Decline Curve Analysis Learning Objectives of Lecture 8:

Importance of decline curves Decline curve models Decline curve plots Applications

Decline Curve Analysis Preliminaries:

MBE analysis without further manipulations allows estimation of only N and m for HCs. Estimation of production rate specially as function of time is also of great importanceUnder natural depletion rate normally declines with recoveryMajority of oil and gas reservoirs show natural production rate decline according to standard trendsUnless natural trend is interrupted (water injection, well shut in) the natural decline trend is expected to continue until abandonment

Decline Curve Analysis

Natural decline trend is dictated by natural drive, rock andfluid properties well completion, and so on. Thus, a majoradvantage of this decline trend analysis is implicit inclusionof all production and operating conditions that wouldinfluence the performance.

The standard declines ( observed in field cases and whosemathematical forms are derived empirically) areExponential declineHarmonic decline Hyperbolic decline

Decline Curve Analysis When the average reservoir pressure decreases with timedue to oil and gas production, this in turn causes the welland field production rates to decrease yielding a rate timerelation similar to that in the following figure.Definition of normalized production rate decline, D:

Decline Curve AnalysisD = continuous production decline rate at time t (1/time)If t = years:Da= annual continuous production decline rate (1/year)If t = months:Dm= monthly continuous production decline rate (1/month)

Unit of q is not important

Decline curve models The general decline curve models is defined accordingto their relation with q as follows:

where n is called as the decline exponent

The three standard decline models (usually observed infield) are defined as follows.

Decline curve models 1. Exponential decline (n=0):

2. Harmonic decline (n=1):

3. Hyperbolic decline

where Di is the initial decline rate

Decline curve models

Producing rate during decline period: substitutingthe decline curve model equation into the normalizedproduction rate decline equation yields:

Separating the variables and integrating:

Decline curve modelsIntegrating and solving for q yields the generalhyperbolic rate decline

substituting n=1 results inthe harmonic rate decline

Setting n=0 and integratingand solving for q yieldsexponential rate decline

Decline curve modelsCumulative production as a function of q for each model are determined as:

exponential decline

harmonic decline

hyperbolic decline

Decline curve modelsTime at abandonment:If we define the economic limit when the production rate is qa then the exponential, harmonic and hyperbolic declines would have the following abandonment times respectively:

Decline curve modelsCumulative production during decline period:

Since q(dt/dq) from decline curve models is

Decline curve modelsCumulative production as a function of q becomes:

which integrates to ( for n0 or 1) to yield the cumulative production expression for hyperbolic decline

Graphical Features of ModelsCartesian plots yields

Graphical Features of ModelsSeilog plots yield

Graphical Features of ModelsCartesian q vs Gp plots yield

Graphical Features of ModelsSemilog q vs Gp plots yield

Decline curve modelsFor harmonic decline n=1, and hence, cumulative production as a function of q:

which integrates to

Decline curve modelsFor exponential decline n=0, Cumulative production as a function of q:

which integrates to

Decline curve modelsTime at abandonment:If we define the economic limit when the production rate is qa then the exponential, harmonic and hyperbolic declines would have the following abandonment times respectively:

Graphical Features of ModelsFor exponential declineone can write:

Therefore a plot of ln(q) vs t gives astraight line withslope equal to (-Di)and intercept equalto (ln(qi)).

Graphical Features of ModelsFor hyperbolic decline no immediate straight form is obtained, therefore a linear plot which allows us to determine two parameters namely Di and qi simultaneously is not available.

In summary : The production plots allows us two determine the nature of decline and then we canobtain the decline model parameters.

Graphical Features of ModelsFor harmonic decline one can write from cum. prod. eqn:

slope=(-Di/qi) intercept=(ln(qi))

Graphical Features of ModelsFor harmonic declineone can also write from decline rate. eqn:

slope = (Di/qi) Intercept=(1/qi).

Graphical Features of ModelsFor hyperbolic decline no immediate straight form is obtained, therefore a linear plot which allows us to determine two parameters namely Di and qi simultaneously is not available.

In summary : The production plots allows us two determine the nature of decline and then we canobtain the decline model parameters.

Production PlotsA plot of log(q) vs t is

Linear if decline is exponentialConcave upward if decline is hyperbolic (n>0) or harmonic

A plot of q vs Np is

Linear if decline is exponentialConcave upward if decline is hyperbolic(n>0) or harmonic

A plot of log(q) vs Np is

Linear if decline is harmonicConcave downward if decline is hyperbolic (n1.

