Abstract

This paper presents simple and practical equations to estimate a priori the decline-exponent for decline-curve analysis of gas wells. The proposed equations are applicable for gas wells with closed boundaries and constant rock properties. Data required includes initial reservoir pressure, bottomhole flowing pressure, and fluid properties. These data are that required in a typical decline-curve analysis (i.e., no extra data is required). A field example is presented to verify the developed concept and to demonstrate the utility.

Introduction

Two practical issues arise in applying the "Fetkovich-Arps type" of type-curves.1–4 First, it is very difficult to match a decline exponent at early depletion stage even with good quality data (e.g., around the region of tdD 1 in the type-curve of Fig. 15). Unfortunately, this is also the time (early depletion) when some key answers (e.g., gas-in-place) are needed most for a future planning. Poor data quality (e.g., scattering) makes the situation even worse. Second, the decline exponent, strictly speaking, is not a constant. (We will show examples later.) Rigorous handling the variable decline exponent is possible. Such an avenue, however, will offset significantly the advantages of using the Fetkovich-Arps approach.

An estimated most-likely or optimal decline exponent before data analysis is desirable to guide the type-curve matching, and to relieve the previously mentioned two difficulties. Future rate-time prediction and fluid-in-place estimation then can be made more confidently and as early as possible. The objective of this study, thus, is to develop a method to estimate or predict the decline exponent of gas wells before data analysis.

Instantaneous Decline Exponent

The instantaneous decline rate, D, is defined as the fractional instantaneous change in flow rate with time,6,7

  • Equation 1

The instantaneous decline exponent, b, is defined as the instantaneous time-rate change of the inverse of D,6,7

  • Equation 2

The definitions of D and b given above are general, valid for any time and any type of flow. The unit of D is that of the reciprocal of time while b is dimensionless. The case of b = 0 (or equivalently, D = constant) would result in exponential decline whereas cases of b > 0 give hyperbolic decline.

Based on gas inflow equation during boundary-dominated flow period and material balance equation, Appendix A derives the following theoretical expressions (pseudopressure, pp, and p-squared, p2, approaches) of post-transient instantaneous b for constant BHP production,

  • Equation 3

where n is the back-pressure curve exponent. Other notations are defined in the Nomenclature. Pore volume was assumed to be constant in deriving Eq. 3.

Pseudopressure-Based Instantaneous b.

Eq. 3 indicates that the instantaneous b, in general (pp-approach), is not a constant; b reflects the change of viscosity-compressibility product µgct as the reservoir pressure changes. Specifically, b is the slope of ln(1/µgct) vs. ln[pp(pR)- pp(pwf)]. If the slope is positive and constant, b is a positive constant and the Arps' decline equations are valid. b is zero only if the µgct product is constant. Appendix B shows that the post-transient rate-time equation (constant BHP) is exponential if the variation of µgct (and Vp if necessary) can be taken into account in the time scale.

Pseudopressure-Based Instantaneous b.

Eq. 3 indicates that the instantaneous b, in general (pp-approach), is not a constant; b reflects the change of viscosity-compressibility product µgct as the reservoir pressure changes. Specifically, b is the slope of ln(1/µgct) vs. ln[pp(pR)- pp(pwf)]. If the slope is positive and constant, b is a positive constant and the Arps' decline equations are valid. b is zero only if the µgct product is constant. Appendix B shows that the post-transient rate-time equation (constant BHP) is exponential if the variation of µgct (and Vp if necessary) can be taken into account in the time scale.

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