(China Youshi University (East China) Petroleum Engineering College, Qingdao, Shandong 266555)
Abstract: The establishment of reasonable similarity criteria is of great guiding significance to the physical simulation of fracture-cave unit in water injection development. In this paper, the similarity criteria group used to guide the physical simulation of fracture-cavity unit in water injection development is derived by equation analysis and dimensional analysis respectively, which further verifies the correctness of the similarity criteria group. Through the screening and combination of the above similarity criteria groups, six similarity criteria are finally obtained, which can reflect the main characteristics of fracture-cavity units in water injection development. It is found that the similarity criteria set obtained by equation analysis can be expressed by the similarity criteria set obtained by dimensional analysis. Finally, the physical meanings of the six similarity criteria are recovery ratio, pressure-to-weight ratio, Reynolds number, cubic law under multi-fracture conditions, fracture-cavity ratio and injection-production ratio.
Keywords: similarity criterion; Slot unit; Water injection development; Equation analysis method; Dimensional analysis method
Establishing similarity criteria to guide physical simulation of water flooding in fractured-vuggy units
Li Aifen, Dong Zhang
(China Shiyou University (East China) Petroleum Engineering College, Qingdao, Shandong 266555)
Abstract: It is of great guiding significance to establish appropriate similarity criteria for physical simulation of water flooding in fractured reservoirs. In this paper, the similarity criteria for guiding the physical simulation of water flooding in fractured-vuggy units are obtained by equation analysis and dimensional analysis respectively. The validity of similarity criterion has been proved. Through the selection and combination of the above similarity criteria, six similarity criteria reflecting the main characteristics of water injection development of fractured-vuggy units are obtained. The results are as follows. The similarity criterion derived from equation analysis can be expressed by the criterion derived from dimensional analysis. These six similarity criteria are: recovery ratio, pressure-to-weight ratio, Reynolds number, cubic law under the condition of multiple fractures, the ratio of fracture number to karst cave number, and the ratio of injection quantity to oil production.
Keywords: similarity standard; Slot-hole unit; Water injection; Equation analysis method; Dimensional analysis method
introduce
Fracture-cave unit is the basic development unit of fracture-cave carbonate reservoir [1 ~ 3], and water injection development has achieved good results in the production process of fracture-cave carbonate reservoir [4, 5]. Therefore, in order to develop fractured-vuggy carbonate reservoirs reasonably and efficiently, we must first understand the development law of fractured-vuggy units by water injection.
Physical simulation is an important method to study the mining law of fractured-vuggy units [6 ~ 8]. Physical simulation must meet the similarity theory to ensure its own scientific nature. It can be considered that similarity criterion is the basis of physical simulation.
At present, many scholars have not considered the similarity criterion [9 ~ 13] in the physical simulation experiment of fracture-cavity unit in water injection development, and the similarity criterion used to guide the physical simulation of fracture-cavity unit in water injection development is even rarer. In this paper, the similarity criteria for physical simulation of fractured-vuggy units in water injection development will be derived by equation analysis and dimensional analysis [14 ~ 16] respectively. On the basis of verifying the correctness of the similarity criteria group, the similarity criteria used to guide the physical simulation of fractured-vuggy units in water injection development are screened out through sorting and analysis.
1 Derivation of similarity criteria group by equation analysis method
Basic assumptions of 1. 1
In order to obtain similarity criteria through equation analysis, a mathematical model describing the simulated object must be established first. Before establishing a mathematical model, make the following basic assumptions.
There is oil-water two-phase flow in (1) reservoir. Because the crude oil in Tahe fractured-vuggy reservoir belongs to low saturation pressure crude oil, the existence of dissolved gas in the reservoir is ignored.
(2) In fractured-vuggy reservoirs, large fractures are the main flow channels, so the influence of capillary force is ignored [17];
(3) It is assumed that the injection-production balance will be maintained during water injection development;
(4) For the time being, the filling conditions in caves and cracks are not considered.
1.2 mathematical model
The mathematical model includes continuity equation [18], motion equation, saturation equation, auxiliary equation, definite solution condition and initial condition.
