CN116861818B - Multilayer gas reservoir pressure-dividing and pressure-mixing test well simulation method under complex condition - Google Patents
Multilayer gas reservoir pressure-dividing and pressure-mixing test well simulation method under complex condition Download PDFInfo
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Abstract
The invention discloses a multi-layer gas reservoir pressure-dividing and pressure-mixing test well simulation method under complex conditions, which comprehensively considers complex conditions that each gas layer is not fully pressure-opened, the lengths of each gas layer pressure crack are equal or unequal, the flow density of each pressure crack can change along with the position and time, the thickness of each gas layer storage layer can be unequal, the outer boundary radius of each gas layer can be unequal, the original stratum pressure of each gas layer can be unequal, the seepage parameters of each gas layer hole are equal or unequal and the influence of gas high-pressure physical properties, creatively establishes a pressure-dividing and pressure-mixing test well model under the complex conditions, successfully solves the problem by a semi-analytic method, and obtains a high-quality bottom hole pressure dynamic curve based on a solving result. Compared with the similar inventions, the invention has more consideration factors, stronger applicability and more compliance with actual complex conditions; compared with the finite element method and the finite difference method, the method does not need to carry out unit dispersion on the reservoir, only needs to carry out small quantity of sectional dispersion on the fracture, and is accurate and efficient in calculation.
Description
Technical Field
The invention belongs to the technical field of well testing simulation methods, and particularly relates to the technical field of multilayer gas reservoir pressure-dividing and pressure-mixing test well simulation methods.
Background
The well testing technology of the oil and gas well is a very important technology for understanding the oil and gas reservoir and is known as an 'eye for developing the oil and gas reservoir' by the petroleum industry. Through well testing, some important dynamic parameters of the oil and gas reservoir buried in the ground, such as stratum permeability, well bore pollution coefficient, skin coefficient, seam half length of a fracture, and the like, can be obtained. Establishing a reasonable unstable well test model and accurately simulating the bottom hole pressure dynamics of an oil and gas well are the basis and premise of a well test analysis technology.
For some gas reservoirs with extremely low permeability, even though the gas reservoirs are fractured, the single-layer productivity is low, and the economic benefit of layered gas production is poor, so that a combined production mode is often adopted. In the fields of analytical methods and semi-analytical methods, the current well test models and corresponding dynamic simulation of bottom hole pressure (i.e., well test simulation) are more for single-layer reservoirs and relatively less for multi-layer reservoirs.
In the existing few well test simulation researches on multi-layer oil and gas reservoirs, most of the well test simulation researches are aimed at uncracked oil and gas wells, and few of the well test simulation researches are aimed at multi-layer oil and gas reservoir fracturing well test simulations. For example Mao Zhenglin ("tight sandstone multi-layer gas reservoir seepage characteristics and yield splitting method research [ D ]", southwest petroleum university's institute of major, 2021:11-20), wang Xiaolu ("complex reservoir fracturing well unstable seepage model and well test analysis method research [ D ]", southwest petroleum university's doctor's institute of major, 2015:32-44), zhang Lu ("multi-layer fracturing well test analysis method research [ D ]", southwest petroleum university's institute of major, 2011:12-21) are studied for multi-layer fracturing gas reservoir fracturing well test simulation, a well test model is established, and bottom hole pressure dynamics thereof are obtained. However, this technique only considers the situation that each layer is fully pressed open, and does not consider the influence of the degree of the pressing open of each layer on the well test simulation, which is not in accordance with some practical situations. In the tight hydrocarbon reservoir well test and productivity prediction method proposed by the congratulation company ("tight hydrocarbon reservoir well test and productivity prediction D". Shuoshi university of science and technology, 2019:31-36), a source function in three directions is established first, then a Newman product formula is utilized to obtain a multi-layer partial pressure combined production well bottom pressure expression under the influence of incomplete pressure opening of each layer, and dynamic calculation and characterization of bottom hole pressure are realized by utilizing the expression. However, this technique still has some of the following disadvantages: (1) The technology does not consider the uneven distribution of the flow density of each laminated crack and the change of the flow density along with time, and is not in accordance with the actual exploitation condition; (2) The final solution obtained by the technology is a combined form formed by an error function, a time domain integral and an infinite series, which is not beneficial to calculation; (3) The technology only considers the situation that the original stratum pressures of all layers are equal, and fails to consider the situation that the original stratum pressures of all layers are not equal in practice. Other researches on multi-layer oil and gas reservoir fracturing well test simulation are carried out, and the bottom hole pressure dynamic state is obtained, but only the situation that each layer is fully fractured is considered, the influence of different fracturing degrees of each layer on the well test simulation is not considered, and the situation is not consistent with many actual exploitation situations.
From the above, there is still a lack of well test simulation methods which are strict, accurate and easy to calculate and are consistent with the actual complex production conditions.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a multi-layer gas reservoir pressure-dividing pressure-testing well simulation method which comprehensively considers the complex fracturing well conditions of incomplete fracturing of each gas layer, unequal lengths of pressure cracks of each gas layer, variable flow densities of each pressure crack along with position and time, unequal thickness of each gas layer reservoir layer, unequal outer boundary radius of each gas layer, unequal original stratum pressure of each gas layer, unequal permeability parameters of each gas layer and influence of gas high-pressure physical properties and can be solved by adopting a semi-analytic method.
