# An exploration of pressure dynamics using differential equations defined on a fractal geometry

## Abstract

Pressure tests have been used to understand the behavior of naturally fractured reservoirs (NFRs) by establishing parameters such as permeability, connectivity of fractures, system compressibility, fracture network parameters, and others associated with the physical properties of the reservoir. The complex fracture network of a NFR produces anomalous fluid flow, which is shown by the pressure behavior at the well. A plausible model to explain this behavior is to assume that it arises from a fractal fracture system. However, to estimate the parameters that define this fractal fracture system, fluid flow models based on the modification of Darcy’s law are used. On the other hand, in this work, a fractal structure with known fractal dimension: Sierpinski gasket (SG) is proposed. The objective of this work is to solve numerically the diffusion equation on SG, and establish a comparison with flow models used in the oil industry to investigate the fractality assumption. This study is focused on the qualitative behavior of the solution of the diffusion equation defined on a SG supported by the Kigami’s theory of differential equations on fractals.

- Authors:

- Programa Académico de Posgrado, Instituto Mexicano del Petróleo. Eje Central Lázaro Cárdenas Norte 152 (Mexico)
- Universidad Nacional Autónoma de México (Mexico)

- Publication Date:

- OSTI Identifier:
- 22769348

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 97 MATHEMATICAL METHODS AND COMPUTING; DIFFUSION EQUATIONS; FLOW MODELS; FLUID FLOW; FRACTURED RESERVOIRS; FRACTURES; MATHEMATICAL SOLUTIONS; PERMEABILITY

### Citation Formats

```
Chaidez-Félix, Jesús M., E-mail: chaidez-07@hotmail.com, and Velasco-Hernández, J. X.
```*An exploration of pressure dynamics using differential equations defined on a fractal geometry*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0398-5.

```
Chaidez-Félix, Jesús M., E-mail: chaidez-07@hotmail.com, & Velasco-Hernández, J. X.
```*An exploration of pressure dynamics using differential equations defined on a fractal geometry*. United States. doi:10.1007/S40314-016-0398-5.

```
Chaidez-Félix, Jesús M., E-mail: chaidez-07@hotmail.com, and Velasco-Hernández, J. X. Tue .
"An exploration of pressure dynamics using differential equations defined on a fractal geometry". United States. doi:10.1007/S40314-016-0398-5.
```

```
@article{osti_22769348,
```

title = {An exploration of pressure dynamics using differential equations defined on a fractal geometry},

author = {Chaidez-Félix, Jesús M., E-mail: chaidez-07@hotmail.com and Velasco-Hernández, J. X.},

abstractNote = {Pressure tests have been used to understand the behavior of naturally fractured reservoirs (NFRs) by establishing parameters such as permeability, connectivity of fractures, system compressibility, fracture network parameters, and others associated with the physical properties of the reservoir. The complex fracture network of a NFR produces anomalous fluid flow, which is shown by the pressure behavior at the well. A plausible model to explain this behavior is to assume that it arises from a fractal fracture system. However, to estimate the parameters that define this fractal fracture system, fluid flow models based on the modification of Darcy’s law are used. On the other hand, in this work, a fractal structure with known fractal dimension: Sierpinski gasket (SG) is proposed. The objective of this work is to solve numerically the diffusion equation on SG, and establish a comparison with flow models used in the oil industry to investigate the fractality assumption. This study is focused on the qualitative behavior of the solution of the diffusion equation defined on a SG supported by the Kigami’s theory of differential equations on fractals.},

doi = {10.1007/S40314-016-0398-5},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}