(Institute of Petroleum Exploration and Development, China Petrochemical Corporation, Beijing 100083)

Abstract: Fracture-cave carbonate reservoir has the characteristics of large scale of reservoir space change, complex medium and various fluid flow modes, which makes the mature numerical simulation theory and technology of sandstone reservoir unable to be applied. Therefore, numerical simulation of fractured-vuggy reservoirs has become a difficult and hot spot in the world at present, which restricts the rational and efficient development of such reservoirs. Therefore, on the scale of fractured-vuggy reservoirs, according to the thought framework of continuum, the dual media is developed, and the equivalent multi-media theory is formed, that is, the multiphase flow problem in fractured-vuggy reservoirs is equivalent to the multiphase flow problem in multiple continuum media, a triple-media continuum model including caves, fractures and dissolved pores is established, the theory of representation unit is studied, and the establishment criteria of the model are put forward. At the same time, in view of the problem that fluid flow in large caves in fractured-vuggy reservoirs needs to be described in detail, a coupled numerical simulation technology is proposed. It mainly includes establishing mathematical models for numerical simulation of porous media area, karst cave area and their interfaces in fractured-vuggy reservoirs, realizing the coupling of Navier-Stokes flow in karst cave and Darcy flow in matrix, solving the problem of oil-water two-phase interface treatment, and forming the interface conditions between karst cave and porous media area. Then, the equivalent multi-media model and numerical algorithm of coupled numerical simulation are studied respectively. Finally, according to the numerical simulation technology of fractured-vuggy reservoir, a three-dimensional three-phase fluid numerical simulator is compiled, and the water flooding process is simulated through physical simulation experiments and numerical simulation experiments. The consistency of the results verifies the correctness of this method.

Key words: fractured-vuggy reservoir; Numerical simulation; Multimedia; Fluid-seepage coupling

Study on numerical simulation technology of fractured-vuggy carbonate reservoir

Zhang Yun Kangzhijiang

(China Petrochemical Exploration and Development Research Institute, Beijing 100083)

Abstract: Fracture-cave carbonate reservoir has the characteristics of different reservoir space scale, complex medium and various fluid flow modes. Mature sandstone reservoir simulation theory and technology cannot be used. Therefore, the numerical simulation method of natural fractured-vuggy carbonate reservoirs is a worldwide difficulty and focus, which restricts the efficient development of such reservoirs. Then, according to the characteristics of fractured-vuggy reservoirs, the equivalent multi-media numerical simulation technology based on dual-media theory is formed. In other words, the multiphase flow problem is equivalent to the multiphase flow problem of continuous media in multiple fractured-vuggy reservoirs. The continuum medium of the triple medium model, including caves and fractures, is established. Then, the representative primary school papers are studied and the model rules are put forward. In order to better describe the fluid flow in large caves in fractured-vuggy reservoirs, a coupled numerical simulation technique is proposed. A mathematical model including porous media area, karst cave area and their interfaces is established, which realizes the coupling of Navier-Stokes flow in karst cave and Darcy flow in matrix, solves the problem of oil-water two-phase interface, and gives the interface conditions for forming karst cave and porous area. Then the numerical algorithms of numerical simulation of multi-media model and coupled model are studied. Finally, the numerical simulator of fractured-vuggy reservoir is developed. The physical simulation and numerical simulation of water injection simulation process are completed. The correctness of the numerical simulation method is verified by it.

Key words: fractured-vuggy reservoir; Numerical simulation; Multimedia; Coupled flow

introduce

More than half of the discovered oil and gas reserves in the world come from carbonate reservoirs [1]. As a special type, fractured-vuggy carbonate reservoirs also account for a large proportion of oil and gas resources in China and even the world. Fracture-cave carbonate reservoir is an unconventional reservoir type with large reserves, which can form large reservoirs and is also an important part of carbonate reservoir production in the world.

In recent ten years, the research object is the numerical simulation of clastic reservoir, and its related theoretical and technical research is based on porous media theory, which has made great development and formed industrial technical application. However, for fractured-vuggy carbonate reservoirs with large spatial variation scale and complex media, the existing theories and technical methods are not applicable in many aspects. Therefore, numerical simulation of fractured-vuggy carbonate reservoirs is carried out, including equivalent multi-media numerical simulation technology [2 ~ 16] and coupled numerical simulation technology [17 ~ 20].

