Dynamic simulation method for variable diversion fracture bottom hole pressure of three-area mixed medium low-permeability reservoir

Abstract

The invention discloses a three-area mixed medium low-permeability reservoir variable diversion fracture bottom hole pressure dynamic simulation method, which is based on a three-area mixed medium, considers a fracturing fracture and a limited closed boundary closer to the actual condition of a reservoir, and also considers uniform fracture conductivity and fracture variable conductivity at the same time. The simulated double logarithmic curve can comprehensively describe the whole unstable seepage process, and the problem of fine description of the dynamic variable flow guide fracture bottom hole pressure of the three-area mixed medium low-permeability reservoir can be effectively solved, so that theoretical support is provided for well test interpretation under the complex condition, and development of the well test analysis theory is promoted.

Description

Technical Field

[0001] The present invention relates to the field of well testing analysis in oil and gas field development engineering, and in particular to a dynamic simulation method for bottom hole pressure in oil and gas well testing analysis. Background Art

[0002] The statements in this section merely provide background information related to the present disclosure and may not constitute prior art.

[0003] Due to the low permeability, low-permeability reservoirs require hydraulic fracturing to achieve economic development. After fracturing, a region with relatively good physical properties will be formed near the wellbore, and the reservoir properties in this region are significantly different from those in the far well area. In addition, actual low-permeability reservoirs often have strong heterogeneity, and the physical properties in different regions of the reservoir vary greatly. Conventional homogeneous reservoir models and two-zone composite reservoir models are difficult to accurately describe this heterogeneity phenomenon, and a three-zone composite reservoir model is required to describe it. In addition, due to the influence of factors such as the expansion law of the fracture and the transport of proppant at different locations, the actual conductivity of the fracture is not equal everywhere, but changes with location, that is, "variable conductivity fracture".

[0004] In addition, after hydraulic fracturing, in addition to the formation of the main fracture near the wellbore (zone 1), the natural fractures with smaller openings and poorer development in the near-wellbore area may be expanded and propped up by the fracturing fluid, forming a natural fracture system with substantial significance for seepage, forming a dual medium together with the original matrix pores in the area. In the area far from the wellbore, that is, the area not acted by the fracturing fluid (zones 2 and 3), the matrix pores are still the main ones, which is the "three-zone mixed medium" mentioned here.

[0005] Accurate simulation of bottom hole pressure dynamics is the basis of oil and gas well test analysis and an important means to obtain reservoir and fracture dynamic parameters. Researchers have established a variety of bottom hole pressure dynamic models for different reservoirs and wells, and successfully solved these models. For example, Liu Pengcheng et al. published an article in 2004 (Liu Pengcheng, Wang Xiaodong, Wan Yujin. Analysis of bottom hole pressure dynamics of vertical fracture wells with limited conductivity in three-zone composite reservoirs [J]. Oil and Gas Well Testing, 2004, 13 (1): 4-7.). The model constructed in the article only considered the case of uniform conductivity of the fracture, but did not consider the variable conductivity of the fracture; all the zones considered were homogeneous reservoirs, not three-zone mixed media, and the simulated double logarithmic curve was not complete in describing the flow stage, and could not fully describe the entire unstable seepage process. Wu Minglu et al. published an article in 2022 (Wu Minglu, Li Tao, Zhao Gaolong et al. Horizontal well test model for dual-porosity medium and three-zone composite reservoir [J]. Oil and Gas Well Testing, 2022, 31(04): 6-12. DOI: 10.19680 / j.cnki.1004-4388.2022.04.002.). In this article, hydraulic fracturing was not considered, and all regions in the reservoir were considered to be dual media rather than three-zone mixed media; and the model considered an infinite outer boundary rather than a finite closed boundary that is closer to the actual reservoir situation.

[0006] In addition, the applicant has found through extensive research on relevant domestic and foreign literature and patent technologies that, up to now, there has been no detailed simulation research on the bottom hole pressure dynamics of variable conductivity pressure fractures in three-zone mixed medium low permeability reservoirs.

[0007] Therefore, studying the bottom hole pressure dynamics of variable conductivity pressure fractures in three-zone mixed medium low permeability reservoirs has not only important theoretical significance, but also important practical significance.

[0008] It should be noted that the information disclosed in the above background technology section is only used to enhance the understanding of the background of the present disclosure, and therefore may contain information that does not constitute prior art. Summary of the invention

[0009] In view of this, the present invention provides a method for simulating the dynamic bottom hole pressure of a low permeability reservoir, so as to solve the problem that there is currently no corresponding simulation method for the bottom hole pressure dynamics of variable conductivity pressure fractures in a three-zone mixed medium low permeability reservoir, thereby failing to accurately characterize the bottom hole pressure dynamics.