The technical scheme of the invention is as follows:
A multi-layer gas reservoir partial pressure fit test well simulation method under complex conditions comprises the following steps:
S1, constructing a multi-layer gas reservoir pressure-separation gas production well physical model under the complex condition (namely, comprehensively considering incomplete fracturing of each gas layer, unequal lengths of pressure cracks of each gas layer, variable flow densities of each pressure crack along with position and time, unequal thickness of each gas layer, unequal radius of the outer boundary of each gas layer, unequal pressure of the original stratum of each gas layer, unequal pore permeation parameters of each gas layer and high-pressure physical influences of gas);
s2, establishing a multilayer gas reservoir dimensionless point sink seepage model aiming at the multilayer gas reservoir split-pressure gas production well physical model established in the S1, wherein the method comprises the following steps of:
s21, aiming at the physical model of the multilayer gas reservoir pressure-dividing gas production well in the S1, adopting a seepage mechanics theory, and establishing a multilayer gas reservoir dimensional point seepage model in a pressure form;
s22, converting the multi-layer gas reservoir with factor point sink seepage model in the pressure form in S21 into a multi-layer gas reservoir with factor point sink seepage model in the pseudo-pressure form;
s23, properly defining a set of dimensionless quantities, and converting the multi-layer gas reservoir dimensionless point sink seepage model in the pseudo-pressure form in S22 into a multi-layer gas reservoir dimensionless point sink seepage model;
S3, aiming at the multilayer gas reservoir dimensionless point sink seepage model in the S2, obtaining a dimensionless pressure solution of the multilayer gas reservoir dimensionless point sink seepage model in a Laplace domain, namely a first dimensionless seepage pressure solution by comprehensively applying Laplace transformation and Fourier finite cosine integral transformation;
S4, based on the first dimensionless seepage pseudo-pressure solution in the S3, integrating along the height direction of each gas layer pressure crack in the multilayer gas reservoir pressure-dividing gas production well physical model by utilizing an integration form superposition principle to obtain a dimensionless pseudo-pressure solution caused by a j-th layer vertical line in the multilayer gas reservoir, namely a second dimensionless seepage pseudo-pressure solution;
s5, based on the second dimensionless seepage pressure-planning solution in S4, integrating the seam length of the pressure crack in the multi-layer gas reservoir pressure-dividing gas production well model by utilizing an integral form superposition principle to obtain a dimensionless pressure-planning solution containing the to-be-determined dimensionless flow density parameter of the multi-layer gas reservoir pressure-dividing gas production well model in Laplace space, namely a third dimensionless seepage pressure-planning solution;
S6, carrying out sectional discrete on the crack along the crack length, carrying out sectional integration on the third dimensionless seepage pseudo-pressure solution in S5 on discrete units, summing up, taking calculated pressure points on each discrete unit node, and simultaneously utilizing the characteristics of infinite diversion crack, thereby obtaining the undetermined dimensionless flow density of the ith discrete unit on the jth crack in Laplace space The j-th gas layer dimensionless bottom hole s pseudo pressure/>(I.e., third dimensionless osmotic bottom-hole sub-pressure) of M x N linear algebraic equations, i.e., a first set of linear algebraic equations;
S7, redefining the j-th gas layer dimensionless bottom hole S quasi-pressure psi wsDj (namely redefining the third dimensionless seepage bottom hole quasi-pressure in S6) by replacing the j-th gas layer original quasi-pressure psi Ij with the reference quasi-pressure psi r, wherein the redefined quasi-pressure is called j-th gas layer dimensionless bottom hole quasi-pressure psi wDj (namely fourth dimensionless seepage bottom hole quasi-pressure), and M-1 linear equations of the third algebraic seepage bottom hole quasi-pressure of different discrete unit nodes of different pressure cracks, namely a second linear algebraic equation set, are obtained according to the relation between the third dimensionless seepage bottom hole quasi-pressure of each discrete unit node of any pressure crack in S6 and the fourth dimensionless seepage bottom hole quasi-pressure of the fourth dimensionless seepage bottom hole quasi-pressure;
S8, obtaining a dimensionless flow density normalization equation of 1 gas well according to the relation between the flow density of the fracturing cracks of different discrete units and the yield of the gas well;
S9, combining the first linear algebraic equation set in S6, the second linear algebraic equation set in S7 and a dimensionless flow density normalization equation of the gas well to obtain a linear sealing matrix for well testing simulation, wherein the linear sealing matrix is composed of (M multiplied by N+M) linear algebraic equations and (M multiplied by N+M) unknowns, and the linear sealing matrix is solved by using Stehfest numerical inversion and Gauss elimination, so that a well testing simulation result of the multi-layer gas reservoir partial pressure gas production well under the complex condition is obtained.
Compared with the prior art, the invention has the following beneficial effects:
(1) According to the invention, various factors such as incomplete pressure opening of each gas layer, unequal lengths of each gas layer pressure crack, changes of flow density of each pressure crack along with position and time, unequal thickness of each gas layer reservoir, unequal radius of the outer boundary of each gas layer, unequal pressure of each gas layer original stratum, unequal pore permeation parameters of each gas layer and influence of gas high-pressure physical properties on simulation of a multi-layer gas reservoir pressure-separation pressure-testing well are comprehensively considered, so that the applicability is stronger, and the multi-layer pressure-separation pressure-testing well is more consistent with actual complex mining conditions;
(2) The method does not need to carry out unit dispersion on the reservoir, only needs to carry out small quantity of sectional dispersion on the fracture, does not have the problems of matrix stability, numerical dispersion and the like of a finite element method and a finite difference method, is accurate and efficient in calculation, and is more suitable for well testing simulation under the complex condition.
Drawings
FIG. 1 is a physical model of a multi-layer gas reservoir split-pressure gas production well according to an embodiment.
FIG. 2 is a schematic illustration of the end points and nodes of the j-th fracture after the fracture is discretized in the embodiment.
FIG. 3 is a log plot of a multi-layer gas reservoir split pressure test well for the complex case obtained in the present embodiment.
Fig. 4 is a schematic diagram of a first linear flow segment according to an embodiment.
Fig. 5 is a schematic diagram of an early vertical pseudo-radial flow segment in an embodiment.
Fig. 6 is a schematic diagram of a second linear flow segment according to an embodiment.
Fig. 7 is a schematic diagram of the late planar quasi-radial flow section in the embodiment.
Fig. 8 is a graph showing the effect of the jth gas layer dimensionless primary pressure-planning psi IDj on a log-log curve in the embodiment.
FIG. 9 is a graph showing the effect of the j-th gas layer pressure opening degree h wj/hj on a double logarithmic curve in the embodiment.
FIG. 10 is a graph showing the effect of the jth gas laminated crack length L fj on a double logarithmic curve in the preferred embodiment.
FIG. 11 is a graph showing the effect of the j-th gas layer thickness h j on a log-log curve in the embodiment.
FIG. 12 is a graph of the effect of formation coefficients χ 1 and χ 3 on a log curve, according to one embodiment.
Fig. 13 is a graph showing the effect of the storage ratios ω 1 and ω 3 on the double logarithmic curve in the embodiment.
Fig. 14 is a graph showing the effect of the storage ratios ω 2 and ω 3 on the double logarithmic curve in the embodiment.
FIG. 15 is a graph showing the effect of the ratio k vj/khj of vertical permeability to horizontal permeability on a log-log curve for the j-th gas layer according to the example.
Detailed Description
The present invention will be described in detail with reference to the following examples and drawings, but it should be understood that the examples and drawings are only for illustrative purposes and are not intended to limit the scope of the present invention in any way. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In some specific embodiments, the multi-layer gas reservoir partial pressure production test well method under the complex condition specifically comprises the following steps:
s1, constructing a complex situation, namely comprehensively considering a multi-layer gas reservoir pressure-dividing gas production well physical model under the influence of incomplete fracturing of each gas layer, unequal length of each gas layer fracture, variable flow density of each fracture along with position and time, unequal thickness of each gas layer reservoir, unequal radius of each gas layer outer boundary, unequal pressure of each gas layer original stratum, unequal pore permeation parameters of each gas layer and high pressure physical property of gas.
