CN114201932A - Well testing simulation method for tight reservoir fracturing well under complex condition - Google Patents
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Abstract
The invention discloses a well testing simulation method for a tight reservoir fracturing well under complex conditions, which comprises the following steps: and constructing a compact reservoir fracturing well test model which simultaneously considers the variable diversion effect of the fracturing fracture, the stress sensitivity effect of the reservoir and the radial heterogeneity of the reservoir and can simultaneously and accurately simulate the bottom hole pressure and the liquid production conditions of different fracture parts. The invention can provide an unstable well testing model and a simulation method which are more comprehensive and can comprehensively consider the influence of various factors for the exploitation of the pressure-sensitive composite compact oil reservoir variable diversion fractures.
Description
Technical Field
The invention relates to the technical field of well testing simulation methods.
Background
Well testing is one of the main means for acquiring dynamic information of oil and gas reservoirs, and is an extremely important technology for recognizing oil and gas reservoirs. By well testing, some important dynamic parameters of the oil and gas reservoir deeply buried underground, such as stratum permeability, shaft pollution coefficient and the like, can be obtained. The method is used for establishing a correct and reliable unstable well testing model and accurately simulating the pressure dynamics of the oil and gas well, and is an operation basis and a necessary premise of a well testing analysis technology.
The tight sandstone type oil and gas well has extremely low oil reservoir permeability, the oil reservoir of the tight sandstone type oil and gas well is not generally provided with industrial productivity without fracturing modification, and the previous research shows that the tight sandstone reservoir also has obvious stress sensitivity effect and heterogeneity.
However, in the deep research on the well testing model and the dynamic pressure characterization of the fractured well in the prior art, the constructed model almost focuses on the fracturing situation with uniform flow conductivity (whether limited flow conductivity or infinite flow conductivity fracturing), the situation that the flow conductivity changes with different parts of the fracturing (i.e. variable flow conductivity) is rarely considered, and a large amount of experiments and application data show that the flow conductivity of the fracture at different parts is not constant but gradually decreases along the fracture extending direction, and the reasons include that: in fracturing, the fracturing fluid first fractures the near-wellbore formation and then extends outward, while the proppant also first enters the near-wellbore fracture and then is transported distally along the fracture, etc.
A few of the prior arts have studied well testing models of variable diversion pressure fractures and dynamic simulation methods of bottom hole pressure, but still have many defects, such as: some model solutions are carried out in a real time domain, and the solution form of the model comprises a two-degree series, a power integral function, an error function and the like, so that the model is too complex and difficult to accurately calculate; in some models, the dimensionless conductivity coefficient changing along with the position is prematurely extracted out of the integral number for processing during solving, so that the theoretical tightness is insufficient; almost all variable diversion fracture well testing models only aim at reservoirs in a single area, and have limited applicability to compact reservoirs with strong heterogeneity; some models combine a dimensionless diversion coefficient with an abscissa by introducing integral, so that the processing of the variable diversion capacity of the fracture is realized, but the construction process is too complicated, and more importantly, the liquid production conditions of different fracture parts cannot be obtained through simulation.
In conclusion, the prior art lacks a compact reservoir fracturing well test model which can be solved by an analytic method or a semi-analytic method, comprehensively and effectively considers the variable flow conductivity characteristics of a fracturing fracture, the stress sensitivity effect of a reservoir and the radial heterogeneous characteristics of the reservoir, and can simultaneously and accurately simulate the bottom hole pressure and the liquid production conditions of different fracture parts, and a simulation method thereof.
Disclosure of Invention
The invention aims to provide a new well testing simulation method which is carried out based on a compact reservoir fracturing well testing model constructed by simultaneously considering fracture variable diversion, reservoir stress sensitivity and reservoir radial heterogeneity, can accurately simulate the fracturing well bottom pressure transient state under the influence of multiple factors, and simulation data such as flow distribution conditions flowing into different parts of a fracture from a stratum at different moments, and provides an important implementation tool for quantitative analysis of fracturing well testing and productivity influence factors thereof under complex conditions.
The technical scheme of the invention is as follows:
a well testing simulation method for a tight reservoir fracturing well under complex conditions comprises the following steps:
constructing a compact reservoir fracturing well test model which simultaneously considers the variable flow guiding effect of a fracturing fracture, the stress sensitive effect of a reservoir and the radial heterogeneity of the reservoir and can simultaneously and accurately simulate the change of the bottom hole pressure and the liquid production conditions of different parts of the fracture;
wherein the content of the first and second substances,
the variable diversion effect means that the permeability of the fracturing fracture changes along with different positions;
the stress sensitive effect means that the permeability of the oil reservoir changes along with the change of pressure;
the radial heterogeneity means that the permeability, the porosity and the comprehensive compression coefficient of an inner region (I region) of the reservoir are different from those of an outer region (II region).
Preferably, the change in bottom hole pressure comprises a change in bottom hole pressure over time for a given production rate for the well, and the fluid production conditions at different locations of the fracture comprise conditions of magnitude of flow from the formation into different locations of the fracture.