[0010] To achieve the above-mentioned purpose of the invention, the method for dynamic simulation of bottom hole pressure of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir comprises:

[0011] Establish a dimensional mathematical model of formation seepage in source wells of three-zone mixed medium finite boundary low permeability reservoir;

[0012] Converting the dimensioned mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage into a dimensionless mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage;

[0013] Solve the dimensionless mathematical model of line source well formation seepage in the three-zone mixed medium finite boundary low permeability reservoir to obtain the line source solution of the three-zone mixed medium finite boundary low permeability reservoir, and perform integral superposition to obtain the pressure response formula containing unknown flow distribution function caused by variable conductivity pressure fractures in the three-zone mixed medium finite boundary low permeability reservoir;

[0014] Establish a dimensional model of seepage in variable conductivity pressure fractures, and transform it into a dimensionless model of seepage in variable conductivity pressure fractures;

[0015] The dimensionless seepage model of the variable conductivity hydraulic fracture is solved to obtain the hydraulic fracture pressure distribution equation, and the pressure response equation containing the unknown flow distribution function is solved in parallel to obtain the integral equation describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability reservoir;

[0016] The integral equation describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability oil reservoir is discretized by units to form M linear algebraic equations, and together with the total flow equation, M+1 linear algebraic equations are formed to obtain a linear matrix describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability oil reservoir.

[0017] In the present disclosure and possible embodiments, the method for establishing a dimensional mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage includes:

[0018] Determine the partial differential equation for seepage in zone 1, the partial differential equation for seepage in zone 2, and the partial differential equation for seepage in zone 3;

[0019] The seepage partial differential equation of zone 1 includes the seepage partial differential equation of the natural fracture system of zone 1 and the seepage partial differential equation of the matrix pore system of zone 1;

[0020] Determine the interface conditions between area 1 and area 2, the interface conditions between area 2 and area 3, the inner boundary conditions and the outer boundary conditions of the line source;

[0021] The outer boundary is a circular closed outer boundary;

[0022] The partial differential equation for seepage in zone 1, the partial differential equation for seepage in zone 2, the partial differential equation for seepage in zone 3, the interface conditions between zone 1 and zone 2, the interface conditions between zone 2 and zone 3, the inner boundary conditions of the line source and the outer boundary conditions constitute a dimensional mathematical model for the line source well formation seepage in the three-zone mixed medium finite boundary low permeability oil reservoir.

[0023] The present disclosure has the following beneficial effects:

[0024] The method for simulating the bottom hole pressure dynamics of a low permeability reservoir disclosed in the present invention is based on a three-zone mixed medium, takes into account the hydraulic fractures and the finite closed boundary that is closer to the actual reservoir situation, and also takes into account the uniform fracture conductivity and the fracture variable conductivity. The simulated double logarithmic curve can comprehensively characterize the entire unstable seepage process. Therefore, it can effectively solve the problem of fine characterization of the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in a low permeability reservoir with a three-zone mixed medium, thereby providing theoretical support for the well test interpretation in this complex situation and promoting the development of the well test analysis theory. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The above and other objects, features and advantages of the present disclosure will become more apparent through the following description of the embodiments of the present disclosure with reference to the accompanying drawings, in which:

[0026] Figure 1 The physical model of variable conductivity pressure fracture of three-zone mixed medium low permeability reservoir in the embodiment;

[0027] Figure 2 A double logarithmic diagram of the bottom hole pressure dynamics of a variable conductivity pressure fracture in a three-zone mixed medium low permeability reservoir in the embodiment;

[0028] Figure 3 A comparison diagram of double logarithmic curves of variable conductivity and uniform flow pressure fractures in a three-zone mixed medium low permeability reservoir in an embodiment;

[0029] Figure 4 is r in the embodiment 1D The influence diagram of double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir;

[0030] Figure 5 is r in the embodiment 2D The influence diagram of double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir;

[0031] Figure 6 is the mobility ratio M in the embodiment 12 The influence diagram of double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir;

[0032] Figure 7 is the mobility ratio M in the embodiment 23 The influence diagram of double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir;

[0033] Figure 8 is r in the embodiment eD The influence diagram of double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir;

[0034] Fig. 9 is the storage capacity ratio ω in the embodiment 1Influence diagram of double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. DETAILED DESCRIPTION

[0035] The present disclosure is described below based on embodiments, but it is worth noting that the present disclosure is not limited to these embodiments. In the detailed description of the present disclosure below, some specific details are described in detail. However, for the parts that are not described in detail, those skilled in the art can also fully understand the present disclosure.

[0036] In addition, those skilled in the art should understand that the drawings are provided only to illustrate the purpose, features and advantages of the present disclosure, and the drawings are not actually drawn to scale. At the same time, unless the context clearly requires, the words "include", "comprise" and the like in the entire specification and claims should be interpreted as inclusive rather than exclusive or exhaustive; that is, "including but not limited to" meaning.

