the
pore volume within the rock matrix, and the fluid content within the void
space. This provides an estimate of the hydrocarbons-in-place, from which
ultimate recovery can be estimated by using an appropriate recovery factor.
Each of the factors used in the calculation have inherent uncertainties that,
when combined, cause significant uncertainties in the reserves estimate. Figure
4 is a typical geological net pay isopach map that is often used in the
volumetric method.
Figure 4: A typical geological net pay
isopach map
The estimated ultimate recovery (EUR) of an
oil reservoir, STB, is given by:
EUR
= N(t) RF
Where
N(t) is the oil in place at time t, STB, and RF is the recovery factor,
fraction. The volumetric method for calculating the amount of oil in place (N)
is given by the following equation:
Where:
N(t)
= oil in place at time t, STB
Vb
= bulk reservoir volume, RB = 7758 A h
7758
= RB/acre-ft
A
= reservoir area, acres
h
= average reservoir thickness, ft
φ
= average reservoir porosity, fraction
So(t)
= average oil saturation, fraction
Bo(p)
= oil formation volume factor at reservoir pressure p, RB/STB
Similarly,
for a gas reservoir, the volumetric method is given by:
EUR
= G(t) RF
Where
G(t) is the gas in place at time t, SCF, and RF is the recovery factor,
fraction. The volumetric method for calculating the amount of gas in place (G)
is given by the following equation:
Where:
G(t)
= gas in place at time t, SCF
Vb
= bulk reservoir volume, CF = 43560 A h
43560
= CF/acre-ft
A
= reservoir area, acres
h
= average reservoir thickness, ft
φ
= average reservoir porosity, fraction
Sg(t)
= average gas saturation, fraction
Bg(p)
= gas formation volume factor at reservoir pressure p, CF/SCF
Note
that the reservoir area (A) and the recovery factor (RF) are often subject to
large errors. They are usually determined from analogy or correlations. The
following examples should clarify the errors that creep in during the
calculations of oil and gas reserves.
Decline Curves
A
decline curve of a well is simply a plot of the well’s production rate
on the y-axis versus time on the x-axis. The plot is usually done on a semilog
paper; i.e. the y-axis is logarithmic and the x-axis is linear. When the data
plots as a straight line, it is modeled with a constant percentage decline “exponential
decline”. When the data plots concave upward, it is modeled with a “hyperbolic
decline”. A special case of the hyperbolic decline is known as “harmonic
decline”.
The
most common decline curve relationship is the constant percentage decline
(exponential). With more and more low productivity wells coming on stream,
there is currently a swing toward decline rates proportional to production
rates (hyperbolic and harmonic). Although some wells exhibit these trends,
hyperbolic or harmonic decline extrapolations should only be used for these
specific cases. Over-exuberance in the use of hyperbolic or harmonic
relationships can result in excessive reserves estimates. Figure 5 is an
example of a production graph with exponential and harmonic extrapolations.
Figure 5: Decline curve of an oil well
Decline
curves are the most common means of forecasting production. They have many
advantages:
·
Data
is easy to obtain,
·
They
are easy to plot,
·
They
yield results on a time basis, and
·
They
are easy to analyze.
If
the conditions affecting the rate of production of the well are not changed by
outside influences, the curve will be fairly regular, and, if projected, will
furnish useful knowledge as to the future production of the well.
Exponential
Decline
As
mentioned above, in the exponential decline, the well’s production data plots
as a straight line on a semilog paper. The equation of the straight line on the
semilog paper is given by:
Where:
q
= well’s production rate at time t, STB/day
qi
= well’s production rate at time 0, STB/day
D
= nominal exponential decline rate, 1/day
t
= time, day
The
following table summarizes the equations used in exponential decline.
