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Oil Field Models, Decline Rates and Convolution
BY D E N N I S C O Y N E POSTED ON J U N E 2 5 , 2 0 1 4
This post is by Dennis Coyne
The eventual peak and decline of light tight oil (LTO) output in the Bakken/ Three Forks play of
North Dakota and Montana and the Eagle Ford play of Texas are topics of much conversation at
the Peak Oil Barrel and elsewhere.
The decline rates of individual wells are very steep, especially early in the life of the well (as
much as 75% in the first year for the average Eagle Ford well), though the decline rates become
lower over time and eventually stabilize at around 6 to 7% per year in the Bakken.
What is not obvious is that for the entire field (or play), the decline rates are not as steep as the
decline rate for individual wells. I will present a couple of simple model to illustrate this concept.
Much of the presentation is a review of ideas that I have learned from Rune Likvern and Paul
Pukite (aka Webhubbletelescope), though any errors in the analysis are mine.
A key idea underlying the analysis is that of convolution. I will attempt an explanation of the
concept which many people find difficult.
At Wikipedia there is a fairly mathematical presentation of the concepts which often confuses
people. There are a couple of nice visuals to convey the concept as well see this page.
In the visual below a function f (in blue) is convolved with a function g (in red) to produce a
third function (in black) which we could call h where h=f*g and the asterisk represents
convolution, just as a + symbol is used to represent addition.
“Convolution of box signal with itself2” by Convolution_of_box_signal_with_itself.gif: Brian
Amberg
derivative work: Tinos (talk) – Convolution_of_box_signal_with_itself.gif. Licensed under CC
BY-SA 3.0via Wikimedia Commons.
I think the best way to present convolution is with pictures. Chart A below shows a relationship
between oil output (in barrels per month) and months from the first oil output for the average
well in an unspecified LTO play.
This relationship is a simple hyperbola of the form q=a/(1+kt), where a and k are constants of
13,000 and 0.25 respectively, t is time in months, and q is oil output.
Chart A is often referred to as a well profile. The values for the constants were chosen to make
the well profile fairly similar to an Eagle Ford average well profile. EUR30 is the estimated
ultimate recovery from this average well over a 30 year well life.
Chart B shows the relationship between the number of new wells that begin producing each
month and the months from the start of production for the entire field.
The convolution of the relationship shown in chart A and the relationship shown in chart B
results in a third relationship shown in Chart C below, oil output vs. months from start of field
output. Output has been converted to kb/d from barrels per month.
It is indeed strange that two very different shapes (a hyperbola and a trapezoid) would combine
to form the shape shown in chart C. A spreadsheet can be downloaded here, with the scenario
above laid out.
What was surprising to me when I first tried this analysis was that a combination of the average
well profile with the number of wells added each month reproduced the oil output data fairly
closely.
To clarify this further, I have created a simple model. As before, we have a hyperbolic well
profile in chart 1 (slightly different than chart A above) and the number of new wells added each
month in chart 2, but in chart 2 this is over a short 6 month period. After that time no more new
wells are added.
In the chart below I show the output for each group of wells that begins production in successive
months. The output from all wells starting production in month 1 are labelled “month 1 wells”,
there are 6 of these groups up to “month 6 wells”. The number of wells added each month is
shown as a dashed line read off the right axis. Remember that 30 wells are added each month
from month 1 to month 6 so output for “month x wells” will be 30 times month 1 of the well
profile in month x and 30 times month 2 of the well profile in month x+1, etc.
The convolution of Chart 1 and Chart 2 results in Simple oil model 1 shown below.
This model is very simple in order to present how the principle works in a clear manner. When
the annual decline rate for the “field” is compared to the average well’s annual decline rate, they
are very similar for this simple 6 month model. More realistic models are presented later for
comparison.
Note that month zero in the chart below is the month of maximum annual decline rate, for the
average well the maximum annual decline rate happens in month 13 and for the field it occurs in
month 18, the curves have been shifted to the left by 13 and 18 months so that the maximum
decline rates match up at month zero for easy comparison.
The spreadsheet for simple model 1 can be downloaded here.
A second simple model with the number of wells added each month rising from 5 new wells per
month to 30 new wells per month over 6 months and then falling back to no wells added by
month 12 is shown below.
