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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

IMPACT FACTOR 2018: 1.005

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Volume 15, Issue 1

# Coupling of two-phase flow in fractured-vuggy reservoir with filling medium

Haojun Xie
• Center of Multiphase Flow in Porous Media, China University of Petroleum (East China), Qingdao, 266580, China
• Other articles by this author:
/ Aifen Li
• Center of Multiphase Flow in Porous Media, China University of Petroleum (East China), Qingdao, 266580, China
• Other articles by this author:
/ Zhaoqin Huang
• Corresponding author
• Center of Multiphase Flow in Porous Media, China University of Petroleum (East China), Qingdao, 266580, China
• Email
• Other articles by this author:
/ Bo Gao
/ Ruigang Peng
• Center of Multiphase Flow in Porous Media, China University of Petroleum (East China), Qingdao, 266580, China
• Other articles by this author:
Published Online: 2017-03-02 | DOI: https://doi.org/10.1515/phys-2017-0002

## Abstract

Caves in fractured-vuggy reservoir usually contain lots of filling medium, so the two-phase flow in formations is the coupling of free flow and porous flow, and that usually leads to low oil recovery. Considering geological interpretation results, the physical filled cave models with different filling mediums are designed. Through physical experiment, the displacement mechanism between un-filled areas and the filling medium was studied. Based on the experiment model, we built a mathematical model of laminar two-phase coupling flow considering wettability of the porous media. The free fluid region was modeled using the Navier-Stokes and Cahn-Hilliard equations, and the two-phase flow in porous media used Darcy's theory. Extended BJS conditions were also applied at the coupling interface. The numerical simulation matched the experiment very well, so this numerical model can be used for two-phase flow in fracture-vuggy reservoir. In the simulations, fluid flow between inlet and outlet is free flow, so the pressure difference was relatively low compared with capillary pressure. In the process of water injection, the capillary resistance on the surface of oil-wet filling medium may hinder the oil-water gravity differentiation, leading to no fluid exchange on coupling interface and remaining oil in the filling medium. But for the water-wet filling medium, capillary force on the surface will coordinate with gravity. So it will lead to water imbibition and fluid exchange on the interface, high oil recovery will finally be reached at last.

PACS: 47.11.-j; 47.54.De; 47.55.-t; 47.61.Jd

## 1 Introduction

For fractured-vuggy carbonate reservoirs in Tahe Oilfield of China, solution caves and fractures are main spaces for fluid storage and flow. But the huge caves are normally filled with collapsing rocks, carried sands or chemical sedimentation, which is shown in Fig 1. Different kinds of filling mediums have different wettability and permeability [1, 2]. The oil-water flow in reservoir is the coupling flow of free flow in un-filled area and porous flow in filling medium. Most wells can only open the top of the cave because of heavy loss of drilling fluid, so we usually use water injection to displace the oil by gravity in the caves. In recent years, many physical and numerical simulations have been conducted for the flow mechanism and remaining oil distribution in fractured-vuggy media.

Figure 1

Field outcrop of filled cave and physical model

For the experiment, P Egermann (2007) used the tight matrix with large vugs to evaluate the matrix-imbibition phenomenon complex water-gas-flow interactions between vugs and micro pores [3]. Kang (2006) and Wang (2011) used glass etched microscopic model to stimulate the water injection process in fracture and vugs in low pressure [4, 5]. Li (2010) built a large size water tank with PVC caves to simulate the fracture-vuggy reservoirs, and considering the filling medium using glass balls without cemented in the cave [6]. After that, Li and Zhang (2012) conducted the elastic recovery experiment in insular cavity using high-pressure container [7]. Although these experiments considering the filling medium, but they use only glass balls or carbonate rocks. Since the parameters of the filling medium that were used were unknown, numerical simulations were not conducted for these experiment results.

Coupled free fluid flow and porous flow systems arise in a wide range of environmental and industrial applications, such as evaporation from soil influenced by atmosphere flow [8], multi-phase fluid flow through fractured-vuggy carbonate reservoir [9, 10] and etc. The key challenge in such flow and transport processes is the coexistence of free flow and porous flow [11-13]. In recent years, two-domain approach is usually used for coupling of free fluid flow and the porous flow. Jackson developed a transition region model that is capable of coupling two-phase and multi-component systems, but the solution is non-trivial and still an open issue [14]. Chen developed a numerical model for the coupling two-phase free flow and porous flow [15]. Their model consists of coupled Cahn-Hilliard and Navier-Stokes (CHNS) equations in the free fluid region and the classical two-phase Darcy equations in the porous medium. Han recently proposed a coupled Cahn-Hilliard and Stokes-Darcy (CHSD) two-phase flow model, which consists the Cahn-Hilliard-Darcy equations system in porous medium and the Cahn-Hilliard-Stokes equations system in the free flow region [16]. More recently, Huang built developed a CHNS model with extended BJS conditions. And the experiment proved it more suitable to capture the macroscopic flow characteristics [17, 18].