According to some preferred embodiments of the present invention, the construction of the tight reservoir fractured well testing model comprises the following setting conditions:
the fracturing well is a production well of a pressure-sensitive composite compact oil reservoir, a stress sensitive effect exists in a reservoir layer of the pressure-sensitive composite compact oil reservoir, and the permeability of the oil reservoir in the reservoir layer follows an exponential change rule along with the change of pressure;
the seepage field of the pressure-sensitive composite compact oil reservoir is an isothermal seepage field;
the variable diversion effect exists in the fracturing fracture formed in the fracturing process of the fracturing well.
According to some preferred embodiments of the present invention, the constructing of the tight reservoir fractured well test model comprises:
establishing a line junction well seepage model of the pressure-sensitive composite compact oil reservoir, which comprises the following steps: giving a line-junction well model solution condition considering the stress sensitivity effect;
solving the seepage model of the junction well by adopting Laplace transformation, Pedrosa transformation, perturbation transformation and Bessel function theory to obtain a perturbation solution of the junction well;
obtaining a pressure response solution of the pressure-sensitive composite compact oil reservoir caused by the fractured well by adopting a superposition principle based on the perturbation solution;
establishing a fracture seepage model considering the stress sensitivity effect and the variable diversion effect based on the flow characteristics in the fracture;
solving the fracture seepage model to obtain an integral equation describing seepage in the fracture model, namely a seepage calculation model in an integral form;
and solving the seepage integral equation by adopting a unit dispersion method and distribution integration to obtain a solution matrix (namely a linear algebraic equation system) for describing the bottom hole pressure and the flow density.
The line junction well is a well which has small borehole size compared with the oil deposit area scale and can be considered as a straight line when the borehole radius approaches 0.
According to some preferred embodiments of the invention, the constructing further comprises: and performing dimensionless processing on the line junction well seepage model and the fracturing fracture seepage model to obtain corresponding dimensionless models, and then solving based on the dimensionless models.
According to some preferred embodiments of the invention, the dimensionless processing is achieved by introducing a dimensionless quantity.
According to some preferred embodiments of the present invention, the solution of the dimensionless model is implemented by a Pedrosa transformation, a perturbation transformation, a Laplace transformation, a bessel function theory, a double integral, a unit dispersion of integral equations, and a fractional integral.
According to some preferred embodiments of the invention, the simulation method further comprises:
solving the solution matrix by a Gaussian elimination method, and inverting the solution of the Laplace domain to a real time domain by a Stehfest numerical inversion method to obtain a solution of the real time domain (including a bottom hole transient pressure solution and a fracture flow density distribution solution);
further, the bottom hole pressure of the real time domain is substituted into the Pedrosa transformation, and the dimensionless bottom hole transient pressure in the real time domain considering the stress sensitivity effect is obtained.
According to some preferred embodiments of the invention, the obtaining of the pressure response solution comprises: and integrating the perturbation solution along the length of the slit by using a pressure drop superposition principle.
According to some preferred embodiments of the present invention, the obtaining of the seepage calculation model comprises: solving a dimensionless model of the obtained fracture seepage model; performing double integration on a second derivative term of the pressure with respect to the abscissa in the solving; and then, the obtained integral expression and the pressure response solution are combined to obtain the seepage calculation model in the integral form.
According to some preferred embodiments of the invention, the solving of the percolation calculation model comprises:
obtaining a discretization model of the seepage calculation model;
the discretization model and a discretization relation between total flow and flow density are combined to obtain a solution matrix;
wherein the discretization comprises a process of cell discretization and distribution integration.
According to some preferred embodiments of the invention, the tight reservoir fracturing well test model comprises one or more of the following models:
a line junction well seepage model in an oil reservoir, comprising:
differential equation of zonal seepage in the reservoir:
differential equation of seepage in the outer region of the reservoir:
inner boundary conditions:
outer boundary conditions:
p2(r,t)|r→∞＝0 (4)，
continuous flow conditions of the inner and outer zone junction surfaces:
and (3) joining surface pressure equality condition:
initial conditions:
p1|t＝0＝p2|t＝0＝pi (7)，
wherein p is1Is an inner zone reservoir pressure, p2The oil reservoir pressure is in the outer region, r is the radial distance from a certain point in the oil reservoir to a line junction well, gamma is the stress sensitivity coefficient of the oil reservoir permeability, and piIs the original formation pressure, phi1Is the porosity of the reservoir in the inner zone2Oil reservoir with external regionPorosity of (C)t1As the combined compressibility of the inner zone reservoir, Ct2The comprehensive compression coefficient of the oil reservoir in the outer region, mu is the viscosity of crude oil, k1iInitial permeability, k, for the inner zone reservoir2iThe initial permeability of the oil reservoir in the outer region, e is a natural index, t is time, xi is infinitesimal quantity,
the yield of the line junction well, B the volume coefficient of the crude oil in the stratum, h the thickness of the oil deposit, pi the circumferential rate, rfThe radial distance of the connecting surface of the inner zone and the outer zone;
the dimensionless model of the seepage model of the line junction well comprises:
wherein the content of the first and second substances,