The physical model of the multilayer gas reservoir partial pressure gas production well is constructed as follows:
Referring to fig. 1, the gas well is located in a multi-layer gas reservoir containing M gas layers, with good barriers between the gas layers, the gas well is drilled through each gas layer, each gas layer is subjected to layered fracturing by the gas well, and then natural gas of each gas layer is produced by the gas well, namely, the partial pressure is produced.
The gas well was produced with a constant surface gas yield q sc and the model had the following characteristics:
(1) Each gas layer can not be completely pressed and cracked, namely the pressing and cracking thickness can be different from the thickness of the gas layer;
(2) The length of each air fracture may be unequal;
(3) The air lamination cracks are not parallel and can form a certain included angle
(4) The thickness of each gas reservoir layer can be unequal;
(5) The j-th gas layer horizontal outer boundary radius r ej (where j=1, 2,3 … M) may not be equal;
(6) Each crack is an infinite guide slit, and the flow density of the crack changes with the position and time;
(7) The temperature of the gas reservoir is T, and isothermal seepage is carried out;
(8) The jth gas formation has an original formation pressure of p Ij, (where j=1, 2,3 … M);
(9) The pore permeation parameters of each air layer can be unequal;
(10) Considering the influence of high-pressure physical properties of gas;
(11) The percolation flow satisfies darcy's law.
Wherein: the thickness of the jth air layer is h j, the radius of the outer boundary of the jth air layer is r ej, the thickness of the jth air layer is h wj, and the vertical coordinate of the horizontal axis which equally divides the height of the jth air layer crack into an upper part and a lower part is z mj.
Preferably, in order to facilitate modeling and solving, a respective independent coordinate system is established for each gas layer, and the specific mode is as follows: each gas formation takes the center position of the well shaft of the gas well as an origin point (x j＝0,yj =0; wherein x j is the x coordinate of any field point in the jth gas formation; y j is the y coordinate of any field point in the jth gas formation), vertically takes the top boundary of each gas formation as the starting point of the vertical direction of the gas well (z j =0; wherein z j is the z coordinate of any field point in the jth gas formation), and vertically downward is the positive direction of z j.
Specifically, referring to fig. 1, in the multi-layer gas reservoir pressure-separation gas production well model, the multi-layer gas reservoir contains M gas layers with thickness h 1,h2,…,hj,…hM respectively, each gas layer comprises a closed top surface, a closed bottom surface, an inner boundary (fracture) and an outer boundary, wherein any gas layer j contains a fracture with pressure-separation thickness, namely, the fracture height h wj, a gas well shaft vertically drills through each gas layer from top to bottom, and forms the fracture symmetrically distributed on two sides of the shaft in each gas layer, the lengths of the different gas lamination fractures can be unequal, a three-dimensional coordinate system is established with the midpoint of the top boundary of each gas layer as the coordinate system origin point j(xj＝0,yj＝0,zj =0 of the gas layer and the vertical direction as the positive direction of the z j axis, and then the coordinate of the fracture end of each gas layer on the x axis is (-L fj,Lfj), namely, the half length of the j lamination fracture is L fj, and the length is 2L fj.
S2, establishing a multilayer gas reservoir dimensionless point sink seepage model aiming at the multilayer gas reservoir split-pressure gas production well physical model established in the S1, wherein the method comprises the following steps of:
s21, aiming at the physical model of the multilayer gas reservoir partial pressure gas production well in the S1, adopting a seepage mechanics theory, and establishing a multilayer gas reservoir dimensionless point seepage model in a pressure form:
Assuming that any point in the j-th gas layer is converged at (x wj,ywj,zwj) in the multi-layer gas reservoir partial pressure gas production well model, the ground gas production rate is According to the seepage mechanics theory, a multi-layer gas reservoir with a pressure form can be established as follows:
① Pressure-form multi-layer gas reservoir seepage differential equation:
Wherein: r j is the distance between any field point in the j-th gas layer and the point sink (x wj,ywj,zwj) in the radial direction of the plane after converting the rectangular coordinate system into the cylindrical coordinate system, M; x j is the x coordinate, m of any field point in the j-th gas layer; y j is the y coordinate, m, of any field point in the j-th gas layer; z j is the z coordinate, m, of any field point in the j-th gas layer; x wj is the x coordinate, m, of any point sink in the j-th gas layer; y wj is the y coordinate, m, of any sink in the j-th gas layer; z wj is the z coordinate, m, of any point sink in the j-th gas layer; p j is the pressure at any field point in the j-th gas layer, pa; k hj is the j-th gas layer horizontal permeability, m 2;kvj is the j-th gas layer vertical permeability, m 2;μ(pj) is the j-th gas layer gas viscosity, pa·s, as a function of the j-th gas layer pressure p j; z (p j) is a jth gas layer gas deviation factor, dimensionless, as a function of a jth gas layer pressure p j; c g(pj) is the j-th gas layer gas compression coefficient as a function of the j-th gas layer pressure p j, pa -1;φj is the porosity of the j-th gas layer, dimensionless; t is the production time, s.
② Initial conditions:
pj(rj,zj,t)|t＝0＝pIj (2)
Wherein: p Ij is the original formation pressure, pa, of the jth gas formation.
③ Outer boundary conditions:
Wherein: r ej is the j-th gas layer horizontal outer boundary radius, m; h j is the j-th gas layer thickness, m.
④ Inner boundary conditions: assume that the ground production at any one of the junctions at point (x wj,ywj,zwj) in the jth zone isThe internal boundary conditions are:
Wherein: epsilon is the vertical infinitesimal length, m; sigma is the radial infinitesimal length, m; p sc is ground atmospheric pressure, pa; t is the gas reservoir temperature, K; t sc is the ground temperature, K; Is the ground production of any point sink where the jth gas layer is located at point (x wj,ywj,zwj), m 3/s.
The formulas (1) - (6) form the multi-layer gas reservoir with the pressure form as a multi-point sink seepage model.
S22, converting the multi-layer gas reservoir with factor point sink and permeate model in the pressure form in S21 into a multi-layer gas reservoir with factor point sink and permeate model in a pseudo-pressure form:
The following pseudo pressures were introduced:
Wherein: psi j is the jth air pressure, pa/s; p 0 is the reference pressure, pa; psi Ij is the original pseudo pressure of the jth gas layer, pa/s.
The multi-layer gas reservoir induced-point sink-flow model of the pressure form can be converted into the multi-layer gas reservoir induced-point sink-flow model of the pseudo-pressure form as follows:
① Layer gas reservoir seepage differential equation:
Wherein: mu gj is the gas viscosity at the original pressure of the jth gas layer, i.e. mu gj＝μ(pIj),Pa·s;Cgj is the gas compression coefficient at the original pressure of the jth gas layer, i.e. C gj＝Cg(pIj),Pa-1.
