Modelling and Simulation in Engineering

Volume 2019, Article ID 4034860, 10 pages

https://doi.org/10.1155/2019/4034860

## Modeling the Effect of Intersected Fractures on Oil Production Rate of Fractured Reservoirs by Embedded Fracture Continuum Approach

University of Transport and Communications, No. 3 Cau Giay Street, Hanoi, Vietnam

Correspondence should be addressed to Hong-Lam Dang; nv.ude.ctu@mal.gnoh.gnad

Received 18 February 2019; Revised 12 June 2019; Accepted 18 June 2019; Published 2 July 2019

Academic Editor: Michele Calì

Copyright © 2019 Hong-Lam Dang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The homogenization of matrix and short fractures is one of the conventional approaches to deal with a plenty of fractures in different scales. However, the accuracy of this approach is still a question when long fractures and short fractures are distributed in the homogenized model. This paper describes a new hybrid method in which the long fractures will be modeled explicitly by the embedded fracture continuum approach and short fractures are considered through the homogenized technique. The author used this hybrid method to demonstrate the effect of fractures which are intersected to the well on the oil production rate as well as the elapsed time of a fractured reservoir in a depletion process. The advantages of the new hybrid method are easy assembling of numerous fractures into the model and incorporation of the complex fracture behaviour into the model.

#### 1. Introduction

Prediction of the oil production rate of fractured reservoirs is still a challenge due to the complexity of fracture distributions as well as their behaviours. The permeability of a fracture is usually much larger than that of the rock matrix. Hence, the fluid will flow mostly through the fracture network if fractures are connected. This implies that the fluid transport in the naturally fractured reservoirs will be governed by the fracture connectivity and their distribution [1, 2]. There are three approaches commonly used in modeling naturally fractured formations: (1) continuum models using effective properties of representative element volume, (2) discrete fractured models in which fractures are introduced explicitly, and (3) hybrid models that combine the above two models.

The first approach to model a fractured reservoir in the literature is the homogenized approach [3–5]. The fact of using the equivalent medium in the modeling means that the porous space and permeability of the rock matrix as well as those of the fracture network are gathered and characterized through only a single porosity and a single permeability represented by the overall (effective) permeability. Thus, the case study (known hereafter as the homogenized model) considers only the effect of the single porosity and single permeability. The advantage and limitation of using the effective single continuum approach were discussed in [3, 4] by comparing with the traditional dual continuum model in [6]. These authors discussed that this approach can capture the baffled flow effect, while the dual continuum approach is better equipped to enhance flow through the fracture system which is however limited for the models they used, in the regular and orthogonal fracture geometry. The unsteady-state flow effects between fracture and matrix (matrix-fracture transfer) are not considered in the effective single continuum approach which however can be more easily addressed in the dual continuum model by using the matrix-fracture transfer functions (sugar cube-like matrix and fracture geometry). Thus, as detailed in [4], the effective single continuum modeling approach can be applied when the unsteady-state between matrix and surrounding fractures is not a dominant feature.

The second approach in this study is that fractures are explicitly introduced in the model. The question of what fracture should be considered explicitly in a model is an open question. In fact, in every modeling approach (and this is valuable for all models not only for fractures), the discontinuities of lower scale than that of observation are neglected. For example, when a rock sample is tested in the laboratory, the pores or cracks not detectable by naked observation are neglected. When cracks in site are observed, the cracks under a given value are simply neglected, and the media under this scale are considered as continuum. Likewise, in our model, only the long fractures with length superior to the representative element volume (REV) size will be considered explicitly. The choice of only long fractures modeled explicitly in this work comes from the fact that long factures play an important role in fractured reservoir behaviour [7]. This concept can be explained by the fact that, in reality, the fractured rock mass is highly heterogeneous, and the assumption of using the uniform aperture of the fracture networks to simplify the modeling by using the equivalent medium through the upscaling approach can over/underestimate the problem. In such heterogeneous media, there is a high probability of the existence of several initial fractures/faults (or even induced fractures during drilling) with a large aperture. The consideration of these fractures in the classical upscaling approach can violate the notion of REV. The scenario proposed in this work can be an alternative approach to simplify the problem and matches well with the idea proposed by some other scholars [2, 4]. In their work, Lee et al. [4] proposed a hierarchical approach for modeling fluid flow in a naturally fractured reservoir with multiple length-scale fractures. They classified the fractures as short, medium, and long fractures. The short and medium fractures were associated with the matrix through the homogenization technique to define the effective porous medium, while the long fractures (considered as the major conduits) are separated and modeled explicitly in the model. The flexibility and performance of this hierarchical approach were demonstrated by these authors in modeling the complex fractured rock masses.

