# An efficient hybrid local nonmatching method for multiphase flow simulations in heterogeneous fractured media

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DOI: 10.1007/s00366-014-0355-0

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- Mustapha, H. Engineering with Computers (2015) 31: 347. doi:10.1007/s00366-014-0355-0

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## Abstract

This paper presents simulation methodology that combines a local nonmatching grid with a discrete fracture model. Designed for 2D and 3D multiphase flow simulations in standard simulators, the method handles matrix–matrix, fracture–fracture, and matrix–fracture connections in the context of an unstructured, local nonmatching grid. The grid is generated at the fracture intersections, enabling accurate modeling of small control volumes between connecting fractures. Grids are obtained simply by redistributing the volume of small control volumes surrounding the small control volumes, making the method computationally efficient. A unified method to calculate the interblock transmissibility is used for both matching and nonmatching mesh. An unstructured finite-volume graph-based reservoir simulator with a two-point flux approximation reads the new grid by making a simple modification to the graph of connections between the control volumes. The method requires no special treatment of fracture–fracture or matrix–fracture transmissibility calculations and has the flexibility to simulate any flow problem efficiently. Several 2D and 3D numerical examples demonstrate the method’s performance and accuracy. Both simple and complex fracture configurations are presented with various levels of geologic and fluid complexity. The numerical results are in good agreement with those of a reference solution obtained on a finely structured grid.

### Keywords

Discrete fracture modelHybrid gridMatching gridNonmatching gridMortar finite volume### List of symbols

## 1 Introduction

Petroleum reservoirs and aquifers are often highly heterogeneous media with preferential flow paths that result from natural fracture networks [1, 2]. Commonly, a reservoir or aquifer is represented by a set of blocks of porous media and a net of fractures [3–5].

Dual-porosity and dual-permeability [6–9], single-porosity [9], and discrete fracture models [10–12] are the most common approaches used in reservoir simulation to model these media. Despite the numerical efficiency of the dual-porosity and dual-permeability approach, it has several limitations related to the accurate definition of the exchange functions between the matrix and the fractures. These functions may not be adequately defined for complex phenomena. The single-porosity model is a more accurate and physics-based approach. In such a model, the fractures are described explicitly in the medium. For example, in 2D porous media a fracture is represented by a very thin rectangle. The large contrast in size of the matrix and fracture grid blocks makes this approach impractical. Another alternative is the discrete fracture model. It is a simplification of the single-porosity model in which the fractures are represented by (*n* − 1)-dimensional elements in an *n*-dimensional domain. This simplification is supported by the fact that the fracture aperture is small compared to that of the matrix scale. As a result, computational efficiency is improved considerably [9].

Various numerical methods based on discrete fracture models have been used to simulate single and multiphase flow in fractured media. A finite-difference (FD) method [13–15] is used for solute transport simulations in 2D- and 3D-fractured media. The Galerkin finite-element (FE) method has been used to model two-phase flow [16, 17]. The control volume finite-element (CVFE) method on its own and combined with a Galerkin FE method have also been used to solve two-phase flow equations in fractured media [18–23]. A combination of the mixed finite-element (MFE) and discontinuous Galerkin (DG) methods has been used for single- and two-phase flow problems in fractured media [24]. The finite-volume (FV) method is a mass conservative method that gives correct location of fronts. In addition, this method performs very well on unstructured grids where the complex fractured media can be represented efficiently. The FV method is as simple as the FD method, and locally has the same level of accuracy as the FE method.

A simplified FV method using two different spaces has been applied in the context of hybrid grid approaches to simulate two-phase flow problems in fractured media [25, 26]. The fractures are modeled using the discrete fracture model to simplify the grid generation and reduce the geometric complexities in a space known as grid space; the fracture and the matrix grid blocks are treated similarly during the simulation process, but in another space called computational space. In addition, the authors have removed the intermediate small control volume at the fracture intersections. An approximation of the transmissibility between connecting fracture grid blocks is also presented. The approximation was based on a star-delta approach [27]. However, this approximation is accurate for single-phase incompressible flow problems only. It is important to mention that the “hybrid-grid” term is to describe the dual representation of the fracture grid blocks in the grid and computational spaces; however, the discrete fracture model denotes this reduction of the fracture dimension by 1.