[0037] This example takes a certain study area as the object, and uses the method disclosed in this disclosure to simulate the bottom hole pressure dynamics of the three-zone mixed medium low permeability reservoir variable conductivity pressure fracture in the study area. The specific process is as follows:

[0038] Figure 1 is a physical model of variable conductivity fracture in a three-zone mixed medium low permeability reservoir in the embodiment; as shown in the attached Figure 1 As shown in the figure, it is assumed that a vertical well in a low permeability reservoir forms a limited conductivity fracture centered on the wellbore in the reservoir after fracturing, and the half-length of the fracture is x F Due to the influence of geological and engineering factors, the reservoir exhibits strong heterogeneity. From the wellbore to the outside, three reservoir areas with different physical properties are formed, which require a three-area composite model to accurately describe. The reservoir thickness is h, and the original formation pressure is p i .

[0039] Other assumptions are as follows:

[0040] (1) Area 1 is a fracture-pore dual medium, which can be described by the Warren-Root dual-pore model;

[0041] (2) 2. Zone 3 is a single-pore medium;

[0042] (3) The conductivity of the hydraulic fracture varies with location;

[0043] (4) Ignore gravity and capillary forces;

[0044] (5) Darcy seepage.

[0045] In order to accurately obtain the bottom hole pressure dynamics of the gas well under the above situation, it is necessary to comprehensively consider the effects of the three-zone mixed medium, variable conductivity, and finite closed outer boundary, establish an analysis model of the bottom hole unstable pressure of variable conductivity fractures in a three-zone mixed medium low permeability reservoir, and successfully solve it. Based on the solution results, a calculation program is compiled to achieve a detailed characterization of the bottom hole pressure dynamics.

[0046] The main steps of performing the dynamic simulation of bottom hole pressure in this embodiment include:

[0047] Step 1: Establish a dimensional mathematical model of the source well formation seepage in a three-zone mixed medium finite boundary low permeability reservoir.

[0048] Step 2: Convert the dimensional mathematical model of line source well formation seepage in three-zone mixed medium finite boundary low permeability reservoir into a dimensionless mathematical model of line source well formation seepage in three-zone mixed medium finite boundary low permeability reservoir.

[0049] Step 3: Solve the dimensionless mathematical model of line source well formation seepage in the three-zone mixed medium finite boundary low permeability reservoir, obtain the line source solution of the three-zone mixed medium finite boundary low permeability reservoir, and perform integral superposition to obtain the pressure response formula containing unknown flow distribution function caused by variable conductivity pressure fractures in the three-zone mixed medium finite boundary low permeability reservoir.

[0050] Step 4: Establish a dimensional model of seepage in variable conductivity pressure fractures and transform it into a dimensionless model of seepage in variable conductivity pressure fractures.

[0051] Step 5: Obtain the fracture pressure distribution equation of the dimensionless model of variable conductivity fracture seepage, and simultaneously establish the pressure response equation containing the unknown flow distribution function obtained in step 3 to obtain the integral equation describing the bottom hole pressure dynamics of variable conductivity fractures in the three-zone mixed medium low permeability reservoir.

[0052] Step 6: Discretize the integral equation obtained in step 5 describing the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir into units to form M linear algebraic equations, and together with the total flow equation, form M+1 linear algebraic equations to obtain the final description linear matrix of the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir.

[0053] Step 7: Solve the linear matrix obtained in step 6, draw the double logarithmic curve of bottom hole pressure dynamics in combination with numerical inversion programming, divide the flow stage, and perform sensitivity analysis on the parameters affecting the curve.

[0054] This embodiment further describes the above main steps, and the specific implementation methods are as follows:

[0055] Step 1: Establish a dimensional mathematical model of the source well formation seepage in a three-zone mixed medium finite boundary low permeability reservoir.

[0056] 1) Partial differential equation for seepage in zone 1

[0057] ① Partial differential equation of seepage in natural fracture system in zone 1:

[0058] By combining the mass conservation equation, motion equation, state equation, and crossflow equation, we can obtain the partial differential equation for seepage in the natural fracture system in zone 1:

[0059]

[0060] Where: p f1 is the natural fracture system pressure in zone 1 of the reservoir; p m1 is the matrix pore system pressure in zone 1; μ 1 is the viscosity of crude oil in zone 1; φ f1 is the porosity of the natural fracture system in zone 1; C tf1 is the comprehensive compression coefficient of the natural fracture system in zone 1; k f1 is the permeability of the natural fracture system in zone 1; α 1 is the shape factor between the natural fractures and matrix pores in zone 1; k m1 is the permeability of the matrix pore system in zone 1; r is the radial distance; and t is the time.

[0061] ② Partial differential equation of seepage in the matrix pore system of zone 1:

[0062]

[0063] Where: φ m1 is the porosity of the matrix pore system in zone 1; C tm1 It is the comprehensive compressibility coefficient of the matrix pore system in zone 1.

[0064] 2) Partial differential equation of seepage in zone 2:

[0065]

[0066] Where: p 2 is the pressure in zone 2 of the reservoir; μ 2 is the viscosity of crude oil in zone 2; φ 2 is the porosity of zone 2; C t2 is the comprehensive compression coefficient of zone 2; k 2 is the zone 2 permeability.