(1) continuity equation
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(2) Equation of motion
When (x, y, z)∈ cracks, the fluid flow can be described in the form of Darcy's law.
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The absolute permeability in Darcy's law can be calculated by the modified cubic law [19].
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When (x, y, z) ∈ karst cave, the fluid flow work can be described by the straight line A of N -S equation [20].
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Where ▽2 Ux, ▽2uy▽2Uz are Laplacian operators.
Multiply the three formulas in formula (7) by dx, dy and dz respectively, and then add them together, considering the oil-water two phases, to obtain:
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(3) Saturation equation
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(4) Auxiliary equation
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Output:
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Attention number:
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Normalization of 1.3
For the convenience of derivation, the above equation is rewritten with normalized saturation and normalized relative permeability.
Normalization of (1) dimensionless term
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(2) Modification of the equation
Substitute equations (14) and (15) into the continuity equation:
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Equations (1.7) and (1.8) replace the equations of human motion:
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Where k * = krowc or k * = krwor.
Saturation equation:
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Parameter description: ρo is the density of oil, g/cm3; ρw is the density of oil, g/cm3; Uo is the oil phase velocity, cm/s; Uw is the water phase velocity, cm/s; Uox is the velocity of oil phase in X direction, cm/s; Uwx is the velocity of water phase in x direction, cm/s; Uoy is the velocity of oil phase in Y direction, cm/s; Uwy is the velocity of water phase in Y direction, cm/s; Uoz is the velocity of oil phase in z direction, cm/s; Uwz is the velocity of water phase in z direction, cm/s; Qo is the mass flow of oil phase inflow (outflow), g/s; Qw is the mass flow of water phase inflow (outflow), g/s; φ is the total porosity of the reservoir; φv is the porosity of the karst cave; φf is fracture porosity; The same is true of oil phase saturation; Sw is the water phase saturation; △S is the saturation of movable fluid; Swc is irreducible water saturation; Sor is residual oil saturation; Is the normalized oil phase saturation; It is standardized water phase saturation; T is time, s; K is absolute permeability, μ m2; Kro is the relative permeability of oil phase; Krw is the relative permeability of water phase; Is the normalized relative permeability of oil phase; Normalize the relative permeability of water phase; Krowc is the relative permeability of oil phase under irreducible water saturation, constant; Krwor is the relative permeability of water phase in residual oil saturation, which is constant; μo is the viscosity of oil phase, mPa·s；; ; μw is the viscosity of water phase, mPa·s；; ; Po is the oil phase pressure,10-1MPa; Pw is the water phase pressure,10-1MPa; G is the acceleration of gravity, m/s2; N is the number of end cracks; H is the height of end face, m; B is the crack opening, in microns; δ is the correction coefficient of cubic law; E is the wall roughness, in microns; L' is the length of the reservoir, km; W is the reservoir width, km; H is the height of the reservoir, km; Nf is the fracture density,1/m; Nv is the density of karst cave,1/m3; Vv is the average volume of karst cave, m3; Lw is the contact area between fracture and fluid (fracture length multiplied by fracture width), m2; D is the half diameter m of the borehole; I is water injection, m3/d.
Establishment of 1.4 similarity criterion
Taking the oil phase equation of formula (19) as an example, this paper introduces the derivation method of similarity criterion.
Divide the first term of equation (19) by the fifth term: (assuming the velocity uo is in the L direction);
Divide the first term of the formula (19) by the fourth term to get:
Divide the fourth term of the formula (19) by the fifth term: the first, second and third dimensions of the formula (19) are the same, so it is not processed;
In this way, three criteria are derived, and other equations are treated in this way, and finally a series of similar criteria are obtained. In addition, dimensionless parameters themselves belong to similarity criteria, such as △S, φv and φ f.
