CN112417737B - Unconventional oil and gas reservoir flow distribution model parameter sensitivity acquisition method and system - Google Patents

Unconventional oil and gas reservoir flow distribution model parameter sensitivity acquisition method and system Download PDF

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CN112417737B
CN112417737B CN202011441076.XA CN202011441076A CN112417737B CN 112417737 B CN112417737 B CN 112417737B CN 202011441076 A CN202011441076 A CN 202011441076A CN 112417737 B CN112417737 B CN 112417737B
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胡义升
庞康
孙艳彬
刘志斌
闵超
丁显锋
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Southwest Petroleum University
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Abstract

The invention discloses a method and a system for acquiring the parameter sensitivity of a flow splitting model of an unconventional oil and gas reservoir, wherein the unconventional oil and gas reservoir model is initialized firstly; specifying model parameters and injection-production arrangement; specifying a function of the model output variable; then carrying out forward simulation on the model parameters; solving a system equation of the unconventional oil and gas reservoir model to obtain a state variable value of the model; then solving an adjoint equation based on the state variable value, and determining the value of the adjoint variable; and finally, calculating the sensitivity of the unconventional oil and gas reservoir flow splitting model parameters through the values of the state variables and the values of the accompanying variables. The scheme provides a new method for acquiring the sensitivity of the unconventional oil and gas reservoir model, and provides a new reference for the analysis and research of the sensitivity of the system output state relative to the system parameters.

Description

Unconventional oil and gas reservoir flow distribution model parameter sensitivity acquisition method and system
Technical Field
The invention relates to the field of petroleum engineering, in particular to a method and a system for acquiring parameter sensitivity of a flow rate model of an unconventional oil and gas reservoir.
Background
Many observation data can be collected in the production process of the unconventional oil and gas reservoir, the data represent main production indexes and reflect the property and state change of the real unconventional oil and gas reservoir, and the unconventional oil and gas reservoir model can be updated by utilizing the data, so that the accuracy of the unconventional oil and gas reservoir model is improved, and the production optimization result has practical significance. This requires calculating the sensitivity variation relationship between the observed data and unconventional reservoir model parameters and using the observed data to estimate the state and parameters of the unconventional reservoir model.
Further exploration and optimization are carried out on the method for acquiring the sensitivity of the parameter of the unconventional oil and gas reservoir flow rate model, so that a better calculation scheme is provided when the method serves the field of petroleum engineering, and the method is a problem to be solved by technical personnel in the field.
Disclosure of Invention
One of the technical problems to be solved by the present invention is to provide a method for obtaining the sensitivity of the split-flow model parameter of the unconventional oil and gas reservoir, and particularly to solve the relationship of the objective function with respect to the unconventional oil and gas reservoir model parameter.
In order to solve the technical problems, the invention provides a method and a system for acquiring the sensitivity of the unconventional oil and gas reservoir flow splitting model parameter, so as to provide a new way and an acquisition method for analyzing and researching the sensitivity of the system output state relative to the system parameter.
The invention is realized by the following technical scheme:
the unconventional oil and gas reservoir flow split model parameter sensitivity acquisition method comprises the following steps:
the method comprises the following steps: obtaining state variables of the unconventional oil and gas reservoir model based on a system equation of the unconventional oil and gas reservoir model;
step two: obtaining a relationship between the state variable and the accompanying variable based on the state variable, thereby determining an accompanying variable value;
step three: and obtaining the sensitivity of the unconventional reservoir flow splitting model parameters through the state variables and the accompanying variables.
Where the model output variable generally refers to the output value of a well model in an unconventional reservoir, the output variable of the well model is the bottom hole flow pressure if the well specifies a flow rate. If the well specifies a bottom hole flow pressure, then the output variable of the well is the total flow rate. These output variable values can be observed during production of unconventional reservoirs;
in unconventional reservoir injection and production systems, the state variables of the system are affected by system parameters in addition to control variables. For an unconventional oil and gas reservoir injection and production system, parameters of the system comprise permeability, porosity and saturation at the initial moment, the change of the parameters can influence the system states such as flow pressure, flow velocity, water phase saturation and the like, and the problem of sensitivity of the system states relative to the parameters can be classified into an optimization problem.
Firstly, determining a system equation and a saturation equation, and constructing a Lagrangian function according to a sensitivity problem form and in order to obtain an optimality condition of an optimization problem;
determining optimality conditions which need to be met by state variables and system parameters corresponding to extreme points of the objective function of the sensitivity problem according to the definition of the Lagrangian function; the optimality condition is obtained by differentiating a Lagrangian function with respect to a state variable; the optimality condition is expressed in the form of an equation, which is a system of equations for accompanying variables, indicating the relationship of state variables and accompanying variables, according to the definition of the Lagrangian function.
Under the condition that the state variables are known, at each time step, firstly solving an adjoint saturation equation to obtain a value adjoint saturation, and then solving the adjoint equation to obtain a value adjoint variable; since the right-hand term of the adjoint equation contains the adjoint variable of the next time step, the adjoint equation needs to be solved in the reverse order of the time steps.
