# Solution for counter-current imbibition of 1D immiscible two-phase flow in tight oil reservoir

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## Abstract

Spontaneous imbibition is an important mechanism for fractured reservoir to enhance oil recovery. Wetting phase enters porous media with the force of capillary pressure and gravity and replaces oil in matrix. To investigate the imbibition of tight reservoirs, on the consideration of tight formation characteristics, this paper derived a one dimension, two phases, counter-current imbibition model, after dimensionless of distance and time, Galerkin method for spatial discretization and time integration, solutions were given, comparisons of conventional sandstone and tight formation were made. The results have indicated that: (1) Imbibition can be divided into gravity assisting, gravity opposing and zero gravity in terms of different gravity conditions. (2) Saturation front of tight formation moves faster than sandstone because of high capillary pressure. (3) Capillary pressure plays the dominant role than gravity in imbibition. Influence of gravity is much greater in high-permeability sandstone than in tight reservoirs. (4) Horizontal well multi-stage fracturing and massive fracturing can increase fracture area and fracture volume, and increase the contact area with wetting phases, this will result in a greater imbibition and a great recovery of oil.

### Keywords

Two-phase flow Counter-current imbibition Capillary pressure Gravity Tight oil reservoir Hydraulic fracturing## Introduction

Imbibition can be defined as the inflow of wetting phase and the displacement of non-wetting phase in the porous media under the forces of capillary pressure, gravity and buoyancy force. Imbibition can be divided into co-current imbibition and counter-current imbibition according to the flow direction (Mattax and Kyte 1962). In co-current imbibition, wetting phase pushes non-wetting phase out of the matrix in the same direction. While in the counter-current imbibition, wetting phase imbibes into matrix, displacing the non-wetting phase in the opposite direction. Co-current imbibition is much faster and more efficiency than the counter-imbibition. However, as a matter of fact, the counter-current imbibition is the main recovery mechanism because only one face of matrix can contact non-wetting phase in most cases.

Experimental and modeling method for imbibition has been studied by many researchers. Ryzhik (1960) derived a 1D self-similar solution for counter-current model by assuming the linear function of capillary pressure and relative permeability with water saturation. Yortsos et al. (1993) obtained an analytical solution by the assumption of some constrained relationship of capillary pressure, relative permeability and water saturation. Reis and Cil (1993) got the approximate solution for 1D counter-current imbibition considering capillary pressure. Kashchiev and Firoozabadi (2003) and Behbahani et al. (2006) solved a counter-current imbibition model of water wetting and fractured reservoirs, and influencing factors of gravity, capillary pressure and viscous resistance . Some scholars also adopted the method of pore scale network (PSN) method (Blunt 2001; Valvatne and Blunt 2004), Lattice Boltzmann (LBM) method (Porter et al. 2009; Galindo-Torres et al. 2013) and computational fluid dynamics (CFD) method (Hammecker et al. 1993; Standnes 2006) to study the mechanism of co-current and counter-current imbibition.

However, some of the above models ignore the influence of gravity (Behbahani et al. 2006; Blunt 2001); some dealing capillary pressure and relative permeability too simple (*J*(*S*) = ln *S*) (Kashchiev and Firoozabadi 2003; Behbahani et al. 2006). Most of all, they are models of fractured reservoir or high-permeability sandstone reservoir, which may make a great difference from tight oil reservoir.

In this paper, we derived a general formula of 1D two immiscible phase flow considering gravity, capillary pressure and buoyancy force considering characteristics of tight oil reservoir. Capillary pressure and relative permeability of tight oil were obtained by matching with experiment data and were substituted into the general formula. After dimensionless, spatial dispersion by Galerkin method and time difference, we solved the high nonlinear partial differential equation. Solutions for three different boundary conditions and gravity conditions were given.

## Derivation of the model

### Hypothesis

(1) Two-phase flow (wetting and non-wetting); (2) one-dimensional flow; (3) inflow velocity of wetting phase equals to the outflow velocity of non-wetting phase, and in opposite direction; (4) no reaction between different fluids, means immiscible flow; (5) fluid is incompressible; (6) isothermal flow and (7) considering effects of gravity. Three different models are shown as follows.

### Derivation

*v*

_{w}is the seepage velocity of wetting phase,

*v*

_{nw}is the seepage velocity of non-wetting phase,

*K*is absolute permeability,

*μ*

_{w}is viscosity of wetting phase,

*μ*

_{nw}is viscosity of non-wetting phase,

*k*

_{rw}is wetting phase relative permeability,

*k*

_{rnw}is non-wetting phase relative permeability,

*ρ*

_{w}is wetting phase density, and

*ρ*

_{nw}is non-wetting phase density.

*φ*is matrix porosity,

*S*

_{w}is water saturation.

This is the general formula of counter-current imbibition of two immiscible phases (considering gravity).

*S*

_{iw}is irreducible water saturation,

*S*

_{nwr}is resistant saturation of non-wetting phase,

*H*is the height of porous media.

### Simplification of the formula

We can find that formula (9) is a high nonlinear partial differential equation. Since non-wetting phase relative permeability (*k* _{ro}), wetting phase relative permeability (*k* _{rw}) and capillary pressure (*P* _{c}) are discontinuous functions of water saturation (*S* _{w}), PDE formula (9) can not be solved directly. Dealing with relative permeability and capillary pressure can be done as follows.

