CN112906178B - Method for calculating pressure field of horizontal water injection well by using point source function - Google Patents
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Abstract
The invention provides a method for calculating a pressure field of a horizontal water injection well by using a point source function, which comprises the following steps: establishing a model and preparing data; the second step is that: determining the calculation range of space and time in the model and determining the calculation precision; the third step: initializing the model established in the first step; the fourth step: performing each iteration step calculation within the space and time range of the model; the fifth step: judging whether the iterative computation reaches a space boundary (xe, ye, ze) and a time cut-off point T, if the four dimensions reach the boundary, ending the computation process, otherwise, repeating the computation process of the fourth step; the method can be used for carrying out pressure distribution calculation and dynamic characteristic analysis on the horizontal water injection well development mode, and provides scientific basis for reasonably formulating a development scheme, producing dynamic analysis and evaluation and dynamic prediction.
Description
Technical Field
The invention relates to a method for calculating a pressure field of a horizontal water injection well by using a point source function.
Background
In the water injection exploitation oil field, the fluid flow in a vertical well or directional well exploitation mode follows the radial flow seepage rule, under the water injection exploitation mode, the potential gradient between a production well and an injection well is the largest, injected water can suddenly enter along the dominant direction, the fingering phenomenon is serious in the water injection and displacement process, the water drive sweep efficiency is low, and the improvement of the recovery ratio is seriously restricted. Due to the inevitable fingering reason of the radial flow development mode, the initial relatively continuous crude oil distribution state is changed into a dispersed state, partial oil-containing regions cannot be displaced, and the plane sweep efficiency is low. With the continuous progress of the development technology of the horizontal well, the horizontal production well is widely applied to the aspect of improving the recovery efficiency, and the horizontal water injection well is gradually applied, but relatively few researches are carried out on the aspect of researching the development technology of the horizontal water injection well. Through research and study, the existing horizontal water injection well development technology mainly focuses on the aspects of numerical model research and physical simulation research, and the physical simulation analysis technology solves the assumption that the crude oil recovery rate can be greatly improved in the development mode. The numerical simulation research solves the influence of dynamic and static parameters in development and production on the recovery ratio and solves the economic problem of the oil displacement efficiency. The research on the seepage theory method of the horizontal water injection well is relatively less, and no effective reference is provided for theoretical understanding and analysis technologies such as the seepage mechanism, the pressure field distribution and the well potential energy evaluation of the horizontal water injection well, so that the development of the quantitative analysis technology of the horizontal water injection well is limited. The theoretical cognitive degree in the development mode is determined by a pressure analysis technology in the oil field dynamic analysis aiming at the horizontal water injection well development well pattern. In order to improve the theoretical cognitive level of a horizontal water injection well displacement development mode and know the dynamic law of the development mode, development and production are guided. Based on the actual application requirements of the oil field and the existing theory and technical level, a pressure dynamic and flow dynamic analysis relational expression under a horizontal water injection well pattern mode is established through a point source function theory, a mirror image reflection principle and a Newman product principle in a seepage theory, and a theoretical basis is provided for a horizontal water injection well pressure field analysis and evaluation method. The scientific and reliable theoretical basis is provided for scientifically formulating development schemes and determining reasonable development and production strategies.
Disclosure of Invention
The invention aims to provide a method for calculating a horizontal water injection well pressure field by using a point source function, which aims to solve the technical problem of establishing a change rule of a space pressure field distribution condition under a horizontal water injection well development mode.