Note that the “month 7 wells” output curve is the same as the “month 5 wells“ output curve, but
shifted 2 months to the right. Likewise month 8 is month 4 shifted 4 months to the right and this
same symmetry is true for months 9 and 3(6 month shift right), months 10 and 2, and months 11
and 1 where the shift right in the curve is equal to the difference in the month when the well
started production (8 months and 10 months for the last two cases respectively).
When all of these 11 curves are added up for each month (the convolution of the “well output of
the average new well” chart and the “number of new wells added per month” chart) we get the
Simple Oil Model 2 chart below.
Simple model 2 can be downloaded here.
I now present a different model with a higher EUR well profile (than in chart A) and a lower rate
of addition of new wells (than in chart B). This model’s well profile is similar to the average
North Dakota Bakken well profile.
The convolution of the two charts above results in the field output shown below.
How does the annual field decline rate compare to the average new well annual decline rate in
this case? In the chart below we see that a slower decrease in the rate that new wells are added
causes the annual field decline rate to be only 22% at most, about 3 times lower than the
maximum annual well decline rate.
The spreadsheet for the model above can be downloaded here.
As this result is rather counterintuitive, I will try another modification to the model. The well
profile remains unchanged, but there is a steeper reduction in the rate that new wells are added to
field production.
Such a scenario could occur if there was a steep drop in oil prices as in the early 1980s. It will
also occur if there is a decrease in new well productivity which will reduce profits and the
incentive to add more wells.
The well profile chart is unchanged, the other two charts are as follows:
Even in this case the maximum annual field decline rates are less than half the maximum well
decline rate. This is because we have almost 15,000 wells added over an 11 year period and their
decline behavior in the aggregate is much different than that of an individual well. See chart
below.
Note that the field decline rate is very high, close to a 30% maximum rate in this scenario. If the
rate that new wells are added drops to zero over a 1 to 2 year period and no further wells are
added, we would expect the field decline to behave like the gray curve in the chart
above. Spreadsheet for the 5.6 Gb scenario can be downloaded here.
Earlier I mentioned that when I first tried this method I was surprised that such a simple model
could accurately match output from the Bakken or Eagle Ford fields.
Using data from the North Dakota Industrial Commission(NDIC) on oil output, the number of
new wells added per month, and individual well data(from Rune Likvern initially and lately from
Enno Peters) I attempted to match scenarios initially presented by Rune Likvern at the Oil Drum.
Below I present the well profile and number of new wells added each month.
When the two charts above are combined (convolved) we get the output curve below.
Note that the sharp drop off in the number of producing wells added each month is not very
realistic and is an artifact of the way I set up these simple models for illustration (they end at 130
months so the number of producing wells had to be ramped down very quickly).
Such a scenario would be more likely if there was a sharp rise in well costs, or a sharp drop in oil
prices or new well productivity (EUR). The field decline rate is somewhat similar to the previous
scenario, rising quickly to a 28% annual decline rate which falls to 10% after 5 years and to 7%
in 8 years.
This simple Bakken model can be downloaded here.
A fairly realistic scenario for the North Dakota Bakken (it is a little on the low end of likely
scenarios) is presented now for comparison to the model above. This scenario has an ERR
(economically recoverable resource) of 5.3 Gb where the more likely range is 7 to 9 Gb, based
on USGS estimates. The average well profile and number of new wells added each month are
below.
When we convolve the two charts above the following model output results. The match to the
data is surprisingly good.
The annual field decline rate and well decline rate are shown below. In this case the maximum
annual field decline is about 16% in 2021 and falls to 8% by 2026 and to 5% in 2031, the
maximum annual well decline rate is 61%, the well decline rate is shown for a well starting
production in Dec 2013.
The spreadsheet with this more realistic model is quite large (18 MB) so those with limited
bandwidth may want to skip it. The realistic Bakken model can be downloaded here.
For the Eagle Ford play I was able to collect data on single well leases from the Railroad
Commission of Texas, data on the number of producing wells in the play and output data. I
developed an average well profile (shown below) and combined it with the number of new wells
added each month to produce an output chart.
Note that the output chart is for crude only and does not include condensate.
The two charts above are combined (or convolved) to give the output chart below.