dimensionless pressure (l ═ 1,2, w represent inner zone, outer zone, and bottom hole, respectively), rD＝r/xfIs dimensionless radial distance, xfIs the half-length of the crack,
for a non-dimensional stress sensitivity coefficient,
in order to have a dimensionless time,
for dimensionless production of well convergence, M12Initial fluidity ratio of inner and outer zones
ω12Is the internal and external storage capacity ratio (phi)1Ct1/φ2Ct2)，rfDThe radius of the interface is distinguished for the inside and the outside;
a perturbative solution for a junction-line well seepage model, comprising:
wherein the content of the first and second substances,
the solution of the dimensionless pressure of the oil deposit in the inner region after the Pedrosa transformation and the Laplace transformation,
the zero order perturbation solution of the dimensionless pressure of the oil deposit in the inner region after the Pedrosa transformation, the Laplace transformation and the perturbation transformation,
the dimensionless yield of the line-junction well after Laplace transformation is K0Is a modified Bessel function of the second kind, K, of order 01Is a modified Bessel function of order 1, second kind, sigma1Is an intermediate variable, σ2Is an intermediate variable, #BIs an intermediate variable, I0Is a 0 th order Bessel function of the first kind, I1Is a first class modified Bessel function of order 1, u is a Laplace variable, xDIs a dimensionless abscissa, x, of the field pointwDDimensionless abscissa, y, of source sinkDAs a dimensionless ordinate, y, of the field pointwDIs a dimensionless ordinate of the source sink;
the pressure response solution:
wherein the content of the first and second substances,
is dimensionless flux density (flowing directly into the fracture from the formation);
a fracture seepage model comprising:
wherein p isfIs the fracture pressure, x is the abscissa, y is the ordinate, WfWidth of the press crack, kfAs the permeability of the fracturing fracture, qfIs the density of the flow (directly into the fracture from the formation);
a dimensionless model of a fracture-infiltration model, comprising:
wherein the content of the first and second substances,
in order to have a dimensionless fracture pressure,
q is dimensionless fracture conductivityfD(xD,tD)＝2qf(x,t)xfQ is dimensionless flow density, q is oil well production;
seepage calculation model:
wherein the content of the first and second substances,
a bottom hole pressure solution of a Laplace space is obtained, alpha is an integral variable, and v is an inner layer integral upper limit in double integration;
discretization model of seepage calculation model:
where N is the number of discrete units of dimensionless fracture half-length, Δ xDFor each discrete element length, j is the discrete element ordinal number, xDjIs the end point of the jth discrete cell,
is the middle point of the j discrete unit, i is the integer pointer variable in the accumulation formula;
the relation model and the discretization model of the total flow and the flow density are as follows:
and (3) final solution model:
[A][X]＝[B] (33)
wherein [ A ] is a coefficient matrix, [ B ] is a constant term array matrix, and [ X ] is an unknown array matrix.
According to some preferred embodiments of the present invention, the simulation process based on the tight reservoir fractured well testing model comprises:
solving the solution model by Gaussian elimination method to obtain transformed parameters
The corresponding parameter G is obtained by the following Stehfest numerical inversion formulawDAnd q isfDi(i＝1,2,…N+1)：
Wherein, VkIs an inversion coefficient, s, in a Stehfest numerical inversion formulakIs a Laplace variable in an inverse formula, and:
substituting the corresponding parameters into the Pedrosa transformation to obtain the dimensionless bottom hole pressure p in the real time domainwDThe following are:
the invention establishes a compact reservoir fracturing well test model which simultaneously considers fracture variable diversion, reservoir stress sensitivity and radial heterogeneity and can simultaneously and accurately simulate the bottom hole pressure and the liquid production conditions of different fracture parts, and provides an efficient and accurate simulation method for reservoir development under complex conditions.
The simulation method can accurately obtain the bottom hole pressure transient state of the fractured well, the flow distribution conditions and the typical curves flowing into different parts of the fracture from the stratum at different moments, and provides an important technical tool for well testing and productivity analysis of the fractured well under the complex conditions.
In the prior art, a Finite Difference Method (FDM), a Finite Element Method (FEM) and other regional pure numerical methods need to perform unit dispersion in the whole oil reservoir region, and due to the fact that the width of a fracturing fracture is extremely small, a large number of grids need to be encrypted near the fracturing fracture or a non-structural grid needs to be adopted, so that the number of needed grid divisions is large, the order of a formed solving matrix is large, and efficient calculation such as computer programming is not facilitated. In some preferred embodiments of the invention, methods such as Pedrosa transformation, Laplace transformation, superposition principle, double integration, fractional integration, unit dispersion, Stehfest numerical inversion and the like are comprehensively adopted, an analytic method is mainly used, numerical values are used as assistance, only unit dispersion is needed on a fracture, the order of the formed solving matrix is greatly reduced, and the simulation efficiency is obviously improved.
The invention can provide an unstable well testing model and a simulation method which are more comprehensive and can comprehensively consider the influence of various factors for the exploitation of the pressure-sensitive composite compact oil reservoir variable diversion fractures.
Drawings
FIG. 1 is a specific tight reservoir fracture well model.
Fig. 2 is a schematic diagram of a fracture cell discretization.
FIG. 3 is a typical curve of a pressure-sensitive composite tight reservoir variable flow guiding fracture well testing model.
FIG. 4 is a graph comparing typical curves for uniform and varying flow fractures.
FIG. 5 shows the parameter γDAnd (3) an influence graph of a pressure-sensitive composite tight oil reservoir well testing typical curve.
FIG. 6 is a drawing showingParameter rfDAnd (3) an influence graph of a pressure-sensitive composite tight oil reservoir well testing typical curve.
