Assessment of diversion performance is key to determining success of stimulation. Doubts remain, however, regarding the evaluation of diversion effectiveness. As diverter enters the formation, a hump in the surface pressure curve usually is expected. It then can be interpreted as supporting evidence for diversion. This, however, is a simplification of the fluid-diversion process. Such a hump may not be observed during a diversion stage even when the process is effective. In the complete paper, the inverse-injectivity method of evaluating matrix-stimulation performance is modified and validated with real data of two matrix-acidizing operations in a gas-condensate field.

## South Pars Gas Field

In 1971, an enormous gas reservoir, the North Dome Field, was found in the Qatar Dome structure of the Persian Gulf. In the years since, surveys have highlighted the existence of abundant amounts of gas and condensate accumulations in a huge dome. This single gas-bearing structure is delineated by the geographical borders between Iran and Qatar in the Persian Gulf. According to one recent estimate, almost 19% of the world’s gas rests in this vast basin.

The field includes two independent gas-bearing formations, Kangan (Triassic) and Upper Dalan (Permian). Each formation is divided into layers separated by impermeable barriers, namely K1, K2, K3, and K4. The K1 and K3 layers mostly are formed of dolomites and anhydrites. Layers K2 and K4 possess better reservoir specifications (i.e., highly permeable and porous, consisting of limestone and dolomite). An anhydrite-composed layer, the Nar Member, separates Layer K4 from the underlying K5 layer, which has nonsatisfactory reservoir qualities.

## Methodology

In this study, the inverse injectivity and its integral and derivative plot are coupled with results of production logs to evaluate diverter-system performance in two real acidizing operations performed in the South Pars field. What makes this work different is the complexity of the treated reservoir in terms of the number of layers (four distinct formations), formation heterogeneity, and long perforation intervals (an approximately 1,000-ft net interval).

The productivity index is a measure of the well’s production potential and is defined as a ratio of the total liquid surface rate to the pressure drop at the midpoint of the producing interval. Because the direction of flow in this case is in a direction opposite that of production, this parameter is considered as the reverse of the productivity index and is called inverse injectivity.

In this study, a program is written to calculate modified inverse injectivity during two real matrix stimulations performed in the field. The equations used in this method are provided in the complete paper.

## Results

**Matrix Stimulation Input Data and Design. **The proposed method was used to evaluate data gathered from two matrix-acidizing operations. The treatment data are processed using both inverse injectivity and a derivative plot. Tables 1 and 2 of the complete paper show a data summary and the pumping schedule of the wells studied in this paper, respectively. **Figs. 1 and 2** of this synopsis show the porosity and permeability, respectively, of Well B to highlight the heterogeneity of this complex reservoir.

**Inverse-Injectivity Analysis.** Results of the methodology are presented in detail in the complete paper to determine the diverter effectiveness for Wells A and B. In Well A, inverse injectivity has dropped from 223 psi/(bbl/min) when preflush reaches perforations to a value near 10 at the end of the job. All stages have shown a clear hump in both surface treating pressure and inverse injectivity. The first hump is approximately 5 psi/(bbl/min); in the rest of the stages, this parameter is approximately 1 to 5 psi/(bbl/min). To make inverse injectivity more meaningful, this parameter was plotted with the derivative and integral function.

No visible change exists in the stages from concavity to convexity in the integral function. Despite negligible changes in surface pressure, the inverse injectivity illustrates vivid diverter humps. This demonstrates that, in cases where monotonic surface pressure is observed, room remains to have an efficient inverse-injectivity curve with several humps.

In Well B, inverse injectivity has reached a value near 10 at the end of the job. All stages have shown a clear hump both in surface treating pressure and inverse injectivity. The first hump is approximately 12 psi/(bbl/min); in the rest of the stages, this parameter is approximately 1 to 4 psi/(bbl/min). To make inverse injectivity more sensible, this parameter was plotted with the derivative and integral function. Again, the authors found no noticeable change in the stages from concavity to convexity in the integral function.

An important consideration is the effect of pumping rate on the performance of the diverter and, consequently, inverse injectivity. In Well A, the pumping rate after the exposure of stimulation fluids is 20 bbl/min, which gradually is increased to 35 bbl/min in the overflush stage. In Well B, immediately after exposure of stimulation fluids to the formation, the pumping rate is increased to 45 bbl/min and is kept until the last stage. Two important points can be inferred:

- In Well A, after the exposure of treating fluids to the formation, the inverse injectivity has a decreasing trend, while, in Well B, the inverse injectivity fluctuates within a very restricted range.
- The magnitude of inverse-injectivity humps in Well B are close to each other, while, in Well A, these humps are sharper; approaching the end of the job, their magnitude decreases.

**Pre- and Post-Treatment Evaluation (Production Log Verification). **The authors had access to production-logging data as supporting evidence. The target of stimulation in the two wells was Layer K4 because of both its high-pressure conditions and high hydrocarbon reserves. The majority of production before stimulation in Well A had been from the bottom of Layer K4 (66.87%), while Layers K1, K2, and K3 had produced 12.37, 5.42, and 15.35%, respectively. After stimulation, considering the absolute value of the production contribution, Layers K1, K2, K3, and K4 produced 20.8, 0.87, 2.6, and 64.99%, respectively. Despite a small change in the production of Layer K4, it is clear that production in this layer has become uniform and has increased, a direct consequence of diversion and stimulation.

When considering Well B in light of these observations, production contribution can be deduced as having been switched from Layer K1 before stimulation to Layer K4 after stimulation. This shows that the primary objective of stimulation in this well—to stimulate Layer K4 despite its high reservoir pressure and low permeability—has been fulfilled, even though production of Layer K1 is decreased. However, this is mainly the result of a difference in formation pressures and their petrophysical characteristics. Layer K4’s pressure was approximately 400 psi greater than that of Layer K1. It is evident that, after the decrease in Layer K4’s reservoir pressure, the contribution of other layers will increase gradually.

## Conclusions

- In Well A, inverse injectivity showed a clear hump while diverters were entering into the reservoir, even if surface treating pressure humps were not sharp.
- A sign change in the derivative curve from negative to positive shows where a diversion happens; this is concurrent with the buildup of inverse injectivity. These two parameters have conforming behavior. However, using a derivative plot can eliminate misleading increases observed in an inverse-injectivity graph. Therefore, it is recommended to refer to a derivative curve for precise identification of diverter performance points.
- Pumping rate has a profound constructive effect on the performance of the diverter.
- Verification through production logs shows that, in Wells A and B, production contribution of nonproductive layers is improved.
- Failure to observe surface pressure humps is not necessarily an indication of diversion deficiency. In many cases in the past, effective diversions may have been considered ineffective.
- Because the nature of fluids injected for stimulation are the same in matrix stimulation of oil and gas wells (liquid), this methodology can be applied generally for oil wells.

This article, written by *JPT* Technology Editor Chris Carpenter, contains highlights of paper SPE 200135, “A Comprehensive Method for Diverter-Performance Evaluation During Stimulation of Long-Interval Heterogeneous Reservoirs: A Case Study” by **A.F. Safari** and **H. Panjalizadeh,** Mehran Engineering and Well Services Company; **M. Pournik,** SPE, The University of Texas Rio Grande Valley; and **H. Jafari **and **A. Zangeneh, **Mehran Engineering and Well Services Company, prepared for the 2020 SPE Conference at Oman Petroleum and Energy Show, Muscat, Oman, 14–16 September. The paper has not been peer reviewed.