1 Establishment of mathematical model

1. 1 model construction standard

Multimedia theory is essentially a continuum theory, and the premise of continuum theory is that it represents the existence of elements. At present, there have been many achievements in the study of single-medium characterization unit [2 1], but there are few studies on the theory of multi-medium characterization unit at home and abroad. This is because the spatial scale of different pore types in complex media is very different, and the flow patterns of multiphase fluids in pores are also diverse, so the characterization units of complex media often do not exist in the scale range we study. In order to solve this problem, we put forward the concept of multiple representation units of complex media.

For complex medium reservoirs, let ω k (x0) be the volume of complex medium region, x0 be the center of mass of the volume ω k (x0), and e be the extension of complex medium (mass, void space, fluid mass passing through per unit time, etc.). ), e is the connotation quantity corresponding to the epitaxial quantity (density, porosity, mass flow, etc.). ). e(ωK(x0)) represents the extension quantity in the volume ωK(x0), eK(x) represents the connotation quantity of point X, and m, f and v are matrix, fracture and karst cave. If:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Then, the denotation unit of the denotation quantity eK corresponding to the connotation quantity EK exists, and the continuous medium method and the multimedia method can be used. Otherwise, it will be treated separately, that is, the discrete method (coupling method) will be adopted. Equations (1) and (2) are the criteria for establishing multimedia models of complex media.

1.2 governing equations in porous media

Karst cave-fractured reservoir is considered to be isothermal and contains oil-water two-phase fluid. The fluid flow equation in the complex medium region is:

Water phase:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Oil phase:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Among them, when the B-phase fluid (W is water; O is oil) is Darcy flow, and its velocity is defined as follows according to Darcy theorem:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Where ρ β is the density of β phase under reservoir conditions; Is to remove the oil phase density of dissolved gas under the condition of oil reservoir; φ is the effective porosity of oil layer; μ β is the viscosity of β phase; Sβ is the saturation of β phase; Pβ is the pressure of β phase; Qβ is the sink/source term of β component per unit volume of stratum; G is the acceleration of gravity; K is the absolute permeability of oil layer; Krβ is the relative permeability of β phase; D is depth.

1.3 governing equations in caves

Navier-Stokes equation is adopted as the governing equation of the free flow region in the tunnel. In the free-flowing area, oil and water are immiscible, forming two fluids. There is an obvious oil-water interface in the hole, which can be clearly shown. The control equations include oil zone control equation, water zone control equation and oil-water interface motion equation.

Firstly, the mass conservation equation and momentum conservation equation are given for the region where oil and water exist.

Oil phase equation:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Water phase equation:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Where fσ stands for surface tension.

This is a two-phase flow equation of slightly compressible fluid.

As a special case, it is assumed that oil and water are incompressible and have the same density. At this point, by eliminating the density constant in the above equation, we can get:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

And (8) and (9) are two sets of standard N-S equations of motion. The difference is that the properties of two-phase fluids, such as density and viscosity, are different at the interface. And the equation of motion of the interface.

Let's consider the equation of motion of the oil-water interface. There are two forms of interface, and different algorithms can choose different forms.

The (1) interface is described by a point set,

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

In this case, the point on the interface moves at the fluid speed according to the following law.

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

(2) The point on the interface is determined by the equation F(x, t) = 0.

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

At this time, F(x, t) is satisfied.

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Where u represent that velocity of fluid movement.

Physical quantities such as pressure and velocity are continuous at the oil-water interface. However, density, viscosity and other physical quantities representing fluid characteristics are different.

1.4 Interface conditions between caves and porous media areas

Interface conditions include concentration continuity, pressure balance and flow balance. Considering the actual situation of the reservoir, the interface conditions can be simplified. Because the pressure of porous medium pd and the pressure of cave ps are very large. Relative to the formation pressure, the velocity and viscosity are very small and can be ignored. At the same time, it can be assumed that there is no slip between the cave flow region and the porous medium region in the tangential direction of the boundary. Under this assumption, the interface condition can be expressed as

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

This set of conditions actually represents the continuity of concentration, pressure and velocity on the interface.

In the actual calculation process, the interface condition (12) is generally easy to use, especially when discretized by finite difference and finite volume. However, it can be directly applied in the process of solving and deducing weak forms by finite element method.

2 numerical algorithm research

2. 1 equivalent multi-media model numerical simulation technology

Reservoir model is considered as isothermal condition, including oil, gas and water. Water and oil, two liquid components, exist in water phase and oil phase respectively, while gas not only exists in gas phase, but also can be dissolved in oil. Each phase fluid flows according to Darcy's theorem under the action of pressure, gravity and capillary force; The flow in and between caves is non-Darcy flow or pipe flow.