FIG. 7 shows the parameter M12And (3) an influence graph of a pressure-sensitive composite tight oil reservoir well testing typical curve.
FIG. 8 is a graph of dimensionless flux density distribution at different locations of the fracture at different times.
Detailed Description
The present invention is described in detail below with reference to the following embodiments and the attached drawings, but it should be understood that the embodiments and the attached drawings are only used for the illustrative description of the present invention and do not limit the protection scope of the present invention in any way. All reasonable variations and combinations that fall within the spirit of the invention are intended to be within the scope of the invention.
According to the technical scheme of the invention, the well testing simulation method of the tight reservoir fracturing well under the specific complex condition comprises the following steps:
s1 referring to fig. 1, a tight reservoir fracture well model under complex conditions is first established, and its settings include:
the fracturing well is a well at the center of a pressure-sensitive composite compact oil reservoir, and the half length of a fracture formed after hydraulic fracturing is xfThe width of the seam is wfVertical hydraulic fracture of (1);
the pressure-sensitive composite tight reservoir can be classified as having a radius r1An inner region (region I) and the remaining outer region (region II), the inner region and the outer region forming a radius reTotal well area of → ∞;
the pressure-sensitive composite compact oil reservoir has the level, equal thickness and original formation pressure of piProducing the oil well with a fixed wellhead yield q;
the reservoir of the pressure-sensitive composite compact oil reservoir has a stress sensitive effect, if the permeability of the reservoir is reduced along with the reduction of pressure, and the corresponding change of the reservoir follows an exponential change rule;
the formed cracks are diversion cracks with permeability changing along with position, namely, the formed cracks meet kf＝kf(x) Wherein k isf(x) Denotes the crack permeability, k, as a function of position xfRepresents the crack permeability;
the seepage field of the pressure-sensitive composite compact oil reservoir is an isothermal seepage field;
gravity and capillary forces were neglected in the model.
S2, based on the fracturing well model, firstly establishing a line junction well seepage model in the oil reservoir considering the pressure-sensitive effect;
more specifically, it may include:
establishing an oil reservoir seepage differential equation which simultaneously establishes a pressure-sensitive equation, a motion equation, a continuity equation and a rock and liquid state equation for the inner region and the outer region respectively;
considering the influence of the stress sensitive effect of the oil reservoir, giving a proper inner oil reservoir boundary condition based on the line junction well, simultaneously giving an outer oil reservoir boundary condition, an engagement surface flow continuous condition, a pressure equal condition and an initial condition, and combining each condition with an obtained oil reservoir seepage differential equation to form a line junction well seepage model in the compact composite oil reservoir.
The method comprises the following specific steps:
and respectively establishing an oil reservoir seepage differential equation which simultaneously establishes a pressure-sensitive equation, a motion equation, a continuity equation and a rock and liquid state equation for the inner region and the outer region:
differential equation of zone I (inner zone) seepage
Differential equation of zone II (outer zone) seepage
Wherein p is1Reservoir pressure (Pa), p in zone I2Reservoir pressure (Pa) in the II region, r is the radial distance (m) from a certain point in the reservoir to the line junction well, and gamma is the stress sensitivity coefficient (Pa) of the permeability of the reservoir-1)，piIs the original formation pressure (Pa), phi1Porosity (decimal) of the reservoir in zone I2Porosity (decimal) of reservoir in zone II, Ct1For compression of I-zone reservoirsCoefficient (Pa)-1)，Ct2Compressibility factor (Pa) for reservoir in II zone-1) Mu is the viscosity (Pa · s) of the crude oil, k1iInitial permeability (m) for a zone I reservoir2)，k2iInitial permeability (m) for a zone II reservoir2) E is a natural index, and t is time(s).
Considering the influence of the stress sensitive effect of the oil reservoir, giving a proper inner oil reservoir boundary condition based on the line junction well, simultaneously giving an outer oil reservoir boundary condition, an engagement surface flow continuous condition, a pressure equal condition and an initial condition, and combining each condition with an obtained oil reservoir seepage differential equation to form a line junction well seepage model in the compact composite oil reservoir:
more specifically, the conditions may be set as follows:
utilizing Darcy's law, the conditions for obtaining the inner boundary of the oil reservoir based on the line junction well are as follows:
where ξ is the infinitesimal quantity (m),
production (m) for a line junction well3And/s), B is the volume coefficient (dimensionless) of the crude oil in the stratum, h is the thickness (m) of the oil reservoir, and pi is the circumferential rate.
The outer boundary of the reservoir is considered to be infinite, and the outer boundary conditions are obtained as follows:
p2(r,t)|r→∞＝0
the continuous conditions of the junction surface flow are as follows:
the conditions for the pressure equality of the interface are as follows:
the initial conditions were as follows:
p1|t＝0＝p2|t＝0＝pi
s3, carrying out dimensionless treatment on the obtained line junction well seepage model of the compact composite oil reservoir;
more specifically, it may include:
obtaining a dimensionless form of the relevant parameter;
and carrying out dimensionless conversion on the line junction well seepage model of the compact composite oil reservoir based on the dimensionless form of the obtained parameters.