In this paper, we built physical and numerical models for filled caves considering the wettability of the filling medium. Experiments verified the accuracy of the numerical models. Based on the results of the experiments and numerical simulations, the influences of wettability on the coupling flow and remaining oil distribution were analyzed. The results indicated that the capillary force determines the fluid exchange on interface and remaining oil in the filling medium.

## 2.1 Physical Model

Visual physical models were built to study the oil-water exchanging mechanism simplified from the field outcrop (Fig. 1). The model used PMMA (polymethyl methacrylate) as transparent boundary to simulate the cave. The glass beads were cemented by epoxy in bottom of the model. to simulate filling medium area in the cave, and the empty part at the top is the un-filled area, which is shown in Fig. 2. Size of the model is 10 cm × 10 cm × 1 cm; filling degree in cave is about 50% percent; the particle size of filling medium is 0.6 mm; porosity of the filling medium is about 35%; permeability is about 1 × 10-10 m2. Inlet and outlet were drilled at two sides on the top of model. The experiment process is shown in Figure 1c.

Figure 2

Fluid saturation at different times in experiment of oil-wet model

## 2.2 Material

In this experiment, saline water (salinity 5000 mg/L, relative density 1.0) and kerosene (relative density 0.8) was used as formation water and oil. And the kerosene was colored red by Sudan III. The interfacial tensions between the saline water and the oil is about σwo = 25 mN/m. The wettability of the pore surface in filling medium was controlled by the epoxy [19].

## 2.3 Experiment procedures

(1) Fully saturate the model with oil, and measure the oil volume in the model by weighing method. Then connect the model to the inlet and outlet pipes, and hold the model vertical as shown in Figure 1.

(2) Inject water into the model with a low flowrate of 0.5 ml/min and produce liquid at atmospheric pressure. Keep water injection for 48 hours and observe the water displacing oil under gravity.

(3) Record fluid flow dynamic and liquid production, then clean and dry the model.

## 3.1 Equations in free flow region

Two-phase flow in free flow region was described by phase filed method, including Cahn-Hilliard and N-S equations, given by

(1) Cahn-Hilliard equation $∂ϕ∂t+uF⋅∇ϕ=∇⋅γβϵ2∇ΨΨ=−∇⋅(ϵ2∇ϕ)+ϕ(ϕ2−1)$(1)

φ is a dimensionless phase field variable, which is equal to 1 in oil and -1 in water; uF is the free flow velocity, [m/s] ; ε is the thickness of interface, [m]; γ is the mobility; β is the magnitude of the mixing energy, [N]; ψ is an auxiliary phase field variable.

(2) N-S equations $∇⋅uF=0$(2) $ρ∂uF∂t+(uF⋅∇)uF=∇⋅(−pFI+τ)+ρg+G∇φ$(3)

where ρ is the fluid density, [kg/m3]; p is the fluid pressure, [Pa]; I is d×d identity tensor, d is the dimensionality; τ = μ[∇uF + (∇uF )T] denotes the viscous part of the stress tensor, [Pa]; μ is the fluid viscosity, [Pa·s]; g is the gravity acceleration vector, [m/s2]; and G is the chemical potential, [N/m2].

## 3.2 Equations in porous flow region

In porous flow region, the mass balance equation is: $φ∂ρlSl∂t+∇⋅ρluPl=ρlql,l=w,o$(4)

where Φ is the porosity of the porous domain; ρl is the l -phase fluid density; Sl is the l -phase fluid saturation; uPl is the l -phase fluid seepage velocity; and ql is the source/sink term of l -phase fluid, [1/s]. The uPl s defined by the Darcy's law: $uPl=−Kpkdμl(∇pPl−ρlg),l=w,o$

where KP is the absolute permeability, [m2]; krl is the relative permeability of l -phase fluid, and 0≤ krl(Sw) ≤1; pl is the l -phase fluid pressure; μl is the 1-phase fluid viscosity; g is the gravity constant; The constraint for the saturation and pressures relationship are: $∑lSl=Sw+So=1;pPcSw=pPo−pPw$(5)

where ppc is capillary pressure, [Pa] ; pp0 and pp0 is the pressure of water and oil, [Pa].