The sensitivity of the objective function with respect to the system parameters is equal to the partial differential of the Lagrangian function with respect to the system parameters. According to the form of the Lagrangian function, a sensitivity expression of system parameters can be written out, and a Jacobian matrix of parameters of accompanying variables and system equations is needed to be calculated;
in the system equation, there are two matrices related to permeability parameters, and in order to obtain the two matrices, it is necessary to assemble together the corresponding sub-matrices on each computing unit. The sub-matrix on each computing unit can be calculated by simulating a finite difference method, and different forms of sub-matrices can be obtained corresponding to different numerical formats.
The scheme adopts a two-point type numerical value format to calculate the submatrix on each unit, and the diagonal submatrix can be obtained by adopting the format. At this time, the two assembled matrices are in a diagonal form, wherein a diagonal element of one matrix is an inverse of a conductivity matrix of each calculation unit, a diagonal element of the matrix is an inverse of mobility of each well, a well model index is required to be used for calculating mobility of the well, and permeability of the well unit is required to be used for calculating the well model index. With respect to each calculation unit, if the conductivity matrix is calculated using a two-point type numerical format, the system equation on each calculation unit is in a linear relationship with the reciprocal of the permeability on that unit; then, combining the form of a system equation, writing a representation form of permeability sensitivity;
calculating a conductivity matrix of each unit by using a two-point type inner product in a simulation finite difference method according to a numerical format of a system equation on each unit, assembling the reverse conductivity matrix on each calculation unit as diagonal elements to obtain a new matrix, and writing a relational expression of the flow of each calculation unit, the pressure of the unit and the pressure on a unit surface; the elements in the permeability matrix on each calculation unit are only related to the reciprocal of the permeability of the unit, and then an expression of the sensitivity of the objective function relative to the permeability of the unit is shown.
The coefficients in the well model due to the reservoir also have a relationship to the permeability of the well unit. Selecting an adopted well model for a specified well; and (3) calculating the well model only by using the permeability on the well unit, writing an expression of the sensitivity of the objective function relative to the permeability of the well unit, and finally adding the permeability sensitivity of the whole system to the flow unit part and the well model part.
And respectively calculating the sensitivities of the output variable function on initial water phase saturation, porosity and permeability by using the obtained state variable value and the accompanying variable value.
The further technical scheme is as follows:
the system for acquiring the parameter sensitivity of the unconventional oil and gas reservoir flow splitting model comprises the following modules:
the first module is used for obtaining state variables of the unconventional oil and gas reservoir model based on a system equation of the unconventional oil and gas reservoir model; a second module, which obtains the relation between the state variable and the accompanying variable based on the state variable, thereby determining the value of the accompanying variable; and the third module is used for obtaining the sensitivity of the unconventional oil and gas reservoir flow splitting model parameters through the state variables and the accompanying variables.
The first module obtains state variables of the unconventional reservoir model according to the following expression:
Figure BDA0002830358050000041
s.t.
G(sn-1，K)xn＝Hun
en(sn，sn-1，vn，φ)＝0
in the formula (I), the compound is shown in the specification,
Figure BDA0002830358050000042
an objective function that represents the problem of sensitivity,
Figure BDA0002830358050000043
representing the system equation, en(sn，sn-1，vnPhi, 0 represents the saturation equation, matrix G(s)n-1K) represents a coefficient matrix of a system equation, and a vector K represents the permeability of the reservoir; vector unRepresenting controlled variables
Figure BDA0002830358050000044
Respectively representing the specified bottom hole flow pressure value and the total flow velocity of the well, and mapping the vector into a right-end term of a system equation by using a matrix H; vector SnRepresenting the water phase saturation on the computing unit, and the vector phi represents the porosity on the computing unit; n-1, 2L N represents a time step; vector xnRepresents a state variable (v)n，qw，n，pn，πn，pw，n) Wherein the state variables respectively represent: flow on cell face, flow rate of well, pressure on cell face, bottom hole flow pressure.
The second module obtains a relationship between the state variable and the accompanying variable to determine the accompanying variable value according to the following expression:
the following system of equations is derived from the Lagrangian function definition and optimality conditions:
Figure BDA0002830358050000051
Figure BDA0002830358050000052
in the formula, GTCoefficient matrix G(s) representing system equationn-1K) since the matrix is symmetrical, G)TG, which means that the coefficient matrix of the system of equations is the same as the system of equations, differing only in the right-hand term;
Figure BDA0002830358050000053
representing a non-linear function en(sn，sn+1，vnPhi) with respect to the state variable sn，vnTranspose of Jacobian matrix of (1);
Figure BDA0002830358050000054
representing a gradient of the objective function with respect to the state variable;
solving adjoint variables from adjoint equations
Figure BDA0002830358050000055
The third module obtains the sensitivity of the unconventional hydrocarbon reservoir split flow model parameters according to the following expression:
jacobian matrices for parameters using adjoint variables and system equations are derived:
the sensitivity relation of the objective function on the initial water phase saturation of the unconventional reservoir flow split model parameter is as follows:
Figure BDA0002830358050000056
the sensitivity relationship of the objective function with respect to porosity is:
Figure BDA0002830358050000057
here matrix P3Is a diagonal element as a vector
Figure BDA0002830358050000058
A diagonal matrix of (a);
the sensitivity of the objective function with respect to the permeability of the cell is related by:
Figure BDA0002830358050000061
here, the
Figure BDA0002830358050000062
Representing corresponding accompanying variables on a surface
Figure BDA0002830358050000063
The vector of values of (a) is,
Figure BDA0002830358050000064
the representation corresponds to a flow variable
Figure BDA0002830358050000065
The companion variable of (a);
Figure BDA0002830358050000066
the elements in (a) represent the flow values at the center of the respective faces of the calculation unit,
Figure BDA0002830358050000067
representing a total mobility value calculated using a previous time step saturation value; n isikArea-weighted normal vector on the representation surface, cikDenotes the direction vector pointing from the center of the cell to the center of the plane, and the subscript ik denotes this magnitude as defined on the kth plane of cell i;
the sensitivity relationship of the objective function with respect to the permeability of the well unit is:
Figure BDA0002830358050000068
in the formula (I), the compound is shown in the specification,
Figure BDA0002830358050000069
representing the total volume flow of the well,
Figure BDA00028303580500000610
indicating the well mobility calculated using the water phase saturation at the previous time step, WI indicating the well index,
Figure BDA00028303580500000611
is indicative of the pressure of the well unit,
Figure BDA00028303580500000612
representing the bottom hole flow pressure,. DELTA.z the thickness of the well element, r0Denotes the equivalent well radius, rwRepresenting the well radius.