More specifically, the dimensionless form of the parameter includes:
dimensionless pressure:
wherein, subscript l is 1,2, w, which respectively represents 1 region, 2 region, bottom hole;
dimensionless radial distance: r isD＝r/xf；
Dimensionless stress sensitivity coefficient:
dimensionless time:
dimensionless well convergence yield:
more specifically, the following line junction well seepage non-dimensional model of the compact composite oil reservoir can be obtained through the non-dimensional conversion:
wherein M is12Is the initial fluidity ratio of the inner and outer zones, rfDTo distinguish the radius of the interface, omega, from the inside12Is the inner-outer zone capacity ratio.
S4, solving the linear well seepage dimensionless model to obtain a perturbation solution (linear solution) of the linear well seepage dimensionless model;
more specifically, the solution is realized by a Pedrosa transformation, a perturbation transformation and a Laplace transformation.
More specifically, the solving process comprises the following steps:
the Pedrosa transformation was introduced as follows:
wherein the content of the first and second substances,
the solution is obtained after the dimensionless pressure of the oil deposit in the region I is subjected to the Pedrosa transformation and the Laplace transformation.
The perturbation transformation was introduced as follows:
taking into account gammaD<<1, the zero order perturbation solution in the concrete implementation can meet the engineering precision requirement.
The Laplace transform is as follows:
wherein u is a Laplace variable,
the solution of dimensionless pressure of the oil deposit in the region I after the Pedrosa transformation and the Laplace transformation
Under the above transformation, the line summary of the dimensionless model in the zone 1 formation can be found as follows:
wherein the content of the first and second substances,
wherein psiBIs an intermediate variable, K1Is a modified Bessel function of the second kind, order 1, K0Is a second class of modified Bessel function of order 0, σ1Is an intermediate variable, σ2Is an intermediate variable, xDIs a dimensionless abscissa, x, of the field pointwDDimensionless abscissa, y, of source sinkDAs a dimensionless ordinate, y, of the field pointwDDimensionless ordinate, I, of source sink0Is a 0 th order Bessel function of the first kind, I1Is a 1 st order modified bessel function.
S5, obtaining a pressure response solution caused by the fractured well by utilizing a pressure drop superposition principle based on the obtained line convergence solution;
more specifically, it may include:
setting the linear density flow on the fracturing fracture to qf(x, t), solving the integral of the length of the seam by using a pressure drop superposition principle to obtain a pressure response expression in the oil reservoir caused by the fractured well, wherein the expression comprises the following steps:
wherein u is a Laplace variable,
is a dimensionless flux density (flowing directly into the fracture from the formation).
S6, considering the stress sensitivity effect and the flow conductivity of the fracture along with the position change, establishing a fracture seepage model of the variable flow conductivity fracture;
more specifically, the fracture seepage model is constructed as follows:
wherein p isfIs the fracture pressure, x is the abscissa, y is the ordinate, WfWidth of the fracturing crack, qfIs the density of the flow (from the formation directly into the fracture).
S7, carrying out dimensionless treatment on the obtained fracturing fracture seepage model to obtain a fracturing fracture seepage dimensionless model;
more specifically, it may include:
obtaining a dimensionless form of the relevant parameter;
the fracture-seepage model is subjected to a dimensionless transformation based on the dimensionless form of the resulting parameters.
More specifically, dimensionless forms of the parameters include:
dimensionless fracture pressure:
the flow conductivity of the dimensionless crack is as follows:
dimensionless flux density: q. q.sfD(xD,tD)＝2qf(x,t)xf/q。
The fracture seepage non-dimensional model obtained by the non-dimensional transformation is as follows:
s8, obtaining a description expression of an integral form of seepage in the model, namely a seepage calculation expression, based on the fracture seepage dimensionless model and the obtained pressure response expression;
more specifically, it comprises: carrying out Pedrosa transformation, perturbation transformation and Laplace transformation on the fracture seepage dimensionless model to obtain a relevant solution;
performing double integration on the correlation solution to obtain a correlation integral expression;
and (3) obtaining the seepage calculation formula of the integral form by combining the obtained correlation integral formula with the reservoir pressure response expression:
wherein the content of the first and second substances,
is a bottom hole pressure solution of Laplace space, alpha is an integral variable, CFDAnd v is the upper limit of the inner layer integral in the double integral for the dimensionless fracture conductivity.
S9, establishing a discretization model of the seepage description formula;
more specifically, the discretization model is obtained based on a cell discretization method and a distribution integral, and further includes:
carrying out dimensionless conversion on the half length of the crack to a range of [0,1 ];
referring to FIG. 2, a dimensionless fracture half-length [0,1]]The interval is divided into N equal parts, i.e. N discrete units, where each equal part, i.e. each unit, has a length Δ xD，xDj(j-1, 2, … N +1) is where the jth endpoint,
is the midpoint of the jth discrete cell;
regarding the linear density flow on the same discrete unit as uniform, the following discretization model is obtained through distribution integration:
wherein i is an integer pointer variable in the accumulation formula;
it can be seen that when j varies from 1 to N, equation (35) represents N linear algebraic equations, with the unknowns in the equations being
(i＝1,2,…N+1)、
And (N +1) the total flow is still unable to be solved directly, and further, an equation is established based on the relation between the total flow and the flow density for solving.
