## Behavior of Arps Equation in Shale Plays

#### Emanuel Martin

Decline curves analysis through the Arps method has been historically the tool more used to make wells reserves forecast, Due to its simplicity and low cost.

The method consist in the graphic extrapolation of the production from the historical production in a semi-log graph (log(q) vs t) until abandonment. Arps found three types of behaviors declination different when there is boundary dominated flow regimen: Exponential, Hyperbolic and Harmonic.

Also he introduced the equation for each type and used the concept of the loss ratio and its derivative to derive the equation. The three declinations have b values between 0 and 1. Where b=0 represent the exponential decline 0<b<1 represent the hyperbolic decline and b=1 represent the harmonic decline.

Where Di is the rate of initial decline, in [days^{-1}], qi is the initial rate in [Mpcd], b is the decline exponent and t is the time in days.

Equations to calculate the rate at a given time and for the calculation of the Estimated Ultimate Recovery (EUR) are shown below.

If we have production data also we can calculate the EUR as follows: we need to know the rate of economic limit, which is determined by an economic evaluation, put it into the equation of accumulated production, along with the coefficient of decline, b parameter and the current production rate. Then add the volume of gas produced to date to the estimated reserves and can get the EUR as shown in the graph.

## Application the Arps method to shale Plays

When we apply the Arps method to conventional gas fields, the majority exhibit a behavior exponential decline during the greater part of their lives, showing a slight variation in its final stages.

When applying Arps in shale plays we find b values greater than 1 that mathematically are possible but it’s not correct to use it to make the reserves forecasting as we will see later.

The following graph shows the application of Arps in a well of the Barnett play, it was taken from a consultant.

As we can see the b value is equal to 1,67 (b=1,67) giving us an estimated well life over 50 years and a EUR equal to 2,6Bcf. However if we only take the final part of the production history where to apply Arps the b value of gives us zero (b=0) and use these to make the well reserves forecasting, it give us a life of 12 years and a EUR equal to 1,3Bcf, it’s to say that the EUR forecasting was decreased to less than one half and the well life has declined by more than 4 times.

The economic impacts of these variations are very large and deserves analyze what is happening; we need to know why while the greater the value of b the well reserves grow dramatically.

### What means b bigger than 1?

If we observe the fundamentals of the Arps method we can see that it was developed to make well production forecasting when this exhibit boundary-dominated flow regime and constant bottomhole pressure, in this way the b exponent is between 0 and 1. In the shale plays due to low permeability and long periods of transient flow the b exponent is normally greater than 1.

Then an b value greater than 1 mean that the flow regime is transient and that there isn't boundary dominated flow regime for which was developed the Arps method, so it is not correct to use it to make reserves forecasting of the field when we find these values in the b coefficient.

In fact Valko and Lee demonstrated that for b values greater than one the accumulated production of the well reach to infinity when the time reach to infinity, for that overestimated exaggeratedly the EURO and life of the well.

## Common features in the shale plays

In the shale plays is frequent to find us with combinations of long periods of transient flow (several wells of Barnett play have been in transition flow regime for more than six years) and relatively short stories of production, where only a small fraction of hydraulically fractured wells have reached the boundary dominated flow regime for which the Arps decline model is appropriate and can be used it to make an appropriate production forecast.

Due to this and with the order to continue using the curves of decline to make production forecasting (as from the Arps equation) some authors have made modifications in the equation to observe that in the shale plays the b parameter is not constant with the time if not variable. The different methods arise to find the function that best matches the behavior of the b parameter and the flow regimes: transient, transition, and boundary dominated flow.

#### The new models are:

- Power-law exponential.
- Duong.
- Stretched Exponential (SEDM)
- Continuous EUR.
- Logistic Growth Model.

In this post we’ll start explaining the Power Law Exponential method and in the following we’ll explain the remaining.

## Power Law Exponential (PLE)

The first proposal to modify the Arps equation was made by ILK in 2008 and was called PLE models the loss ratio, its foundations is that the loss ratio follows an exponential function in the early times of production and converts into a constant function in the late times. This model of the loss ratio is substituted in the original definition of the loss ratio, integrated it and solving for the rate (q) as we can see below:

PLE definiton of the Loss Ratio:

Deriving we obtain:

Substituting it in the original definition of the Loss Ratio (equation 1) and solve it to q we obtain:

Which reduces in form to the power law loss-ratio rate decline relation as defined by Ilk.

Where:

- qi = Rate "intercept" defined by Eq. 6 [q(t=0)].
- D1 = Decline constant "intercept" at 1 time unit defined by Eq. 3 [D(t=1 day)].
- D∞ = Decline constant at "infinite time" defined by Eq. 3 [D(t=8)].
- Di = Decline constant defined by Eq. 6.
- n = Time "exponent" defined by Eq. 3.

In the next q-D-b graph we can see the PLE and Hyperbolic method:

In order to validate the new method we will apply it to an example made with simulator, looking for compare the forecast resulting to apply the new method with the input values used in the simulator and then, we will apply it to examples of the Barnett Play to compare the results obtained with the traditional model of Arps.

#### Case 1: Horizontal Gas Well with Multiple Transverse Fractures. (1)

We have the simulation of a horizontal well with multiple fractures centered in a rectangular reservoir (2x1). Below is shown a table with the fluid and reservoir properties:

When observing the q-D-b graph we can see that the boundary dominated flow regime has been reached after of the 100 days and boundary effects are observed during late times of the production.

It also suggests to us a strong behavior power-law when the D parameter is inspected visually together with the decreasing trend of the parameter b.