Compared with the prior art, the invention has the following advantages and beneficial effects:
firstly, initializing an unconventional oil and gas reservoir model; specifying model parameters and injection-production arrangement; specifying a function of the model output variable; then carrying out forward simulation on the model parameters; solving a system equation of the unconventional oil and gas reservoir model to obtain a state variable value of the model; then solving an adjoint equation based on the state variable value, and determining the value of the adjoint variable; and finally, calculating the sensitivity of the unconventional oil and gas reservoir flow splitting model parameters through the values of the state variables and the values of the accompanying variables. The scheme provides a new method for acquiring the sensitivity of the unconventional oil and gas reservoir model, and provides a new reference for the analysis and research of the sensitivity of the system output state relative to the system parameters.
Drawings
The accompanying drawings, which are included to provide a further understanding of the embodiments of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principles of the invention. In the drawings:
FIG. 1 is a schematic flow chart of a method for obtaining sensitivity of a parameter of a flow rate model of an unconventional hydrocarbon reservoir in one embodiment of the following embodiments;
fig. 2 is a schematic structural diagram of a parameter sensitivity acquisition system of an unconventional reservoir flow-splitting model in one specific implementation of the following examples.
Detailed Description
The method and system for obtaining the sensitivity of the unconventional reservoir split flow model parameter of the invention are explained in detail in the following with reference to the attached drawings of the specification.
Example 1:
as shown in fig. 1, the method for obtaining the sensitivity of the parameter of the unconventional reservoir split-flow model includes the following steps:
the method comprises the following steps: obtaining state variables of the unconventional oil and gas reservoir model based on a system equation of the unconventional oil and gas reservoir model;
step two: obtaining a relationship between the state variable and the accompanying variable based on the state variable, thereby determining an accompanying variable value;
step three: and obtaining the sensitivity of the unconventional reservoir flow splitting model parameters through the state variables and the accompanying variables.
The model output variable here generally refers to the output value of a well model in an unconventional reservoir, which is the bottom hole flow pressure if the well specifies a flow rate. If the well specifies a bottom hole flow pressure, then the output variable of the well is the total flow rate. These output variable values can be observed during production of unconventional reservoirs;
in unconventional reservoir injection and production systems, the state variables of the system are affected by system parameters in addition to control variables. For an unconventional oil and gas reservoir injection and production system, parameters of the system comprise permeability, porosity and saturation at the initial moment, the change of the parameters can influence the system states such as flow pressure, flow velocity, water phase saturation and the like, and the problem of sensitivity of the system states relative to the parameters can be classified into an optimization problem.
Firstly, determining a system equation and a saturation equation, and constructing a Lagrangian function according to a sensitivity problem form and in order to obtain an optimality condition of an optimization problem;
determining optimality conditions which need to be met by state variables and system parameters corresponding to extreme points of the objective function of the sensitivity problem according to the definition of the Lagrangian function; the optimality condition is obtained by differentiating a Lagrangian function with respect to a state variable; the optimality condition is expressed in the form of an equation, which is a system of equations for accompanying variables, indicating the relationship of state variables and accompanying variables, according to the definition of the Lagrangian function.
Under the condition that the state variables are known, at each time step, firstly solving an adjoint saturation equation to obtain a value adjoint saturation, and then solving the adjoint equation to obtain a value adjoint variable; since the right-hand term of the adjoint equation contains the adjoint variable of the next time step, the adjoint equation needs to be solved in the reverse order of the time steps.
The sensitivity of the objective function with respect to the system parameters is equal to the partial differential of the Lagrangian function with respect to the system parameters. According to the form of the Lagrangian function, a sensitivity expression of system parameters can be written out, and a Jacobian matrix of parameters of accompanying variables and system equations is needed to be calculated;
in the system equation, there are two matrices related to permeability parameters, and in order to obtain the two matrices, it is necessary to assemble together the corresponding sub-matrices on each computing unit. The sub-matrices on each computing unit can be calculated by simulating a finite difference method, and different forms of sub-matrices can be obtained corresponding to different numerical formats.
