Applications of Rate Normalization for Well Performance Analysis (II)

"I don't believe in exotic explanations. I just don't" (M. Fetkovich, 1993)

💠Introduction

Part I of this series introduced the concept of normalization as a means to approximate the convolution integral, as defined by Duhamel’s principle, to obtain a usable plotting function for analyzing well performance. It was demonstrated that normalization enabled the conversion of variable-rate and variable-pressure production histories into an equivalent, simpler form, such as a constant-rate drawdown pressure response. The focus of Part I was on the use of the post-transient model. This article expands the use of the latter, showing 3 practical applications useful for reservoir surveillance and monitoring, namely:

✅Investigated In-Place

✅Rate Estimation

✅ Reservoir pressure estimation

💠Theory

The rate-normalized pressure model can be expressed in compact form as

The terms are defined as follows:

If q-pwf data correspond to the PSS regime, a Cartesian plot of the rate-normalized pressure drop, ΔP/q, versus tmb produces a straight line with slope m_∆𝐏/𝐪 and intercept bₚₛₛ. The investigated in-place, Ninv, can be calculated sequentially as time progresses. Differentiating the rate-normalized pressure model with respect to tmb leads to,

which offers a way to monitor the behavior of the investigated in-place, Ninv. During transient conditions (t<tₚₛₛ), Ninv is less than N, and stabilizes once PSS conditions are reached (t>tₚₛ). Once the slope and the intercept are determined, the rate can be back-calculated from

Thus, comparing the calculated rate to the measured one serves as an essential reality check for the straight-line analysis. Additionally, the Cartesian analysis delivers another valuable byproduct: a continuous estimate of the average reservoir pressure. Starting with the expression for the pressure drop from the material balance model for a subsaturated oil reservoir (Dake 1978),

On the other hand, the pressure drop from the PSS relationship is

eliminating pi,

Hence, pwf can be readily converted into p once boundary-dominated conditions are achieved. The calculated values can be compared with available pressure measurements as a consistency check. Note this formulation assumes sub-saturated conditions. Multiphase flow will be treated later in this series.

💠Application

For the field case used in Part I, Figure 1 shows that the investigated in-place exhibits a rapid rise during transient conditions, which are short-lived, stabilizing at around 2.9 MMstb once PSS develops.

The post-transient model accurately reconstructed the rate history with an absolute average error of only 4.4% (Figure 2). The model proved to be robust even during the rapidly changing 0-200 hr period; while the deviation is slightly higher, the model-generated rate still matched the field data without significant error

The reservoir pressure profile is generated using the Cartesian analysis output, and a continuous estimation is obtained (Figure 3). Such estimation can be compared with existing static pressure measurements for verification purposes. Furthermore, the PSS-derived pressure may also be used for material balance studies or for calibrating numerical simulation models.

💠Takeaways

✔ The post-transient model is a powerful, simple tool that provides valuable insight by augmenting the value of already acquired data. The foundations are straightforward relationships, based on first principles, that deliver a practical understanding of the well and reservoir system

✔ Made possible by normalization, this method offers a cost-effective alternative to dedicated well tests (like BUPs), thus avoiding unnecessary deferred production. This approach is particularly relevant given the wealth of continuous rate and pressure measurements available from permanent gauges

💠References

Dake, L P. 1978. Fundamentals of Reservoir Engineering. 1st. Amsterdam: Elsevier Science.

Part I: https://www.garudax.id/posts/dwindtadolfo_first-article-in-a-series-on-rate-normalization-activity-7391611659171069952-7uk0?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAEVlxkBlJl8TpuI_R8dvlYFQBq8ncGT24k

Fetkovich, M.J (1993). https://www.ipt.ntnu.no/~curtis/courses/DCA/Fetkovich-DCA-Course-1993-05-(24-26)/