When apply the Power Law Exponential relation we see that it get an excellent matches of the production data, D parameter and b parameter through all the flow regimes. When apply the hyperbolic rate-decline relation with a value of b parameter equal to 1 as suggest the tendency of the data to us, we get a poor matches of the production rate and of the D parameter in the early times of production.

The reserves estimated by the new method are consistent with the simulator inputs giving the following values:

- EUR (PLE with D∞≠0) = 2,67BSCF.
- EUR (PLE with D∞=0) =2,94BSCF.
- EUR (Arps with b=1) =7,14BSCF.

The values obtained with de Power Law Method are practically the same with D∞≠0 and D∞=0, only an 10% of difference while the EUR calculated with the hyperbolic model is almost 2,7 times greater than the estimated by the Power Law function.

Below are shown a Log-Log graph and semi-Log graph with the rate and accumulated production for this well.

We can conclude to depart of this example that the Power Law Exponential Relation is a Very efficient tool to make the reserves forecasting in a horizontal gas wells with multiple fractures and that the Arps method has overestimated the reserves almost 3 times even with b values set in 1.

#### Field Case 1: Barnett Shale. (2)

In this example we have almost 6 years of monthly production. We observe that the boundary dominated flow regime has not been established yet.

The q-D-b graphic verify the Power Law behavior in the D parameter already that the tendency of t D graph, it’s essentially a straight line on the log-log graph. Consequently, computed b-parameter data also follows a decreasing trend the new function achieves good matches of the production and of the D parameter during all flow regimes.

When apply the hyperbolic rate-decline relation with a value of b parameter equal to 1,4 as it suggest to us by the data tendency, we get a good matches in the rate of production except in the early times in the same way that the D parameter that doesn't get a good fit up to 200 days.

The estimated reserves are:

- EUR (PLE with D∞≠0) = 1,2BSCF
- EUR (PLE with D∞=0) =1,4BSCF
- EUR (Arps with b=1,4) = 57 BSCF

The values obtained with de Power Law Method are practically the same with D∞≠0 and D∞=0, only a difference of 17% while the EUR calculated with the hyperbolic model is 48 times bigger than the estimated by the Power Law function. This unreal value given by the Arps model is should to use it to make reserves forecasting when we have transition flow.

Below are shown a Log-Log graph and semi-Log graph with the rate and accumulated production for this well.

#### Field Case 2: Barnett Shale. (3).

In this example of the Barnett play we have approximately 7 years of production data.

When we observe the graph q-D-b it can be suggested us that the well is in transition to boundary dominated flow regime.

Once more the trend of the D parameter data is essentially a straight line in a log-log graph verifying this the applicability of the PLE method. The computation of the b-parameter data trend is affected by data noise and end-point effects. We can see that the matches achieved by the new method are satisfactory for the D parameter and the flow along all the flow regimes.

When apply the hyperbolic rate-decline relation with a value of b parameter equal to 1,4 again as it suggest to us by the data tendency; the matches of the D parameter is quite poor achieving matches only the data at late times in the well history production but without achieving matches the early times again.

The estimated reserves are:

- EUR (PLE with D∞≠0) = 2,7BSCF.
- EUR (PLE with D∞=0) =16BSCF .
- EUR (Arps with b=1,4) = 151 BSCF.

This time we find a EUR nearly 6 times higher for the PLE function with D∞≠0 and a reserve value totally absurd to apply the Arps method, which gives us a EUR 56 times greater than the forecasting by the new function. This large difference is due to having used Arps when the value of b is bigger than 1 in transition flow regime to make the reserves forecasting.

Below are shown a Log-Log graph and semi-Log graph with the rate and accumulated production for this well.

## Conclusion:

As we can see the PLE method give to us a good matches and a reliable production forecast, Than proven with several examples made with simulator it give the correct value of reserves, showing to good model the three types of flow: transient, transition, and boundary dominated flow data, and proven it with real examples of Barnett play also achieving consistent and reasonable reserves estimates.

Even so the PLE method doesn’t must be used as unique method to make the reserves forecasting, getting advice comparing the values obtained with other DCA method, reservoir simulated and most of all with the Time-Rate-Pressure analysis as we'll show in the next post.

#### Cons:

Parameter regressions can be unrealistic/unstable in early time if not Constrained

1,2,3-This examples were taken from SPE 119897.

#### AUTHOR: Emanuel Martin, Petroleum Engineer.

#### Written by the author:

- BARNETT TECHNICAL INFORMATION
- WHAT IS A SHALE?
- WHAT IS HYDRAULIC FRACTURING?
- WHERE IS THE GAS STORED?
- HOW TO EVALUATE A SHALE PLAY?
- FLOW MECHANICS IN SHALE GAS
- EXCEL WORKSHEET TO CALCULATE THE PERMEABILITY IN SHALE GAS FIELDS
- BEHAVIOR OF ARPS EQUATION IN SHALE PLAYS
- U.S. Dry Shale Gas Production
- U.S. Tight Oil Production
- Argentina Shale Production

## Bibliography:

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- A NEW SERIES OF RATE DECLINE RELATIONS BASED ON THE
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- Experimental investigation of matrix permeability of gas shales. Authors: Rob Heller, John Vermylen, and Mark Zoback. Date:11-2013. https://pangea.stanford.edu/researchgroups/stress/sites/default/files/Heller%20et%20Zoback%202014.PDF
- PRESSURE NORMALIZATION OF PRODUCTION RATES IMPROVES FORECASTING RESULTS, Author: JUAN MANUEL LACAYO ORTIZ. Date: August 2013, Texas A&M University.
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