[0067] 3) Partial differential equation of seepage in three zones:

[0068]

[0069] Where: p 3 is the pressure of zone 3 in the reservoir; μ 3 is the viscosity of crude oil in zone 3; φ 3 is the porosity of zone 3; Ct3 is the comprehensive compression coefficient of the three zones; k 3 is the zone 3 permeability.

[0070] 4) Interface conditions between zone 1 and zone 2:

[0071] p f1 (r 1 ,t)=p 2 (r 1 ,t) (5)

[0072]

[0073] Where: r 1 is the radius corresponding to the interface between area 1 and area 2; M 12 is the mobility ratio of the natural fracture system in zone 1 to that in zone 2,

[0074] 5) Interface conditions between zone 2 and zone 3:

[0075] p 2 (r 2 , t) = p 3 (r 2 , t) (7)

[0076]

[0077] Where: r 2 is the radius corresponding to the interface between zone 2 and zone 3; M 13 is the mobility ratio of the natural fracture system in zone 1 to that in zone 3,

[0078] 6) Line source internal boundary conditions:

[0079]

[0080] Among them: ξ is a radial infinitesimal quantity; is the line source surface production; π is the circumference of a circle; h is the gas layer thickness; B is the crude oil volume coefficient.

[0081] 7) External boundary conditions (circular closed external boundary):

[0082]

[0083] Where: r e is the outer border radius.

[0084] Formulas (1) to (10) constitute the dimensionless mathematical model of the source well formation seepage in a three-zone mixed medium finite boundary low permeability reservoir.

[0085] Step 2: Convert the dimensional mathematical model of line source well formation seepage in three-zone mixed medium finite boundary low permeability reservoir into a dimensionless mathematical model of line source well formation seepage in three-zone mixed medium finite boundary low permeability reservoir.

[0086] 1) Introduce dimensionless quantities suitable for this model

[0087] Dimensionless pressure of natural fracture system in zone 1:

[0088] Dimensionless pressure of matrix pore system in zone 1:

[0089] Dimensionless pressure in zone 2:

[0090] Dimensionless pressure in zone 3:

[0091] Dimensionless bottom hole flowing pressure:

[0092] Dimensionless radial distance: r D =r / x F (Where: x F is the half length of the fracture, m)

[0093] Dimensionless Time:

[0094] Storage capacity ratio of natural fracture system in zone 1:

[0095] Storage capacity ratio of zone 1 to zone 2:

[0096] Storage capacity ratio of zone 1 to zone 3:

[0097] Crossflow coefficient in zone 1: (where: j = 1, 2, 3)

[0098] Dimensionless interface radius r j : (where: j = 1, 2)

[0099] Dimensionless outer boundary radius r e :

[0100] Dimensionless time:

[0101] Dimensionless line source output:

[0102] Flow ratio between zone 1 and zone 2:

[0103] Flow ratio between zone 1 and zone 3:

[0104] Flow ratio between zone 2 and zone 3:

[0105] 2) Dimensionless mathematical model of formation seepage in source wells of three-zone mixed medium finite boundary low permeability reservoir

[0106] Using the above dimensionless quantities, the model composed of equations (1) to (10) can be transformed into:

[0107]

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117] Equations (11) to (20) constitute the dimensionless mathematical model of line source well formation seepage in a three-zone mixed medium finite boundary low permeability reservoir.

[0118] Step 3: Solve the dimensionless mathematical model of line source well formation seepage in the three-zone mixed medium finite boundary low permeability reservoir, obtain the line source solution of the three-zone mixed medium finite boundary low permeability reservoir, and perform integral superposition to obtain the pressure response formula containing unknown flow distribution function caused by variable conductivity pressure fractures in the three-zone mixed medium finite boundary low permeability reservoir.

[0119] Introduce the following Laplace transform:

[0120]

[0121]

[0122]

[0123]

[0124] in: They are p Df1 、p Dm1 、p D2 、p D3 The form after Laplace transformation; s is the Laplace variable.

[0125] After transformation, the line source solution of the dimensionless mathematical model of formation seepage of line source wells in three-zone mixed medium finite boundary low permeability reservoir in Laplace space is obtained as follows:

[0126]

[0127] Where: I 0 (x), K 0 (x) are the first and second type zero-order imaginary Bessel functions, respectively; I 1 (x), K 1 (x) are the first-order imaginary Bessel functions of the first and second kinds respectively; the expression of β is as follows:

[0128]

[0129] in:

[0130] f 1 =(a 23 g 2 -a 24 g 1 ) 11 -(a 13 -a 14 g 1 ) 21

[0131] f 2 =(a 23 g 2 -a 24 g 1 ) 12 -(a 13 -a 14 g 1 ) 22

[0132] g 1 =(a 45 a 56 -a 46 a 55 ) 33 a 56 -(a 35 a 56 -a 36 a 55 ) 43 a 56

[0133] g 2 =(a 45 a 56 -a 46 a 55 ) 34 a 56 -(a 35 a 56 -a 36 a 55 ) 44 a 56

[0134]

[0135]

[0136]

[0137]

[0138]

[0139] (s is a Laplace variable)

[0140] Formula (25) is the line source solution of the dimensionless mathematical model of line source well formation seepage in the three-zone mixed medium finite boundary low permeability reservoir in Laplace space.