In order to achieve the purpose, the specific technical scheme of the method for calculating the pressure field of the horizontal water injection well by using the point source function is as follows:
a method for calculating a pressure field of a horizontal water injection well by using a point source function comprises the following steps:
the first step is as follows: establishing a model and preparing data; constructing a horizontal water injection well on one side of a closed fault, wherein the boundary conditions are the fixed injection strength of the horizontal water injection well and the closed fault, and establishing a calculation model of pressure field distribution in the space by taking the fixed injection strength and the closed fault as conditions; acquiring a horizontal water injection well, reservoir and fluid parameters and horizontal water injection well injection parameters corresponding to the model;
the second step is that: determining the calculation range of space and time in the model and determining the calculation precision; determining a calculation range, a calculated space step length and a calculated time step length according to the requirements of practical application; determining a calculation range through a space boundary and a time boundary, and controlling calculation precision through setting a space step length and a time step length; calculating the pressure value of each point in the space in the region and each moment in the time range; the spatial range is defined within a calculation spatial range from the spatial origin coordinate (0,0,0) to the spatial coordinate (xe, ye, ze); the time range is limited from the initial 0 moment to the termination T moment;
the third step: initializing the model established in the first step; carrying out model initialization calculation by using a formula (6), substituting the data prepared in the first step and the second step into the formula (6), calculating the pressure change value at the position of the space coordinate origin (0,0,0) at the initial time t being 0, and recording the calculation result in a four-dimensional array delta P [ x, y, z, t ], so as to obtain the pressure change at the space coordinate origin at the initial time, which is expressed as delta P [0,0,0,0 ];
the fourth step: performing each iteration step calculation within the space and time range of the model; the calculation sequence carries out single-step calculation in a space-first time-second time sequence, the single-step calculation is carried out sequentially from the dimensions of x, y, z and t according to the step length set in the second step by using a formula (6), and the iterative calculation result of each step is stored in a four-dimensional array;
the fifth step: judging whether the iterative computation reaches a space boundary (xe, ye, ze) and a time cut-off point T, if all four dimensions reach the boundary, ending the computation process, otherwise, repeating the computation process of the fourth step; the stored calculation result is data of the pressure field distribution and the change trend of the horizontal water injection well, and can reflect the relation between the pressure distribution in the space and the pressure change along with time under the development mode of the horizontal water injection well.
In the method for calculating the pressure field of the horizontal water injection well by using the point source function, the formula (6) involved in the third step and the fourth step is an analytical expression of pressure change, and a fixed integral relation in which a linear source function relation of the horizontal water injection well is used as a kernel function and upper and lower integral limits are time is used.
According to the principle of doherty superposition and the theory of green's function, the theoretical analysis of the spatial pressure field distribution is based on the form of the following solution:
the horizontal well in the model can be decomposed into a stripe source with one side closed fault and width L in the x direction, and the corresponding point source function is expressed as S h,x The expression is as follows:
the point source function of the horizontal well in the y direction in the model can be expressed as 2 x y in the infinite horizon by using the mirror image reflection principle w Two linear sources are point sources S with the same intensity h,y Can be derived by using the superposition principle, and the expression is as follows:
the point source function of the horizontal well in the z direction in the model is a straight line source in the top and bottom closed boundaries, and the point source function is expressed as S h,z
Synthesizing formulas (2) to (4), and deriving a horizontal well three-dimensional source function expression according to a Newman product principle, wherein the expression is shown in formula (5):
when formula (5) is substituted into formula (1) and q (τ) is a constant value independent of time, the pressure field comprehensive expression of the mathematical model can be expressed as formula (6)
The above formula (6) is an expression of a point source function integral form of the spatial pressure field distribution of the horizontal water injection well, wherein the key core content is the establishment of a point source function S (x, y, z, t); in order to calculate the pressure value at the time t of each spatial point (x, y, z) in the formula (6), a constant integral formula of the spatial point at the time t needs to be calculated; solving can be carried out by using a numerical calculation method; the pressure field calculated by the mathematical model is that the unsteady state pressure change value is calculated under the condition of constant flow rate, so that the pressure field distribution under the water injection development mode is obtained, and the integral calculation is performed once at each time step, so that the pressure value at the moment is obtained.
The formula notation explains: in equation (1): p (x, y, z, t) is a formation pressure value (atm) at a space coordinate (x, y, z) point and time t, Pi is an original formation pressure (atm), S (x, y, z, t-tau) is a model source function, q (tau) is unit impact strength of a source in a mathematical model, namely unit length flow, a horizontal water injection well is a unit horizontal water injection section length water injection quantity, the unit is cm3/(S cm), phi is rock porosity (decimal), and ct is a rock and fluid comprehensive compression coefficient (1/atm).
