SPE Paper 139357x

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Traditional Material-Balance Analysis from a Stochastic Point of View:Predicting Reservoir PerformanceRodolfo Anthony Galecio, UNI. /Víctor A. Huerta, Repsol.

Copyright 2010, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Latin American & Caribbean Petroleum Engineering Conference held in Lima, Peru, 1–3 December 2010.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewedby the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, ormembers. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print isrestricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

We see traditional material balance calculations as an old technique when we face the challenging issue of forecasting reservoirperformance, numerical simulation is by far the selected method used by reservoir engineers; but, what happen in an early

development phase of an oilfield, when we have scarcely data and a high degree of uncertainty associated?. The aim of this work is

to show that methods based on MBE can be successfully integrated with Monte Carlo technique in order to predict reservoir

performance under these circumstances.

We have selected two methodologies; the first one was proposed by Tarner, designed to predict the Cumulative Oil production

as a function of the average reservoir pressure (reservoir depletion). Tarner uses simultaneously the MBE and the GOR definition

to estimate the value of the Cumulative Gas Production, through the convergence of both values. The second method was proposed

by Tracy, in which a trial and error procedure is followed to converge to the instantaneous Gas Oil Rate, which in turns, allows us

to estimate the incremental cumulative oil production. Special attention was given to the ratio of relative permeability, because

both methods are quite sensitive to it, and usually this ratio is calculated by correlations (in absence of special core analysis). Flow

rate forecasts were calculated using Vogel approach and regarding decline rates of local oilfields.

A macro in excel was plugged into a commercial risk analysis software in order to combine the material balance approach with

Monte Carlo simulation, so as to address simultaneously to reservoir properties uncertainty and limited production records, insteadof getting on with complex numerical simulation models. A small mature area in Talara Basin producing from Mogollón formation

was successfully tested with this integrated tool; in contrast to the conventional decline curve forecasts, a sound set of analytical

predictions weighed by certainty were prepared, which is indeed more reliable for a decision-making process.

IntroductionDuring early stages of development of a leisure area, we have to face the challenge of dealing with scarce and poor quality data;

the full dataset required for numerical simulation is often unavailable until further and expensive studies have been made on the

reservoir. Under this scenario, quantifying the uncertainty related to the reservoir’s performance is a “must” in order to provide

reliable and responsible answers which help us to conceptualize the whole project. An inexpensive and quickly answer to this

problem can be provided by traditional material balance calculations, when they are used in conjunction with stochastic techniques

designed to model the uncertainty inherent to any forecasting process. The way we use the MBE depends on the methodology

selected (excellent references can be found in modern reservoir engineering textbooks); therefore, we have limited the scope of

this paper to describe fundamental relationships which help us to understand the whole process.

We have selected two forecasting methodologies: the Tarner’s approach, which is based on a simplified version of the MBE

designed to estimate the Cumulative Gas Production, as a function of the declining reservoir pressure. In order to solve this

equation we have to estimate in advance the depletion’s degree of the reservoir at the desired pressure, in other words, a first

approach of the Cumulative Oil Production is needed before the calculation itself could be performed; a second value of this

variable is calculated through the equation which defines the Cumulative Gas Production, as an average of two different moments

in the production life of the reservoir; this expression, as we will see, is a function of the Instantaneous Gas Oil Rate, then we need

also to calculate this value in advance from an equation derived from Darcy’s law, this is a very important relationship since it is

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strongly affected by the relative permeability ratio between oil and gas. Finally, both values are compared, if the difference is

within certain predefined tolerance, our first estimate of the Cumulative Oil Production will be considered essentially right,

otherwise the entire process is repeated until the desire level of accuracy is reached.

The second approach belongs to Tracy, who has reorganized the classical MBE using groups of PVT properties in order to

obtain a convenient and meaningful expression from which we can derive the incremental Cumulative Oil Production as a function

of the Instantaneous Gas Oil Rate, similarly to the former method. In order to solve this equation, we need to make again a first

assumption, this time on the Initial Instantaneous GOR; this value will be compared with the calculated GOR obtained from theDarcy`s relationship mentioned above. Tracy has proposed to use a comparison ratio between this two GORs; its value must be

between 0.999 and 1.001 to validate our initial assumption, again an iterative process is necessary to reach the desired accuracy.