[0141] Assume that the flow density of the fracture is q F (x, t), and by using the superposition principle, integral superposition is performed on equation (25) to obtain the pressure response equation containing unknown flow distribution function caused by variable conductivity pressure fractures in a three-zone mixed medium finite boundary low permeability reservoir:

[0142]

[0143] in: Yes FD (x wD ,t D )’s Laplace transform; q FD (x wD ,t D ) is q F The dimensionless quantity (x, t) is defined as q FD (x D ,t D )=2q F (x,t)x F / q;q F (x,t) is the flow density; x wD is xw The dimensionless quantity x w is the horizontal coordinate of the line source at any position.

[0144] When the position point is taken on the hydraulic fracture, equation (27) becomes:

[0145]

[0146] Step 4: Establish a dimensional model of seepage in variable conductivity pressure fractures and transform it into a dimensionless model of seepage in variable conductivity pressure fractures.

[0147] Due to the influence of factors such as fracture aperture and proppant delivery, the conductivity of the fracture is not uniform but variable. Here, the change of fracture conductivity with position is uniformly equivalent to the fracture permeability k F Varies with position, i.e. k F is a function of x:

[0148]

[0149]

[0150]

[0151]

[0152] Where: p F is the fracture pressure; p f1 is the natural fracture system pressure in Zone 1.

[0153] Equations (29) to (32) constitute the dimensional seepage model of variable conductivity pressure fractures.

[0154] The following dimensionless quantities are further defined:

[0155] Dimensionless pressure of fracture:

[0156] Dimensionless fracture conductivity:

[0157] Dimensionless flow density of hydraulic fracture: q FD (x D ,t D )=2q F (x,t)x F / q

[0158] Under the above definition of dimensionless variables, the dimensioned seepage model of variable conductivity pressure fractures can be transformed into the following dimensionless seepage model of variable conductivity pressure fractures:

[0159]

[0160]

[0161]

[0162]

[0163] Step 5: Obtain the fracture pressure distribution equation of the dimensionless model of variable conductivity fracture seepage, and simultaneously establish the pressure response equation containing the unknown flow distribution function obtained in step 3 to obtain the integral equation describing the bottom hole pressure dynamics of variable conductivity fractures in the three-zone mixed medium low permeability reservoir.

[0164] The dimensionless seepage model of the variable conductivity hydraulic fracture is solved to obtain the hydraulic fracture pressure distribution equation:

[0165]

[0166] Combining the pressure response equation (27) with unknown flow distribution function caused by variable conductivity pressure fracture in three-zone mixed medium finite boundary low permeability reservoir obtained above, equation (37) can be transformed into:

[0167]

[0168] The above equation is the integral equation describing the bottom hole pressure dynamics of variable conductivity fractures in three-zone mixed medium low permeability reservoir. This integral equation cannot be solved directly and needs further processing.

[0169] Step 6: Discretize the integral equation obtained in step 5 describing the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir into units to form M linear algebraic equations, and together with the total flow equation, form M+1 linear algebraic equations to obtain the final description linear matrix of the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir.

[0170] The hydraulic fracture is discretized into segments, and the half fracture length is divided into M segments. The length of each discrete segment is

[0171] △x=x F / M. The flow density on the same discrete unit can be considered uniform, and equation (38) can be changed to:

[0172]

[0173] Where: △x D is the dimensionless length of the discrete segment, which is the dimensionless form of △x and is defined as

[0174] △x D =△x / x F .

[0175] When j changes from 1 to M, equation (39) represents M linear algebraic equations, and the unknowns in the equations are and The number of unknown variables is M+1, which cannot be solved yet, and other equations need to be supplemented.

[0176] According to the relationship between total flow and flow density, we have:

[0177]

[0178] Using the dimensionless flow density q of the fracture FD and the dimensionless length of the discrete segment △x D The definition of , equation (38) can be transformed into:

[0179]

[0180] Performing Laplace transform on the above formula, we get:

[0181]

[0182] Equations (39) and (42) represent M+1 linear algebraic equations, and the unknowns in the equations are and There are also M+1 unknowns, which is equal to the number of linear algebraic equations, so a successful solution can be achieved.

[0183] Step 7: Solve the linear matrix obtained in step 6, draw the double logarithmic curve of bottom hole pressure dynamics in combination with numerical inversion programming, divide the flow stage, and perform sensitivity analysis on the parameters affecting the curve.