② Initial conditions:
ψj(rj,zj,t)|t＝0＝ψIj (9)
③ Outer boundary conditions:
④ Inner boundary conditions:
the equations (8) - (13) form a pseudo-pressure multi-layer gas reservoir induced point sink-seepage model.
S23, properly defining a set of dimensionless quantities, and converting the multi-layer gas reservoir dimensionless point sink-seepage model in the pseudo-pressure form in S22 into a multi-layer gas reservoir dimensionless point sink-seepage model:
The dimensionless quantity is specifically defined as follows:
(1) The j-th air layer dimensionless s-pseudo pressure ψ sDj (i.e., the dimensionless pseudo pressure defined on ψ j by the original pseudo pressure ψ Ij of each layer, which can be referred to herein as first/second/third dimensionless seepage pseudo pressure):
Wherein: Δψ sj is the corresponding s-pseudo pressure drop of ψ j, Δψ sj＝ψIj-ψj; (khh) t represents the total formation factor, And ψ Ij is the original pseudo pressure of the j-th layer.
(2) The j-th gas layer dimensionless bottom hole s-pressure-fit wsDj (i.e., the original pressure-fit psi Ij of each layer is used to define a dimensionless pressure-fit for psi j, which herein may represent a first/second/third dimensionless seepage bottom-hole pressure-fit):
Wherein: Δψ wsj is the bottom hole s pseudo pressure drop corresponding to ψ wj and Δψ wsj＝ψIj-ψwj;ψwj is the wellbore pseudo pressure of the j-th layer.
(3) The j-th gas layer dimensionless pseudo pressure ψ Dj (i.e., the dimensionless pseudo pressure defined on ψ j with reference pseudo pressure ψ r, i.e., the fourth dimensionless seepage pseudo pressure):
Wherein: Δψ j is the bottom hole pseudo pressure drop corresponding to ψ j; Δψ j＝ψr-ψj,ψr is the reference pressure, taken as the highest value max (ψ Ij) in the original pseudo pressures of each gas layer.
(4) The j-th gas layer dimensionless bottom-hole pseudo pressure ψ wDj (i.e., the dimensionless bottom-hole pseudo pressure defined for ψ wj by using the reference pseudo pressure ψ r, that is, the fourth dimensionless seepage bottom-hole pseudo pressure):
Wherein: Δψ wj is the bottom hole pseudo pressure drop corresponding to ψ wj; Δψ wj＝ψr-ψwj,ψr is the reference pressure, taken as the highest value max (ψ Ij) in the original pseudo pressures of each gas layer.
(5) Dimensionless quantity x Dj of x j:
xDj＝xj/Lfj
(6) Dimensionless quantity x wDj of x wj:
xwDj＝xwj/Lfj
(7) Dimensionless quantity y Dj of y j:
yDj＝yj/Lfj
(8) Dimensionless quantity y wDj of y wj:
ywDj＝ywj/Lfj
(9) Dimensionless quantity z Dj of z j:
zDj＝zj/hj
(10) Dimensionless quantity z wDj of z wj:
zwDj＝zwj/hj
(11) z mj (midpoint coordinates of the jth gas layer pressure-open thickness segment), dimensionless quantity z mDj:
zmDj＝zmj/hj
(12) Dimensionless quantity r Dj of r j:
(13) The dimensionless quantity r eDj of r ej (j-th gas layer horizontal outer boundary dimensionless radius):
(14) Dimensionless quantity of σ σ D:
σD＝σ/Lfj
(15) Dimensionless quantity of epsilon D:
εD＝ε/Lfj
(16) Dimensionless quantity L fDj of L fj:
(17) Dimensionless quantity h Dj of h j:
(where r w is the wellbore radius, m)
(18) Dimensionless quantity h wDj of h wj:
(19) Dimensionless quantity t D of t (dimensionless production time t D):
Wherein: (phi hC g)t represents the total stored Rong Neng force, Mu g1 is the gas viscosity at the original pressure of the 1 st gas layer, pa.s.
(20)Dimensionless quantity/>
(21) The formation coefficient of the j-th gas layer is x j:
(22) The j-th air layer elastic storage capacity ratio omega j:
(23) The j-th air layer viscosity ratio mu Rj:
the subscript "j" in the above expression represents the jth gas layer.
By properly defining the set of dimensionless numbers, the correctness of the model is ensured, and meanwhile, the model is simplified and solved.
Under the above dimensionless quantity definition mode, the formulas (8) - (13) are converted into the following multilayer gas reservoir dimensionless point sink-seepage model:
Equations (14) - (19) are multi-layer gas reservoir dimensionless point sink-flow models.
S3, aiming at the multilayer gas reservoir dimensionless point sink seepage model in S23, obtaining a dimensionless pressure solution of the multilayer gas reservoir dimensionless point sink seepage model in a Laplace domain, namely a first dimensionless seepage pressure solution by comprehensively applying Laplace transformation and Fourier finite cosine integral transformation;
The following Laplace transform was introduced:
Wherein: s is a Laplace variable based on t D, dimensionless; a Laplace transform representing ψ sDj; /(I) The Laplace transform of ψ Dj is represented.
The following Fourier finite cosine integral transform was introduced with respect to z Dj:
fourier limited cosine integral positive transform:
wherein: f is a Fourier finite cosine integral positive transform operator; Representing/>, after Fourier finite cosine integral positive transformation Cos () is a cosine function; n is a non-negative integer, n=0, 1,2,3, …; pi is the circumference ratio.
Fourier limited cosine integral inverse transform:
Wherein F -1 is a Fourier finite cosine integral inverse transformation operator; n (N) is an integral function related to a non-negative integer N, expressed as follows:
After Laplace transformation and Fourier finite cosine integral transformation, solving dimensionless fit pressure solutions of the multi-layer gas reservoir dimensionless point sink-flow model (namely formulas (14) to (19)) in a double transformation domain:
Wherein: Is transformed by Laplace/> K 0 () is a second class zero-order deformed bessel function; k 1 () is a second class of first order deformed bessel function; i 1 () is a first-order deformed bessel function; i 0 () is a zero-order deformed bessel function of the first type; alpha n,j is an intermediate variable, whose expression is: /(I)(Where n=0, 1,2,3, …).
Inverse transform is performed on equation (25) using Fourier limited cosine integral inverse transform equation (23), to obtain:
equation (26) is a first dimensionless seepage pseudo-pressure solution of the multi-layer gas reservoir dimensionless point sink seepage model in the Laplace domain.