The method for calculating the pressure field of the horizontal water injection well by using the point source function has the beneficial effects that: the horizontal water injection well considers that the pressure field distribution of one side under the condition of a closed fault is more consistent with the actual condition of the oil field, and is suitable for the fault block oil field. The condition that the constant pressure boundary analyzed by the existing pressure technology is single is overcome. The calculation space range and the calculation time range can be subjected to applicable adjustment, the calculated space and time boundaries can be expanded and reduced according to the pressure recognition and the evaluation requirement of the displacement area, and the calculation accuracy is controllable. The calculation method considers the three-dimensional anisotropy of the oil reservoir space and the heterogeneity of the oil reservoir space, and is wide in application range.
Drawings
FIG. 1 is a block diagram of the steps performed in the method of calculating the pressure field of a horizontal water injection well using a point source function according to the present invention.
FIG. 2 is a schematic diagram of a mathematical model of the present invention showing top and bottom closure of a homogeneous oil layer, horizontal water injection wells and a closed fault distribution on one side; the notation in the figure is: 1 oil layer one side closed fault plane distributed along x-z plane, 2 horizontal water injection well midpoint space coordinate (x) w ,y w ,z w )3, the horizontal length L of the horizontal water injection well, 4, the vertical thickness ze of the oil layer and 5, and calculating the boundary (xe, ye, ze) in a three-dimensional space.
Detailed Description
In order to better understand the purpose, structure and function of the present invention, the following describes the method for calculating the pressure field distribution of the horizontal water injection well in detail with reference to fig. 1 and fig. 2.
As shown in the attached figure 1, the method for calculating the pressure field distribution of the horizontal water injection well comprises the following steps:
the first step is as follows: establishing a model and preparing data; the model is built according to the schema of fig. 2, and basic data required by calculation are prepared. FIG. 2 illustrates the creation of a model in three-dimensional space: the horizontal water injection well is arranged at one side of the closed fault, and the boundary conditions of the model are that the fixed injection strength of the horizontal water injection well and the closed fault surface at one side of the oil layer are distributed 1 along the x-z plane, and the fixed injection strength and the closed fault surface are taken as stripsEstablishing a calculation model of pressure field distribution in the space; acquiring parameters of a horizontal water injection well corresponding to the model: including the spatial coordinates (x) of the midpoint of the horizontal water injection well w ,y w ,z w )2, the length (L) of a horizontal section of the horizontal water injection well and the vertical thickness (ze) of an oil layer are 3 and 4 respectively; the oil layer physical property parameters comprise: oil layer porosity phi and oil layer comprehensive compression coefficient c t (ii) a The fluid parameters include fluid viscosity μ; the injection parameters of the horizontal water injection well comprise: the injection quantity q of the horizontal section with unit length;
x w horizontal water injection well midpoint x-direction coordinate, unit: cm;
y w horizontal water injection well midpoint y-direction coordinate, unit: cm;
z w horizontal water injection well midpoint z-direction coordinate, unit: cm;
l-length of horizontal section of horizontal water injection well, unit: cm;
z e -oil layer vertical thickness, unit: cm;
phi-reservoir porosity, unit: a decimal number;
c t -integral compressibility of the reservoir, in units: atm -1 ；
μ -fluid viscosity, unit: mPas;
q-horizontal segment injection amount per unit length, unit: cm 2 /s。
The second step is that: determining the calculation range of space and time in the model and determining the calculation precision; determining a calculation range, a calculated space step length and a calculated time step length according to the requirements of practical application; determining a calculation range through a space boundary and a time boundary, and controlling calculation precision through setting a space step length and a time step length; calculating the pressure value of each point in the space in the region and each moment in the time range; the spatial range is defined within a calculation spatial range from the origin coordinate to a spatial coordinate (xe, ye, ze) 5; the time range is limited from the initial 0 moment to the termination T moment; the space calculation step length can be set as Δ x ═ Δ y ═ Δ z ═ 100cm, the time step length is set as Δ t ═ 600s, and the step is essentially to discretize the continuous space and time in the calculation range, so as to prepare the data for the subsequent calculation at each discrete space point and time point;
x e model x-direction computation boundaries, in units: cm;
y e model y-direction computation boundaries, in units: cm;
the space calculation step length in the delta x-x direction has the unit: cm;
the space in the delta y-y direction calculates the step size, unit: cm;
step size is calculated in the space of the delta z-z direction, and the unit is: cm;
Δ t-time step, unit: s;
t-calculation termination time, unit: and s.