#### Relative permeability

*k*

_{rw}

^{0}is water relative permeability at residual oil saturation,

*k*

_{rnw}

^{0}is oil relative permeability at irreducible water saturation,

*S*

_{iw}is irreducible water saturation,

*S*

_{rnw}is residual oil saturation,

*S*is normalized water saturation, and

*m*and

*n*is relative permeability index for oil phase and water phase which depends on formation rock’s pore scale structure and wettability (Fig. 2).

Parameters used in the model

K (×10 | | | | | | |
---|---|---|---|---|---|---|

0.2 | 0.7 | 2 | 20 | 0.3 | 0.25 | 3 |

| | | | | | |
---|---|---|---|---|---|---|

0.2 | 1 | 0.8 | 1.0 | 0.8 | 1 | 1.5 |

#### Capillary pressure

*J*(

*S*) function to describe two-phase flow.

*J*(

*S*) as ln

*S.*This kind of simplification does not correspond to tight oil formation. In this paper, we match the capillary pressure with experimental result by an exponentially fitted method as shown in Fig. 3. This capillary pressure curve was substituted into formula (9). From Fig. 3, we can find that capillary pressure of high-permeability sandstone is much smaller than tight oil. This is a result of the nano-scale pores and throats distributed in tight formation.

## Solution for the model

*Z*and dimensionless time

*T*to make this PDE to be dimensionless.

*z*is imbibition height, m;

*H*is the total model height, m;

*Z*is the dimensionless height.

*S*and

*T*as formula (21–23), respectively.

*S*) and dimensionless time (

*T*), dimensionless height (

*Z*).

NB^{−1} derived in this paper is the same as Schechter et al. (1991) result. It can be described as the ratio of capillary pressure and gravity, a parameter to describe the contribution of capillary pressure and gravity to imbibition. Schechter has a conclusion that capillary pressure is dominated when NB^{−1} is larger than 5, while gravity is dominated when NB^{−1} is less than 0.2.

## Model verification and discussion

*x*= 0. Grid 2 ~100 is non-wetting phase to represent the water saturation at the position of

*x*> 0. Relative permeability and capillary pressure are the same as our model. Some other parameters are shown in Table 1.

*model b*) in high-permeability sandstone and tight formation. We can find that: (1) Saturation front of tight oil reservoir moves faster than conventional high-permeability sandstone because of high capillary pressure, which indicates a greater imbibition of tight formation. (2) Gravity shows an obvious influence in conventional high-permeability sandstone, which shows little influence in tight reservoir. (3) Influence of gravity is not obvious at the beginning, which makes a highlight as time goes on (1 h, 20 days, 80 days and 400 days).

*model c*) in tight formation and high-permeability sandstone. Except above conclusions, we can also find that: (1) Influence of capillary pressure is much greater than gravity both for gravity assisting imbibition and gravity opposing imbibition. (2) Saturation front moves faster in gravity assisting imbibition than in gravity opposing imbibition in both kinds of rocks.

In 2012, after 6 months’ shut-in after hydraulic fracturing, a shale gas well in Marcellus (Cheng 2012) extracted a great amount of gas while a little amount of water. This has inspired researcher to investigate the imbibition mechanism during shut-in periods. We have find that, (1) Imbibition distance of conventional high-permeability sandstone is about 2, 16 and 34 cm for 1, 20 and 80 days. Oil recovery via imbibition in this kind of formation is little because of short imbibition distance. (2) While for tight formation, imbibition distance is about 10, 54 and 106 cm in 1, 20 and 80 days. This is a much longer distance and will result in a great oil recovery.

Horizontal well multi-stage fracturing and massive fracturing method not only increase fracture area and fracture volume but also increase the contact area between formation and the wetting fluids injected. These injected chemical fluids can result in a change of interfacial tension and wettability, utilization of imbibition mechanism caused by them will result in a good EOR performance.

## Conclusion

- 1.
Considering characteristics of tight formation, a one-dimensional two-phase counter-current imbibition model for tight formation was derived; after dimensionless, the partial differential equation was solved by Galerkin spatial dispersion and temporal difference. Solution for three different boundary conditions and gravity conditions were given.

- 2.
Imbibition can be divided into gravity assisting, gravity opposing and zero gravity in terms of different gravity conditions. Imbibition of tight formation is much greater than sandstone because of the high capillary pressure.

- 3.
Capillary pressure plays the dominant role in imbibition. Influence of gravity is much greater in high-permeability sandstone than in tight formation.

- 4.
Horizontal well multi-stage fracturing and massive fracturing can increase fracture area and fracture volume and increase the contact area with wetting phase, which may result in a greater oil recovery with the utilization of imbibition mechanism. Imbibition distance may be a reference for engineers to design fracture spacing in horizontal well’s hydraulic fracturing.

## Notes

### Acknowledgments

This work is supported by the National Science and Technology Major Project (No. 2016ZX05023). The authors would express their appreciation to the project for the contribution of research fund. We also thank Mr. XX for the assistance of language issues.

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