ISSN 2658–5782
DOI 10.21662
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Shagapov V.Sh., Bashmakov R.A., Fokeeva N.O., Shammatova A.A. Fluid pressure dynamics in a hydraulic fracture during transient-well-operation mode. Multiphase Systems. 18 (2023) 1. 1–8 (in Russian).
2023. Vol. 18. Issue 1, Pp. 1–8
DOI: 10.21662/mfs2023.1.001
Fluid pressure dynamics in a hydraulic fracture during transient-well-operation mode
Shagapov V.Sh., Bashmakov R.A.∗∗, Fokeeva N.O.∗,∗∗∗, Shammatova A.A.∗∗∗∗
Mavlyutov Institute of Mechanics of UFRC RAS, Ufa, Russia
∗∗Ufa University of Science and Technology, Ufa, Russia
∗∗∗RN-BashNIPIneft, Ufa, Russia
∗∗∗∗Ufa State Petroleum Technological University, Ufa, Russia


Hydraulic fracturing is one of the most effective methods for intensifying hydrocarbon production. The paper describes the transient-well-operation modes of the well, establishes a connection between the changing fluid flow in the well and the pressure in the fracture and reservoir. With this purpose, a solution is found to the integro- differential equation that describes the change in pressure in the hydraulic fracture with a known change in pressure in the well or at a given flow rate. Based on the bottomhole pressure change data for a short-term pump operation that maintains a constant flow rate and subsequent pump shutdown (Π-shaped flow rate change), it is convenient to determine the characteristics of a hydraulic fracture using the formulae proposed in the paper. The results of the work can be used in hydrodynamic studies of wells. The article shows the application of the obtained equations for determining bottomhole pressure when solving a practical problem with field data, and also presents a formula that allows calculating its conductivity without taking into account the permeability and width of the hydraulic fracture.


hydraulic fracturing,
oil well,
hydraulic fracture pressure dynamics,
transient-well-operation modes

Article outline

In 2023, it is planned to test the first native hydraulic fracturing at the field in Khanty-Mansi Autonomous Okrug. The prototype consists of 12 units and was developed with the participation of the state corporation «Roscosmos», which confirms the demand for technology. The aim of the article is to analyze the way of accuracy improving for the fluid flow model in a well with a hydraulic fracture in the case of a transient operation mode considering fluid leakage into the formation. The article deals with analytical solutions of the integro-differential equation, which allow determining the flow rate or bottomhole pressure according to a given law of pressure change in the bottomhole or well flow rate, as well as the evolution of pressure in a hydraulic fracture. A comparison of pressure change curves and the results of actual well operation data is given.


Hydraulic fracturing has been used for over 70 years. During this period, many studies have been carried out. Cinco's works describe a bilinear flow, Kanevskaya's books told about stationary filtration and some researchers from Ufa have been studied unsteady flows in recent years. The article describes a method for obtaining analytical solutions to the system of fluid filtration equations in formations subjected to hydraulic fracturing in a non-stationary case.

Methods and approaches

Despite a lot of scientific works, there are issues that are not sufficiently covered, for example, the task of describing a piecewise constant change in flow rate/pressure in a well with hydraulic fracturing. Several steps have been taken to achieve this goal. An analysis of a number of works on modeling pressure changes in a hydraulic fracture was carried out. A bilinear scheme was considered: the fluid propagates from the well along the fracture held by the propant, and then enters the formation, where it flows perpendicular to the fracture plane. A theoretical study of fluid filtration in the formation surrounding the fracture was carried out, as well as a study of the dynamics of pressure and flow rate changes in non-stationary well operation.


The result of comparing the calculations with the actual data is sufficient to draw a conclusion that the constructed mathematical model is capable of reproducing the fluid filtration in a well with a hydraulic fracture during transient operation modes with high accuracy. Thus, using these formulas, and knowing the law of change in flow rate and pressure in the well, it is possible to determine the characteristics of the hydraulic fracture, its conductivity and reservoir.


  1. [And the «oil» on the shoulder. Diversification in the space industry] I neftajnka po plechu. Diversifikazia v kosmicheskoy otrasli. Russian Space Magazine. 2022. No. 04(38). Pp. 64–67 (in Russian).
  2. Gringarten A.C., Ramey H.J. Jr. Unsteady-State Pressure Distributions Created by a Well With a Single Horizontal Fracture, Partial Penetration, or Restricted Entry // SPE J. 1974. V. 14. Pp. 413–426.
    DOI: 10.2118/3819-PA
  3. Heber Cinco L., F. Samaniego V., N. Dominguez A. Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture // SPE J. 1978. V. 18. Pp. 253–264.
    DOI: 10.2118/6014-PA
  4. Khabibullin I.L., Khisamov A.A. Unsteady Flow through a Porous Stratum with Hydraulic Fracture // Fluid Dynamics. 2019. V. 54. Pp. 594– 602.
    DOI: 10.1134/S0015462819050057
  5. Shagapov V.S., Nagaeva Z.M. On the theory of seepage waves of pressure in a fracture in a porous permeable medium // J. Appl. Mech. Tech. Phy. 2017. V. 58. Pp. 862–870.
    DOI: 10.1134/S0021894417050121
  6. Nagaeva Z.M., Shagapov V.Sh. Elastic seepage in a fracture located in an oil or gas reservoir // Journal of Applied Mathematics and Mechanics. 2017. V. 81, issue 3. Pp. 214-222.
    DOI: 10.1016/j.jappmathmech.2017.08.013
  7. Khabibullin I.L., Khisamov A.A. Modeling of unsteady fluid filtration in a reservoir with a hydraulic fracture // J. Appl. Mech. Tech. Phy. 2022. V. 63. Pp. 652–660.
  8. Bashmakov R.A., Fokeeva N.О., Shagapov V.Sh. Some features of fluid filtration in a hydraulic fracture under transient well operation conditions // Mechanics of Solids. 2022. V. 57, No. 8. Pp. 1923–1935.
    DOI: 10.3103/S0025654422080040
  9. Charnyi I.A. [Underground Hydrogasdynamics] Podzemnaya gidrogazodinamika. Moscow, Gostoptekhizdat. 1963. P. 397 (in Russian).
  10. Shagapov V.S., Nagaeva Z.M. Approximate Solution of the Problem on Elastic-Liquid Filtration in a Fracture Formed in an Oil Stratum // J. Eng. Phys. Thermophy. 2020. V. 93. Pp. 201–209.
    DOI: 10.1007/s10891-020-02109-4