Logarithm Model for Decline Curve Analysis

Introduction

A typical decline curve analysis (DCA) model expresses flow rate q as a function of time f(t) so that q = f(t) with a first order time derivative ( ) df t dq dt dt = . Arps JJ [1] decline curves are solutions of the equation

1 n dq aq dt + = − (1)

The factors a and n are empirically determined and are constant with respect to time t. The empirical constant n ranges from 0 to 1. The shape of the decline curve depends on the value of n as shown in Table 1. The term qi is initial flow rate.

Decline curve analysis provides information which can be used to estimate cumulative production N. If we assume a functional relationship ( ) q f t = is known, cumulative production is the integral over flow rate q for a period of time that ranges from initial time t0 to final time t, thus ( ) 0 0 t t t t N qdt f t dt = = ∫ ∫ (2) As an example, cumulative production N for Arps exponential decline equation is − = = ∫ (3) t iq q N qdt a

0 where flow rate is integrated over time from initial rate qi at initial time t0 = 0 to rate q at time t. Time periods such as day, month and year need to be consistently applied. For more discussion of Arps equation and decline curve analysis, see Lee [2], Sun [3], Fanchi and Christiansen [4], Ezekwe [5], Panja and Wood [6].

Another way to infer the time dependence of flow rate is to consider different types of flow. Flow regime models provide different relationships between pressure drop and time. As a first approximation, well flow rate during primary depletion is proportional to pressure drop p ∆ where pressure drop is the difference in pressure i wf p p p ∆ = − between initial reservoir pressure ip and wellbore flowing pressure wf p . The relationship between flow rate and time can be obtained by writing productivity index / J q p = ∆

as q J p = ×∆ and replacing pressure drop with its time- dependent form for different flow models. Table 2 shows relationships between flow rate q and time t for some flow regime examples.

or The purpose of this paper is to develop a decline curve analysis model for flow rate proportional to logarithm of time ln t. This model is referred to here as the logarithm model (LNDM).

Logarithm Model (LNDM)

The radial flow relationship between flow rate and time in Table 2 suggests that flow rate q is proportional to the logarithm of time ln t in the form ln q a t b = + (4) where a, b are constant with respect to time and n = 1. The differential equation for the rate-time dependence in Equation (4) is obtained by calculating the time derivative ( ) ln d a t b dq a dt dt t + = = (5) Equation (5) is the differential equation for the logarithm model LNDM and is the analog of Equation (1). The general solution of Equation (5) is ln + where = -a ln q a t b b λ = (6) We have introduced the parameter λ to show how to make physical units consistent. To demonstrate this, substitute -a ln b λ = into Equation (6) to obtain ( ) a ln t a ln ln / q a t λ λ = − = (7) The units are consistent when λ has the unit of time so that / t λ is dimensionless. The slope a has the same unit as flow rate q and 0 a < for a flow rate that declines as time increases. The parameter λ is expressed in terms of a, b by ( ) exp / b a λ = − .

Cumulative Production and Economic Limit

Cumulative production for the logarithm model is given by the integral ( ) 0 ln N a t b dt τ τ = + ∫ (8) for the time interval 0 t τ τ ≤ ≤ . We solve the integral in Equation (8) using ln ln axdx x ax x = − ∫ to find

0 0 ln N a t t t bt τ τ τ τ   = − +   (9)

( ) ( ) ( ) 0 0 0 0 ln ln N a a b τ τ τ τ τ τ τ τ = − − − + − (10)

The maximum production time occurs when flow rate 0 q → since flow rate 0 q < when t τ > . We have the condition ( ) 0 ln q t a b τ τ = = = + (11) Solving for τ gives exp b a τ   = −     (12) The economic limit occurs at time econ τ τ ≤ .

An alternative economic limit can be set for a non-zero economic rate econ q using the following procedure. The time corresponding to the non-zero economic rate econ q is found using the equation ( ) ln econ econ econ q t t q a t b = = = + (13) The time corresponding to the economic limit econ t is then ( ) exp / econ econ t q b a   = −   (14)

Initial Condition of LNDM Model

The logarithm model LNDM requires a careful interpretation of volumetric flow rate as a function of time when t goes to 0. The usual initial condition for models such as the familiar Arps models in Table 1 uses flow rate q(t) at the beginning of the time period, that is flow rate at q(t = 0). In the case of the LNDM model, we cannot use ln(q(t)) at time t = 0.

An alternative approach that is suitable for the LNDM model is to recognize that volumetric flow rate data are typically reported as the average volume of fluid produced in a period of time. For example, the first volumetric flow rate may be the volume produced in a day, a month, or a year. If we choose to view the flow rate as the volume produced throughout the time period rather than an instantaneous rate at the beginning of the time period, then we can define an integral that begins at the completion of time period 1. In practice, we can use daily, monthly, or annual rates of production versus time and integrate over time beginning at the completion of the first time period rather than at the start of the first time period. Initial production rate, or IP, is the volume produced during the first time period divided by the duration of the time period.

Conclusions

A decline curve analysis model called the logarithm model (LNDM) was developed for flow rate proportional to the logarithm of time. The functional relationship between flow rate and logarithm of time was inferred from a study of flow regimes. We show how to use the rate-time relationship to calculate cumulative production at a specified economic limit for the LNDM model.

The use of the logarithm of time requires a careful interpretation of volumetric flow rate at the initial condition. The initial condition for the LNDM model states that the initial flow rate is the volume of production during the first period of time divided by the first period of time.

The LNDM model was illustrated by applying it to shale gas production and shale oil production decline curves. The decline curves represent a selection of North American shales.