This paper aims at presenting a more efficient solution to overcome the drawback of the method discussed above with emphasis on the treatment of the cells at connecting fractures. In contrast to star-delta approach, the proposed method consists of generating a local nonmatching grid (LNMG) around the connecting fractures, followed by a local mortar FV [28–30] method. The proposed method consists of automatically redistributing the volume of some fracture grid blocks to increase the size of the intermediate control volume between connecting fractures. The method is very simple in the sense that no change is applied to the other grid blocks, and the graph of connections between the control volumes can simply be updated to account for some new connections. This approach is very flexible and can be used by any reservoir simulator without any specific numerical treatment. In addition, the LNMG generation does not require a specific grid generator. The grid generation is achieved through a simple rearrangement of any existing matching grid. The INTERSECT^{1} scalable FV-based commercial simulator was used for this study [31–34]. The simulator has been employed on structured matching grids and on the unstructured LNMG method. The paper is organized as follows. First, the proposed grid method is introduced. Then, a description of the numerical method and the transmissibility between nonmatching control volumes are presented. Several numerical examples are then given to demonstrate the performance of the proposed method. The examples are in 2D and 3D, and include some level of geometric and fluid complexity such as large-scale properties, gravity, and capillary pressures.

## 2 Geometrical discretization

^{2}in a 1-mm fracture thickness system. Because of the time-step limitations, this small control volume may potentially increase the computational run time. To avoid this drawback, Karimi-Fard et al. [25] proposed removing this small control volume in the context of the HG approach just described.

*L*

_{1}and

*L*

_{2}, three new control volumes are created each of length [(

*L*

_{1}+

*L*

_{2}+

*e*)/3], where

*e*is the thickness of the fracture being modified. It is important to note that the LNMG technique should be applied only in one of the intersecting fractures. A simple search of the larger fracture control volumes surrounding the intermediate control volume can be used to decide the fracture to be locally modified. As a result, comparable fracture control volumes are obtained using LNMG. Combining the LNMG and HG methods results in a more efficient representation through HLNMG (Fig. 7).

Note that the matrix control volume 17 is connected to fracture control volumes 12, 13, and 18 using HLNMG, but is connected only to fracture control volumes 12 and 18 using HMG. For a general-purpose two-point flux approximation (TPFA) reservoir simulator, the graph of connections between control volumes can be slightly modified to account for the new connections obtained from the local modification of the intermediate fracture control volumes. The HLNMG method does not impose any additional condition on the connectivity of other control volumes.

## 3 Numerical method

The proposed HLNMG method is valid for any general-purpose reservoir simulator. However, for the sake of completeness, we briefly present the system of equations and the numerical method employed in the simulator used in this study.

*w*

_{i},

*x*

_{i},

*y*

_{i},

*i*= 1,…,

*n*

_{c}},

*p*, and {

*S*

_{α},

*α*= oil, gas and water} to avoid conversion to mass and for more flexibility with respect to the level of implicitness [32, 37]. Here,

*w*,

*x,*and

*y*are the component mole fractions in the water, oil, and gas phases, respectively. The

*n*

_{c}denotes the number of hydrocarbon components,

*p*is the phase pressure, and

*S*is the phase saturation. Conservation of mass, Darcy’s law, thermodynamic equilibrium between phases, a saturation constraint, and a mole fraction constraint are the governing equations of our physical problem. Applying the divergence theorem, we can write the component

*i*material balance equation for a general control volume

*j*as

In the left side of Eq. 1, the first term denotes the rate of change of the number of moles of component *i* in control volume *j*, the second term is the sum of intercell flows of component *i* into control volume *j* from the *n*_{j} connected control volumes *k*, and the third term represents the sum of flows of component *i* into control volume *j* from external *n*_{s} sources or sinks, *s*. It is very clear that for each control volume *j*, Eq. 1 accounts for fluxes coming from all control volumes sharing with *j* a part or a complete facet. Numerically, the number of these control volumes may vary with respect to the grid shapes. For example, *n*_{17} is equal to 4 for the structured MG in Fig. 3, and *n*_{17} is equal to 5 for the LNMG method in Fig. 6.