S4, based on the first dimensionless seepage simulated pressure solution in the S3, integrating along the height direction of each gas layer pressure crack in the multilayer gas reservoir pressure-dividing gas production well physical model by utilizing an integration form superposition principle to obtain a dimensionless simulated pressure solution caused by a j-th layer vertical line in the multilayer gas reservoir, namely a second dimensionless seepage simulated pressure solution;
Based on the obtained first dimensionless seepage pseudo-pressure solution, namely formula (26), integrating along the height direction of each gas layer pressure crack in the multilayer gas reservoir pressure-dividing gas production well physical model by utilizing the superposition principle of an integral form to obtain a dimensionless pseudo-pressure solution caused by a j-th layer vertical line in the multilayer gas reservoir, namely a second dimensionless seepage pseudo-pressure solution:
Wherein: the ψ sDj is the j-th air layer dimensionless s-pseudo pressure, which is a dimensionless version of ψ sj where it represents the second dimensionless osmotic pressure pseudo pressure defined for ψ j using the original pseudo pressures of the layers ψ Ij; z mj is a vertical coordinate of a horizontal axis which equally divides the height of the fracture into an upper part and a lower part on the j-th fracture, and m; is transformed by Laplace For the yield of the j-th gas layer vertical manifold, m 3/s;a1 is an intermediate variable, dimensionless, expressed as follows:
The integration result of formula (27) is:
Wherein: z mDj is the dimensionless quantity, dimensionless, of z mj.
Equation (28) is a second dimensionless simulated pressure solution for seepage caused by the vertical manifold of the jth gas layer in the multi-layer gas reservoir.
S5, based on the second dimensionless seepage pressure-planning solution in S4, integrating the seam length of the pressure crack in the multi-layer gas reservoir pressure-dividing gas production well model by utilizing an integral form superposition principle to obtain a dimensionless pressure-planning solution containing the to-be-determined dimensionless flow density parameter of the multi-layer gas reservoir pressure-dividing gas production well model in Laplace space, namely a third dimensionless seepage pressure-planning solution;
The obtained third dimensionless osmotic pseudo-pressure solution is as follows:
Wherein: Qfj (xwDj, tD) after Laplace transformation, qfj (xwDj, tD) is the flow density on the j-th fracture, m 3/(m.s);a2 is an intermediate variable, dimensionless, expressed as follows:
Definition of fracture dimensionless flow density (in Laplace space):
Wherein: qfDj (xwDj, tD) after Laplace transformation; qfDj (xwDj, tD) is the dimensionless flow density on the j-th fracture.
Selecting a position point (field point) of the required calculated pressure on a fracture surface, taking the characteristics of infinite diversion fracture into consideration, and taking the formula (30) into the formula (29), and converting the formula (29) into:
wherein: c 1 is an intermediate variable, dimensionless, expressed as follows:
since the dimensionless flow density within the same layer is symmetric about the wellbore, equation (31) can be rewritten as:
Wherein: c 2 is an intermediate variable, dimensionless, expressed as follows:
/>
equation (33) is also a third dimensionless osmotic pseudo-pressure solution. It can be seen that the formula contains the dimensionless flow density of the jth crack in Laplace space Due to/>Is an unknown function of dimensionless coordinates xwDj, so the bottom hole pressure cannot be completely solved by using the solution.
S6, carrying out sectional discrete on the crack along the crack length, carrying out sectional integration on the third dimensionless seepage pseudo-pressure solution in S5 on discrete units, summing up, taking calculated pressure points on each discrete unit node, and simultaneously obtaining the undetermined dimensionless flow density of the ith discrete unit on the jth crack in Laplace space by utilizing the characteristics of infinite diversion crackThe j-th gas layer dimensionless bottom hole s pseudo pressure/>(I.e., third dimensionless osmotic bottom sub-pressure) of M x N linear algebraic equations, i.e., a first set of linear algebraic equations:
Referring to fig. 2, each gas-layer fracture half length is discretized into N discrete units, each fracture half length has n+1 end points and N nodes, each node refers to a midpoint corresponding to each discrete unit, (K=1, 2, … N, j=1, 2, … M) represents the dimensionless abscissa of the kth node on the jth fracture, x Dk,j (k=1, 2, … n+1, j=1, 2, … M) represents the dimensionless abscissa of the kth endpoint on the jth fracture, and the dimensionless length Δx D =1/N of each segment. Assuming that the flow density distribution is uniform over the same discrete unit, equation (33) may be transformed into a discrete summation form as follows:
wherein: c 3 is an intermediate variable, dimensionless, expressed as follows:
/>
Wherein: Is Laplace transform of q fDi,j; q fDi,j is the dimensionless flow density on the fracture discrete units (i, j); x Di,j and x Di+1,j are the dimensionless x coordinates of the left and right endpoints of the ith discrete element of the jth gas laminated slit, respectively.
When the field point in the formula (35) is taken at the node of the fracture, there is:
Wherein: Is Laplace transform of ψ wsDj; psi wsDj is the jth gas formation dimensionless bottomhole s-pressure defined by the jth gas formation original pressure, here the third dimensionless seepage bottomhole pressure.
From the formulae (35) and (37)
Wherein: c 3 is an intermediate variable, dimensionless, and has the same expression as before.
K in formula (38) and intermediate variable c 3 is the node number on each fracture, k=1, 2,3, … N; j is the number of each crack, j=1, 2,3, … M, so equation (38) represents m×n linear algebraic equations, i.e. the first linear algebraic equation set. Unknowns in equation (38)And/>(Where i=1, 2,3, … N; j=1, 2,3, … M) total (mxn+m), and since the number of linear algebraic equations is smaller than the number of unknowns, the equations are temporarily not solved, and other equations have to be combined.
S7, redefining the j-th layer of dimensionless bottom hole S quasi-pressure psi wsDj (namely redefining the third dimensionless seepage bottom hole quasi-pressure in S6) by replacing each layer of original quasi-pressure psi Ij with a reference quasi-pressure psi r, wherein the redefined quasi-pressure is called j-th layer of dimensionless bottom hole quasi-pressure psi wDj (namely fourth dimensionless seepage bottom hole quasi-pressure), and obtaining M-1 linear equations of the third dimensionless seepage bottom hole quasi-pressure of different discrete unit nodes of different pressure cracks according to the relation between the third dimensionless seepage bottom hole quasi-pressure of each discrete unit node of any pressure crack in S6 and the fourth dimensionless seepage bottom hole quasi-pressure of the fourth dimensionless seepage bottom hole quasi-pressure, namely a second linear algebraic equation set, wherein the fourth dimensionless seepage bottom hole quasi-pressure of different discrete unit nodes of different pressure cracks in infinite diversion mode is equal:
The original pseudo-pressure psi Ij of each layer is replaced by a reference pseudo-pressure psi r to redefine the third non-dimensionless seepage bottom-hole pseudo-pressure psi wsDj, the redefined pseudo-pressure is called a fourth non-dimensionless seepage bottom-hole pseudo-pressure psi wDj, and the relation between the redefined pseudo-pressure and the redefined pseudo-pressure is:
ψwDj＝ψIDj+ψwsDj (39)
wherein, ψ IDj is the j-th layer dimensionless original pseudo pressure.