The scheme adopts a two-point type numerical value format to calculate the submatrix on each unit, and the diagonal submatrix can be obtained by adopting the format. At this time, the two assembled matrices are in a diagonal form, wherein a diagonal element of one matrix is an inverse of a conductivity matrix of each calculation unit, a diagonal element of the matrix is an inverse of mobility of each well, a well model index is required to be used for calculating mobility of the well, and permeability of the well unit is required to be used for calculating the well model index. With respect to each calculation unit, if the conductivity matrix is calculated using a two-point type numerical format, the system equation on each calculation unit is in a linear relationship with the reciprocal of the permeability on that unit; then, combining the form of a system equation, writing a representation form of permeability sensitivity;
calculating a conductivity matrix of each unit by using a two-point type inner product in a simulation finite difference method according to a numerical format of a system equation on each unit, assembling the reverse conductivity matrix on each calculation unit as diagonal elements to obtain a new matrix, and writing a relational expression of the flow of each calculation unit, the pressure of the unit and the pressure on a unit surface; the elements in the permeability matrix on each calculation unit are only related to the reciprocal of the permeability of the unit, and then an expression of the sensitivity of the objective function relative to the permeability of the unit is shown.
The coefficients in the well model due to the reservoir are also related to the permeability of the well units. Selecting an adopted well model for a specified well; and (3) calculating the well model only by using the permeability on the well unit, writing an expression of the sensitivity of the objective function relative to the permeability of the well unit, and finally adding the permeability sensitivity of the whole system to the flow unit part and the well model part.
And respectively calculating the sensitivities of the output variable function on initial water phase saturation, porosity and permeability by using the obtained state variable value and the accompanying variable value.
Example 2:
fig. 2 is a schematic structural diagram of a parameter sensitivity acquisition system for a unconventional reservoir flow splitting model according to the present invention, as shown in fig. 2, including the following modules:
the first module is used for obtaining state variables of the unconventional oil and gas reservoir model based on a system equation of the unconventional oil and gas reservoir model; a second module for obtaining a relationship between the state variable and the accompanying variable based on the state variable, thereby determining an accompanying variable value; and the third module is used for obtaining the sensitivity of the unconventional oil and gas reservoir flow splitting model parameters through the state variables and the accompanying variables.
The first module obtains state variables of the unconventional reservoir model according to the following expression:
Figure BDA0002830358050000101
s.t.
G(sn-1，K)xn＝Hun
en(sn，sn-1，vn，φ)＝0
in the formula (I), the compound is shown in the specification,
Figure BDA0002830358050000102
an objective function that represents the problem of sensitivity,
Figure BDA0002830358050000103
express system equation, en(sn，sn-1，vnPhi, 0 represents the saturation equation, matrix G(s)n-1K) represents a coefficient matrix of a system equation, and a vector K represents the permeability of the reservoir; vector unRepresenting controlled variables
Figure BDA0002830358050000104
Respectively representing the specified bottom hole flow pressure value and the total flow velocity of the well, and mapping the vector into a right-end term of a system equation by using a matrix H; vector SnRepresenting the water phase saturation on the computing unit, and the vector phi represents the porosity on the computing unit; n-1, 2L N represents a time step; vector xnRepresents a state variable (v)n，qw，n，pn，πn，pw，n) Wherein the state variables respectively represent: flow on cell face, flow rate of well, pressure on cell face, bottom hole flow pressure.
The second module obtains a relationship between the state variable and the accompanying variable according to the following expression, thereby determining the accompanying variable value:
the following system of equations is derived from the Lagrangian function definition and optimality conditions:
Figure BDA0002830358050000105
Figure BDA0002830358050000106
in the formula, GTCoefficient matrix G(s) representing system equationn-1K) since the matrix is symmetrical, G)TG, which means that the coefficient matrix of the system of equations is the same, differing only in the right-hand term;
Figure BDA0002830358050000111
representing a non-linear function en(sn，sn+1，vnPhi) with respect to a state variable sn，vnTranspose of Jacobian matrix of (1);
Figure BDA0002830358050000112
representing a gradient of the objective function with respect to the state variable;
solving adjoint variables from adjoint equations
Figure BDA0002830358050000113
The third module obtains the sensitivity of the unconventional hydrocarbon reservoir split flow model parameters according to the following expression:
jacobian matrices for parameters using adjoint variables and system equations are derived:
the sensitivity relation of the objective function on the initial water phase saturation of the unconventional reservoir flow split model parameter is as follows:
Figure BDA0002830358050000114
the sensitivity relationship of the objective function with respect to porosity is:
Figure BDA0002830358050000115
here matrix P3Is a diagonal element as a vector
Figure BDA0002830358050000116
A diagonal matrix of (a);
the sensitivity of the objective function with respect to the permeability of the cell is related by:
Figure BDA0002830358050000117
Figure BDA0002830358050000118
here, the
Figure BDA0002830358050000119
Representing corresponding accompanying variables on a surface
Figure BDA00028303580500001110
The vector of values of (a) is,
Figure BDA00028303580500001111
the representation corresponds to a flow variable
Figure BDA00028303580500001112
The companion variable of (a);
Figure BDA00028303580500001113
the elements in (a) represent the flow values at the center of the respective faces of the calculation unit,
Figure BDA00028303580500001114
representing a total mobility value calculated using a previous time step saturation value; n is a radical of an alkyl radicalikRepresenting area-weighted normal vectors on a surface, cikRepresenting a direction vector pointing from the center of the cell to the center of the plane, and the subscript ik indicating that this magnitude is defined at the celli on the kth face;
the sensitivity relationship of the objective function with respect to the permeability of the well unit is:
Figure BDA0002830358050000121
Figure BDA0002830358050000122
in the formula (I), the compound is shown in the specification,
Figure BDA0002830358050000123
representing the total volumetric flow rate of the well,
Figure BDA0002830358050000124
indicating the well mobility calculated using the water phase saturation at the last time step, WI indicating the well index,
Figure BDA0002830358050000125
is indicative of the pressure of the well unit,
Figure BDA0002830358050000126
representing the bottom hole flow pressure,. DELTA.z the thickness of the well element, r0Denotes the equivalent well radius, rwRepresenting the well radius.