Hyperbolic
Decline
Alternatively,
if the well’s production data plotted on a semilog paper concaves upward, then
it is modeled with a hyperbolic decline. The equation of the hyperbolic decline
is given by:
Where:
q
= well’s production rate at time t, STB/day
qi
= well’s production rate at time 0, STB/day
Di
= initial nominal exponential decline rate (t = 0), 1/day
b
= hyperbolic exponent
t
= time, day
The
following table summarizes the equations used in hyperbolic decline:
Harmonic
Decline
A
special case of the hyperbolic decline is known as “harmonic decline”,
where b is taken to be equal to 1. The following table summarizes the equations
used in harmonic decline:
Material Balance Calculations for Oil Reservoirs
A
general material balance equation that can be applied to all reservoir types
was first developed by Schilthuis in 1936. Although it is a tank model
equation, it can provide great insight for the practicing reservoir engineer.
It is written from start of production to any time (t) as follows:
Expansion of oil in the oil
zone +
Expansion of gas in the gas
zone +
Expansion of connate water in
the oil and gas zones +
Contraction of pore volume in
the oil and gas zones +
Water influx + Water injected
+ Gas injected =
Oil produced + Gas produced +
Water produced
Mathematically,
this can be written as:
Where:
N
= initial oil in place, STB
Np
= cumulative oil produced, STB
G
= initial gas in place, SCF
GI
= cumulative gas injected into reservoir, SCF
Gp
= cumulative gas produced, SCF
We
= water influx into reservoir, bbl
WI
= cumulative water injected into reservoir, STB
Wp
= cumulative water produced, STB
Bti
= initial two-phase formation volume factor, bbl/STB = Boi
Boi
= initial oil formation volume factor, bbl/STB
Bgi
= initial gas formation volume factor, bbl/SCF
Bt
= two-phase formation volume factor, bbl/STB = Bo + (Rsoi - Rso) Bg
Bo
= oil formation volume factor, bbl/STB
Bg
= gas formation volume factor, bbl/SCF
Bw
= water formation volume factor, bbl/STB
BIg
= injected gas formation volume factor, bbl/SCF
BIw
= injected water formation volume factor, bbl/STB
Rsoi
= initial solution gas-oil ratio, SCF/STB
Rso
= solution gas-oil ratio, SCF/STB
Rp
= cumulative produced gas-oil ratio, SCF/STB
Cf
= formation compressibility, psia-1
Cw
= water isothermal compressibility, psia-1
Swi
= initial water saturation
Δpt
= reservoir pressure drop, psia = pi - p(t)
p(t)
= current reservoir pressure, psia
The
MBE as a Straight Line
Normally,
when using the material balance equation, each pressure and the corresponding
production data is considered as being a separate point from other pressure
values. From each separate point, a calculation is made and the results of
these calculations are averaged. However, a method is required to make use of
all data points with the requirement that these points must yield solutions to
the material balance equation that behave linearly to obtain values of the
independent variable. The straight-line method begins with the material balance
written as:
Defining
the ratio of the initial gas cap volume to the initial oil
volume as:
and
plugging into the equation yields:
Let:
Thus
we obtain:
The
following cases are considered:
1.
No gas cap, negligible compressibilities, and no water influx
2.
Negligible compressibilities, and no water influx
Which
is written as y = b + x. This would suggest that a plot of F/Eo as the y
coordinate versus Eg/Eo as the x coordinate would yield a straight line with
slope equal to mN and intercept equal to N.
3.
Including compressibilities and water influx, let:
Dividing
through by D, we get:
Which
is written as y = b + x. This would suggest that a plot of F/D as the y
coordinate and We/D as the x coordinate would yield a straight line with slope
equal to 1 and intercept equal to N.
Drive
Indexes from the MBE
The
three major driving mechanisms are:
1. Depletion drive (oil zone oil
expansion),
2. Segregation drive (gas zone gas
expansion), and
3.
Water drive (water zone water influx).
To
determine the relative magnitude of each of these driving mechanisms, the
compressibility term in the material balance equation is neglected and the
equation is rearranged as follows:
Dividing
through by the right hand side of the equation yields:
The
terms on the left hand side of equation (3) represent the depletion drive index
(DDI), the segregation drive index (SDI), and the water drive index (WDI)
respectively. Thus, using Pirson's abbreviations, we write:
DDI + SDI + WDI = 1
The
following examples should clarify the errors that creep in during the
calculations of oil and gas reserves.