After spatial discretization by finite volume method and time discretization by backward first-order difference, the internal equation of element I after discretization can be obtained as follows:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Where m is the mass of the beta phase; Superscript n represents the quantity at the previous moment; The superscript n+ 1 indicates the quantity at the current moment; Vi is the volume of unit I (matrix, crack or cave); △t is the time step; ηi is the set of units j connected with unit i; Fβ and ij are the mass flow terms of β phase between I unit and J unit; Qβi is the source term and sink term of β phase in unit I. The flow terms f β, ij between multimedia units I and J can be expressed as:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Where λ β, ij+ 1/2 is the fluidity of β phase,

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

In order to describe various flow patterns in complex media, in the multi-media model of complex media, the flow between cells is divided into seepage (Darcy flow, non-Darcy low-speed flow or non-Darcy high-speed flow), one-dimensional pipe flow, two-dimensional flow on fracture surface (non-Darcy high-speed flow) and three-dimensional "cave flow" in unfilled caves.

2.2 Coupled numerical simulation technology

Aiming at the problem of fluid flow in large caves of fractured-vuggy reservoirs, based on the theory of Navies-Stokes equation, a mathematical model of oil-water two-phase immiscible slightly compressible flow is established, and the numerical simulation technology of Navies-Stokes flow and seepage coupling in complex media reservoirs is realized. The establishment of its numerical model includes two steps: discretization of the solution region and discretization of the equation. The discretization of the solution region produces a numerical description of the solution region, including the location and boundary description of the solution point. Space is divided into finite discrete areas, called control volume or volume grid. For transient problems, the time interval is also divided into finite time steps. The discretization of the equation transforms the terms of the control equation into algebraic expressions.

For any physical quantity φ, its transfer equation can be written as:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

The finite volume method requires the integral form of the control equation in the control volume VP based on P:

Proceedings of the International Conference on Unconventional Oil and Gas Exploration and Development (Qingdao)

Because the diffusion term is the second derivative of φ, in order to ensure consistency, the discrete order in a finite volume must be equal to or greater than the order of the equation.

The accuracy of the discrete method depends on the assumed variation function φ = φ (x, t) of the space-time position near point P.

To obtain the discrete form of the transfer equation, the key lies in the values of the interface F and its vertical gradient, namely φf and S (▽φ) f, and the values of the surface located on the boundary of the region are calculated by the boundary conditions.

3 Verification of reservoir numerical simulation method

According to the research results of reservoir numerical simulation method, the corresponding numerical simulation software is compiled. In order to verify the correctness of the numerical simulation method, the physical experiment of water flooding was carried out. In the experiment, the filler is 5mm white marble in the right half and 3mm white marble in the left half, and the water injection rate is 0.9L/min. The model is full of oil, and the oil shifts from left to right. The experimental results are shown in figure 1. The same parameters are used for numerical simulation, and the results are shown in Figure 2. By comparison, it can be concluded that the numerical simulation experiment is consistent with the physical experiment, thus verifying the correctness of the method.

Figure 1 Physical experiment of water flooding phenomenon

Fig. 2 Oil saturation diagram of reservoir numerical simulation experiment

4 conclusion

(1) dual media theory is widely used in fractured reservoirs, which solves the problem of large fluid flow difference between matrix and fracture. Experts have carried out triple medium numerical simulation research on small karst caves, but there is no relevant report on karst cave-type fractured-vuggy reservoirs. Through research, a set of adaptive implicit numerical simulation method based on multi-media fractured-vuggy reservoir is formed, which can deal with karst caves.

(2) Two key problems, the interface calculation of two-phase flow in the tunnel and the coupling calculation of porous media in the tunnel, are studied. Combined with the physical experiment of oil-water two-phase flow in cave, the technical problem of oil-water two-phase interface is solved, the numerical simulation method of coupled reservoir is formed and the numerical solution is determined.

(3) According to the numerical simulation method, the corresponding numerical simulation software is compiled, and the correctness of the method is verified by experiments.

refer to

Li Ke, Liu Yun, Liu Ming. Study on calculation method of fractured-vuggy carbonate reservoir reserves [J]. Petroleum Drilling and Production Technology, 2007,29 (2):103 ~104.

H. Kazemi, L.S. Merrill Jr., K.L. porterfield. Numerical simulation of water-oil flow in naturally fractured reservoirs. Reservoir dynamic simulation [C]. Los Angeles, 1976.