To complete our deterministic preliminary model, we have to include a relationship to estimate the evolution of the Cumulative

Oil Production through the life time; this is accomplished by using an Inflow Performance Relationship, which reflects the

productivity of the well. Several empirical approaches are available to model present and future performance; in particular, we

have chosen the classical Vogel’s approach.

The next step implies the use of the Monte Carlo simulation to quantify the uncertainty related to our model; as we have seen,

both methodologies share an iterative character, which in turn makes them suitable for computer programming. To simplify the

process, we have employed a spreadsheet controlled by a Macro; this is a convenient arrangement because many commercial risk

analysis programs are designed as an add-in which allows us to set when and how to include a Macro in the simulation process.

Some questions arise at this stage: Which of the random variables in our model contain significant uncertainty? Are they related to

each other? Which of them have a substantial impact in our results? The answers to these questions demand an interdisciplinary

point of view; for example, some random variables like the Initial Oil in Place can be precisely calculated from static models

provided by Geologists, Geophysics and Reservoir Engineers, others, such as, Relative Permeability Ratio, in absent of field data,

may require the use of correlations designed to mimic its behavior as a function of fluid saturations. Finally, a statistic expertise is

needed to design correlation studies and sensitive analysis, which are required to determine the way these random variables are

related and their relative weight in the model. Once the shape and limits of the input distributions have been defined, we can

proceed with the simulation itself, our goal is to obtain the probability distribution for the selected output variable; the way we use

these results is beyond the scope of this article, but it can be pointed out that not only these results are a reliable description of the

uncertainty associated to our predictions, but also, they are used as an useful tool in decision-making processes, according to the

Expected Value theory. Indeed an output distribution could be easily used as an input distribution for complex financial

simulations.

We have selected a solution gas drive reservoir for our case study, the Mogollón formation in Talara basin. This is a mature

area; therefore, in order to test our procedure, an uncertainty scenario was defined according to our initial assumptions aboutquality and quantity of data. As we will see, the proposed procedure offers valid results in a relatively easy and quickly way.

Finally, two remarks, first these methodologies have been developed to be used with saturated reservoirs, above the bubble point

pressure an iterative process is not needed, a direct material balance calculation can be performed to forecast the performance of an

undersaturated reservoir, if this is the case the cumulative production of the undersaturated period must be included in our total

report below the bubble point pressure; second, although both methodologies were primarily designed to deal with dissolved gas

drive reservoirs, their application is not limited at all, indeed there are theoretical adjustments designed to take into account other

primary drive mechanisms, for example a water drive reservoir can be modeled adjusting the relationship used to describe the oil

saturation and using some of the available mathematical models designed to predict the water influx in the reservoir.

A theoretical overview, some basic relationshipsThe relationships that govern the methodologies employed are going to be introduced in the same order in which they appears in

the correspondent workflow, let’s start with the Tarner method: first of all an initial assumption must be made about the

Cumulative Oil Production at the future reservoir pressure considered, in other words a first estimated value of the reservoirdepletion has to be stated in order to solve the following equation which is derived from the MBE for a volumetric and

undersaturated reservoir:

[1]

As we can see this equation relates, in order to calculate our first estimated value of the Cumulative Gas Production, an initial

reservoir condition which is setting as a reference point and the future reservoir condition, both defined in terms of their

correspondent PVT properties. If these PVT properties are considered representative and reliable, the accuracy of the prediction

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will depend on the estimated value of the Initial Oil in Place and our first assumption about the future Cumulative Oil Production.

The next step is to obtain the Oil Saturation at the assumed future reservoir pressure using an equation derived from the

relationship between the remaining oil volumes after producing N p stock - tank barrels of oil and the reservoir pore-volume

definition:

[2]

The uncertainty related to this expression is also affected by the same considerations mentioned above plus the effect of an

imperfect knowledge of the initial water saturation in the reservoir, the resulted Oil Saturation is added to the Initial WaterSaturation to get the Total Liquid Saturation which is often correlated with the Relative Permeability Ratio (RPR) between Oil and

Gas, this is a key parameter necessary to compute the Instantaneous Gas Oil Rate from the next expression derived from Darcy’s

law:

[3]