In this paper, the embedded fracture continuum (EFC) approach [7–9] will be used to model a hypothetical reservoir whose natural fractures coincide with that of the Sellafield site [10]. The objective is to illustrate the effects of intersected fractures on the oil production rate of a fractured reservoir. The choice to use the data of Sellafield is motivated only by the completeness of the data. To demonstrate the effect of intersected fractures on the oil production rate of a fractured reservoir, two scenarios are studied: the first scenario is the homogenized approach as described above; the other is explicitly a fracture approach as explained in the preceding paragraph. The second scenario will be separated in two cases which aim to investigate in more detail the effect of these fractures on a hypothetical well behaviour (well production notably). In the first case, we consider that at least one of these long fractures intersects the well (known hereafter as the conductive fracture model), and in the other case, any long fractures intersect the well (known as the nonconductive fracture model).

The configuration of problems considered here is quite similar as the one studied in the work of [2]. In that work, the author would like to elucidate the influence of well and long fracture intersection on the productivity of the well. However, in comparison with that contribution [2], there are some essential differences in the present study. In effect, to consider the communication between porous matrix and fractures, as well as fractures and well, Li and Lee [2] used the transport index between matrix-fracture transfer and the fracture transmissibility to the well. The later parameter is determined by using directly Peaceman’s productivity index (PI) which is largely used in practice for the well drilled in the medium without fracture intersection. The author assumed that the pressure drop along the fracture inside the well block is negligible, and the productivities from the fracture and well are superposed. This assumption means that the fractures connected to the well will become part of extended well geometry. However, as the fracture surface is much larger than the well surface, the production from the fracture surface will be much larger than that from the well surface. Inversely, in the present work, we interest only the production calculated from the well surface. This well surface is explicitly considered in our model through which the production rate (fluid flux) will be calculated directly while the drop of pressure in the fractures is decided by the transient flow in their corresponding fracture cells.

The implementation of these fractures in the model at the present stage seems easy by using the EFC approach based on the fracture cell concept [7–9]. Note that each fracture cell in the model represents a porous medium which has its own porosity and permeability and is different from those of the fractured matrix. These models represent in effect the double porosity and double permeability medium to distinguish with the initial scenario based on the single effective medium. Otherwise, as mentioned above, in all calculations, we neglected the unsteady-state feature between matrix and fractures in this fractured matrix. This EFC approach was implemented with the help of an open source code deal.II [7–9, 11, 12].

#### 2. Equivalent Elastic Properties and Permeability of the Fracture Cell in the Embedded Fracture Continuum Approach

##### 2.1. Equivalent Elastic Properties of the Fracture Cell

The first work in this approach was reported by Figueiredo et al. [13, 14] in which they considered that the elastic properties of all fracture cells generated in the whole medium are isotropic and are characterized by the same Young’s modulus calculated from the fracture cell intersected by a horizontal fracture. Otherwise, Poisson’s ratio is equal to one of the intact rock:where *E*_{m}, , *k*_{n}, and *h* indicate, respectively, Young’s modulus, Poisson’s ratio of the intact matrix, the normal stiffness of fracture, and the element size.

More precisely, in these last works, one does not distinguish fracture cells intersected by one or many fractures (Figure 1(a)). It also means that this approximation is only applicable for fractures owing to the same mechanical properties.