Several alternative time-discretization schemes are supported by the simulator, including implicit pressure and saturation (IMPSAT) and adaptive implicit (AIM), to provide numerical stability at reduced computational cost [38]. The IMPSAT approaches [39] and an AIM approach based predominantly on IMPSAT [40] can be used, in which the time-step selection is made according to the stable time-step criteria [41]. The split of primary and secondary variables, as defined by [37], is configurable at run time according to the phases present and the state of the fluid model. In this paper, we used an AIM method that is suitable for grids with large contrasts in the element control volume sizes; i.e., fracture and matrix grid blocks.

In this study, a cell-based FV method was implemented in the simulator. A graph of connections can simply be supplied to describe the connectivity between the control volumes. This serves to approximate the flow rates coming into a control volume from connecting control volumes (Eq. 1). In this paper, we used a standard local mortar FV method to account for the HLNMG. The theoretical background [30] is not discussed here, and only the numerical perspective is presented. It is important to note that the graph of connections can be locally modified to account for the new connections imposed by the changes made over the small control volume at fracture intersections.

*i*at time

*τ*between two connected control volumes

*j*and

*k*can be approximated using a standard TPFA method,

*T*

_{jk}denotes the transmissibility constant between control volumes

*j*and

*k*. The variables

*λ*,

*b,*and Φ are the phase mobility, molar density, and potential, respectively. The phase potential includes the phase pressure and the capillary pressure and gravity forces. The superscript * denotes the upstream direction for each phase and for each connection. Note that

*χ*is the component mole fraction in the related phase. The transmissibility calculation between NMG control volumes is described as follows: In an MG, the transmissibility between any two control volumes

*j*and

*k*(Fig. 8), generalized [25] from the transmissibility calculation for corner-point systems [42], can be written as

In Eqs. 3 and 4, *A*_{c,jk} is the contact area between control volumes *j* and *k*; *A*_{jk} is the complete area of each neighbor; *K*_{j} is the control volume *j* permeability; *H*_{jk} is the distance between the centroid of the control volume *j* and the centroid of the interface between the control volumes *j* and *k*; n_{j} is the unit normal vector to the interface inside block *j*, and u_{jk} is the unit vector along the line joining *O*_{j} the block *j* center and centroid of the interface. Note that, for adjacent control volumes in a fully conformed mesh, the interface areas *A*_{jk}, *A*_{jk} and *A*_{c,jk} are equal.

*j*side belongs to block

*k*, and another part belongs to block l. To generalize the transmissibility expression in Eq. 3, the variables have been bi-indexed to localize the interface between any two control volumes. Thus, this formula is also valid for the HLNMG method proposed, and there is no need to introduce any other transmissibility calculation formula.

## 4 Numerical examples

Several numerical examples are presented here to demonstrate the performance and accuracy of the HLNMG method. Although only examples of 2D and 3D two-phase flow problems are presented, use of the HLNMG method is not restricted to a specific flow problem. First, we consider simple configurations of fractures to validate the HLNMG method. Additional model complexities including gravity, capillary pressure forces, and large-scale properties have also been added to better illustrate the efficiency of the method. The HLNMG results on an unstructured grid using a local mortar FV method are compared to reference solutions achieved using a FV method on a finely structured MG. The hybrid MG graph of connections is only updated in the HLNMG case, and the commercial simulator is used on both grid types with the same physical parameters. An AIM scheme is employed for the time discretization, and a cell-based FV and MFV for the space discretization.