Again we have supposed that reliable data about the viscosity of the fluids in the reservoir are available in order to simplify the

uncertainty modeling process which will be focused on the Original Oil in Place and the Relative Permeability Ratio as key

parameters. Now we can calculate the second estimated value of the Cumulative Gas Production at the future reservoir pressure

considered as follows:

[4]

This expression comes from the integral which defines the incremental cumulative gas produced:

between two states: an initial well known current state and a future unknown state, when we use the trapezoidal rule to approach

the integral; this suggests an essential weakness of the method, as our desired future state is far away from the present conditions

the accuracy of our prediction will be accordingly affected. Finally both estimates of the Cumulative Gas Production must be

compared and an agreement should be found within an accepted tolerance, then a relative error is defined with respect to the

average between both estimates:

[5]

If its value is below our predefined tolerance, the process will have concluded and the reported value will be the calculated average

otherwise a new assumption must be made on the future Cumulative Oil Production, as we will see this iterative process implies

the use of a preset increment in the predicted N p.

As we have mentioned Tracy reorganized the classical MBE using functions of PVT properties, these functions have to becalculated at the desired future reservoir pressure before starting the iterative process itself, for a saturated and volumetric

reservoir:

[6]

[7]

Both expressions are referred to an initial reservoir condition which as in the former method has to be explicitly defined; the next

step is to estimate the Instantaneous Gas Oil Rate at the selected new reservoir pressure, which we will call GOR1, the average

between this estimate and the current GOR allows us to solve the equation derived from Tracy’s rearrangement of the MBE for the

Incremental Cumulative Oil Production:

[8]

It has to be pointed out that in this expression the Cumulative Production variables are referred to a current state instead of a fixed

initial state, this is a key difference between both methodologies Tracy proposes that after each successful iteration the forecasted

values of the Cumulative Production become the new reference point, the current state of the reservoir. The future Cumulative Oil

Production is then calculated from:

[9]

Once we get this value the process is similar to the Tarner method, the future Oil Saturation is calculated through the equation 1.2

and the second value of the future Gas Oil Rate can be obtained from Equation 1.3, which we will call GOR2. Finally in order to

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validate our initial assumption the tolerance of a comparison ratio is set forth as follows:

[10]

Therefore if an adequate agreement between both values is achieved the iterative process will be considered finished otherwise a

new GOR assumption has to be made until the accepted tolerance has been reached, the future Cumulative Gas Production is

calculated from:

[11]

As we mentioned above the forecasted Cumulative Production obtained becomes the next known current condition to beconsidered for the next forecasting cycle. At this point we have gotten just only the relationship between the average reservoir

pressure and the Cumulative Production but how to relate this performance to time? The answer is provided by the Inflow

Performance Relationship concept.

Reservoir Performance vs. TimeSimply stated the Inflow Performance Relationship is the graphical representation of the relationship between the bottom-hole

flowing pressure of a well and the Oil Flow Rate at a given average reservoir pressure, this is a practical way to reflect the

operational condition of a well, its completion, damage or associated mechanical problems, on the well’s ability to produce. There

are several available empirical methodologies to calculate the IPR of a well able to characterize vertical and horizontal wells, how

do we select a particular method depends on many factors but at the end the matching’s degree with the real well behavior is by far

the most important criteria to be considered. We have to remark that these methods are designed to deal with the IPR non linear

behavior observed below the bubble point pressure, above this point it is enough to use the Productivity Index J to model the

resulting straight line relationship between both variables. We have selected the classical Vogel method to model the present andfuture IPR of a vertical well, based on at least one measure of a stabilized bottom-hole pressure and its correspondent rate the

method use a simple empirical equation:

[12]

Where the Absolute Open Flow AOF is the theoretical rate obtained when the wellbore pressure is equal to zero; if the reservoir is

saturated the equation is used almost directly to calculate the present oil flow rate attached to a specific operative bottom-hole

pressure in the well but if the reservoir is undersaturated is convenient to use an improvement proposed by Beggs, he sets forth the

procedure to be follow according to the relative position of the recorded stabilized bottom-hole pressure with respect to the bubble

point:

1. If the production test gives us a stabilized bottom-hole pressure above or equal to the bubble point, the productivity index

is then calculated from:

[13]

This calculated productivity index is enough to describe the linear relationship of the IPR above the bubble point

pressure:

[14]

But if we consider operating the well using a bottom-hole pressure below the bubble point, Beggs proposes an expression

derived from Equation 1.12 to calculate the present oil flow rate:

[15]

Where Qob is just the oil flow rate at the bubble point pressure previously obtained from:

2. A stabilized bottom-hole pressure recorded below the bubble point requires a somewhat more complex expression for the

productivity index:

[16]

Again the present oil flow rate is determined using the linear relationship described by the Equation 14 if our operative

bottom-hole pressure is above the bubble point, below this point Equation 15 must be used to describe the IPR.