## 3.3 Interface conditions for coupling

Continuity of normal flux is given by: $uF⋅nPF=uPw+uPo⋅nPF=uP⋅nPF$(6)

Continuity of normal stress is: $nPF⋅−pFI+τnPF=−1+φ2pPo+1−φ2pPw=−pPw+1+φ2pPc$(7)

The tangential jump condition used the extended two-phase BJS condition [16] which was developed by: $tPF⋅(−p+FI+τ)nPF=−ημKptPF⋅uFb$(8)

## 3.4 Initial conditions

Initial conditions for two domains are: $pFx,t=0=pF0xuFx,t=0=uF0xφx,t=0=φ0x,inΩF$(9) $pPlx,t=0=pPl0xSlx,t=0=Sl0x,inΩP$(10)

The CHNS equations are implemented by using the laminar two-phase flow (phase field) model and the two-phase Darcy equations are modeled by using coefficient form PDE model in COMSOL Multiphysics V5.0. The coupled flow system is solved using the direct solver MUMPS (multi-frontal massively parallel sparse direct solver).

## 4.1 Comparison between experiments and numerical simulations

(1) Results of oil-wet model

Fluid saturations in experiments and numerical simulations at three different times were given in Figure 3 and Figure 4.

Figure 3

Fluid saturation at different times in numrical simulation of oil-wet model

Figure 4

Fluid saturation at different times in experiment of water-wet model

In the experiment results, oil-phase is red and water-phase is white. While the color code represents relative density in numerical simulation results. Considering the micro roughness of porous media in experiment, the water phase movement at beginning in Figure 2a and Figure 3a was a little different, and it was reasonable. The oil recovery of physical experiment and numerical simulation is 72.8% and 75.4%, so the experiment results have a good agreement with the numerical simulations comparing Figure 3 and 4. At the beginning, the water will first drop on the surface of the porous media, and then the water phase will cover the surface. With injection continued, the oil-water interface will slowly raise to the top of the model. Capillary force in porous media will block the gravity differentiation, so the water will not flow across the interface and get into the porous media in the whole process.

(2) Results of water-wet model

For the water-wet model, the oil recovery of physical experiment and numerical simulation is 94.3% and 95.8%, so the experiment results still have a good agreement with the numerical simulations comparing Figure 3 and 4. The injected water will first drop on the surface of the porous media, and then the water phase will cover the surface. The capillary force has the same direction with gravity for the water-wet model. Under the cooperating effect of capillary force and gravity, the water will flow across the coupling interface with water injection continued. And the oil in the porous media will flow upward as a droplet, which can be seen on the interface in both Figure 4b and Figure 5b. So the oil in porous media will be displaced at last.

Figure 5

Fluid saturation at different times in numrical simulation of water-wet model

## 4.2 Fluid exchange at interface

Fluid velocity distributions at 30 min were calculated in numerical simulations to analyze the fluid exchange at coupling interface, which is shown in Figure 6.

Figure 6

Fluid velocity distribute at 30 min

Compared with free flow, the viscous resistance of Darcy flow is very high, so the velocity in porous media was really low in both two models and even can't be reflected in Figure 6. But the velocity field above the interface is different for two models. In oil-wet model, there is nearly no fluid exchange at the interface, so the velocity above the interface is very low and the direction is mostly horizontal. The oil-water interface increases slowly to the top of the model, and the oil flow to the outlet due to the driven pressure and gravity. The upper part of water phase flow to the outlet due to driven pressure, but it will flow downward at the right boundary because of gravitational differentiation, which lead to vorticity in Figure 6a.

While for the water-wet model, there is obvious fluid exchange at the interface. So the velocity above the interface in higher than oil-wet model, and its direction are somehow vertical. Water flows across the interface made the vorticity more complex than oil-wet model. Mean-while, the oil drop generated on the interface flows upward to the top of the model, and it also lead to vorticity in oil phase. The injected water and oil flows across the interface are discrete drops, the velocity field and vorticity are not very stable and ideal.

## 5 Conclusion

(1) The validation of the coupling model is verified by comparing numerical simulations with experimental results. So this method is suggested in the future works.

(2) Wettability of filling medium determines the remaining oil and oil recovery in filling medium. For the coupling flow, capillary pressure in porous media are relatively high compared with the driving pressure difference of free flow. Injected water can't flow into the oil-wet porous media and there is nearly no fluid exchange at the interface. But capillary force cooperated with gravity will lead to water imbibition in water-wet filling medium and obvious fluid exchange on interface. Thus, high oil recovery will be reached.

(3) Capillary force in the main reason for low oil recovery in fractured-vuggy reservoirs with oil-wet filling medium. So surfactant injection is an ideal method to decrease the capillary force and increase oil recovery, which needs a further experimental and numerical study on it.

## Acknowledgement

National Science and Technology Major Project of China (2016ZX05014-003-002 and 2016ZX05009-001-007) supported this work.

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Accepted: 2016-12-13

Published Online: 2017-03-02

Citation Information: Open Physics, Volume 15, Issue 1, Pages 12–17, ISSN (Online) 2391-5471,

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