Note that there is about 20% of Eagle Ford output that is condensate, when this condensate is
added to the URR above for crude only we get a URR of 5.1 Gb of C+C.
As in the case of the North Dakota Bakken/Three Forks the match between the model and data is
surprisingly good considering the simplicity of the model and the complexity of the real world.
Summary
Oil field output can be simulated with the convolution of the average well profile of newly added
wells and the number of new wells added each month. I presented several simple models to
demonstrate this concept. An obvious weakness for any attempt at forecasting is that the future
average well profile may change over time and the number of new wells added in any future
month is unknown.
The decline rate of a field of wells will tend to be considerably lower than the decline rate of the
individual well. The field decline rate depends on several factors: the decline rate of individual
wells, the total number of wells in the field, the period of time over which these older wells were
added (whether the period was long or short), and finally the rate at which the number of new
wells added decreases as the field begins to decline.
Several models were presented showing how the field decline rate might vary under differing
circumstances.
The concepts presented were applied to scenarios which simulated both the North Dakota
Bakken and Eagle Ford shale plays with fairly good precision.
In a future post I plan to show how the convolution of two mathematical functions is used to
develop the Oil Shock Model.
Modelling PVT properties of reservoir hydrocarbon fluids
while oil field development planning
A.I. Brusilovsky and A.N. Nugayeva
Sibneft
Understanding and development of approaches to reservoir fluids description are among the
most topical issues for the gas sector: typically, companies acquire both oil and gas fields. This
paper focuses on description of an efficient approach to preparation of data used to generate
reservoir fluid PVT relationships in "black oil" hydrodynamic models. Case history of using this
technique in Yarainerskoye field (West Siberia) is discussed here.
Several alternative models differing in PVT properties formation of reservoir fluids are used for
hydrodynamic simulation of hydrocarbons field development.
The use of "black oil" models presumes that properties of phases depend only on pressure at constant
temperature. Thus, two types of models are distinguished: the first one considers the solubility of oil in gas
phase (i.e. wet gas), and the second - that oil is missing in the gas phase (i.e. dry gas). Models of the first
type are used more widely while forecasting the development of oil/condensate deposits and
gas/condensate fields under natural depletion conditions.
Multi-component fluid flow models are basically applied when it is necessary to forecast reservoir properties
for the development of gas/condensate, oil-and-gas, and oil deposits using external gas drive characterised
by intensive inter-phase mass transfer. For these models, reservoir fluid properties could not be represented
(even approximately) as exclusively pressure-dependent, since at the same pressure gas phases of different
composition can be present within the reservoir bed.
I. Data used for PVT models development. Adequate reasoning using per-bed reservoir fluid properties
is required for reserves estimation and field development planning. The uniform (in terms of both area and
volume) characterisation of reservoir beds by representative fluid sampling is one of the most important
requirements. Thus, well-grounded presentation of reservoir fluid composition could be established, as well
as its physicochemical properties. In practice, two basic techniques are used to produce representative fluid
samples:


Downhole sampling
Separator sampling.
Now it would be pertinent to consider basic conditions underlying the representative reservoir fluid
samples.1
Downhole sampling. At least three samples should be recovered to provide adequate analysis of reservoir
fluid PVT properties. Pressure at the sampling point should exceed saturation pressure, i.e. the recovered
sample should be a single-phase one.
Separator sampling. When downhole sampling is not possible, examination of the recombined specimen is
carried out, which has been mixed using oil and gas samples recovered while separation of the produced
fluid. The measured GOR value is used while recombining. In case that the recombined sample at reservoir
conditions (P = const, T = const) forms a two-phase gas-liquid system, the liquid phase (under-gas zone)
and the gas phase (gas cap) should be examined separately.
II. Theoretical basis of PVT properties modelling. The "black oil" model considers reservoir fluid as a
two-component mix. The "oil" component - stock tank oil, and the "gas" component - the dissolved gas.
According to Gibbs rule of phases, due to pseudo-binarity of the mixture, it would be sufficient to use
relationships between properties of fluid and vapour phases on pressure at reservoir temperature. Density is
set under standard conditions for both "oil" and "gas" components. The input data set on reservoir fluid
properties used in "black oil" models, includes formation volume factor, gas content and dynamic viscosity
vs pressure relationships. The structure of input data concerning vapour phase properties depends on
whether dry gas or wet gas model is applied. In both cases pressure relationships are set vs pressure,
formation volume factor and dynamic viscosity. In case of wet gas model it is necessary to set data for "oil"
component solubility in the vapour phase.