Material
Balance Calculations for Gas Reservoirs
Material
balance calculation is an excellent tool for estimating gas reserves. It is
based on the non-ideal gas law, PV = ZnRT. If a reservoir comprises a closed
system and contains single-phase gas, the pressure in the reservoir will
decline proportionately to the amount of gas produced. Unfortunately, sometimes
bottom water drive in gas reservoirs contributes to the depletion mechanism,
altering the performance of the non-ideal gas law in the reservoir. Under these
conditions, optimistic reserves estimates can result. Figure 8 is a typical
material balance plot for a tank-type reservoir.
Figure 8: P/z plot of a gas reservoir
For a single-phase gas reservoir, the MB
equation takes the form:
The
following cases are considered:
1. Volumetric (Closed) Gas
Reservoirs, Neglecting Compressibilities
2. Gas Reservoirs with Water Influx,
Neglecting Compressibilities
3. Gas Reservoirs with Water influx,
Including Compressibilities
4. Abnormally High-Pressure Gas
Reservoirs
5. Abnormally High-Pressure Gas
Reservoirs, Including Dissolved Gas in Water (Fetkovitch Technique)
6. Wet Gas Reservoirs
7.
Tight Gas Reservoirs
Volumetric
(Closed) Gas Reservoirs, Neglecting Compressibilities
The
material balance equation for gas reservoirs, when water and rock
compressibilities are neglected, is given by:
Since
And
Plugging
into the above equation, we obtain:
Which
can arranged in a straight line form as follows:
This
suggests plotting Gp on the x-axis versus p/z on the y-axis and drawing the
best fit line through the set of points. The slope of the best fit line would
be -pi/(zi G) and the y-intercept would be pi/zi.
Gas
Reservoirs with Water Influx, Neglecting Compressibilities
The
material balance equation for gas reservoirs with water influx and negligible
water and rock compressibilities, is given by:
Rearranging
and dividing all through by Bg yields:
Substituting
for Bg = C z/p on the LHS and dividing by G yields:
Solving
for p/z yields:
Which
suggests plotting (Gp - (We - Wp Bw)/Bg) on the x-axis versus p/z on the y-axis
and drawing the best fit line through the set of points. The slope of the best
fit line would be -pi/(zi G) and the y-intercept would be pi/zi. The following
example would clarify those concepts.
Gas
Reservoirs with Water Influx, Including Compressibilities
The
material balance equation for gas reservoirs with water influx and water and
rock compressibilities is given by:
Let
Dividing
all through by Bg and rearranging yields:
Substituting
for Bg = C z/p yields:
Rearranging
in a straight line form yields:
Which
suggests plotting (Gp - (We - Wp Bw)/Bg) on the x-axis versus p/z(1-CeΔpt) on
the y-axis and drawing the best fit line through the set of points. The slope
of the best fit line would be -pi/(zi G) and the y-intercept would be pi/zi.
The following example will clarify this method of solution.
Abnormally
High-Pressure Gas Reservoirs
In
abnormally high-pressure gas reservoirs, Cf is a strong function of pressure.
Thus we set:
Therefore
the equation is written as follows:
This
is in the form:
y
= b + mx
Where:
Abnormally
High-Pressure Gas Reservoirs, Including Dissolved Gas in Water (Fetkovitch
Method)
For
high-pressure gas reservoirs with dissolved gas in water, Fetkovitch et al.
expressed the material balance equation as follows:
Where:
"M"
is defined as the associated water volume ratio. The associated water
and pore volumes external to the net pay include Non-Net Pay (NNP) such
as inter-bedded shales and dirty sands plus external water volume found in Aquifers.
This volume is expressed as a ratio relative to the pore volume of the net-pay
reservoir; i.e.