Example 3:
embodiment 3 is a specific implementation manner provided on the basis of embodiment 1, and is shown in fig. 1:
in order to quantitatively investigate the change in the system state, the problem of sensitivity of the system state with respect to the parameters can be ascribed to the following optimization problem.
Figure BDA0002830358050000127
s.t.
G(sn-1，K)xn＝Hun (6-1-2)
en(sn，sn-1，vn，φ)＝0 (6-1-3)
Equations (6-1-2) and (6-1-3) represent the system equation and the saturation equation, respectively. Matrix G(s)n-1K) represents the coefficient matrix, the vector K represents the permeability of the reservoir, the vector x representsnRepresents a state variable (v)n，qw，n，pn，πn，pw，n). Respectively, flow on the face of the computational cell, flow velocity of the well, pressure on the computational cell, pressure on the face of the computational cell, bottom hole flow pressure. Vector unRepresenting controlled variables
Figure BDA0002830358050000128
Representing the specified bottom hole flow pressure value and total well flow velocity, respectively, the matrix H maps this vector to the right-hand term of equation (6-1-2). Vector snIndicating the water phase saturation on the computing unit and the vector phi indicates the porosity on the computing unit. n is 1, and 2L N represents a time step. Representing an objective function related to the state variables. It is to be noted that the permeability appears only in the system equation (6-1-2), and the porosity and the initial time saturation appear in the system equation (6-1-3).
To get the optimality condition for this optimization problem, the following Lagrangian function is constructed, depending on the form of the sensitivity problem.
Figure BDA0002830358050000131
Here, the
Figure BDA0002830358050000132
The representation corresponds to a state variable xn，snIs used as a companion variable. According to the definition of the Lagrangian function, the corresponding state variables and system parameters at the extreme point of the sensitivity problem objective function (6-1-1) need to meet the following optimality condition.
Figure BDA0002830358050000133
Figure BDA0002830358050000134
Figure BDA0002830358050000135
The optimality conditions (6-1-5) are derived from the differentiation of the Lagrangian function with respect to the state variable. These two conditions can be expressed in the following equation form according to the definition of Lagrangian function.
Figure BDA0002830358050000136
Figure BDA0002830358050000137
Where G isTCoefficient matrix G(s) representing system equation (6-1-2)n-1K) since the matrix is symmetrical, G)TThis means that the coefficient matrix of the system of equations is the same, differing only in the right-hand term.
Figure BDA0002830358050000138
Representing a non-linear function en(sn，sn+1，vnPhi) with respect to the state variable sn，vnTranspose of the Jacobian matrix of (a).
Figure BDA0002830358050000139
Representing the gradient of the objective function with respect to the state variable.
Equations (6-1-8) - (6-1-9) are a set of equations for the companion variable, indicating the relationship of the state variable and the companion variable, also referred to as the companion set of equations. The adjoint equations are linear equations due to the differential operation. Note the two adjoint equationsWhen the state variable is known, at each time step, the accompanying saturation equation (6-1-9) is firstly solved to obtain the accompanying saturation
Figure BDA0002830358050000141
Then solving the adjoint equation (6-1-8) to obtain the adjoint variable
Figure BDA0002830358050000142
The value of (c). Since the right-hand term of the adjoint equations (6-1-9) contains the adjoint variable of the next time step, it is necessary to solve the adjoint equations in reverse order of the time steps. Let the accompanying variable at time N +1 be zero, i.e. zero
Figure BDA0002830358050000143
It is noted that the adjoint equations (6-1-8) - (6-1-9) are formally identical to the adjoint equations obtained when solving the production optimization problem, except that the objective functions of the sensitivity problem and the production optimization problem are different, which causes the right-hand terms of the adjoint equations for the two problems to be different.
The optimality conditions (6-1-6) are derived from the differentiation of the Lagrangian function with respect to the accompanying variable. The equations corresponding to this condition are the system equations (6-1-2) and (6-1-3). The optimality conditions (6-1-7) are derived from the differentiation of the Lagrangian function with respect to the system parameter variables. The sensitivity of the objective function with respect to the system parameters is equal to the partial differential of the Lagrangian function with respect to the system parameters. The sensitivity of the system parameters can be expressed as the following expression according to the form of the Lagrangian function.