S10, the discretization model and a discretization relation between total flow and flow density are combined to obtain a solution model;
more specifically, the obtaining of the discretized relationship between total flow and flow density includes:
establishing a relation between total flow and flow density on the whole fracturing fracture as follows:
after the discretization of the cells in the discretization model similar to the obtained seepage description, the following discretized relation is obtained:
it can be seen that the discretization relational expression obtained in the step S10 and the discretization model obtained in the step S9 represent (N +1) linear algebraic equations, and the unknown number in the equations is
(i＝1,2,…N+1)、
Is also (N +1), so closed solution is possible.
Further, the discretization model of the seepage description formula and the discretization relational expression of the total flow and the flow density are combined to obtain a solution model in a matrix form as follows:
[A][X]＝[B]
wherein [ A ]]Is a coefficient matrix, [ B ]]Is a matrix of constant terms, [ X ]]Is a matrix of unknown arrays, and
s11, solving the solution model to obtain the needed analog quantity;
wherein the desired analog quantity may be, for example, a dimensionless bottom hole pressure p in the real time domainwDFlow distribution q on fracturing fracturefD(xD,tD) And the like.
More specifically, the solving process may include:
solving the solution model by Gaussian elimination method to obtain parameters
(i＝1,2,…N+1)；
By the following Stehfest numberParameter G is obtained by inversionwD、qfDi(i＝1,2,…N+1)：
Wherein, VkIs an inversion coefficient, s, in a Stehfest numerical inversion formulakIs a Laplace variable in an inverse formula, and:
substituting into the Pedrosa transformation to obtain dimensionless bottom hole pressure p in the real time domainwDThe following are:
s12, drawing a simulation process based on each analog quantity, and carrying out simulation analysis;
preferably, the method comprises the steps of drawing a dimensionless bottom hole pressure curve and a derivative curve of the pressure-sensitive composite compact oil reservoir, dividing flowing stages, and analyzing the influence of variable diversion, stress sensitivity, the radius of an internal and external distinguishing interface, the fluidity of an internal and external area on the curve, the flow distribution characteristics of uniform flow and variable diversion and the like.
The invention further provides the following simulation experiment conditions:
setting a parameter rfD＝5,γ D0 or 0.03 or 0.06, [ omega ] 1, M [ 3 ], CfD(xD)＝18－12xD。
Based on the process of the specific embodiment, a typical curve of the pressure-sensitive composite compact reservoir variable diversion fracture well testing model can be drawn by utilizing numerical inversion programming, and the typical curve is composed of the pressure and derivative curves of the variable diversion fracture, as shown in the attached figure 3.
As can be seen from fig. 3, the seepage process of the diversion-induced fracture can be divided into 6 stages: (1) a section of bi-linear flow, the pressure of the sectionAnd the derivative curve appears as a straight line with a slope of "1/4"; (2) an inner region linear flow segment, the pressure and derivative curve of which is represented by a straight line with a slope of "1/2"; (3) a transition section; (4) the inner zone is a pseudo-radial flow segment (pseudo-radial flow around the fracture), the pressure derivative curve of which is a horizontal straight line segment with the height of 0.5; (5) the inner zone simulates a transition section of radial flow to the outer zone simulating radial flow; (6) the outer region simulates a radial flow section, when no stress sensitive effect is influenced, the pressure derivative curve of the section is a horizontal straight line section, and the height of the straight line section depends on the fluidity ratio M of the inner region and the outer region12The size of (2).
The typical curve obtained by the present invention was compared with the typical curve of a uniform flow pressure fracture, and the results are shown in fig. 4. Wherein, for the convenience of comparison, the dimensionless conductivity coefficient C of the non-uniform flow pressure fracturefDFrom the near well end (x)D0) linearly down to the far well end (x)D6 in 1) and an average conductivity of 12, and C of uniform flow pressure fracturefDThe values are equal, and the specific variation is as follows: cfD(xD)＝18－12xD(wherein: x)D0 to 1). As can be seen from the figure: the main difference of the typical curves in both cases is mainly represented by the bilinear flow phase, in which the variable conductance is shifted down with respect to the uniform flow, whether it be a pressure curve or a derivative curve.
According to the simulation condition, the influence of different parameters on the well testing model is further analyzed, and the following steps are carried out:
(1) influence of stress sensitivity
With reference to FIG. 5, the dimensionless stress sensitivity coefficient γ is shownDThe influence graph of the well testing typical curve of the composite tight oil reservoir can show that: the stress sensitive effect mainly affects the outer quasi-radial flow section, and when the stress sensitive effect exists, the pressure derivative of the section is expressed as an upward inclined straight line section, gammaDThe larger the slope of the straight line segment, the higher the position.
(2) Inside and outside distinguishing boundary radius rfDInfluence of (2)
Referring to FIG. 6, the inner and outer boundary radius rfDThe influence graph of the well testing typical curve of the composite tight oil reservoir can show that:rfDthe larger the transition from inner region pseudo-radial flow to outer region pseudo-radial flow occurs later.
(3) Ratio of fluidity between inner and outer regions M12Influence of (2)
Referring to the inside-outside zone fluidity ratio M shown in FIG. 712Influence diagram on the well test typical curve of the composite tight reservoir (for the convenience of analysis, influence of stress sensitive effect is neglected here), it can be seen that: m12When the flow rate is more than 1, the position of the outer zone quasi-radial flow section is higher than 0.5, and M12The larger, the higher the location; otherwise, M12When < 1, the position of the outer quasi-radial flow section is lower than 0.5, and M12The smaller, the lower the position; m12The outer zone pseudo-radial flow section is also a 0.5 height horizontal line at 1.