## 5 Example 1: 2D-fractured porous media

^{5}md. A bottom-hole pressure (BHP) is imposed at the oil producer, and a 0.01 PV/D of water has been injected. Capillary pressures are neglected in this example.

Relevant data for Examples 1 and 2

Domain dimensions | 1 m × 1 m |

Matrix | \(\phi_{\text{ma}} = 0.2,K_{\text{ma}} = 1\,{\text{md}}\). |

Fracture | \(\phi_{\text{f}} = 1,K_{\text{f}} = 10^{5} \,{\text{md}},e = 0.1\,{\text{mm}}\) |

Fluid | \(\mu_{\text{o}} = 0.45\,{\text{cp}},\mu_{\text{w}} = 1\,{\text{cp}}\) |

Injection rate | 0.01 PV/D |

Grid | Example 1 |

Structured MG: 1,600-square control volumes | |

Unstructured HLNMG: 2,711 2D triangle and 1D fracture control volumes | |

Example 2 | |

Structured MG: 8,000-cube control | |

Unstructured HLNMG: 10,940 3D prism and 2D fracture control volumes |

## 6 Example 2: 3D-fractured porous media

The commercial simulator’s computational time is compared using both methods for nonzero capillary pressures. The HLNMG method is about five times faster than the MG method in this example, as shown in Fig. 9. Despite the greater number of unstructured control volumes, this figure shows that the simulator is about five times faster on an unstructured HLNMG than on a structured MG. This result is supported by two key elements, discussed previously, in the representation of both methods: The HLNMG removes the intermediate control volumes at the connecting fractures and this provides more flexibility to the simulator to increase the time step; and the linear system is sparser and better conditioned.

## 7 Example 3: 3D field-scale problem

Relevant data for Example 3

Domain dimensions | 3,000 m × 3,000 m × 270 m |

Matrix | \(\phi_{\text{ma}} = 0.2,K_{\text{ma}} = 10\,{\text{md}},1\,{\text{md and }}15\,{\text{md in layer }}1,2,{\text{ and }}3,{\text{ respectively}}\) |

Fracture | \(\phi_{\text{f}} = 1,K_{\text{f}} = 10^{5} \,{\text{md}},e = 1\,{\text{mm}}\) |

Fluid | \(\mu_{\text{o}} = 0.45\,{\text{cp}},\mu_{\text{w}} = 1\,{\text{cp}}\) |

Injection rate | 10 |

Grid | Example 3 |

Unstructured HLNMG: 164,556 3D prism and 2D rectangular fracture control volumes |

## 8 Conclusions

In this paper, we have presented a flexible method for modeling fracture–fracture and matrix–fracture flow interactions. The method uses a local nonmatching grid technique to generate equal fracture grid block sizes around connecting fractures. This increases the size of the intermediate control volumes at the fracture intersections, and has the effect of reducing computational time and providing an accurate approximation of the fracture–fracture flow. The method can be applied to any physical problem and grid elements of any shape including squares and triangles in 2D, cubes, and tetrahedra and prisms in 3D. The method is ideal for graph-based reservoir simulators that describe the connections between control volumes through a list. This list or graph can be updated to account for the newly created nonmatching grid elements. The method has been tested with several examples and the results have shown that in 2D and 3D reservoirs with orthogonal fracture configurations, the method is very accurate compared to structured fine-grid reference solutions; and this is true for both zero and nonzero capillary pressure cases. The computational time and performance are better than simulations using the structured grid cases. Finally, a 3D large-scale problem has been solved to further demonstrate the capability of the method proposed.

## Acknowledgments

We thank the Editor-in-Chief of Engineering with Computers handling our manuscript and the anonymous reviewers for their detailed comments that have helped improve the manuscript. The authors thank Schlumberger for the support and the permission to publish this work.