Summary
Isochronal testing is commonly used to evaluate the performance of gas wells. This paper proposes a new technique to estimate the value of the turbulence coefficient based on isochronal tests. The proposed method is easy to apply and evaluate. Further, the method also provides a value of bg under stabilized conditions which can be used to predict the performance of gas wells under stabilized conditions.
The proposed method is validated using field data under a variety of operating conditions. The values of the turbulence coefficient based on the field data can differ significantly compared to the literature correlations. This further shows the importance of obtaining appropriate reservoir parameters based on the field rather than the laboratory data.
Introduction
The use of isochronal or modified isochronal testing is well established in the gas industry. These tests are common for gas wells which take a long time to reach a stabilized rate. A common example would be a low permeability, fractured reservoir. Instead of testing these wells until a stabilized rate is reached, the wells are tested for a fixed period of time and the bottomhole pressure is measured. For isochronal testing, the well is then shut in until it reaches a stabilized pressure and the procedure is repeated for a different rate. For modified isochronal testing, the well is shut in for a fixed period of time, and the shut-in pressure is measured at the end of that period. The procedure is then repeated at other rates.
By repeating this procedure for different time intervals, we can gather information about rate vs. pressure drop in the formation for these time intervals. Ultimately, using this information, our goal is to establish an appropriate rate vs. pressure drop relationship under stabilized conditions.
Two procedures are commonly used to establish the equation for rate vs. pressure drop. An empirical method states that
q g = C ( p ￣ 2 − p w f 2 ) n . ( 1 )
We can write a simpler equation in terms of pseudo-real pressures as
q g = C [ m ( p ￣ ) − m ( p w f ) ] n . ( 2 )
Under transient conditions, the value of C is not constant. Instead, we can write Eq. 2 as
q g = C ( t ) [ m ( p ￣ ) − m ( p w f ) ] n , ( 3 )
where C(t) represents a term which is a function of isochronal interval t. In the literature, methods are proposed to estimate the value of C corresponding to the stabilized rate based on the transient state information ?C(t) For example, Hinchman et al.1 propose that 1/C(t)1/n be plotted as a function of log t, and the line be extrapolated until t is equal to the time it takes to reach the stabilized state period. In their method, they assume that n is constant, where n is an inverse of slope when log[m(p¯)−m(pwf)] is plotted as a function of qg. Although we get different straight lines corresponding to different t, the authors assume that the slopes are approximately constant.
Another commonly used approach in analyzing isochronal tests is to use an equation,
m ( p ￣ ) − m ( p w f ) = a g q g + b g q g 2 . ( 4 )
A similar equation can also be written in terms of pressure squared terms. Eq. 4 is derived starting from Forchheimer's equation.
Under transient conditions, we can rewrite Eq. 4 as
m ( p ￣ ) − m ( p w f ) = a g ( t ) q g + b g q g 2 , ( 5 )
where ag(t) is a function of isochronal interval, and bg is assumed to be constant. A commonly used technique is to plot ag(t) vs. log (t) and extrapolate ag(t) corresponding to a value of t which represents the time required to reach a stabilized rate.2–4
In using both Eqs. 3 and 5, we have assumed that the contribution due to the non-Darcy effect is not affected during the transient conditions. For example, in applying Eq. 3, we assume that n is constant during the transient period, and in applying Eq. 5, we assume that bg is constant during the transient period. Both n and bg represent the relative contributions of the non-Darcy flow. n will approach 0.5 as the non-Darcy effect becomes dominant, and bg becomes larger as the non-Darcy effect becomes significant. However, by assuming that n and bg are constant during the transient periods, we are ignoring the changes in the relative contributions due to the Darcy and non-Darcy terms.
In this article, we extend the previous analysis to account for changes in the non-Darcy term during the transient period. Further, by proper analysis, we propose a method to estimate the value of the turbulence coefficient based on the evaluation of the transient period data.
Approach
In our approach, instead of using the empirical equation (Eq. 3), we will begin with Forchheimer's equation, where the pressure gradient in a radial reservoir is calculated by
∂ p ∂ r = μ g k v + β ρ g v 2 . ( 6 )
The permeability (k) of the reservoir may be established based on well test data or core information. The turbulence coefficient is difficult to estimate. Although literature correlations5,6 exist to calculate the value of ? based on the laboratory experiments, field evidence7 indicates that the ? values in the field are significantly greater than the laboratory experiments.