A plot of 1/q vs t is

Linear if decline is harmonicConcave downward if decline is hyperbolic (n1.

Hyperbolic decline analysisSince no wells have declines where n=0 or 1 exactly it is more appropriate to use a regression technique to determine all three parameters namley Di, qi and n simultaneously. Two approaches are suggested by Towler:

An iterative linear regressionNonlinear regression

Towler also pointed out that linear regression impose more weight on smaller values of production rates as it involves logs of variables. Furthermore, the two suggested procedures on linear regression do not produce equivalent results.

Therefore, he suggests nonlinear regression as a method whichproduces repeatable results, and weights the production rates equally.

Hyperbolic decline analysis:nonlinear regression steps in excelFrom production data generate a spreadsheet with a column of oil production q vs production time tIdentify the part where finite acting period starts this is the data to be curve fitted by nonlinear regressionRewrite the time and q columns for the curvefit data only.Set up cells for n, Di, and qi.

Hyperbolic decline analysis:nonlinear regression steps in excelSet up a cell to calculate average oil production rate from qSet up a column containing the hyperbolic decline curve equation to fit the production q*Set up a column to calculate errors squared from (q-q*)2.Set up a cell that calculates sum of errors squared, SSE from step 7Set up a column to calculate the total errors squared from (q-qave)2.

Hyperbolic decline analysis:nonlinear regression steps in excelSet up a cell that calculates the sum of the total errors squared, SST, from step 9Set up a cell to calculate the square of the regression coefficient, R2, from R2=1-SSE/SSTInitialize the solver to optimize the contents of the cells that contain n, Di and qi by maximizing the cell that contains R2Note alternative to maximizing R2 is to minimize the sum the squares of the residuals SSE which gives the same result.

Example. Exponential decline

Caution for applicabilityThe emprical decline curve equations assume that the well/field analyzed is produced at constant BHP. If the BHP changes, the character of the well's decline changes.They also assume that the well analyzed is producing from an unchanging drainage area (i.e., fixed size) with no-flow boundaries, If the size of the drainage area changes (e.g., from relative changes in reservoir rates), the character of the well's decline changes. If, for example, water is entering the well's drainage area, the character of the well's decline may change suddenly, abruptly, and negatively.

Caution for applicabilityThe equation assumes that the well analyzed has constant permeability and skin factor. If permeability decreases as pore pressure decreases, or if skin factor changes because of changing damage or deliberate stimulation, the character of the well's decline changes.It must be applied only to boundary-dominated (stabilized) flow data if we want to predict future performance of even limited duration.

Example. Exponential declineSlope=-D 1/quarter year

Example. Exponential decline

Example. Harmonic decline

Example. Harmonic decline

Example. Harmonic decline

Example. Hyperbolic decline

Example. Hyperbolic decline

Chart2

88408768.4032081164

43204784.0425196885

38003084.1248582462

20002185.3402416706

15501645.7880790998

14301293.3753507873

10001048.8992644534

760871.4536550656

710738.0419797386

570634.8656313462

days

q STB/D

Hyperbolic Decline curve

Answer Report 1

Microsoft Excel 11.0 Answer Report

Worksheet: [Decline curve nonlinear regression.xls]Sheet1

Report Created: 18/04/2006 04:24:40

Target Cell (Max)

CellNameOriginal ValueFinal Value

$D$5R2-0.5235280.969890

Adjustable Cells

CellNameOriginal ValueFinal Value

$A$5n0.500003.02681

$B$5Di0.100000.12564

$C$5qi1520.001560.50

Constraints

NONE

Sheet1

Decline curve analysis with nonlinear regression

nDiqi

0.621940.0452223450.000.9863502498.0081493859701960

Monthq STB/monthtqq*

33208.0213

33239.01033timerate

33270.0152030884087685126402209640.0035119418

33298.01453604320478421533533196840.0054821482

33329.01334903800308451247716952040.005937353

33359.0128412020002185343512480040.0088503534

33390.010771501550164691758987040.0103707031

33420.01081180143012931866611406240.0109036868

33451.0105721010001049239122440040.0136201821

33482.09732407608711242230206440.0161550978

33512.096227071073878631969440.0168535406

33543.0924300570635420837171840.0193203001

Sheet1

days

q STB/D

Hyperbolic Decline curve

Sheet2

Sheet3

Example. Hyperbolic declinen=0.622qi-n=1.51e-3nDiqi-n=5.81e-5Compare with the previously obtained values from nonlinear regression