Wet
Gas Reservoirs
In
wet gas reservoirs, the stock tank barrels of the liquid condensate produced
must be accounted for in the cumulative gas production. This is done as
follows:
Where:
(Gp)eff
= effective cumulative gas produced, SCF
Gp
= cumulative gas produced, SCF
GLc
= cumulative liquid condensate produced, STB
KLc
= liquid condensate conversion factor, SCF/STB, which is given by:
SGLc
= Specific gravity of the liquid condensate which is given by:
MWLc
= Molecular weight of the liquid condensate which is given by:
Therefore
the effective gas production would be given by:
Tight
Gas Reservoirs
The
key assumption involved in the P/z plot for gas reservoirs is that the
reservoir behaves as a tank. In tight gas reservoirs, however, this
assumption is violated; thus the plot fails and can incur greater than 100%
error in estimating GIIP. In such a case, the tank assumption does not apply by
definition and the method leads to substantial pressure gradients. These
gradients manifest themselves in terms of scattered, generally curved, and
rate-dependent P/z behavior.
The
Communicating Reservoir (CR) model for tight gas reservoirs as suggested
by Payne (1) will be considered here. This model consists of subdividing the
reservoir into a number of tanks that are allowed to communicate. Such tanks
can either be depleted directly by wells, or indirectly via other tanks. Flow
rates between tanks are set proportionally to either the difference in the
square of tank pressures or the difference in pseudo-pressures. In terms of
pressure squared, the flow between two tanks x and y is determined as:
where
Cxy is known as the communication factor and qxy is the rate of flow between
the two tanks x and y.
Individual
tank pressures are determined by assuming straight line P/z versus G behavior
where G includes both Gp (gas produced by wells in the tank) and Ge ( gas efflux
or influx from connected tanks).
Simulation
The
process of simulating petroleum reservoirs has been thoroughly described
by Peaceman (24) as the:
“process
of inferring the behavior of a real reservoir, the prototype system,
from the performance of a model of that reservoir. The model may be
physical, such as a scaled laboratory model, or mathematical. A
mathematical model of a real reservoir is a set of partial differential
equations, together with an appropriate set of boundary conditions, which are
believed to adequately describe the significant physical processes taking place
in the real reservoir, see Figure 14. The processes taking place in a real
reservoir are basically fluid flow and mass transfer. Up to three immiscible
phases (gas, oil, and water) may flow simultaneously where gravity, capillary,
and viscous forces play an important role in the flow process. Mass transfer
may take place between the phases (chiefly between gas and oil phases).
The
model equations must account for all forces, and should also take into account
an arbitrary reservoir description with respect to heterogeneity and geometry.
The equations are obtained by combining the mass conservation equation with
the equation of motion (Darcy's law).
To
use the mathematical model for predicting the behavior of a real reservoir, it
is necessary to solve the model equations subject to the appropriate boundary
conditions. The methods of solution are basically divided into two main
methods, analytical and numerical. Analytical methods are
applicable only to the simplest cases involving homogeneous reservoirs and very
regular boundaries. Numerical methods, on the other hand, are extremely general
in their applicability and have proved to be highly successful for obtaining
solutions to very complex reservoir situations. A numerical model of a
reservoir, then, is a computer program that uses numerical methods to obtain an
approximate solution to the mathematical model.”
Figure 14: Process of modeling
petroleum reservoirs
Due
to its simplicity, the finite difference method is the most widely used
numerical technique in petroleum industry. Finite differences are easy to
understand and to program. In addition, less input is required to construct a
finite difference grid.
There
are numerous models that were coded using this method. The use of finite
difference grids is restricted by the fact that they can not accurately
approximate the reservoir boundary as shown by Figure 15.
Figure
15: Finite
Difference Method
Reserve
estimates are only by-products of the reservoir simulation process. At the end
of each time step, the simulator calculates the volume of oil and gas in each
cell. Fluid saturations along with pressure values in each cell are updated for
the next time step. The process continues until the end of simulation time. The
amount of oil and gas in the reservoir is calculated by the summation of oil
and gas in all cells.
References:
- S. Naji, Hassan Dr., 2004, Petroleum Reserves Estimation Methods, A Report Submitted to the Energy Studies Department OPEC Secretariat
- www.petrobjects.com
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