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 Modes. Multiphase Systems. 17 (2022) 1–2. 113–121 (in Russian).
2022. Vol. 17. Issue 1, Pp. 113–121
DOI: 10.21662/mfs2022.1.009
Fluid Pressure Dynamics in a Hydraulic Fracture during Transient-Well-Operation Modes
Shagapov V.Sh., Bashmakov R.A.∗∗, Fokeeva N.O.∗,∗∗, Shammatova A.A.∗∗∗∗
Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia
∗∗Bashkir State University, Ufa, Russia
∗∗∗LLC ”RN-BashNIPIneft“, Ufa, Russia
∗∗∗∗Ufa State Petroleum Technological University, Ufa, Russia


At present, 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.


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

Article outline

The low accuracy of conductivity determination and bad fluid flow description in a well with a hydraulic fracture leads to the decrease of the efficiency of classical models. 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. This paper discusses, analytic solutions of the integro-differential equation are obtained for stepwise and continuous changes of the flow rate. The solution makes it possible to determinate the flow rate or bottom hole pressure according to the given pressure changing law in the bottom or well flow rate, as well as pressure evolution in the hydraulic fracture. The article provides a good comparison of the theoretical pressure change curves constructed from the flow rate change data and the results of this observations.


Hydraulic fracturing is one of the most important methods for intensifying oil production. A lot of scientific works have been written in this area for more than 70 years of application. 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. This article describes the way to obtain analytical solutions of a system of equations for fluid filtration in formations subjected to hydraulic fracturing under transient well operation mods. 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.

Materials and Methods

There are many methods that solve any problems of hydraulic fracturing, but the problem of describing the piecewise constant change in the flow rate/pressure in a well with hydraulic fracturing has not been solved yet. Four steps were taken to achieve the goal. An analysis of a number of works on modeling pressure changes in a hydraulic fracture was carried out. An analytical solution of the integro-differential equation with a stepwise fluid flow of limited use in the well was obtained. The Theoretical study of the fluid filtration in the hydraulic fracture surrounding the formation, as well as the dynamics of pressure and flow rate changes during transient well operation were made. Our approach can be seen as the following two-stage process the fluid in the reservoir flows perpendicular to the fracture plane and only then in flows along the fracture from the well (bilinear scheme).


To check the calculations, the authors were provided with the data of the operation of a real industrial vertical well where the flow rate and pressure changes over time. The result of the calculations with reliable data allows us to conclude that the constructed mathematical model of the impact with high accuracy present fluid filtration in a well with a hydraulic fracture during transient operation modes. 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.


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