When creating an Arps projection in PHDwin, the following equations are used to solve for the decline, rate, or volumes from the projection given the other variables. The "type of decline" listed at the top section of the table varies based on the b-factor used in the projection.

Variables

b = Hyperbolic Decline Exponent factor

D = Nominal Decline rate - flat over time for exponential curves

Di = Initial Nominal Decline [Fraction / Unit Time]. The decline rate entered in the ARPs projection will have units next to it. If the decline was entered as an effective decline it needs to be converted from effective to nominal using the equations in this section. Once the annual nominal decline is determined, it can be converted to a decline with units of volume per month by dividing by 12.  

Dt = Decline at time t

e = exponential constant, mathematical standard = 2.7182818284590452...

Np = Cumulative volume produced until time t

qi = Initial Instantaneous Production Rate (time 0) [Volume / Time Unit]

qt = Instantaneous Rate at time t [Volume / Time Unit]

t = Standard Time in units consistent with the Time Units for q (rate) and d (decline). This should be the cumulative time from the first date of the projection until the date to solve for a variable. The time (t) does not have to be an integer. If decimal values are used, months or years with varying day counts are easily supported.

*time units need to be consistent for all of the inputs

All Arps calculations in PHDwin are done using Standard Time, which is represented in the formulas as the variable "t". This allows for smooth and continuous reporting of rates and decline across months with non-equal day counts. However, since PHDwin also honors actual day count, a conversion is done to determine the actual time elapsed in standard time. We can assume that:

•Days in a Standard Year = 365.25

•Days in a Standard Month = 30.4375 days

•PHDwin Standard month = (Number of days in that month x 12)/365.25  =  Number of days in that month /30.4375

oFor example: January = 31 Days/(30.4375 Days in a Std. Month) = 1.01848 Standard Months (slightly more than February, or a month with less days)

oFor example: February (on a non-leap year) = 28 Days/(30.4375 Days in a Std. Month) = .91992 Standard Months

PHDwin allows users to enter the decline rate for Arps projections as tangent effective or nominal. All formulas use the nominal decline (as a Fraction/Time Unit).

To convert from tangent effective to nominal:

To convert from nominal to tangent effective:

De = Effective decline rate

D = Nominal Decline rate