eISSN 2658–5782

DOI 10.21662/mfs

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2026. Vol. 21. Issue 2, Pp. 59–64
URL: https://multiphasesystems.online/mfs2026.2.011
DOI: https://doi.org/10.21662/mfs2026.2.011
Modelling of well tests with stepwise pressure changes in low-permeability reservoirs with a hydraulic fracture
A.Ya. Timershaehova, M.R. Gubaidullin🖂, A.Ya. Davletbaev, Z.S. Mukhametova
Ufa University of Science and Technology, Ufa, Russian Federation

Abstract

The paper examines how the choice of the type of boundary conditions in a mathematical model affects the results of interpretation of data from well tests with a stepwise change in bottom-hole pressure, analyzed by the method of constructing an indicator diagram(ID). This task is especially relevant for low-permeability reservoirs with fractured hydraulic fracturing. The purpose of the work is to determine the features of the application of the ID method under different conditions at the boundaries of the reservoir and to select the optimal duration of the stages of the well test, in order to increase the reliability of the interpretation. When modeling the flow of liquid into a producing well with a fractured hydraulic fracturing, seven stages of well operation were calculated at a bottom-hole pressure from 5 to 6.5 MPa and the duration of each stage from 5 to 30 days. The results showed that if constant pressure is maintained at the reservoir boundary, the duration of the modes has almost no effect on the results of the interpretation of the well test by the ID method. In a reservoir with an impenetrable boundary, an increase in the duration of the stages leads to a decrease in the reliability of the approximation of the pressure dependence on the inflow due to the influence of the boundaries of the model. To eliminate the effect of boundaries for wells in formations with impenetrable boundaries, it is recommended to choose the minimum duration of the stages at which the flow of liquid into the well becomes steady. These conclusions are useful when planning a stepwise pressure change in a well.

Received: 7.05.2026
Accepted: 15.06.2026
Published: 3.07.2026
Citation

Timershaehova AYa, Gubaidullin MR, Davletbaev AYa, Mukhametova ZS. Modelling of well tests with stepwise pressure changes in low-permeability reservoirs with a hydraulic fracture. Multiphase Systems. 2026;21(2):59–64 (in Russian).

Keywords

well test;
indicator diagram;
boundary conditions;
reservoir pressure;
radius of study;
productivity coefficient

Article outline

Well testing (well testing) is a key tool for monitoring reservoir development. It allows evaluation of the flow properties of productive formations, the condition of the near‑wellbore zone, and the influence of reservoir boundaries and neighbouring wells.

One of the common low‑cost well testing methods is the step‑rate pressure test, with subsequent data interpretation using the Inflow Performance Relationship (IPR) method. This approach helps determine well productivity and reservoir pressure. However, in low‑permeability reservoirs developed with hydraulic fracturing (HF), difficulties arise because fluid flow requires the use of specialized mathematical models. The aim of this work is to investigate how reservoir boundary conditions affect the interpretation of well test data using the IPR method.

A production well located in the centre of a reservoir of dimensions is simulated. The well is hydraulically fractured with a half‑length Xf, fracture width wf, fracture permeability kf, and the fracture fully penetrates the formation of thickness h. Oil flow in the low‑permeability reservoir and the fracture is described by the two‑dimensional diffusivity equation. Numerical modelling was performed for no‑flow and constant‑pressure boundary conditions.

For the constant‑pressure boundary case, it is established that the duration of the test steps does not significantly affect the interpretation results nor the quality of fitting the pressure‑rate relationship (coefficient of determination ). The only exception is the stage duration of 5 days, which shows the largest deviation. This is consistent with conventional understanding: increasing the duration of each step in a step‑rate test improves interpretation accuracy because a linear relationship between bottomhole pressure and flow rate is valid under steady‑state flow conditions.

Analysis of productivity indices for different hydraulic fracture half‑lengths (10-150 m) shows that the classical formulas of Economides (for finite‑conductivity fractures) and Prats (for infinite‑conductivity fractures) agree well with reference values over a certain range of fracture lengths. For very long fractures, deviations appear because the fracture approaches the reservoir boundary, distorting the classical analytical solutions.

For the no‑flow (closed) boundary case: In a closed system (impermeable reservoir boundaries), fluid withdrawal causes a continuous decline in average reservoir pressure. Consequently, the IPR deviates from linearity, indicating unsteady‑state (transient) flow. It is important to emphasise that under such conditions, the traditional interpretation of step‑rate tests using the steady‑state IPR method becomes invalid - it leads to systematic errors in evaluating well productivity and other reservoir parameters. A similar situation can arise due to interference from neighbouring wells or in areas with a dense well spacing, where conditions approach those of a closed reservoir. Therefore, when planning tests in closed reservoirs or dense well patterns, special interpretation techniques adapted to transient flow should be used (e.g., rate‑transient analysis, multi‑well deconvolution). Observed non‑linearity of the IPR may serve as an indication of the absence of support from adjacent injectors, the presence of non‑linear flow, or the presence of no‑flow boundaries.

Conclusion

Based on numerical simulation of step‑rate well tests, it is found that in closed systems, prolonged step durations lead to non‑linear distortion of the IPR. This is caused by the continuous decline in average reservoir pressure. Under such conditions, the conventional steady‑state interpretation of the IPR becomes incorrect. It is shown that increasing the step duration under no‑flow boundary conditions amplifies the deviation from linearity, thereby reducing the reliability of productivity estimates. The obtained results should be taken into account when planning and interpreting IPR‑based step‑rate well tests in low‑permeability reservoirs, as well as in areas with dense well spacing, non‑linear flow regimes in low‑permeability reservoirs, etc. In such cases, it is advisable to apply interpretation methods adapted to transient conditions (e.g., rate‑transient analysis considering reservoir boundaries, multi‑well deconvolution).

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