Figure BDA0002830358050000144
Figure BDA0002830358050000145
Figure BDA0002830358050000146
Here, (6-1-10) - (6-1-12) can be regarded as expressions for calculating the sensitivity of the objective function with respect to the system parameters. The computation requires a Jacobian matrix of adjoint variables and system equations for the parameters. Can be obtained by solving the adjoint equation, and the process is basically consistent with the process of obtaining adjoint variables in the production optimization problem, and the difference is only on the construction of a right-end term of the adjoint equation. The following studies the form of the Jacobian matrix of the system equations with respect to the parameters by analyzing the structure of the system equations.
The matrix form of the system equations in chapter iii is reviewed below.
Figure BDA0002830358050000151
Figure BDA0002830358050000152
Here matrix Br(sn-1，K)，Bw(sn-1K) is related to permeability and the matrix P (phi) is related to porosity. The expression (6-1-10) can be written in the following form according to the form of the system equation.
Figure BDA0002830358050000153
Therein use is made of
Figure BDA0002830358050000154
Where I denotes an identity matrix. In the saturation equations (6-1-14), the matrix P (φ) represents diagonal elements of
Figure BDA0002830358050000155
Where Δ t represents the time interval and | E | represents the volume of the computational cell, so that the saturation equation for each computational cell is only related to the porosity of that cellφiIn relation, the sensitivity matrix of the system equation with respect to porosity is a diagonal matrix. The expression (6-1-11) can be written in the following form.
Figure BDA0002830358050000156
Here matrix P1Is a diagonal element of
Figure BDA0002830358050000157
Of a diagonal matrix, matrix P2Is a diagonal element as a vector
Figure BDA0002830358050000158
The diagonal matrix of (a). According to the form of this expression, when the saturation equation is established, the sensitivity of the objective function with respect to porosity can also be written in the following form.
Figure BDA0002830358050000159
Here matrix P3Is a diagonal element as a vector
Figure BDA00028303580500001510
The diagonal matrix of (a).
In System equation (6-1-13), matrix BrAnd matrix BwIn relation to the permeability parameter, in order to obtain these two matrices, it is necessary to assemble together the corresponding sub-matrices on each computing unit. The sub-matrices on each computing unit can be calculated by simulating a finite difference method, and different forms of sub-matrices can be obtained corresponding to different numerical formats. The two-point type numerical format is adopted to calculate the sub-matrix on each unit, and the diagonal sub-matrix can be obtained by adopting the format. Now assembled matrix BrAnd matrix BwIs also in diagonal form, matrix BrThe diagonal elements of (a) are the inverses of the conductivity matrix of each computing unit, the diagonal elements of the matrix being of each wellThe inverse of mobility requires the use of a well model index to calculate the mobility of the well, which requires the use of the permeability of the well unit. Specifically to each calculation unit, if the conductivity matrix is calculated using a two-point type numerical format, the system equation on each calculation unit is linear with the inverse of the permeability on that unit. Through such analysis, in combination with the form of system equation (6-1-13), the permeability sensitivity is expressed in the following form.
Figure BDA0002830358050000161
Here, the
Figure BDA0002830358050000162
The representation corresponds to a flow variable
Figure BDA0002830358050000163
The accompanying variable(s) of (a),
Figure BDA0002830358050000164
the representation corresponds to well flow velocity qw，nIs used as a companion variable.
Figure BDA0002830358050000165
Representing the portion of each calculation unit that has a relationship to permeability,
Figure BDA0002830358050000166
representing the portion of each well model that has a relationship to permeability. Here the structure of the two matrices needs to be specified.
The numerical format of the system equation on each cell is described in chapter ii, the conductivity matrix of each cell is calculated using the two-point type inner product in the analog finite difference method, and then the inverse conductivity matrix M on each cell is calculatediAssembled together as diagonal elements to give matrix Br. That is, the flow rate of each calculation unit and the pressure of the unit, the pressure on the unit surface satisfy the following relational expression.
Figure BDA0002830358050000167
Mi＝diag(L，|cik|2/(nik·Kicik)，L) (6-1-20)
Here vector
Figure BDA0002830358050000168
The elements in (a) represent the flow values at the center of the respective faces of the calculation unit,
Figure BDA0002830358050000169
representing the total mobility value calculated using the previous time step saturation value. MiRepresenting the inverse conductivity matrix, σ, on the computing unitiVectors, scalars, representing elements all 1
Figure BDA0002830358050000171
Representing pressure values, vectors, on a computing unit
Figure BDA0002830358050000172
Elements of
Figure BDA0002830358050000173
Indicating the pressure values at the center of the respective faces of the calculation unit. n isikArea-weighted normal vector on the representation surface, cikRepresenting a direction vector pointing from the center of the computing element to the center of the plane, and the subscript ik indicates that this magnitude is defined on the kth plane of computing element i. Assuming that the calculation unit is a two-dimensional rectangle and no well model exists on the calculation unit, the inverse conductivity matrix MiIs a 4 x 4 diagonal matrix, defining an area-weighted normal vector of n on each faceik＝(nk1，nk2)TDefining a direction vector c on each faceik＝(ck1，ck2)TThe permeability on the calculation unit is
Figure BDA0002830358050000174
The elements on the diagonal of the inverse conductivity matrix can be represented as
Figure BDA0002830358050000175
Thus the elements in the permeability matrix on each computational cell have a relationship only to the reciprocal of the permeability of the cell, and the sensitivity of the objective function to the permeability of the cell can be expressed in the following form.