[3] Rosen. Simulation of natural fractured reservoirs with semi-implicit source terms. Proceedings of the 4th National Conference on Numerical Simulation of Reservoir Performance [C]. Los Angeles, 1976.

[4] Sadie. Simulation of natural fractured reservoirs. Workshop on Reservoir Simulation [C]. San Francisco, California, zip code 1983.

[5]A.C.Hill, G.W.Thomas. A new method to simulate complex fractured reservoirs. American Society of Petroleum Engineers 1985 Middle East Petroleum Technology Conference and Exhibition [C]. Bahrain: SPE 13537, 1985.

[6]F.Sonier, P.Souillard, F.T.Blaskovich. numerical simulation of natural fractured reservoirs. 1986 annual technical conference and exhibition of American society of petroleum engineers [C]. new Orleans: SPE 15627, 1986.

[7] Ishimoto. Improved matrix, fracturing fluid transfer function in dual pore model. American Society of Petroleum Engineers International Petroleum Engineering Conference [C]. Tianjin, China: SPE 17599, 1988.

Abbas Firuzabadi, L Kent Thomas Sixth American Society of Petroleum Engineers Comparative Solution Project: Dual Pore Simulator. 1989 American Society of Petroleum Engineers Reservoir Simulation Seminar [C]. Houston: SPE 1874 1, 1989.

Karimi-Failde, M., Gong, B., dulov, L. J. ... generated a coarse-scale continuous flow model from detailed fault characteristics. Water resources [J] .42, 2006.

[10] Wu Yushu, Qin Guan, Richard e Ewing, et al. Multi-continuum method for simulating multiphase flow in natural fractured porous reservoirs. 2006 International Oil and Gas Conference and Exhibition of American Society of Petroleum Engineers [C]. Beijing, China: American Society of Petroleum Engineers 104 173, 2006.

[1 1] Kang, Wu, Li et al. Multiphase flow simulation of natural fractured-vuggy reservoirs. 2006 Annual Meeting and Exhibition of American Society of Petroleum Engineers [C]. Texas, USA: American Society of Petroleum Engineers 102356, 2006.

[12]S.Gσmez, G.Fuentes, R.Camacho, et al., Application of Evolutionary Algorithm in Well Test Characterization of Karst Reservoir with Natural Fractures. The first international petroleum conference and exhibition [C]. Mexico: 2006.

[13] Peter popov, LAM Raymond Bi, Yarchin effendi Yev, et al. Multi-physical and multi-scale methods for simulating fluid flow in natural fractured-vuggy carbonate reservoirs. The 15t American Society of Petroleum Engineers Middle East Oil and Gas Exhibition and Conference [C]. Kingdom of Bahrain.

Wu Yusheng, Ge Junlin. Unstable seepage in natural fractured reservoirs with three pores [J]. Journal of Mechanical Engineering, 1983,15 (1): 81-85.

Abodah Sa and Elshagis I .. represent the triple pore system of natural fractured reservoirs [C].SPE form. Eval, 1986, 1,13-127.

Wu Yusheng, Liu Honghai, Bodwasson. A triple continuum method for simulating the flow and migration process of fractured rock mass [J]. Journal of Pollutant Water Literature, 2004a, 73: 145- 179.

[17] Kang, Zhijiang,,, Wu Yongchao. Simulation of multiphase flow in natural fractured porous reservoirs. 2006 annual technical conference and exhibition of American Society of Petroleum Engineers [C]. San Antonio, Texas, 24-27, 2006.

Lin Chia en, Yang. Preliminary study on coupled flow theory of pipe flow and seepage [J]. Journal of Youshi University (Natural Science Edition), 2007,22 (2):1~12.

[19] Wu Yongping Geng keqin. Coupling analysis of mechanics and seepage between arch dam and abutment rock mass [J]. Journal of Rock Mechanics and Engineering, 1997,16 (2):125 ~131.

Geng keqin. Study on seepage, mechanics and their coupling analysis of complex rock foundation and its engineering application [D]. Doctoral thesis of Tsinghua University Engineering. 1994.

[2 1] Study on prediction and control of water cone in fractured-vuggy carbonate reservoir [D]. Beijing: Youshi University, China, 2009.

Zhou Chuangbing, Chen Yifeng and Jiang Qinghui. Rock mass characterization unit and rock mass mechanical parameters [J]. Journal of Geotechnical Engineering, 2007,29 (8):1135 ~1142.