In the infinite diversion mode of the pressure cracks, the fourth dimensionless seepage bottom pseudo pressures of different discrete unit nodes of different pressure cracks are all equal, so that:
ψwD1＝ψwD2＝…＝ψwDM (40)
And (3) a combination formula (39) and a formula (40) to obtain:
ψID1+ψwsD1＝ψID2+ψwsD2＝…＝ψIDM+ψwsDM (41)
Laplace transformation is performed on the formula (41) to obtain:
Wherein: (where j=1, 2, …, M) is ψ wsDj after Laplace transform.
Equation (42) represents (M-1) linear algebraic equations, i.e., the second set of linear algebraic equations.
S8, obtaining a dimensionless flow density normalization equation of 1 gas well according to the relation between the flow density of the fracturing cracks of different discrete units and the yield of the gas well;
The relationship between the different discrete unit fracture flow densities q fi,j and the gas well surface gas production q sc is as follows:
Wherein: q fi,j is the flow density on the fracture discrete element (i, j); Δl fj is the length of the fracture discrete element (i, j).
Formula (43) can be in dimensionless form:
Wherein: q fDi,j is the dimensionless flow density on the fracture discrete unit (i, j), which is defined by formula (45); deltaL fDj is the dimensionless length of the fracture discrete element (i, j), which is defined as formula (46)
Performing Laplace transformation on the product (44) to obtain
Wherein: is a Laplace transform of qfDi, j.
Equation (47) is a dimensionless flow density normalization equation for the gas well, which represents 1 linear algebraic equation. Thus, equation (42) and equation (47) together represent M linear algebraic equations.
S9, combining the first linear algebraic equation set in S6, the second linear algebraic equation set in S7 and a dimensionless flow density normalization equation of the gas well to obtain a linear sealing matrix for well testing simulation, wherein the linear sealing matrix consists of (M multiplied by N+M) linear algebraic equations and (M multiplied by N+M) unknowns, and solving the linear sealing matrix by using Stehfest numerical inversion and Gauss elimination to obtain a well testing simulation result of the multilayer gas reservoir split-pressure gas production well under the complex condition:
1) The first linear algebraic equation set (35), the second linear algebraic equation set (39) and the dimensionless flow density normalization equation (44) of the gas well are combined to obtain a linear closed matrix for well testing simulation, wherein the linear closed matrix consists of (M multiplied by N+M) linear algebraic equations and (M multiplied by N+M) unknowns:
The unknowns in the above closed matrix are And/>(Where i=1, 2,3, … N; j=1, 2,3, … M), a total of (mxn+m), solving the linear closed matrix using Gauss's elimination to obtain/>Then, substituting formula (39) to obtain/>According to formula (40), layers/>Equal, so here remove/>Is abbreviated as/>Representing the fourth dimensionless osmotic bottom hole pressure after Laplace transformation.
Above-mentionedThe effects of well reservoir effects and skin effects are not considered. Further, consider the effect of dimensionless well storage coefficient C D and skin coefficient S according to the VAN EVERDINGEN formula: /(I)
Wherein: is the fourth dimensionless bottom hole pressure in Laplace space taking into account dimensionless well storage coefficients and skin coefficients.
The fourth dimensionless bottom hole pressure psi wDS in the real time domain, which takes into account the dimensionless well storage coefficients and the skin coefficients, can be obtained by numerical inversion with Stehfest.
2) Obtaining well test simulation results of multi-layer gas reservoir pressure-separated gas production well under complex conditions
And solving the constructed closed matrix, programming and drawing a well testing simulation result of the multi-layer gas reservoir pressure-dividing gas production well under the complex condition, namely opening degree, dimensionless original simulated pressure, fracture length, thickness, stratum coefficient ratio, storage Rong Bi, an influence diagram of vertical and horizontal permeability ratio on a double logarithmic curve, and analyzing the well testing curve according to curve morphology and seepage mechanism.
FIG. 3 shows a double log curve of a multi-layer gas reservoir pressure-sensitive test well in this complex case, with the pressure curve above and the derivative curve below.
As can be seen from fig. 3, in this embodiment, the percolation process can be divided into 7 percolation stages: stage I is a pure well Chu Duan, where the pressure curve and derivative curve are both straight lines with slope 1; stage II is a transition section, and the derivative curve is an upward hump; stage III is a first linear flow segment, the pressure derivative curve of the segment is a 1/2 slope straight line, and reflects parallel line flow perpendicular to the fracture surface and parallel to the stratum when the pressure wave does not reach the top and bottom boundary yet, and the streamline diagram is shown in FIG. 4; stage IV is an early vertical quasi-radial section, the pressure derivative curve of the section becomes gradually gentle, if the opening degree of each layer is smaller, the section becomes horizontal, and the streamline schematic diagram is shown in fig. 5; the V stage is a second linear flow stage, the pressure derivative curve of the stage is also a 1/2 slope line, the pressure derivative curve reflects parallel line flow which is perpendicular to the fracture surface and parallel to the stratum when the pressure wave reaches the top and bottom boundary, and the streamline schematic diagram is shown in figure 6; stage VI is a late planar quasi-radial flow section, the pressure derivative curve of which is a horizontal line with a height of 0.5, reflecting that the pressure wave has reached the top and bottom boundaries and has propagated far on the reservoir plane, and the surrounding fluid flows to the fracture in a quasi-radial flow manner from the periphery around the fracture as the center, and the streamline schematic diagram is shown in fig. 7; segment VII is the right straight boundary reflection segment closest to the well, the segment pressure derivative curve rising; segment VIII is a reflection segment when the pressure wave propagates not only to the right nearest boundary but also to the upper and lower boundaries. The section of pressure derivative curve continues to rise; section IX is the entire outer boundary reflection section where the pressure wave has propagated to the furthest boundary, where the pressure curve and pressure derivative curve are upturned, in a straight line with a slope of "1".
Fig. 8 is a graph showing the effect of the dimensionless raw pseudopressure psi IDj on a log-log curve. The third layer is chosen to have the highest pressure, and, as defined by the dimensionless pressure, ψ ID3 =0. From this figure, it can be seen that the larger the dimensionless original pseudo pressures ψ ID1 and ψ ID2, the higher the hump position of the transition on the derivative curve, and the higher the pressure curve position, which is similar to the effect of the skin coefficient S on the curve.
FIG. 9 is a graph showing the effect of the opening degree h wj/hj of each layer on a double logarithmic curve. From this figure, it can be seen that the smaller the layer opening h wj/hj, the higher the first linear flow section position and the earlier the early vertical radial flow section occurs, the longer the duration.
FIG. 10 is a graph showing the effect of the crack length L fj on the log-log curve. From this figure, it can be seen that the longer each fracture length L fj, the lower the first linear flow section, the earlier vertical radial flow section, and the second linear flow section are, the less pronounced the horizontal section of the later planar quasi-radial flow (i.e., the later occurs and the shorter the duration).