The third step: initializing the model established in the first step; carrying out model initialization calculation by using a formula (6), substituting the data prepared in the first step and the second step into the formula (6), calculating the pressure change value at the position of the initial time t being 0 and the origin of the space coordinate (0,0,0), and recording the calculation result in a four-dimensional array delta P [ x, y, z, t ], so as to obtain the pressure change at the initial time and the origin of the space coordinate, which is expressed as delta P [0,0,0,0 ];
the fourth step: performing each iterative step calculation within the spatial and temporal range of the model; the calculation sequence carries out single-step calculation in a space-first time-second time sequence, the single-step calculation is carried out sequentially from the four dimensions of x, y, z and t according to the step length set in the second step by using a formula (6), the formula (6) relates to constant integral calculation, the calculation is carried out by using a numerical method, and the iterative calculation result of each step is stored in a four-dimensional array;
the fifth step: judging whether the iterative computation reaches a space boundary (xe, ye, ze) and a time cut-off point T, if the four dimensions reach the boundary, ending the computation process, otherwise, repeating the computation process of the fourth step; the stored calculation result is data of pressure field distribution and variation trend of the horizontal water injection well, and can reflect the relation between the pressure distribution in space and the pressure variation with time under the horizontal water injection well development mode.
The content that is not described in this embodiment is the prior art, and therefore, is not described again.
The invention discloses a core invention point principle of a pressure field distribution calculation method of a horizontal water injection well, which comprises the following steps: the core invention point principle of the method is to utilize a point source function theory, a Newman product principle, a mirror image reflection principle in a seepage theory and a superposition principle in a mathematical physics method. And taking one horizontal water injection well in the space as a straight line source, and determining a point source function of the model by taking the boundary condition as a side closed fault. The point source function reflects the spatial pressure field distribution of the linear source under the boundary condition of the model, namely the establishment of the point source function of the horizontal linear source under the boundary condition is the core content. In the invention, the Newman product principle, the mirror image reflection principle and the superposition principle are utilized to carry out spatial three-dimensional decomposition on the linear source, and the basic point source function in the seepage theory is utilized to finally establish the linear source point source function of the horizontal water injection well. And establishing an unsteady line source mathematical model under the condition that the boundary is a closed fault under the development mode that the horizontal water injection well provides energy drive.
The invention establishes a method for calculating the pressure field distribution and the change trend in the closed fault area on one side of the infinite plane under the condition of constant flow of the horizontal water injection well, and solves the problem of calculating the space pressure distribution and the change trend of the horizontal water injection well. The method is based on a point source function theory and a Newman product principle in a seepage basic theory to deduce a comprehensive pressure distribution relational expression; and then, calculating the fixed integral by using a numerical calculation method to obtain a result, and controlling the calculation precision and range according to the actual application requirement. The calculation method can be flexibly applied, and pressure distribution calculation and dynamic characteristic analysis can be carried out on the horizontal water injection well development mode through the method; the actual production data is dynamically fitted to evaluate and predict dynamics. Can provide scientific basis for reasonably formulating development schemes, producing dynamic analysis and evaluation and dynamic prediction.
It is to be understood that the present invention has been described with reference to certain embodiments, and that various changes in the features and embodiments, or equivalent substitutions may be made therein by those skilled in the art without departing from the spirit and scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.