At this point we just get the current description of the well behavior, its present IPR, but we are working on a forecast therefore we

need to predict the future IPR at the selected average reservoir pressure, if our reservoir is saturated the procedure is reduced to the

estimation of the future AOF through the following two approaches:

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[17]

[18]

The former equation was proposed by Fetkovich and both are used in conjunction with the Equation 1.12 in order to predict the

future IPR of a well in a saturated reservoir, as we mentioned above this assumption simplifies the forecasting procedure, although

we could follow the procedure proposed by Beggs for an undersaturated reservoir, the future IPR calculation relies on the future

Productivity Index which is not easy to predict, in order to make feasible its estimation we have to suppose that the reservoir is

under pseudosteady-state which in turn implies a constant drawdown, under these circumstances we can use as a roughly

approximation the same value obtained at the present conditions for the Productivity Index assuming an almost constant behaviorof this variable otherwise a trend able to depict its behavior must be found. One final important remark, all this procedure is

intended to be used if our primary target is to keep a constant drawdown in order to preserve the drive-energy of the reservoir, but

if it is needed a constant Oil Flow Rate to meet come contract specifications we will use the IPRs predictions just only to

determine the feasibility of such demands.

Once we get all this information the answer to our primary question about the relationship between the Cumulative Oil Production

and the time which is needed to produce it, is provided by the following equation:

[19]

This simple relationship relates two different hard-to-find results in a certain pressure drop interval, from our MBE calculations the

incremental Cumulative Oil Production between two different reservoir pressures and from our IPR estimates an average of the Oil

Flow Rate able to be maintained during this interval, the accuracy of this expression will be obviously affected by the extension of

the considered interval.

The Relative Permeability issueAccording to Tarek Ahmed & Paul D. McKinney perhaps the most influential parameter driving the oil recovery is the relative

permeability ratio between Oil and Gas but what happen if we only have a limited production record to estimate its

behavior’s trend or even no data is available? In such cases the use of appropriate correlation is a must, there are several

correlations designed to match a specific geological environment then its selection should be aid by an experience professional

able to foresee even the limits of its applying, this is particularly useful when we face the challenging task of modeling a

probability distribution for this parameter. Most of these empirical relationships were calculated to take into account key

parameters like residual and initial Saturations and Capillary Pressure data which depicts important characteristics of the formation

as a whole, the hysteresis phenomena is also considered and analytical expressions are often used in numerical simulators;

although it is possible to include all these options in our workflow we will focus on two simple correlations:

1. The Torcaso & Wyllie correlation is used for sandstones and proposes the following simple relationship:

[20]

Where S *

is the Effective Phase Saturation which is used in several correlations as a correlating parameter, in particular

for the Oil phase:

[21]

2. The Corey correlation provides good results in drainage processes like the flowing of Oil in a dissolved gas drive

reservoir, it is expressed as a function of the effective gas saturation:

[22]

[23]

Of course all these correlations provide a single, deterministic estimate of what the relative permeability ratio is, but if we want to

assess the uncertain related to this variable, we will need to vary their estimates within certain predefined range, in other words a

random sampling experiment is required to take us to the next step, the Monte Carlo simulation.

Adding a Stochastic PerspectiveOur laborious forecasting report based on an appropriate theoretical framework and supported by history matching process can’t

help us to answer a simple but decisive question: How sure are you about your results? In other words: What is the uncertainty

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attached to your predictions? The answer relies in a well known technique, the Monte Carlo simulation. Briefly stated by James R.