[0184] Figure 2 It is a double logarithmic diagram of the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir. It can be seen from the figure that the whole flow process can be divided into 10 stages: (1) Bilinear flow section. The pressure derivative of this section is a straight line with a slope of "1 / 4". (2) Linear flow section. The pressure derivative of this section is a straight line with a slope of "1 / 4". (3) Transition section. (4) Zone 1 crossflow section. The pressure derivative curve of this section shows a downward "dent", reflecting the crossflow from the matrix pore system in Zone 1 to the natural fracture system. (5) Zone 1 quasi-radial flow section. The pressure derivative curve of this section is a horizontal section with a height of "0.5". (6) Transition section. (7) Zone 2 quasi-radial flow section. The height value of the pressure derivative curve of this section is equal to "0.5M 12 ”. (8) Transition section. (9) Pseudo-radial flow section in zone 3. The height of the pressure derivative curve in this section is equal to “0.5M 13 ”.

[0185] (10) In the finite closed outer boundary reflection section, the pressure and pressure derivative curves are upward straight lines with a slope of 1.

[0186] Figure 3 This is a comparison of the double logarithmic curves of variable conductivity and uniform flow pressure fractures in three-zone mixed medium low permeability reservoirs. In the example, the dimensionless conductivity coefficient change of variable conductivity pressure fractures satisfies the relationship R FD (x D )=15-8x D , due to x D The range of variation is 0 to 1, so its mean is 11, and R FD (x D ) varies from 7 to 15, with an average value of 11, which is equal to the dimensionless conductivity of uniform flow fractures. As can be seen from the figure, the difference between the double logarithmic curves in the two cases is mainly in the bilinear flow and linear flow stages. The pressure and pressure derivative curves in the variable conductivity case are slightly lower than those in the uniform flow in these two stages.

[0187] Figure 4 is the radius of the interface between zone 1 and zone 2 1D The influence diagram of the double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. It can be seen from the figure that r 1D The larger the value is, the longer the duration of the pseudo-radial flow section in zone 1 is, and the shorter the duration of the pseudo-radial flow section in zone 2 is; that is, r 1D The bigger, Figure 2 The longer the duration of segment (7) is, the shorter the duration of segment (9) is.

[0188] Figure 5 is the radius of the interface between zone 2 and zone 3 2D The influence diagram of the double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. It can be seen from the figure that r 2D The larger the value is, the longer the duration of the pseudo-radial flow section in zone 2 is, and the shorter the duration of the pseudo-radial flow section in zone 3 is; that is, r 2D The bigger, Figure 2 The longer the duration of segment (5) is, the shorter the duration of segment (7) is.

[0189] Figure 6 is the flow ratio between zone 1 and zone 2, M 12 The influence diagram of the double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. It can be seen from the figure that M 12 The larger the value, the higher the position of the pseudo-radial flow section in zone 2 on the pressure derivative curve; that is, M 12 The bigger, Figure 2 The higher the position of segment (7) on the medium pressure derivative curve.

[0190] Figure 7 is the mobility ratio between zone 2 and zone 3, M 23The influence diagram of the double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. It can be seen from the figure that M 23 The larger the value is, the higher the position of the pseudo-radial flow section in zone 3 on the pressure derivative curve is; that is, M 23 The bigger, Figure 2 The higher the position of segment (9) on the medium pressure derivative curve.

[0191] Figure 8 is the dimensionless outer boundary radius r eD The influence diagram of the double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. It can be seen from the figure that r eD The larger the value is, the later the outer boundary reflection segment on the pressure and pressure derivative curve rises; that is, r eD The bigger, Figure 2 The later the (10) section rises.

[0192] Fig. 9 is the storage capacity ratio ω 1 The influence diagram of the double logarithmic curve of variable conductivity pressure fracture in three-zone mixed medium low permeability reservoir. It can be seen from the figure that ω 1 The smaller it is, the higher the position of the linear flow and bilinear flow sections on the pressure and pressure derivative curves, and the deeper the concave of the crossflow section derivative curve; that is, ω 1 The smaller, Figure 2 The higher the positions of sections (1) and (2) on the medium pressure and pressure derivative curves, the deeper the concave section (4) on the pressure derivative curve.

[0193] The above-described embodiments are only embodiments of the present disclosure, and the descriptions thereof are relatively specific and detailed, but they cannot be construed as limiting the scope of the present disclosure. It should be noted that, for a person of ordinary skill in the art, without departing from the concept of the present disclosure, several variations, equivalent substitutions, improvements, etc. may be made, all of which fall within the scope of protection of the present disclosure. Therefore, the scope of protection of the present disclosure shall be subject to the attached claims.

Claims

1. A dynamic simulation method for bottom hole pressure of variable conductivity fracture in three-zone mixed medium low permeability reservoir, It is characterized in that include: Establish a dimensional mathematical model of formation seepage in source wells of three-zone mixed medium finite boundary low permeability reservoir; Converting the dimensioned mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage into a dimensionless mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage; Solve the dimensionless mathematical model of line source well formation seepage in the three-zone mixed medium finite boundary low permeability reservoir to obtain the line source solution of the three-zone mixed medium finite boundary low permeability reservoir, and perform integral superposition to obtain the pressure response formula containing unknown flow distribution function caused by variable conductivity pressure fractures in the three-zone mixed medium finite boundary low permeability reservoir; Establish a dimensional model of seepage in variable conductivity pressure fractures, and transform it into a dimensionless model of seepage in variable conductivity pressure fractures; The dimensionless seepage model of the variable conductivity hydraulic fracture is solved to obtain the hydraulic fracture pressure distribution equation, and the pressure response equation containing the unknown flow distribution function is solved in parallel to obtain the integral equation describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability reservoir; The integral equation describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability oil reservoir is discretized by units to form M linear algebraic equations, and together with the total flow equation, M+1 linear algebraic equations are formed to obtain a linear matrix describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability oil reservoir.