Claims (2)
1. A method for calculating a pressure field of a horizontal water injection well by using a point source function is characterized by comprising the following steps of:
the first step is as follows: establishing a model and preparing data; constructing a horizontal water injection well on one side of a closed fault, wherein the boundary conditions are the fixed injection strength of the horizontal water injection well and the closed fault, and establishing a calculation model of pressure field distribution in the space by taking the fixed injection strength and the closed fault as conditions; acquiring a horizontal water injection well, reservoir and fluid parameters and horizontal water injection well injection parameters corresponding to the model;
the second step is that: determining the calculation range of space and time in the model and determining the calculation precision; determining a calculation range, a calculated space step length and a calculated time step length according to the requirements of practical application; determining a calculation range through a space boundary and a time boundary, and controlling calculation precision through setting a space step length and a time step length; calculating the pressure value of each point in the space in the region and each moment in the time range; the spatial range is defined within a calculation spatial range from the spatial origin coordinate (0,0,0) to the spatial coordinate (xe, ye, ze); the time range is limited from the initial 0 moment to the termination T moment;
the third step: initializing the model established in the first step; carrying out model initialization calculation by using a formula (6), substituting the data prepared in the first step and the second step into the formula (6), calculating the pressure change value at the position of the initial time t being 0 and the origin of the space coordinate (0,0,0), and recording the calculation result in a four-dimensional array delta P [ x, y, z, t ], so as to obtain the pressure change at the initial time and the origin of the space coordinate, which is expressed as delta P [0,0,0,0 ];
the fourth step: performing each iteration step calculation within the space and time range of the model; the calculation sequence carries out single-step calculation in a space-first time-second time sequence, the single-step calculation is carried out sequentially from the dimensions of x, y, z and t according to the step length set in the second step by using a formula (6), and the iterative calculation result of each step is stored in a four-dimensional array;
the fifth step: judging whether the iterative computation reaches a space boundary (xe, ye, ze) and a time cut-off point T, if the four dimensions reach the boundary, ending the computation process, otherwise, repeating the computation process of the fourth step; the stored calculation result is data of the pressure field distribution and the change trend of the horizontal water injection well, and can reflect the relation between the pressure distribution in the space and the pressure change along with time under the development mode of the horizontal water injection well.
2. The method for calculating the pressure field of the horizontal water injection well by using the point source function according to claim 1, wherein the formula (6) involved in the third step and the fourth step is an analytical expression of pressure change, and is a fixed integral relational expression in which a linear source function relational expression of the horizontal water injection well is taken as a kernel function, and upper and lower integral limits are time;
according to the duhamei superposition principle and the theory of the green function, the theoretical analysis of the spatial pressure field distribution is based on the form of the following solution:
the point source function of the horizontal well in the x direction in the model is expressed as S h,x The expression is as follows:
the point source function of the horizontal well in the y direction in the model can be expressed as 2 x y in the infinite layer by using the mirror image reflection principle w Two straight line sources are point sources S with the same intensity h,y Can be derived by using the superposition principle, and the expression is as follows:
the point source function of the horizontal well in the z direction in the model is a straight line source in the top and bottom closed boundaries, and the point source function is expressed as S h,z
Synthesizing formulas (2) to (4), deriving a horizontal well three-dimensional source function expression according to a Newman product principle, and showing in formula (5):
when formula (5) is substituted into formula (1) and q (τ) is a constant value independent of time, the pressure field comprehensive expression of the mathematical model can be expressed as formula (6)
The above formula (6) is an expression of a point source function integral form of the spatial pressure field distribution of the horizontal water injection well, wherein the key core content is the establishment of a point source function S (x, y, z, t); in order to calculate the pressure value at the time t of each spatial point (x, y, z) in the formula (6), a constant integral formula of the spatial point at the time t needs to be calculated; solving can be carried out by using a numerical calculation method; the pressure field calculated by the mathematical model is that the unsteady state pressure change value is calculated under the condition of constant flow rate, so that the pressure field distribution under the water injection development mode is obtained, and the integral calculation is performed once at each time step, so that the pressure value at the moment is obtained.
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