Evans & David L. Olson:

“Monte Carlo simulation is basically a sampling experiment whose purpose is to estimate the distribution of an outcome variable

that depends on several probabilistic input variables”

The proposed definition suggests that a primary task to achieve is identifying the relevant probabilistic input variables in our

model, this is by far the most demanding stage in Monte Carlo process, and it requires not only sensitive analysis which are used to

measure the relative weight of the variables involved in our model but also an experience professional judgment able to shape the

probability distributions which better describes the input variables. Correlations between input variables must be observed in theaim of discarding certain unlikely scenarios and an enough number of sampling experiments must be set forth in order to assure

meaningful results. We are not going to describe in detail the Monte Carlo simulation actually there are a lot of related literature

about it and our approach is essentially an integrated point of view which use this technique as a tool, in this sense fortunately the

sampling process has been almost automatized by commercial software and even improved by using sound variations for the

sampling technique like the Latin Hypercube method, most of these programs have been designed as an add-in which indeed

facilitates their using in conjunction with common spreadsheets, the ability to link the Macro which controls the spreadsheet with

the sampling process is essential and allows us to reduce the time required to build a complete forecasting model.

In our particular case we have chosen two input variables to be modeled: the Initial Oil in Place and the Relative Permeability

Ratio. It is well known that the Initial Oil in Place probability distribution has a lognormal shape; a clear explanation for this is

given by James Murtha:

“Regardless of the shapes of the inputs to a volumetric model – be they skewed right, or left, or symmetric the – output will still be

skewed right, thus approximately lognormal. In fact, as is well known, the Central Limit Theorem guarantees this. The argument is

straightforward: the log of a product (of distributions) is a sum of the logs (of distributions), which tends to be normal. Thus the

product, whose log is normal, satisfies the definition of a lognormal distribution.”

Then once we set forth the upper and lower limits of this distribution and a numerical glimpse about its dispersion the distribution

description is complete; the other variable, the Relative Permeability Ratio, can be described by empirical correlations when there

is a lack of field or laboratory data, as a function of fluid saturations the typical associated range of values for this variable is

ample and requires a semilog scale representation, therefore we will introduce a variation-factor against the original correlation

function, for the case of Torcaso & Wyllie correlation: then varying this α-factor we can

“move” the function over the vertical axe without changing essentially its shape keeping the trend behavior originally depicted by

the correlation. From this a question arises: Why don`t we just assign a specific probability distribution to each depletion stage for

this variable instead of this cumbersome approach? Because assigning a probability distribution demands certain allocation

parameters which are indeed unknowledge until production data becomes avalaible, in this sense our experience advisor could

provide more easily a range of variation for this variable rather than specific fixed statistic parameters, we hope that the case study

will clarify this point.A well designed Monte Carlo simulation can provide the answer to the uncertainty assessment problem, the probability

distributions of predefined output variables, in our case the Cumulative Oil and Gas Production, relate a specific range of possible

outcomes to an also specific uncertainty level, but beyond this simple interpretation is the wide range of applications of this

concept what makes it an invaluable tool, not only does it permit a formal stochastic reserves definition but also it is useful in the

decision making process through the Expected Value concept. The aim of the following sections is showing how a deterministic

forecasting methodology could work in a stochastic way using a simple but effective Macro design.

The Workflow DiagramBoth predictive methods share an iterative character that makes them suitable for programming, and the process is greatly

simplified since some functions like the Oil Saturation and the Instantaneous Gas Oil Rate are also common to both methods; a

structured or modular programming approach has been followed pursuing an understandable and adaptable code. The macro was

written in Visual Basic as a way to use the default capability included in an Excel spreadsheet but the selected programming

language depens heavily on your preference and the degree of complexity to manage. Although the workflow diagram is easy-to-

follow we pointed out that it was necessary to define some variables which are designed to control the process itself, their

description next:

1. Tarner method

a. Dep: this is our initial assumption about the depletation’s degree in the reservoir at the first drawdown

considered.

b. DepVar: this is the amount which must be added to the former parameter until the iteration process has reached

the desired accuracy.

c. RelError: this parameter sets out the accepted tolerance for the iterative process.

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2. Tracy method

a. GorEst: this is the initial estimated value for the instantaneous GOR at the selected new reservoir pressure;

although we can settle the Gas Solubility as a default value we must be aware that until the gas saturation has not

reached its critical value within the reservoir, the GOR will be below this assumption.

b. GorVar: adding a fixed value to the GorEst the iterative process mimics the GOR behavior trend before the

blown-down period.

c. LowerLimit, UpperLimit : these parameters are related to the desired accuracy of the iterative process.