2. The method for dynamic simulation of bottom hole pressure according to claim 1, It is characterized in that The method for establishing a dimensional mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage includes: Determine the partial differential equation for seepage in zone 1, the partial differential equation for seepage in zone 2, and the partial differential equation for seepage in zone 3; The seepage partial differential equation of zone 1 includes the seepage partial differential equation of the natural fracture system of zone 1 and the seepage partial differential equation of the matrix pore system of zone 1; Determine the interface conditions between area 1 and area 2, the interface conditions between area 2 and area 3, the inner boundary conditions and the outer boundary conditions of the line source; The outer boundary is a circular closed outer boundary; The partial differential equation for seepage in zone 1, the partial differential equation for seepage in zone 2, the partial differential equation for seepage in zone 3, the interface conditions between zone 1 and zone 2, the interface conditions between zone 2 and zone 3, the inner boundary conditions of the line source and the outer boundary conditions constitute a dimensional mathematical model for the line source well formation seepage in the three-zone mixed medium finite boundary low permeability oil reservoir.

3. The method for dynamic simulation of bottom hole pressure according to claim 2, Features: The partial differential equation for the seepage of the natural fracture system in zone 1 is: Where: p f1 is the natural fracture system pressure in zone 1 of the reservoir; p m1 is the matrix pore system pressure in zone 1; μ 1 is the viscosity of crude oil in zone 1; φ f1 is the porosity of the natural fracture system in zone 1; C tf1 is the comprehensive compression coefficient of the natural fracture system in zone 1; k f1 is the permeability of the natural fracture system in zone 1; α 1 is the shape factor between the natural fractures and matrix pores in zone 1; k m1 is the permeability of the matrix pore system in zone 1; r is the radial distance; t is the time; The partial differential equation for seepage in the matrix pore system of zone 1 is: Where: φ m1 is the porosity of the matrix pore system in zone 1; C tm1 is the comprehensive compressibility of the matrix pore system in zone 1; The two-zone seepage partial differential equation is: Where: p 2 is the pressure in zone 2 of the reservoir; μ 2 is the viscosity of crude oil in zone 2; φ 2 is the porosity of zone 2; C t2 is the comprehensive compression coefficient of zone 2; k 2 is the permeability of zone 2; The three-zone seepage partial differential equation is: Where: p 3 is the pressure of zone 3 in the reservoir; μ 3 is the viscosity of crude oil in zone 3; φ 3 is the porosity of zone 3; C t3 is the comprehensive compression coefficient of the three zones; k 3 is the permeability of zone 3; The interface conditions between zone 1 and zone 2 are: p f1 (r 1 ,t)=p 2 (r 1 ,t); Where: r 1 is the radius corresponding to the interface between area 1 and area 2; M 12 is the mobility ratio of the natural fracture system in zone 1 to that in zone 2, The interface conditions between zone 2 and zone 3 are: p 2 (r 2 ,t)=p 3 (r 2 ,t); Where: r 2 is the radius corresponding to the interface between zone 2 and zone 3; M 13 is the mobility ratio of the natural fracture system in zone 1 to that in zone 3, The line source internal boundary condition is: Among them: ξ is a radial infinitesimal quantity; is the line source surface production; π is the circumference of a circle; h is the thickness of the gas layer; B is the volume coefficient of crude oil; The outer boundary conditions are: Where: r e is the outer border radius.

4. The method for dynamic simulation of bottom hole pressure according to claim 1, It is characterized in that The method for converting the dimensioned mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage into the dimensionless mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage, comprising: Introduce dimensionless quantities, which are: Dimensionless pressure of natural fracture system in zone 1: Dimensionless pressure of matrix pore system in zone 1: Dimensionless pressure in zone 2: Dimensionless pressure in zone 3: Dimensionless bottom hole flowing pressure: Dimensionless radial distance: r D =r / x F , where: x F is the half length of the fracture, m; Dimensionless time: Storage capacity ratio of natural fracture system in zone 1: Storage capacity ratio of zone 1 to zone 2: Storage capacity ratio of zone 1 to zone 3: Crossflow coefficient in zone 1: Where: j = 1, 2, 3; Dimensionless interface radius: Where: j = 1, 2; Dimensionless outer boundary radius: Dimensionless time: Dimensionless line source output: Flow ratio between zone 1 and zone 2: Flow ratio between zone 1 and zone 3: Flow ratio between zone 2 and zone 3: By using the dimensionless quantity, the dimensioned mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage is converted into the dimensionless mathematical model of the three-zone mixed medium finite boundary low permeability reservoir line source well formation seepage composed of the following functional formula:

5. The method for dynamic simulation of bottom hole pressure according to claim 1, It is characterized in that The method for solving the dimensionless mathematical model of line source well formation seepage in a three-zone mixed medium finite boundary low permeability reservoir to obtain a line source solution for a three-zone mixed medium finite boundary low permeability reservoir comprises: Introduce the following Laplace transform: in: They are p Df1 、p Dm1 、p D2 、p D3 The form after Laplace transformation; s is the Laplace variable; After transformation, the line source solution of the dimensionless mathematical model of line source well formation seepage in the three-zone mixed medium finite boundary low permeability reservoir in Laplace space is obtained as follows: Where: I 0 (x), K 0 (x) are the first and second kind of zero-order imaginary Bessel functions, respectively; I 1 (x), K 1 (x) are the first-order imaginary Bessel functions of the first and second kinds respectively; the expression of β is as follows: Among them: f 1 = (a 23 g 2 -a 24 g 1 )a 11 -(a 13 -a 14 g 1 )a 21 f 2 =(a 23 g 2 -a 24 g 1 )a 12 -(a 13 -a 14 g 1 )a 22 g 1 =(a 45 a 56 -a 46 a 55 )a 33 a 56 -(a 35 a 56 -a 36 a 55 )a 43 a 56 g 2 =(a 45 a 56 -a 46 a 55 )a 34 a 56 -(a 35 a 56 -a 36 a 55 )a 44 a 56 6. The method for dynamic simulation of bottom hole pressure according to claim 5, It is characterized in that The method of integrating and superimposing the line source solutions of the three-zone mixed medium finite boundary low permeability reservoir to obtain the pressure response formula containing unknown flow distribution function caused by variable conductivity pressure fractures in the three-zone mixed medium finite boundary low permeability reservoir comprises: Assume that the flow density of the fracture is q F (x, t), using the superposition principle, the three-zone mixed medium finite boundary low permeability reservoir line source solution is integrated and superimposed, and the pressure response formula containing unknown flow distribution function caused by variable conductivity pressure fracture in the three-zone mixed medium finite boundary low permeability reservoir is obtained: in: Yes FD (x wD ,t D )’s Laplace transform; q FD (x wD ,t D ) is q F The dimensionless quantity (x, t) is defined as q FD (x D ,t D )=2q F (x,t)x F / q;q F (x,t) is the flow density; x wD is x w The dimensionless quantity x w is the horizontal coordinate of the line source at any position; When the position point is taken on the hydraulic fracture, the pressure response equation becomes:

7. The method for dynamic simulation of bottom hole pressure according to claim 6, It is characterized in that The method for establishing a dimensioned model of seepage in a variable conductivity pressure fracture comprises: The variation of hydraulic fracture conductivity with position is uniformly equivalent to the hydraulic fracture permeability k F Varies with position, i.e. k F is a function of x: Where: p F is the fracture pressure; p f1 is the natural fracture system pressure in zone 1; The above four functional formulas constitute the dimensional seepage model of the variable conductivity pressure fracture; and / or, The method for converting the dimensioned seepage model of the variable conductivity pressure fracture into a dimensionless seepage model of the variable conductivity pressure fracture comprises: Define the following dimensionless variables: Dimensionless fracture pressure: Dimensionless fracture conductivity: Dimensionless flow density of hydraulic fracture: q FD (x D ,t D )=2q F (x,t)x F / q; Under the definition of dimensionless variables, the dimensioned seepage model of variable conductivity pressure fractures is transformed into the dimensionless seepage model of variable conductivity pressure fractures:

8. The method for dynamic simulation of bottom hole pressure according to claim 7, Features: The dimensionless model of variable conductivity hydraulic fracture seepage is solved, and the hydraulic fracture pressure distribution equation obtained is: and / or, The integral equation describing the bottom hole pressure dynamics of the variable conductivity hydraulic fracture in the three-zone mixed medium low permeability reservoir obtained by combining the pressure response equation containing the unknown flow distribution function with the hydraulic fracture pressure distribution equation is:

9. The method for dynamic simulation of bottom hole pressure according to claim 8, It is characterized in that The method of performing unit discretization on the integral equation describing the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir to form M linear algebraic equations includes: The hydraulic fracture is discretized into segments, and the half fracture length is divided into M segments. The length of each discrete segment is △x=x F / M; Assuming that the flow density on the same discrete unit is uniform, the integral equation describing the bottom hole pressure dynamics of the variable conductivity pressure fracture in the three-zone mixed medium low permeability reservoir is formed by the discretization of the unit to form the M linear algebraic equations: Where: △x D is the dimensionless length of the discrete segment, which is the dimensionless form of △x, and its definition is △x D =△x / x F .

10. The method for dynamic simulation of bottom hole pressure according to claim 9, Features: The total flow equation is:

Images