Referring to the dimensionless flow density profiles from the formation into different locations of the fracture at different times shown in fig. 8, it can be seen that: at different times, the dimensionless flow density distribution at different parts of the fracture is obviously different. At parameter setting rfD＝5,γD＝0.03,ω＝1,M＝3，T1＜T2＜T3Variable flow guide RfD(xD)＝18－12xDUniform flow RfD＝12，xDUnder 0-1, in the initial stage of flow, the flow density in the middle of the fracturing fracture is obviously higher than that in the two sides, but the specific situation is related to the type of fracturing flow guide, for uniform flow, the flow density is monotonically decreased from the middle of the fracturing fracture to the two ends, and for graphic flow guide, the flow density is first decreased from the middle of the fracturing fracture to the two ends and then slightly increased. Along with the lapse of flow time, the flow density in the middle of the fracturing seam is gradually reduced, and the flow density on both sides is gradually increased, so the difference between the flow density in the middle and the flow density on both sides is gradually reduced, and the flow distribution gradually tends to be balanced.
According to the simulation result, the method can accurately calculate the bottom hole pressure transient state of the fracturing well under the comprehensive consideration of the influence of variable diversion pressure cracks, reservoir stress sensitivity and radial heterogeneity on seepage, can obtain the flow distribution and the like flowing into different parts of the cracks from the stratum at different moments, and provides an important tool for fracturing well testing and productivity analysis under complex conditions.
The above examples are merely preferred embodiments of the present invention, and the scope of the present invention is not limited to the above examples. All technical schemes belonging to the idea of the invention belong to the protection scope of the invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention, and such modifications and embellishments should also be considered as within the scope of the invention.
Claims (10)
1. A tight oil reservoir fracturing well testing simulation method under complex conditions is characterized by comprising the following steps:
constructing a tight reservoir fracturing well test model simultaneously considering a fracturing fracture variable diversion effect, a reservoir stress sensitive effect and radial heterogeneity of a reservoir, and obtaining simulation results of bottom hole pressure change and liquid production conditions of different parts of the fracture through the well test model;
wherein the content of the first and second substances,
the variable diversion effect is a phenomenon that the permeability of the fracturing fracture changes along with different positions;
the stress sensitive effect refers to the phenomenon that the permeability of the oil reservoir changes along with the change of pressure;
the radial heterogeneity refers to the phenomenon that the permeability, porosity and overall compressibility of the inner region of the reservoir differ from those of the outer region.
2. The simulation method of claim 1, wherein the construction of the tight reservoir fracturing well test model comprises the following setting conditions:
the fracturing well is a production well of a pressure-sensitive composite compact oil reservoir, a stress sensitive effect exists in a reservoir layer of the pressure-sensitive composite compact oil reservoir, and the permeability of the oil reservoir in the reservoir layer follows an exponential change rule along with the change of pressure;
the seepage field of the pressure-sensitive composite compact oil reservoir is an isothermal seepage field;
the variable diversion effect exists in the fracturing fracture formed in the fracturing process of the fracturing well.
3. The simulation method of claim 1, wherein the constructing of the tight reservoir fractured well test model comprises:
establishing a line junction well seepage model of the pressure-sensitive composite compact oil reservoir, which comprises the following steps: giving a line-junction well model solution condition considering the stress sensitivity effect;
solving the seepage model of the junction well to obtain a perturbation solution of the junction well;
obtaining a pressure response solution of the pressure-sensitive composite compact oil reservoir caused by the fractured well by adopting a superposition principle based on the perturbation solution;
establishing a fracture seepage model considering the stress sensitivity effect and the variable diversion effect based on the flow characteristics in the fracture;
solving the fracture seepage model to obtain an integral equation describing seepage in the fracture model, namely a seepage calculation model in an integral form;
and solving an integral equation of the seepage by adopting a unit discrete method and a distributed integral method to obtain a solution matrix for describing the bottom hole pressure and the flow density.
4. The simulation method of claim 3, wherein the constructing further comprises: and performing dimensionless processing on the line junction well seepage model and the fracturing fracture seepage model to obtain corresponding dimensionless models, and then solving based on the dimensionless models.
5. The simulation method of claim 4, wherein the dimensionless model is solved by one or more of Pedrosa transformation, perturbation transformation, Laplace transformation, Bessel theory, double integration, integral equation unit dispersion, and fractional integration.
6. The simulation method of claim 4, further comprising:
solving the solution matrix through a Gaussian elimination method, and inverting the solution of the Laplace domain to a real time domain through a Stehfest numerical inversion method to obtain a solution of the real time domain;
further, the bottom hole pressure of the real time domain is substituted into the Pedrosa transformation, and the dimensionless bottom hole transient pressure in the real time domain considering the stress sensitivity effect is obtained.
7. The simulation method of claim 4, wherein the obtaining of the pressure response solution comprises: integrating the perturbation solution along the slit length by using a pressure drop superposition principle, and/or obtaining the seepage calculation model comprises the following steps: solving a dimensionless model of the obtained fracture seepage model; performing double integration on a second derivative term of the pressure with respect to the abscissa in the solving; and then, the obtained integral expression and the pressure response solution are combined to obtain the seepage calculation model in the integral form.