Figure 1. Reservoir oil saturation pressure vs. binary interaction coefficient between CH4 and С7 + (a); stock
tank
oil density vs. "shift parameter" of С7 + (b); reservoir oil formation volume factor vs. methane "shift
parameter" (c);
dynamic viscosity of reservoir oil vs. Vcr,C7+ parameter (d):
pc, ω of С7+: 1 - obtained by solution of the equation of state and Edmister relationship; 2 - obtained
from
correlations
Cubic equations of state are widely used in models applied to estimate PVT properties of natural
hydrocarbon systems. Peng-Robinson (PR) equation of state is the most spread.2,3,4
The use of PR equation of state to describe density of fluid phase in natural hydrocarbon mixtures results in
major inaccuracy. At present, the approach based on "splitting" of phase equilibrium simulation and phases
density adjustment is widely used. This method is based on the correction evaluation for the molar volume
of the phase v calculated using the equation of state by value of c, i.e. the "correct" molar volume is
estimated as
. For mixtures, this parameter is calculated using the following rule:
where N - number of components in the phase; xi - molar percentage of the i component in the phase; ci constant parameter for "pure" icomponent.
Please note that the phase volume correction keeps the result of modelled equilibrium phases composition
constant, i.e. the phase equilibrium should be calculated first, and then the values for phase molar volumes
are to be re-estimated.
Using the PR equation, Jhavery and Youngren5 recommend to calculate the value ci with the help of the socalled "shift parameter", Si which is related to ci via bi factor in the following way: ci = Sibi.
Two methods are used to estimate critical pressure pc, critical temperature Tc, and "acentric factor" ω of
hydrocarbon fraction CN+ (where N = 6, 7,...):
1. Boiling point and density correlations under standard conditions. Here, Kessler and Lee correlations are
mostly used.
2. Equation of state to identify critical pressure. The idea is to use the available density data for the given
fraction under standard conditions psc, and its molar mass, M, with reference to the specific equation of
state. In other words, fraction density calculated according to the equation of state at ρsc= 0.101 325 MPa
and Tsc = 293.1 К (or 288.6 К in compliance with SPE standards), should be equal to its measured value.
This means that to estimate only two out of three values (i.e. Tc, pc, ω), empirical equations should be
applied, and the third value is defined from the equation of state. Kessler-Lee and Edmister correlations can
produce sharp "acentric factor" values' estimates for all groups of hydrocarbons. Correlations for
pcestimation were obtained for hydrocarbons < C20, thus, their extrapolation is less reliable than the use of
corresponding relationships for Tc. Therefore, the Tc estimation is carried out by applying one of the above
correlations. Both Edmister formula and the mentioned Kessler-Lee correlation can be used to calculate the
"acentric factor". Pc and ю values are also calculated from the following system of equations: the equation of
state for psc, Tsc, vsc = M/psc, and the relationship used for "acentric factor" estimation.
III. Reservoir fluid PVT properties modelling while field development planning. For oil reserves
estimation, the values of formation volume factor (FVF) at initial reservoir pressure and the density of stock
tank oil (STO) obtained after multistage separation are used. The significant task here implies the exact
reconstruction of these estimation parameters in the set of data required for hydrodynamic modelling of
field development. Besides, it is necessary to reproduce the experimental values of reservoir fluid saturation
pressure and its dynamic viscosity at initial formation pressure. Sequential "tuning" of modelling parameters
is performed to solve these problems.
1. The reservoir fluid saturation pressure matching.
The solution can be achieved by two methods:


using a unique relationship between saturation pressure and binary interaction coefficient
methane CN+ (N = 7,...) fraction KCN4-CN+: the increase of this value results in growing reservoir
fluid saturation pressure
using unique relationship between mixture saturation pressure and the CN+ (N = 7,...) fraction
boiling point: reservoir fluid saturation pressure grows with the increase in fraction's boiling point
temperature.
