Mathematical research helps increase the precision of oil fracking

Employees of the Institute of Mathematics and Mechanics (Kazan Federal University) proposed and described an algorithm for solving the problem of interpreting tracer studies to determine the lengths of multi-zone hydraulic fracturing cracks using a flow tube filtration model. With a high level of accuracy, the method reduces the computational time by hundreds of times.

In the development of oil and gas fields, especially low-permeability reservoirs, such a method of intensifying hydrocarbon production as multi-zone hydraulic fracturing (MHF) in horizontal wells is used. In this case, the length of the wells is hundreds and thousands of meters, along which up to several dozen cracks (stages) of hydraulic fracturing are performed, the length of each can reach hundreds of meters, as a result of which the area of low pressure in the vicinity of the well increases many times.

For long-term planning and management of the reservoir development process, knowledge of the sizes and filtration parameters of cracks at each MHF interval is required, but these parameters remain unknown and cannot be directly measured. For this purpose, indirect measurement methods are being developed, such as tracer studies.

New solutions require the development of ultra-precise and high-speed calculation methods, which cannot always be implemented, but one of the promising methods was found by scientists from the Institute of Mathematics and Mechanics—Head of the Aerohydromechanics Department Konstantin Potashev, Professor Alexander Mazo, and first-year Ph.D. student Almaz Uraimov.

To reduce the time for calculations, the scientists propose using simplified physical and mathematical models of the process that do not lead to a significant loss of accuracy. Each hydraulic fracturing interval is defined by a single crack with effective properties, and the spatial filtration model of transport in the reservoir is decomposed into a set of problems of reduced dimensionality in separate fixed current tubes. Due to such decomposition, explains Dr. Potashev, there is a significant acceleration of calculations with savings in computing resources. Moreover, these tasks can be solved independently of each other, which allows using parallel computing algorithms for additional multiple acceleration of calculations.

"The main advantage of the methods we use is a fundamental increase in the calculation speed while maintaining high accuracy of the solution. The fact is that traditional three-dimensional models simulate the processes of multiphase multicomponent filtration near hydraulic fracturing cracks for so long that a multivariate high-precision solution of inverse problems with their help is measured in weeks and months. Since such terms significantly exceed the permissible time for making technological decisions, traditional models actually do not allow solving such problems. Therefore, the method we propose makes it fundamentally possible to reliably interpret the results of tracer studies using the most justified approach—numerical modeling of filtration flows," comments Potashev.

In order to optimize the algorithm for selecting crack parameters and increase its stability, the researchers studied the behavior of the target functional and identified a way to reduce the degree of its gullying, which always complicates the search for optimal solutions.

The accuracy of calculations is achieved by using high-resolution computational grids with a spatial step of only a few centimeters. In a three-dimensional model, grids of this level of detail would inevitably lead to a huge number of computational blocks, which would increase the machine time of modeling several times.

"In our case, we are talking about problems of reduced dimensionality along the flow tubes, which quite accurately reproduce the structure of the filtration flow, that is, three-dimensional problems are decomposed into a set of two-dimensional problems in vertical sections of the flow tubes or two-dimensional problems into a set of problems in one-dimensional flow tubes. Such a decomposition with a decrease in the dimensionality of problems allows us to reduce the number of unknown grid blocks by orders of magnitude," adds Dr. Potashev. "In addition, the conditionally impenetrable lateral boundaries of the current tubes make it possible to independently solve problems in each of them. This provides excellent opportunities for implementing a parallel computing apparatus and provides additional acceleration of calculations, which is limited only by the number of computing cores of the processor."

The research was carried out within the framework of a state assignment with financial support from the Kurchatov Institute. The results were published in Matematicheskoe Modelirovanie.

More information:
ОПРЕДЕЛЕНИЕ ДЛИН ТРЕЩИН МНОГОЗОННОГО ГИДРОРАЗРЫВА ПЛАСТА C ПОМОЩЬЮ МОДЕЛИ ФИЛЬТРАЦИИ В ТРУБКАХ ТОКА
www.researchgate.net/publicati … RACII_V_TRUBKAH_TOKA

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