8. The simulation method of claim 3, wherein the solving of the percolation calculation model comprises:
obtaining a discretization model of the seepage calculation model;
the discretization model and a discretization relation between total flow and flow density are combined to obtain a solution matrix;
wherein the discretization comprises a process of cell discretization and distribution integration.
9. The simulation method of any one of claims 1 to 8, wherein the tight reservoir fracturing well test model comprises one or more of the following models:
a line junction well seepage model in an oil reservoir, comprising:
differential equation of zonal seepage in the reservoir:
differential equation of seepage in the outer region of the reservoir:
inner boundary conditions:
outer boundary conditions:
p2(r,t)|r→∞＝0 (4)，
continuous flow conditions of the inner and outer zone junction surfaces:
and (3) joining surface pressure equality condition:
initial conditions:
p1|t＝0＝p2|t＝0＝pi (7)，
wherein p is1Is an inner zone reservoir pressure, p2The oil reservoir pressure is in the outer region, r is the radial distance from a certain point in the oil reservoir to a line junction well, gamma is the stress sensitivity coefficient of the oil reservoir permeability, and piIs the original formation pressure, phi1Is the porosity of the reservoir in the inner zone2Outer region of porosity, Ct1As the combined compressibility of the inner zone reservoir, Ct2The comprehensive compression coefficient of the oil reservoir in the outer region, mu is the viscosity of crude oil, k1iInitial permeability, k, for the inner zone reservoir2iThe initial permeability of the oil reservoir in the outer region, e is a natural index, t is time, xi is infinitesimal quantity,
the yield of the line junction well, B the volume coefficient of the crude oil in the stratum, h the thickness of the oil deposit, pi the circumferential rate, rfThe radial distance of the connecting surface of the inner zone and the outer zone;
the dimensionless model of the seepage model of the line junction well comprises:
wherein the content of the first and second substances,
for dimensionless pressure, l ═ 1,2, w represent inner zone, outer zone, and bottom hole, respectively, and r representsD＝r/xfTo dimensionless radial distance、xfIs the half-length of the crack,
for a non-dimensional stress sensitivity coefficient,
in order to have a dimensionless time,
for dimensionless production of well convergence, M12Initial fluidity ratio of inner and outer zones
ω12Is the internal and external storage capacity ratio (phi)1Ct1/φ2Ct2)，rfDThe radius of the interface is distinguished for the inside and the outside;
a perturbative solution for a junction-line well seepage model, comprising:
wherein，
The solution of the dimensionless pressure of the oil deposit in the inner region after the Pedrosa transformation and the Laplace transformation,
the zero order perturbation solution of the dimensionless pressure of the oil deposit in the inner region after the Pedrosa transformation, the Laplace transformation and the perturbation transformation,
the dimensionless yield of the line-junction well after Laplace transformation is K0Is a modified Bessel function of the second kind, K, of order 01Is a modified Bessel function of order 1, second kind, sigma1Is an intermediate variable, σ2Is an intermediate variable, #BIs an intermediate variable, I0Is a 0 th order Bessel function of the first kind, I1Is a first class modified Bessel function of order 1, u is a Laplace variable, xDIs a dimensionless abscissa, x, of the field pointwDDimensionless abscissa, y, of source sinkDAs a dimensionless ordinate, y, of the field pointwDIs a dimensionless ordinate of the source sink;
the pressure response solution:
wherein the content of the first and second substances,
dimensionless flux density flowing directly into the fracture from the formation;
a fracture seepage model comprising:
wherein p isfIs the fracture pressure, x is the abscissa, y is the ordinate, WfWidth of the press crack, kfAs the permeability of the fracturing fracture, qfIs the density of flow directly into the fracture from the formation;
a dimensionless model of a fracture-infiltration model, comprising:
wherein the content of the first and second substances,
in order to have a dimensionless fracture pressure,
q is dimensionless fracture conductivityfD(xD,tD)＝2qf(x,t)xfQ is dimensionless flow density, q is oil well production;
seepage calculation model:
wherein the content of the first and second substances,
a bottom hole pressure solution of a Laplace space is obtained, alpha is an integral variable, and v is an inner layer integral upper limit in double integration;
discretization model of seepage calculation model:
where N is the number of discrete units of dimensionless fracture half-length, Δ xDFor each discrete element length, j is the discrete element ordinal number, xDjIs the end point of the jth discrete cell,
is the middle point of the j discrete unit, i is the integer pointer variable in the accumulation formula;
the relation model and the discretization model of the total flow and the flow density are as follows:
and (3) final solution model:
[A][X]＝[B] (33)
wherein [ A ] is a coefficient matrix, [ B ] is a constant term array matrix, and [ X ] is an unknown array matrix.
10. The simulation method of claim 9, wherein the simulation process based on the tight reservoir fracturing well test model comprises:
solving the solution model by Gaussian elimination method to obtain transformed parameters
The corresponding parameter G is obtained by the following Stehfest numerical inversion formulawDAnd q isfDi(i＝1,2,…N+1)：
Wherein, VkIs an inversion coefficient, s, in a Stehfest numerical inversion formulakIs a Laplace variable in an inverse formula, and:
substituting the corresponding parameters into the Pedrosa transformation to obtain the dimensionless bottom hole pressure p in the real time domainwDThe following are:
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