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This paper presents a new method to give an analytical solution in Laplace domain directly that is used to describe pressure transient behavior of partially penetrating hydraulic fractures in a box-shaped reservoir with closed boundaries. The basic building block of the method is to solve diffusivity equation with the integration of Dirac function over the distance that is presented for the first time. Different from the traditional method of using the source solution and Green’s function presented by Gringarten and Ramey, this paper uses Laplace transform and Fourier transform to solve the diffusivity equation and the analytical solution obtained is accurate and simple. The effects of parameters including fracture height, fracture length, the position of the fracture, and reservoir width on the pressure and pressure derivative are fully investigated. The advantage of the analytical solution is easy to incorporate storage coefficient and skin factor. It can also reduce the amount of computation and compute efficiently and quickly.

Hydraulic fracturing technology has been a common application in the oil and gas industry during the last two decades. More and more attentions were focused on the study of pressure transient behavior of hydraulically fractured wells. In most published literatures, hydraulic fractures were assumed to be fully penetrating the formation. Limited efforts have been made to investigate the effects of partially penetrating fracture height on the performance of wells. In practice, fully penetrating fractures may lead to an early or immediate water or gas breakthrough in a reservoir with bottom water or gas cap in contact, whereas partially penetrating fractures may be the only way to prevent the early breakthrough [

No matter the problem of wells with or without hydraulic fractures, most scholars considered the fully penetrating wells or fully penetrating hydraulic fractures. However the issue of partial penetration is always ignored. In the early time, some scholars presented some methods to study partially penetrating wells. Muskat, Nisle, Brons and Marting, and Papatzacos used the method of images [

To solve the unsteady-state flow problem of fractures in the reservoir, most solutions were presented based on the using of the source solution and Green’s function provided by Gringarten and Ramey [

These previous solutions were quite significant to the later analysis of the pressure behavior of the partially penetrating fractures. Valkó and Amini [

This study attempts to give some new insights in understanding the partially penetrating hydraulic fractures in a box-shaped reservoir. This paper presents an analytical solution that describes pressure transient behavior of partially penetrating fractures in a box-shaped reservoir and is successfully applied to examine effects of fracture half height, fracture half length, and reservoir width on performance of a fracture in a reservoir with closed boundaries based on pressure and pressure derivative concepts. Moreover the effect of the vertical position of the fracture on the pressure and pressure derivative is fully investigated. More specifically, the diffusivity equation is presented for the first time and the analytical solution of pressure transient behavior in Laplace domain is derived by using Laplace transform and Fourier transform. Then the bottomhole pressure in the real time domain can be obtained by using the inverse Laplace algorithm as proposed by Stehfest [

Consider a partially penetrating hydraulic fracture in a closed homogenous box-shaped reservoir as shown in Figure

Schematic diagram of partially penetrating fracture.

As shown in Figure

The analytical model for the pressure behavior of a fracture in a box-shaped reservoir can be derived based on the solution for the diffusivity equation in the porous media. The diffusivity equation that governs the flow is

Reservoir pressure is initially constant

To simplify the problem we define dimensionless variables as the follows:

Using the dimensionless variables (

The Laplace transform with respect to time is defined as

Fourier cosine transform with respect to

Based on the concept of Fourier cosine transform, the Fourier transform of the

Equation (

Kuchuk and Brighan [

Model validation between this study and literature results.

Based on the analytical solution for a partially penetrating fracture in a box-shaped reservoir presented in the previous part, a sensitivity study for the parameters affecting the pressure and pressure derivative in the model is carried out. The intention of this study is to show the effect of each of these parameters on the dynamic behavior of a partially penetrating fracture in a box-shaped reservoir. We evaluate the pressure transient solution by varying the values of four parameters including the fracture half height, the fracture half length, the fracture position, and the reservoir width. As shown on the plots, the pressure and pressure derivative have different shapes for each combination of fracture height, fracture length, fracture position, and reservoir width. Dimensionless basic parameters used for simulating pressure transient response are presented in Table

Basic data of the system.

Dimensionless parameter | Value |
---|---|

Reservoir length |
4000 |

Reservoir width |
4000 |

Reservoir height |
1 |

Half fracture length |
1 |

Half fracture height |
0.5 |

Variable |
1 |

Fracture position |
(2000, 2000, 0.5) |

Figure

The effect of fracture height for fracture in the center.

Schematic diagram of partially penetrating fracture.

Figure

The effect of fracture height for fracture not in the center.

Figure

The effect of fracture length at

Figure

The effect of fracture length at

The effect of the fracture half length on pressure and pressure derivative when the fracture with large half penetration ratio is in the center of the reservoir is shown in Figure

The effect of fracture length at

Figure

The effect of fracture length at

Figure

The effect of reservoir width.

Type-curve matching is a quick method to estimate reservoir and fracture parameters. The following procedures illustrate how type curve matching is used to calculate reservoir and fracture characteristics such as permeability, fracture half length, and fracture half height.

Plot pressure change (

Obtain the best match of the data with one of the type curves.

Read from an match point:

Calculate

Calculate

Calculate

Calculate the fracture half height:

Calculate the fracture half length:

Giving the reservoir and well data,

Plot pressure change (

Pressure and pressure derivative plot for example.

Obtain the best match of the data with one of the type curves as shown in Figure

Type-curve matching plot for example (

Read from an match point:

Calculate

Calculate

Calculate

Calculate the fracture half height from (

Calculate the fracture half length from (

Results of example.

Parameter | Input value | Calculated value by type curve matching technique |
---|---|---|

Horizontal permeability |
2 | 2 |

Vertical permeability |
2 | 2 |

Fracture half length |
400 | 400 |

Fracture half height |
10 | 10 |

A detailed step by step procedure for solving the analytical solution of a partially penetrating hydraulic fracture in a box-shaped reservoir by using Fourier cosine transform and Laplace transform is presented. The solution can be used to investigate the pressure transient behavior. In this paper, we validated it available with the published analytical solution for a relative simple system. Sensitivity analyses about the effects of the main parameters including fracture height, fracture length, and reservoir width on type curves are also presented in detail. Moreover the effect of the vertical position of the fracture on the pressure and pressure derivative is fully investigated. And an example is used to illustrate that the type curves can be used to analyze transient well test analysis for partially penetrating fracture in closed reservoirs. The merit of the solution is that it can also reduce the amount of computation and compute efficiently and quickly. The solution can be further developed as its great applicability.

The authors declare that there is no conflict of interests regarding the publication of this paper.