Of course we could vary the by-default values of the variables related to the accuracy of the iterative process but we have toconsider that a more demanding degree of accuracy might end in unacceptable slowly time-processing, in the other hand if we

want to set a larger variation on the accumulator variables, the iterative process couldn`t be able to find the right answer entering in

an annoying infinite loop, finding an alternative configuration is always a trial and error process.

As we have mentioned above both methodologies were originally designed to be used with saturated reservoirs but in order to get

a more functional model we have included the necessary code to handle an undersaturated reservoir condition, two remarks at this

respect:

1. Above the bubble point the MBE does not need an iterative process to be evaluated and the calculus can be reduced to

the next expressions:

[24]

[25]

The results provided by these equations must be added to the obtained at saturated condition at the final stage of the

process.2. Although according to Steffenson Tracy’s method can be used for the entire pressure range by redefining the initial oil

volume formation factor at undersaturated condition in the denominator of the Tracy PVT functions we have preferred

for the sake of simplicity to use the former equation for both processes above the bubble point pressure.

Once these iterative processes are linked to the Monte Carlo simulation each cycle is repeated until the pre-settle number of

simulations has been reached and the probability distributions of the outcomes variable have been accomplished, the way we do

this depends largely on the particular risk-analysis software selected. The workflow diagrams are showed in Figures 1, 2 and 3.

Case Study: The Mogollón FormationOur case study is considered a mature area: the Mogollón formation is part of the Talara basin, which is located in the northwest of

Perú, it exhibits a wide range of permeability and porosity ranging correspondingly from 0.01 mD to 2 mD and from 3% to 20%,

but its main attractive as a producer area is the Gross Thickness which averages 2000 ft. General data is provided in conjunction

with the necessary PVT results wich are listed in Table 1: the initial reservoir pressure is 1777 psia, the bubble point pressure

which is 2500 psia, the volumentric estimate of the OOIP is equal to 10 MMSTB and the irreducible water has been estimated in33 percent. This is an ex post facto study then we need to recreate the initial uncertainty scenario, in order to do this the available

production history will be limited to the first two years as we listed in Table 2, this limited history allow us first to check if our

assumption about the recovery mechanism, internal gas drive, was correct, second to compare the volumetric estimate of the OOIP

with the result of the MBE approach this in turn gives us a first glimpse about the uncertainty related to this variable and finally

through a somewhat rudimentary matching process to get the range covered by the relative permeability ratio in this early stage of

production, all of these need expansion.

First of all we must be sure about the primary mechanism which drives the reservoir how we use the MBE depends on it; hence

some useful relations must be recalled from Havlena and Odeh (1963, 1964), for a volumetric saturated oil reservoir:

[26]

Where:

[27]

[28]

[29]

As we see all the variables involved in the calculus of these equations are available and a linear relationship from Equation 25 is

expected, if our assumption about the primary recovery mechanism was right, a graphical display of the pertinent results is showed

in Figure 4. The slope of the resulting straight line should be equal to the volumetric estimate of the OOIP, the observed difference

reflects the inherently uncertainty attached to these measurements and further helps to shape the probability distribution of this

variable, the proposed distribution shaped by this roughly approach is a truncated Lognormal distribution presented in Figure 5.

As it is well known the MBE can be used to determine the relative permeability ratio at each depletion stage from the available

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production history data, reorganizing Equation 3 we can easily derive the next expression: which

in turn requires the following basic definitions to handle the usual way production data is presented:

[30]

[31]

Of course the involved variables into these definitions are potentially sources of uncertainty; the recorded oil and gas flow rates

might not be accurate and as we take longer periods of time to calculate the cumulative oil production the used average oil flow

rate might not represent adequately the trend behavior of this variable. However and for the sake of simplicity we will consider this

dataset as reliable in order to focus our attention on the RPR as a key variable in the matching process using the Torcaso and

Wyllie correlation, indeed after a reasonable history match was achieved a clearly trend in the material balance derived RPR is

observed allowing us to set a central tendency statistic for our proposed correlation variation-factor which in turn helps us to define

a triangular probability distribution for this allocation variable within certain limits as is illustrated in Figure 6; the results of the

history matching process using the Tarner method are presented graphically in Figure 7. Table 3 and Figure 8 show the relative

permeability ratio results.

