# Upscaling in Vertically Fractured Oil Reservoirs Using Homogenization

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## Abstract

Flow modeling in fractured reservoirs is largely confined to the so-called sugar cube model. Here, however, we consider vertically fractured reservoirs, i.e., the situation that the reservoir geometry can be approximated by fractures enclosed columns running from the base rock to the cap rock (aggregated columns). This article deals with the application of the homogenization method to derive an upscaled equation for fractured reservoirs with aggregated columns. It turns out that vertical flow in the columns plays an important role, whereas it can be usually disregarded in the sugar cube model. The vertical flow is caused by coupling of the matrix and fracture pressure along the vertical faces of the columns. We formulate a fully implicit three-dimensional upscaled numerical model. Furthermore, we develop a computationally efficient numerical approach. As found previously for the sugar cube model, the Peclet number, i.e., the ratio between the capillary diffusion time in the matrix and the residence time of the fluids in the fracture, plays an important role. The gravity number plays a secondary role. For low Peclet numbers, the results are sensitive to gravity, but relatively insensitive to the water injection rate, lateral matrix column size, and reservoir geometry, i.e., sugar cube versus aggregated column. At a low Peclet number and sufficiently low gravity number, the effective permeability model gives good results, which agree with the solution of the aggregated column model. However, ECLIPSE simulations (Barenblatt or Warren and Root (BWR) approach) show deviations at low Peclet numbers, but show good agreement at intermediate Peclet numbers. At high Peclet numbers, the results are relatively insensitive to gravity, but sensitive to the other conditions mentioned above. The ECLIPSE simulations and the effective permeability model show large deviations from the aggregated column model at high Peclet numbers. We conclude that at low Peclet numbers, it is advantageous to increase the water injection rate to improve the net present value. However, at high Peclet numbers, increasing the flow rate may lead to uneconomical water cuts.

## Keywords

Upscaling Fractured reservoirs Homogenization Waterflooding Oil recovery## List of Symbols

*A*Horizontal cross-section

*B*3D domain

**c**Vector of the matrix-cell center

*d*Size of a grid cell

*D*_{cap}Capillary-Diffusion coefficient

**e**Unit normal vector

*F*Nonlinear fracture function

*F*_{1,2}Fracture set

*g*Gravity acceleration

*H*Height of the reservoir

**I**Unit tensor

*k*Permeability

*k*_{f}Effective fracture permeability

*k*_{r}Relative permeability

*l*Local scale (lateral matrix column size)

*L*Global scale/length of the reservoir

*M*Nonlinear matrix function

**n**Unit normal vector

*N*_{f}Number of fracture grid cells

*N*_{G}Gravity number

*N*_{m}Number of matrix grid cells

*N*_{zf}Number of fracture grid cells in the vertical direction

**p**Vector of the fracture-cell center

*P*Pressure

*p*_{c}Capillary pressure

**p**′Vector of the fracture-cell center on a horizontal cross-section

*Pe*Peclet number

- PV
Pore volume

**q**Any parameter

*Q*3D domain

*q*_{ext}External (injection/production) rates

*q*_{w}Water injection rate

**r**Coordinate vector

*R*Real

*S*Saturation

*S*_{or}Residual oil saturation

*S*_{wc}Connate water saturation

- SU
Small unit

*t*Time

*u*Velocity

**u**Velocity vector

*W*Width of the reservoir

*x**x*-coordinate**x**_{b}Global coordinate

**x**_{p}Center of a grid cell

**x**_{s}Local coordinate

*x*′Horizontal cross-section position

*y**y*-coordinate*z*Vertical upward direction

*Z*1D domain

## Greek

*α*Phase (oil/water)

*ε*Scaling ratio

*λ*Mobility

*μ*Viscosity

*ξ*Potential/saturation indicator

*ρ*Density

*σ*Coordinates of the boundary

*φ*Porosity

- Φ
Potential

- Ω
Horizontal cross-section domain

*ω*Auxiliary function

## Math Signs and Operator

- 〈 〉
Average sign over volume

- ||
Absolute value of volume

- ≈
Almost equal to

- ⊗
Dyadic product

- \({\surd}\)
Square root

- \({\smallint}\)
Integral

- →
Vector sign

- d
Differential

- ∂
_{π} Partial differential with respect to

*π*- ∂
Boundary of

- ∇
Del (gradient operator)

- Δ
Delta (difference operator)

- ∇
Divergence operator

## Subscripts

- b
Global (big) index

- D
Dimensionless

- f
Fracture

- m
Matrix

- o
Oil phase

- r
Relative

- R
Reference

- s
Local (small) index

- w
Water phase

*z**z*-coordinate (vertical direction)*α*Oil/water index

*ζ*Fracture/matrix index

## Superscripts

^{*}Local fracture index

- (0)
Zeroth-order index

- (1)
First-order index

- (2)
Second-order index

## Notes

### Acknowledgments

We thank Statoil-Hydro for supporting our research on oil recovery from fractured reservoirs. We also thank Sharif University of Technology for their steady collaboration. We acknowledge Maryam Namdar Zanganeh, William R. Rossen, Stefan M. Luthi, and Giovanni Bertotti for many useful discussions and comments. We also acknowledge SPE for granting the Nico van Wingen Memorial Graduate Fellowship in Petroleum Engineering to Hamidreza Salimi.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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