Similarity criteria obtained by combination, such as:
Author:
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Finally, the similarity criteria obtained by equation analysis are as follows:
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The following 33 physical quantities are needed to describe the oil-water two-phase flow in the fractured-vuggy unit:
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These variables include three basic dimensions: P, L and T. According to π theorem of similarity theory [7], there should be 33-3 = 30 similarity criteria, which shows that four similarity criteria are omitted in the derivation of equation analysis method. The missing similarity criteria can be supplemented by dimensional analysis.
Derivation of similarity criteria by two-dimensional analysis method
Basic dimensions include pressure ρ, length l and time t. The variables ρ, U and L including three basic dimensions are selected as the basic parameter group. The physical quantities and their sizes involved in the simulation of two-phase flow in fractured-vuggy elements are shown in the following table 1.
For time t, ρo, uo and l are selected as basic parameters.
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Let the exponent of each basic dimension be zero, and get the homogeneous equations, and get a = 0, b = 1, c =- 1, thus finding the first similarity criterion:
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In the same way, similarity criteria corresponding to each dimension variable can be obtained.
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For combinations of physical quantities with the same dimension, there are the following similarity criteria:
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Other dimensionless parameters are themselves similarity criteria:
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Through dimensional analysis, 30 similarity criteria are obtained. Through comparative analysis, it is found that the similarity criterion obtained by equation analysis method lacks four similarity criteria. In this way, the similarity criterion derived from the equation analysis method is completed.
3 mutual verification of two sets of similarity criteria
The similarity criteria used to guide the simulation of two-phase fluid flow in the tank unit are derived by the above two methods. The similarity criterion obtained by equation analysis has clear physical significance, but the similarity criterion derived by this method is often not comprehensive enough. Generally speaking, the similarity criteria obtained by dimensional analysis will not be omitted, but this method combines all physical quantities with basic parameters to obtain similarity criteria, so that they are forced to be 0, so the similarity criteria obtained by this method often lack physical meaning. Similarity criteria are derived by two methods respectively, and comprehensive and accurate similarity criteria can be obtained by learning from each other's strengths.
3. 1 verification method
Because they are also similar standards to guide the simulation of two-phase fluid flow in slotted elements, the two sets of standards should be completely consistent. If each similarity criterion in a set of similarity criteria can be expressed by another set of similarity criteria, it can be considered that the two sets of similarity criteria are completely consistent.
The similarity criterion obtained by dimensional analysis is used to verify the similarity criterion obtained by equation analysis.
Table 1 physical quantities involved in two-phase flow simulation of fractured-vuggy element and their dimensions
sequential
3.2 Verification process
Firstly, two methods of obtaining similarity criteria groups are listed. In order to distinguish two groups of similarity criteria, the similarity criteria obtained by dimensional analysis are marked (for example).
Similarity criteria set obtained by equation analysis;
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Similarity criteria set obtained from dimensional analysis;
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Through deduction, it is found that:
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This group of similar quasi-A is completely consistent, and the similar quasi-A obtained by the above method is correct.
4 Determination of similarity criteria for physical simulation
Fracture-cave carbonate reservoir has complex reservoir structure and serious heterogeneity, so its physical simulation experiment can not completely simulate the actual situation of oil field. In the process of research, we should grasp the main characteristics of things. According to the requirements of geometric similarity, dynamic similarity and motion similarity, the above similarity criteria groups are screened, sorted and analyzed, and finally six similarity criteria that can reflect the main characteristics of fracture-cave unit water injection development are obtained, as shown in Table 2.
Table 2 Main similarity criteria of physical simulation
sequential
5 conclusion
In this paper, a similarity criterion for guiding the physical simulation of fractured-vuggy elements in water injection development is derived, and the conclusions are as follows:
The similarity criteria set obtained by (1) equation analysis can be expressed by the similarity criteria set obtained by dimensional analysis, which verifies the correctness of the above two similarity criteria sets.
(2) Finally, six similar quasi-A's which can reflect the main characteristics of A-hole unit in water injection development are obtained:
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Their physical meanings are: recovery ratio, pressure-to-weight ratio, Reynolds number, cubic law under multiple fractures, fracture-cavity ratio and water injection-production ratio in turn. refer to
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