Although at this point it is perfectly possible to run a Monte Carlo simulation relating only the cumulative production versus

pressure drawdown the uncertainty attached to the well behavior must be addressed in order to complete the forecasting procedure,

this a special hard-to-do enterprise even at appraisal stage we are not 100 percent sure about the completion procedure to be follow

and despite future well performance improvements able to influence meaningfully the IPR we have to assume at least twoconstraints:

1. The vertical well is is producing under primary recovery mechanism; secondary or terciary recovery requires special

teothetical treatment beyond the scope of this study.

2. There are several factors which determine the way we operate a well, a constant flow rate would be needed to meet

fiscal and contractual exigencies or on the other hand preserving the reservoir drive energy could lead us to set up a

constant drawdown despite a decreasing flow rate. Obviously these over-simplified operational schemes are proposed

in order to handle the complexities of a more realistic situation; in our particular case we have chosen a constant

drawdown throughout the operational life of the well.

Unfortunately production test data was not avalaible therefore an ad-hoc solution was necessary, the irregular trend behavior

exhibited by the flow rate in the matching process was mimicked by an exponential function and a similar approach to the RPR

correlation procedure was used to control its behavior, Figure 9.

What does the output of a MonteCarlo simulation look like? Amog a lot of detailed statistical reports the main result is often

presented as a probability distribution representation of the target variable, our model was extended over the whole productive life

of the reservoir therefore the probability distributions representing both the final cumulative oil production and the final

cumulative production time were obtained and depicted in Figures 10 y 11, as we can see a P90 estimation can be easily made

from these. Another useful representation relates the observed variable trend behavior to the reservoir pressure drawdown: Figures

12 and 13 shows that data is indeed within our forecasted estimates.

Outlining some ConclusionsHow to manage the inherently uncertainty related to forecasts is still an open question, the way we shape the input data and how to

interpret and use the outputs of a complex and always imperfect representation of reality set multiple challenges to our

understanding which range from avoiding the natural bias in our judgements shaping input variables to the properly and coherent

applying of decision-making criteria like the one proposed by Paul D. Newendorp in his classic work about risk analysis in the

petroleum industry: “When choosing among several mutually exclusive decision alternatives select the alternative having the

highest positive Expected Monetary Value”.

This is a statement easy to understand but hard to be follow, since it demands an absolutely independence from moneyconsiderations; however and despite its shortcomings, it would be a better starting point for the decision making process than

judgments based on biased expertise or manager’s rules of thumb.

To sum up this discussion, some conclusions have been written down to inspire more thoughtful research:

1. The way we use the MBE has been improved far beyond its original form; not only there are some modifications to fluid

saturation calculation, but also its traditional domain have been extended to predict water influx effects.

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2. The most important key-variable in the material balance forecasting technique is the relative permeability (RPR). The

RPR is indeed a complex variable by far better characterized by a Dyad: a tensor of rank 2, which implies one

magnitude and two associated directions; our roughly approach to model its behavior through correlation variations is

only intended to simplify its scalar treatment and perhaps to promote surely better proposals.

3. Finding the best methodology to get the present and future IPR of a well has become an ad-hoc issue; indeed both vertical

and horizontal wells can be modeled by semi-empirical relationships, which are suitable to deal with three-phase flow

(with Wiggings correlation) or pseudosteady-state condition (predicted with Fetkovich). Despite these improvements,we should be conscious of the uncertainty linked to the variables of IPR relationships.

4. Adding a stochastic perspective to material balance forecasting methods provides quick and reliable results, allowing us

to deal with high-uncertainty scenarios in a non-expensive way; in addition, our reservoir understanding is improved, an

essential requirement before using expensive and fancy numerical simulation.

5. It can be argued that high-uncertainty scenarios could lead to probability distributions with unpractical range of results.

After all, if all possible answers are considered, we are not in fact answering the question, then why bother all these

cumbersome procedures? This a common widespread misunderstood about the role of the probability approach, a Monte

Carlo simulation can’t reduce the uncertainty related to our project, a wide range of possible outcomes only evidences

the necessity of better information to deal with. A well-informed decision supported by stochastic studies is not always

the rigth one; it is just the most careful of them.