FIG. 11 is a graph showing the effect of the thickness h j of each layer on a log-log curve. From this figure, it can be seen that the smaller each thickness h j, the higher the position of the early vertical radial flow section, and thickness h j has substantially no effect on other stages.
FIG. 12 is a graph of the relative magnitudes of χ 1 and χ 3 versus a log curve with the formation factor ratio χ 2 unchanged. From this figure, it can be seen that the greater χ 1 (i.e., the smaller χ 3), the lower the transition hump position on the derivative curve, and the higher the early vertical radial flow segment position.
Fig. 13 is a graph of the relative magnitude of ω 1 and ω 3 versus the log-log curve for a second layer with a constant storage capacity ratio ω 2. From this graph, it can be seen that the greater ω 1 (i.e., the smaller ω 3), the higher the transition hump position on the derivative curve, the more regular the first and second linear flow segments are, but the smaller the amplitude of the change is; the early vertical radial flow section changes little with the parameter; furthermore, the larger ω 1 (i.e., the smaller ω 3), the earlier the outer boundary reflects the derivative curve of the segment.
Fig. 14 is a graph of the relative magnitude of ω 2 and ω 3 versus the log-log curve for a first layer with a constant storage capacity ratio ω 1. From the graph, the larger omega 2 (namely the smaller omega 3), the higher the hump position of the transition section on the derivative curve, the change rule of the first linear flow section and the second linear flow section is also that the change amplitude is small; the influence of the parameter on the early vertical radial flow section is small; furthermore, the larger ω 1 (i.e., the smaller ω 3), the earlier the outer boundary reflects the derivative curve of the segment.
FIG. 15 is a graph showing the effect of the ratio k vj/khj of vertical to horizontal permeability on a double logarithmic curve for each layer. From this figure, it can be seen that the larger k vj/khj, the lower the early vertical radial flow segment position on the derivative curve, and the other phases are substantially unaffected by this parameter.
The above examples are only 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 concept of the invention belong to the protection scope of the invention. It should be noted that modifications and adaptations to the present invention may occur to one skilled in the art without departing from the principles of the present invention and are intended to be within the scope of the present invention.
Claims (7)
1. The multilayer gas reservoir partial pressure fit test well simulation method under the complex condition is characterized by comprising the following steps of:
s1, constructing a physical model of the multilayer gas reservoir partial pressure gas production well under the complex condition;
Wherein, the complex condition refers to: comprehensively considering the conditions that each gas layer is not fully fractured, the fracture lengths of each gas layer are equal or unequal, the flow densities of each fracture can change along with the position and time, the reservoir thicknesses of each gas layer are equal or unequal, the outer boundary radiuses of each gas layer are equal or unequal, the original formation pressures of each gas layer are equal or unequal, the pore permeability parameters of each gas layer are equal or unequal and the gas high-pressure physical property is influenced;
s2, establishing a multilayer gas reservoir dimensionless point sink seepage model aiming at the multilayer gas reservoir split-pressure gas production well physical model established in the S1, wherein the method comprises the following steps of:
s21, aiming at the physical model of the multilayer gas reservoir pressure-dividing gas production well in the S1, adopting a seepage mechanics theory, and establishing a multilayer gas reservoir dimensional point seepage model in a pressure form;
s22, converting the multi-layer gas reservoir with factor point sink seepage model in the pressure form in S21 into a multi-layer gas reservoir with factor point sink seepage model in the pseudo-pressure form;
s23, defining a set of dimensionless quantities, and converting the multi-layer gas reservoir dimensionless point sink seepage model in the pseudo-pressure form in S22 into a multi-layer gas reservoir dimensionless point sink seepage model;
S3, aiming at the multilayer gas reservoir dimensionless point sink seepage model in the S2, obtaining a dimensionless pressure solution of the multilayer gas reservoir dimensionless point sink seepage model in a Laplace domain, namely a first dimensionless seepage pressure solution by comprehensively applying Laplace transformation and Fourier finite cosine integral transformation;
S4, based on the first dimensionless seepage pseudo-pressure solution in the S3, integrating along the height direction of each gas layer pressure crack in the multilayer gas reservoir pressure-dividing gas production well physical model by utilizing an integration form superposition principle to obtain a dimensionless pseudo-pressure solution caused by a j-th layer vertical line in the multilayer gas reservoir, namely a second dimensionless seepage pseudo-pressure solution;
s5, based on the second dimensionless seepage pressure-planning solution in S4, integrating the seam length of the pressure crack in the multi-layer gas reservoir pressure-dividing gas production well model by utilizing an integral form superposition principle to obtain a dimensionless pressure-planning solution containing the to-be-determined dimensionless flow density parameter of the multi-layer gas reservoir pressure-dividing gas production well model in Laplace space, namely a third dimensionless seepage pressure-planning solution;
S6, carrying out sectional discrete on the crack along the crack length, carrying out sectional integration on the third dimensionless seepage pseudo-pressure solution in S5 on discrete units, summing up, taking calculated pressure points on each discrete unit node, and simultaneously utilizing the characteristics of infinite diversion crack, thereby obtaining the undetermined dimensionless flow density of the ith discrete unit on the jth crack in Laplace space The j-th gas layer dimensionless bottom hole s pseudo pressure/>M x N linear algebraic equations of the third dimensionless osmotic bottom-hole pseudo pressure, i.e., the first set of linear algebraic equations;
S7, the original pseudo pressure psi Ij of the jth gas layer is replaced by the reference pseudo pressure psi r to redefine the jth gas layer dimensionless bottom hole S pseudo pressure psi wsDj in S6, namely redefine the third dimensionless seepage bottom hole pseudo pressure in S6, and the redefined pseudo pressure is called the jth gas layer dimensionless bottom hole pseudo pressure psi wDj, namely the fourth dimensionless seepage bottom hole pseudo pressure; according to the relation between the third dimensionless seepage bottom quasi-pressure of each discrete unit node of any fracture and the fourth dimensionless seepage bottom quasi-pressure of the discrete unit node, and the condition that the fourth dimensionless seepage quasi-pressures of different discrete unit nodes of different fractures are equal in an infinite diversion mode, obtaining M-1 linear algebraic equations, namely a second linear algebraic equation set, of the third dimensionless seepage quasi-pressures of different discrete unit nodes of different fractures;