Figure BDA0002830358050000176
Here, the
Figure BDA0002830358050000177
Representing corresponding accompanying variables on a surface
Figure BDA0002830358050000178
The value of (c). When the computational cell is square, the above expression can be further simplified, since the sides of the squares are equal, n is assumed to be parallel to the y-directionk1＝2ck1，nk2＝2ck2When the coefficient of the above expression is reduced to 0
Figure BDA0002830358050000179
When K isx＝KyWhen K, the sensitivity of the objective function with respect to permeability may be expressed as
Figure BDA00028303580500001710
The coefficients in the well model of the reservoir are also related to the permeability of the well elements. For a given well, the following well model is used.
Figure BDA0002830358050000181
Figure BDA0002830358050000182
Here, the
Figure BDA0002830358050000183
Representing the total volume flow of the well,
Figure BDA0002830358050000184
indicating the well mobility calculated using the water phase saturation at the previous time step, WI indicating the well index,
Figure BDA0002830358050000185
is indicative of the pressure of the well unit,
Figure BDA0002830358050000186
representing the bottom hole flow pressure,. DELTA.z the thickness of the well element, r0Denotes the equivalent well radius, rwRepresenting the well radius. The calculation of the well model uses only the permeability on the well units, and therefore the sensitivity of the objective function with respect to the permeability of the well units can be expressed in the following form.
Figure BDA0002830358050000187
Here, the
Figure BDA0002830358050000188
The accompanying variables corresponding to the well equations are represented. Eventually the permeability sensitivity of the entire system requires the addition of the flow cell section (6-1-21) and the well model section (6-1-24).
Using the obtained state variable values and accompanying variable values, sensitivities of the output variable functions with respect to initial water phase saturation, porosity and permeability are calculated according to expressions (6-1-15), (6-1-17), (6-1-21) and (6-1-24), respectively.
The above-mentioned embodiments, objects, technical solutions and advantages of the present invention are further described in detail, it should be understood that the above-mentioned embodiments are only examples of the present invention, and are not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (2)

1. The method for acquiring the parameter sensitivity of the unconventional oil and gas reservoir flow splitting model is characterized by comprising the following steps of:
the method comprises the following steps: obtaining state variables of the unconventional oil and gas reservoir model based on a system equation of the unconventional oil and gas reservoir model;
obtaining state variables of the unconventional hydrocarbon reservoir model according to the following expression:
Figure FDA0003653701020000011
s.t.
G(sn-1,K)xn＝Hun
en(sn,sn-1,vn,φ)＝0
in the formula (I), the compound is shown in the specification,
Figure FDA0003653701020000012
an objective function that represents the problem of sensitivity,
Figure FDA0003653701020000013
representing the system equation, en(sn,sn-1,vnPhi, 0 represents the saturation equation, matrix G(s)n-1K) represents a coefficient matrix of a system equation, and a vector K represents the permeability of the reservoir; vector unRepresenting controlled variables
Figure FDA0003653701020000014
Respectively representing the specified bottom hole flow pressure value and the total flow velocity of the well, and mapping the vector into a right-end term of a system equation by using a matrix H; vector snIndicating the water phase saturation at the calculation unit, the vector phi indicates the calculation unitPorosity of the substrate; n is 1,2 … … N represents time step; vector xnRepresents a state variable (v)n,qw,n,pn,πn,pw,n) Wherein the state variables respectively represent: flow on the cell face, flow velocity of the well, pressure on the cell face, bottom hole flow pressure;
step two: obtaining the relation between the state variable and the accompanying variable based on the state variable, thereby determining the accompanying variable value;
the following system of equations is derived from the Lagrangian function definition and optimality conditions:
Figure FDA0003653701020000015
Figure FDA0003653701020000021
in the formula, GTCoefficient matrix G(s) representing system equationn-1K) since the matrix is symmetrical, G)TG, which means that the coefficient matrix of the system of equations is the same, differing only in the right-hand term;
Figure FDA0003653701020000022
representing a non-linear function en(sn,sn+1,vnPhi) with respect to the state variable sn,vnTranspose of Jacobian matrix of (1);
Figure FDA0003653701020000023
representing a gradient of the objective function with respect to the state variable;
solving adjoint variables from adjoint equations
Figure FDA0003653701020000024
Step three: sensitivity of unconventional reservoir split-flow model parameters by state variables and accompanying variables
Jacobian matrices for parameters using adjoint variables and system equations are derived:
the sensitivity relation of the objective function on the initial water phase saturation of the unconventional reservoir flow split model parameter is as follows:
Figure FDA0003653701020000025
the sensitivity relationship of the objective function with respect to porosity is:
Figure FDA0003653701020000026
here matrix P3Is a diagonal element as a vector
Figure FDA0003653701020000027
A diagonal matrix of (a);
the relationship of the sensitivity of the objective function with respect to the permeability of the computational unit is:
Figure FDA0003653701020000028
Figure FDA0003653701020000029
here, the
Figure FDA00036537010200000210
Representing corresponding accompanying variables on a surface
Figure FDA00036537010200000211
The vector of values of (a) is,
Figure FDA00036537010200000212
the representation corresponds to a flow variable
Figure FDA00036537010200000213
The accompanying variable of (a);
Figure FDA0003653701020000031
the elements in (b) represent flow values at the center of the respective faces of the computing unit,
Figure FDA0003653701020000032
representing a total mobility value calculated using a previous time step saturation value; n iskiRepresenting area-weighted normal vectors on a surface, ckiDenotes a direction vector pointing from the center of the computing unit to the center of the plane, and the subscript ki denotes that this magnitude is defined on the kth plane of the computing unit i;
the sensitivity relationship of the objective function with respect to the permeability of the well unit is:
Figure FDA0003653701020000033
Figure FDA0003653701020000034
in the formula (I), the compound is shown in the specification,
Figure FDA0003653701020000035
representing the total volume flow of the well,
Figure FDA0003653701020000036
indicating the well mobility calculated using the water phase saturation at the previous time step, WI indicating the well index,
Figure FDA0003653701020000037
is indicative of the pressure of the well unit,
Figure FDA0003653701020000038
representing the bottom hole flow pressure,. DELTA.z the thickness of the well element, r0Denotes the equivalent well radius, rwRepresenting the well radius.