Nomenclatureµ g = gas viscosity, cp

µ o = oil viscosity, cp

AOF = absolute open flow, STB/day

Bg = gas formation volume factor, bbl/scf

Bo = oil formation volume factor, bbl/STB

Boi = initial oil formation volumen factor, bbl/ST

cf = formation rock compressibility, psi-1

co = oil compressibility, psi-1

cw = water compressibility, psi-1

GOR = instantaneous gas oil ratio, scf/STB

Gp = cumulative gas production, scf

J = productivity index, STB/day/psi

k rg = relative pewrmeability to gask ro = relative permeability to oil

N = initial oil in place, STB

Np = cumulative oil production, STB

Pb = bubble point pressure, psig

pr = average reservoir pressure, psig

pwf = bottom-hole flowing pressure, psig

Qo = oil flow rate, STB/day

Rsi = initial gas solubility, scf/STB

Rsi = initial gas solubility, scf/STB

So = oil saturation

Swi = connate or initial water saturation

Φg = gas PVT function

Φo = oil PVT function

ReferencesAhmed, Tarek and McKinney, Paul D. 2005. Advance Reservoir Engineering. Gulf Professional Publishing, Elsevier Inc.

Dake, L. P. 1978. Fundamentals of Reservoir Engineering. Elsevier Inc.

Evans, James R. and Olson, David L. 2002. Introduction to Simulation and Risk Analysis. Prentice Hall.

Hornapour, Mehdi et al. Relative Permeability of Petroleum Reservoirs. CRC Press Inc.

Jelen, Bill and Syrstad, Tracy. 2004. VBA and Macros for Microsoft Excel. Pearson Education Inc.

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Mian, M. A. 2002. Project Economics and Decision Analysis.Volume II: Probabilistic Models.Penn Well Corporation.

Newendorp, Paul.2000. Decision Analysis for Petroleum Exploration. Planning PressTM

.

Wyss, Gregory D. and Jorgensen, Kelly H. A User’s Guide to LHS: Sandia’s Latin Hypercube Sampling Software. Risk

Assessment and Systems Modeling Department Sandia National Laboratories.

Table 1 PVT Data

Table 2 Mogollón Production History

The last column sho

the ratio between

underground

withdrawal and the

combined effect of t

expansión of oil andoriginal dissolved g

the expansion of the

initial water and the

reduction in the pore

volume has been

neglected; this ratio

must be equal to the

initial oil in place, th

average obtained of

this limited product

history is 18.6

MMSTB. The value

used in the model fothis variable was 14

MSTB, an average

between the former

estimation and the

volumetric OOIP.

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Table 3 Material Balance Relative Permeability Ratio

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Figure 1

General description of the

whole process, the darker

boxes denote sub-processes

and the yellow box points

out and important differencebetween both predictive

procedures, as a result of

each complete set of

iterations using Tracy

method a “new knowledge

current condition” is defined

whereas Tarner sets it once at

the beginning of his

procedure.

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SPE-139357-PP 13

Figure 2

Workflow diagrams for

Tarner and Tracy methods;

the parallel description has

been used to emphasize

shared relationships which

are handled as a Public

Functions in the Macro

design. The relative

permeability ratio routine

is not further decribed

because it depends on the

selected correlation to be

used or even a manual

entry option.

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14 SPE-139357-PP

Figure 3

Figure 4

Vogel procedure. Its applying depends on the

production test parameters, the current reservoir

pressure (Test_RP) under which was conducted

the test defines the way we will work. It wasassumed that future IPR calculations are below

the bubble point pressure in order to use

perhaps an over-simplified approach; again this

is only a teorethical first glimpse to a subject

which requires indeed a deeper look.

Underground withdrawal versus

expansion of oil plus its dissolved gas.

The straight line obtained confirms the

assumed driving force in the reservoir:

internal gas drive. The water and rock expansion term was neglected.

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Figure 5

9 1 0

1 1

1 2

1 3

1 4

1 5

1 6

1 7

1 8

1 9

Figure 6

4 0

4 1

4 2

4 3

4 4

4 5

4 6

4 7

4 8

4 9

5 0

Figure 7

Figure 8

Figure 9

0 .

3

0 .

4

0 .

5

0 .

6

0 .

7

0 .

8

0 .

9

1 .

0

Figure 10

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Figure 11

Figure 12

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Figure 13

SPE Paper 139357x

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