2. The STO density matching (a reserves estimation parameter).
Here authors employed the following methods:


using unique relationship between STO density and "shift parameter" value of CN+ fraction Sr,:
increase in "shift parameter" value results in augmentation of STO density
using unique relationship between STO density and CN+ (N = 7,...) fraction density under standard
conditions: the STO density grows with the increase in fraction's density.
3. The reservoir oil FVF at initial formation pressure matching (a reserves estimation parameter). Since
methane is dominating the dissolved gas, its "shift parameter" can be used to match the reservoir oil FVF.
4. The reservoir fluid dynamic viscosity under initial formation pressure matching. To produce relationships
of reservoir fluid dynamic viscosity vs. pressure at various gas content, which are used in modern
hydrodynamic simulators, it is necessary to apply an integrated approach towards both experimental and
calculated data. The Lohrenz-Bray-Clark (LBC) method6 is most commonly employed for such estimates. The
obtained results strongly depend on critical molar volumes, Vcr, for the CN+ (N = 7,...) group of
components. Thus the Vcr CN+ value is a reliable parameter which allows to adequately describe the
dynamic viscosity of reservoir fluid.
Figure 2. Relationships of reservoir fluid volume
factor at various values
ol gas content vs. reservoir pressure (a); saturated
reservoir
luid gas content vs pressure (b); and reservoir fluid
dynamic viscosity
vs. pressure at various gas contents (c):
p and p - initial formation pressure and saturation
pressure at reservoir temperature;
experimental data are marked by circles
IV. Case history. As an example, let us discuss the data obtained while of reservoir fluid properties and
STO density matching for one of the bed of the Yarainerskoye field. The reservoir fluid composition (molar
percentage): C02 - 0.01; N2 - 0.55; CH4 - 34.07; C2H6 - 2.04; C3H8 - 3.20; i-C4H10 - 2.35; n-C4H10 - 2.87; iC5H12 - 2.01; n-C5H12 -2.11; C6 - 4.09; C7+ - 46.70. Reservoir pressure and temperature are 24 MPa and 78
°C, respectively. The experimental value of reservoir fluid saturation pressure at this temperature is 12.7
MPa.
Figure 1 (a-g) shows the relationships illustrating the influence of various modelling parameters on
estimated reservoir fluid properties and STO density. The value points shown in these graphs correspond to
accurate matching of experimental saturation pressure at reservoir temperature, volume factor and dynamic
viscosity of reservoir fluid at initial formation pressure, as well as STO density.
Figure 2 shows calculated relationships of volume factor, gas content, and dynamic viscosity of reservoir
fluid vs.pressure, which could be used in hydrodynamic simulation of reservoir depletion using the "black oil"
model.
Now the following conclusions can be made:


The PVT relationships construction method for "black oil"-type hydrodynamic simulators has been
developed, which is based on experimental and theoretical data generalisation
The algorithm for sequential identification of parameters for the calculated reservoir fluid model was
proposed.
The practical importance of this research lies in the precise matching of reserves estimation parameters: the
Formation Volume Factor of reservoir oil, and the Stock Tank Oil density.
Literature Cited:
1. "Petroleum. Standard for Reservoir Fluids and Separator Oils PVT Studies." OST 153-39.2-048-2003. K.D.
Ashmyan, Yu.A. Shmelev, V.V. Getmanenko, et al., Moscow, VNIIneft, 2003, 85 pp.
2. Brusilovsky, A.I. Phase Behavior of Hydrocarbon Fluids while Oil and Gas Fields Development, Moscow,
Graal, 2002, 575 pp.
3. Whitson C.H. and Brule M.R. Phase Behavior, SPE Monograph Series, Richardson, Texas, 2000, 233 pp.
4. Brusilovsky, A.I. "Cubic Equations of State for Modelling Natural Gas/Condensate Mixtures", Gazovaya
Promyshlennost, 2004, No. 4, pp.16-19.
5. Jhavery, B.S. and Yungren G.K. "Three-parameter modification of the Peng-Robinson equation-of-state to
improve volumetric predictions," SPE Reservoir Engin., 1988, V. 3, N 3 (Aug.), pp. 1033 - 1040.
6. Lohrenz, J.; Bray, B.G.; and Clark, C.R. "Calculating viscosity of reservoir fluids from their composition,"
J.P.T., 1964, p. 1171.