2. The unconventional oil and gas reservoir flow distribution model parameter sensitivity acquisition system is characterized by comprising the following modules:
the first module is used for obtaining state variables of the unconventional oil and gas reservoir model based on a system equation of the unconventional oil and gas reservoir model;
the first module obtains state variables of the unconventional reservoir model according to the following expression:
Figure FDA0003653701020000039
s.t.
G(sn-1,K)xn＝Hun
en(sn,sn-1,vn,φ)＝0
in the formula (I), the compound is shown in the specification,
Figure FDA00036537010200000310
an objective function that represents the problem of sensitivity,
Figure FDA00036537010200000311
express system equation, en(sn,sn-1,vnPhi, 0 represents the saturation equation, matrix G(s)n-1K) represents a coefficient matrix of a system equation, and a vector K represents the permeability of the reservoir; vector unRepresenting controlled variables
Figure FDA0003653701020000041
Respectively representing the specified bottom hole flow pressure value and the total flow velocity of the well, and mapping the vector into a right-end term of a system equation by using a matrix H; vector snIndicating the water phase saturation on the computing unit, the vector phi indicates the computationPorosity on the cell; n is 1,2 … … N represents time step; vector xnRepresents a state variable (v)n,qw,n,pn,πn,pw,n) Wherein the state variables respectively represent: flow on the cell face, flow rate of the well, pressure on the cell face, bottom hole flow pressure;
a second module for obtaining a relationship between the state variable and the accompanying variable based on the state variable, thereby determining an accompanying variable value;
the second module obtains a relationship of the state variable and the accompanying variable according to the following expression, thereby determining an accompanying variable value:
the following system of equations is derived from the Lagrangian function definition and optimality conditions:
Figure FDA0003653701020000042
Figure FDA0003653701020000043
in the formula, GTCoefficient matrix G(s) representing system equationn-1K) since the matrix is symmetrical, G)TG, which means that the coefficient matrix of the system of equations is the same, differing only in the right-hand term;
Figure FDA0003653701020000044
representing a non-linear function en(sn,sn+1,vnPhi) with respect to the state variable sn,vnTranspose of Jacobian matrix of (1);
Figure FDA0003653701020000045
representing a gradient of the objective function with respect to the state variable;
solving adjoint variables from adjoint equations
Figure FDA0003653701020000046
A third module for obtaining the sensitivity of unconventional reservoir split-flow model parameters through state variables and accompanying variables
Jacobian matrices for parameters using adjoint variables and system equations are derived:
the sensitivity relationship of the objective function on the initial water saturation of the unconventional reservoir flow split model parameter is as follows:
Figure FDA0003653701020000051
the sensitivity relationship of the objective function with respect to porosity is:
Figure FDA0003653701020000052
here matrix P3Is a diagonal element as a vector
Figure FDA0003653701020000053
A diagonal matrix of (a);
the sensitivity of the objective function with respect to the permeability of the cell is related by:
Figure FDA0003653701020000054
Figure FDA0003653701020000055
here, the
Figure FDA0003653701020000056
Representing corresponding companion variable on a surface
Figure FDA0003653701020000057
The vector of values of (a) is,
Figure FDA0003653701020000058
the representation corresponds to a flow variable
Figure FDA0003653701020000059
The accompanying variable of (a);
Figure FDA00036537010200000510
the elements in (a) represent the flow values at the center of the respective faces of the calculation unit,
Figure FDA00036537010200000511
representing a total mobility value calculated using a previous time step saturation value; n is a radical of an alkyl radicalkiRepresenting area-weighted normal vectors on a surface, ckiDenotes the direction vector pointing from the center of the computing unit to the center of the plane, and the subscript ki denotes that this magnitude is defined on the kth plane of the computing unit i;
the sensitivity relationship of the objective function with respect to the permeability of the well unit is:
Figure FDA00036537010200000512
Figure FDA00036537010200000513
in the formula (I), the compound is shown in the specification,
Figure FDA00036537010200000514
representing the total volume flow of the well,
Figure FDA00036537010200000515
indicating the well mobility calculated using the water phase saturation at the previous time step, WI indicating the well index,
Figure FDA00036537010200000516
is indicative of the pressure of the well unit,
Figure FDA00036537010200000517
representing the bottom hole flow pressure,. DELTA.z the thickness of the well element, r0Denotes the equivalent well radius, rwRepresenting the well radius.
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