S8, obtaining a dimensionless flow density normalization equation of 1 gas well according to the relation between the flow density of the fracturing cracks of different discrete units and the yield of the gas well;
S9, combining the first linear algebraic equation set in S6, the second linear algebraic equation set in S7 and a dimensionless flow density normalization equation of the gas well to obtain a linear sealing matrix for well testing simulation, wherein the linear sealing matrix consists of M multiplied by N+M linear algebraic equations and M multiplied by N+M unknowns, and solving the linear sealing matrix by using Stehfest numerical inversion and Gauss elimination method to obtain a well testing simulation result of the multi-layer gas reservoir partial pressure production gas well under complex conditions;
Wherein the pressure-form multi-layer gas reservoir dimensionless point sink-flow model comprises the following formulas (1) - (6):
① Pressure-form multi-layer gas reservoir seepage differential equation:
Wherein: r j is the distance between any field point in the j-th gas layer and the point sink (x wj,ywj,zwj) in the radial direction of the plane after converting the rectangular coordinate system into the cylindrical coordinate system, M; x j is the x-coordinate of any field point in the j-th gas layer; y j is the y coordinate of any field point in the j-th gas layer; z j is the z coordinate of any field point in the j-th gas layer; x wj is the x coordinate of any point sink in the j-th gas layer; y wj is the y coordinate of any sink in the j-th gas layer; z wj is the z coordinate of any point sink in the j-th gas layer; p j is the pressure at any field point in the j-th gas layer, pa; k hj is the j-th gas layer horizontal permeability, m 2;kvj is the j-th gas layer vertical permeability, m 2;μ(pj) is the j-th gas layer gas viscosity, pa·s, as a function of the j-th gas layer pressure p j; z (p j) is a jth gas layer gas deviation factor, dimensionless, as a function of a jth gas layer pressure p j; c g(pj) is the j-th gas layer gas compression coefficient as a function of the j-th gas layer pressure p j, pa -1;φj is the porosity of the j-th gas layer, dimensionless; t is the production time, s;
② Initial conditions:
pj(rj,zj,t)|t＝0＝pIj (2)
Wherein: p Ij is the original formation pressure of the jth gas formation, pa;
③ Outer boundary conditions:
Horizontal direction:
jth gas layer top boundary:
jth gas bed bottom boundary:
Wherein: r ej is the horizontal outer boundary radius of the jth gas layer, m; h j is the gas layer thickness of the j-th gas layer, m;
④ Inner boundary conditions: assume that the ground production of a point sink at the j-th gas layer at point (x wj,ywj,zwj) is The internal boundary conditions are:
Wherein: epsilon is the vertical infinitesimal length, m; sigma is the radial infinitesimal length, m; p sc is ground atmospheric pressure, pa; t is the gas reservoir temperature, K; t sc is the ground temperature, K; The ground production, m 3/s, of any point sink where the jth gas layer is located at point (x wj,ywj,zwj);
the pseudo-pressure form multi-layer gas reservoir dimensionality point sink flow model comprises the following formulas (7) - (12):
① Multi-layer gas reservoir seepage differential equation:
Wherein: psi j is the jth air pressure, pa/s; mu gj is the gas viscosity at the original pressure of the jth gas layer, pa.s; c gj is the gas compression coefficient at the original pressure of the jth gas layer, pa -1;
② Initial conditions:
ψj(rj,zj,t)|t＝0＝ψIj (8)
Wherein: psi Ij is the original pseudo pressure of the jth gas layer, pa/s;
③ Outer boundary conditions:
Horizontal direction:
jth gas layer top boundary:
jth gas bed bottom boundary:
④ Inner boundary conditions:
the multi-layer gas reservoir dimensionless point sink seepage model comprises the following formulas (13) - (18):
1) Multi-layer gas reservoir seepage differential equation:
Wherein: chi j is the formation coefficient ratio of the j-th gas layer, and has no dimension; the ψ sDj is the j-th air layer dimensionless s-fitting pressure, namely the original fitting pressure ψ Ij of each layer is used for defining the dimensionless fitting pressure for the ψ j, which can represent the first/second/third dimensionless seepage fitting pressure in the text, and represents the first dimensionless seepage fitting pressure in the text; r Dj is the dimensionless quantity, dimensionless, of r j; l fDj is the dimensionless quantity, dimensionless, of L fj; l fj is the j-th gas layer crack half-length, m; z Dj is the dimensionless quantity of z j, dimensionless; omega j is the elastic storage ratio of the j-th gas layer, and has no dimension; mu Rj is the j-th gas layer viscosity ratio, dimensionless; h Dj is the dimensionless quantity of h j, dimensionless; t D is the dimensionless quantity of the production time t, i.e. the dimensionless production time;
2) Initial conditions:
3) Outer boundary conditions:
Horizontal direction:
The j-th layer top boundary:
Layer j bottom boundary:
wherein: r eDj is the dimensionless quantity of r ej, i.e., the j-th gas layer horizontal outer boundary dimensionless radius;
4) Inner boundary conditions:
Wherein: epsilon D is the dimensionless quantity of epsilon; σ D is the dimensionless quantity of σ; omega j is the elastic storage ratio of the j-th gas layer, and has no dimension; Is/> Dimensionless numbers of (2); z wDj is the dimensionless quantity of z wj.
2. The well test simulation method according to claim 1, wherein in S1:
the multilayer gas reservoir separate pressing gas production well physical model comprises: each gas layer comprises a closed top surface, a closed bottom surface, a crack inner boundary and a closed outer boundary, wherein any gas layer j comprises a crack with a crack opening thickness, namely a crack height h wj, a gas well shaft vertically drills through each gas layer from top to bottom, cracks symmetrically distributed on two sides of the shaft are formed in each gas layer, the lengths of the cracks of different layers are equal or unequal, a three-dimensional coordinate system is established by taking the midpoint of the top boundary of each gas layer as the origin of the coordinate system of the gas layer, and the vertical direction is the positive direction of a z j axis, so that the abscissa of the crack end of each gas layer on the x axis is-L fj and L fj, namely the half length of a j-th laminated crack is L fj, the length of the gas well shaft is 2L fj, and the vertical coordinate of the horizontal axis of the j-th gas layer crack is z mj;
the partial pressure combined mining comprises: each layer is respectively fractured, but natural gas is produced through the same gas well;
The well test simulation method comprises the following steps: establishing a seepage model, solving the seepage model, and calculating and drawing a well test curve.
3. The well test simulation method according to claim 1, wherein in S3, the first dimensionless pseudo-pressure solution of Lapalace fields obtained by comprehensively using Laplace transform and Fourier finite cosine integral transform is as follows:
Wherein: Is transformed by Laplace/> S is a Laplace variable based on t D, dimensionless; k 0 () is a second class zero-order deformed bessel function; k 1 () is a second class of first order deformed bessel function; i 1 () is a first-order deformed bessel function; i 0 () is a zero-order deformed bessel function of the first type; pi is the circumference ratio; cos () is a cosine function; n is a non-negative integer, n=0, 1,2,3, …; alpha n,j is an intermediate variable, whose expression is: /(I)Where n=0, 1,2,3, ….
4. A well test simulation method according to claim 3, wherein in S4, a dimensionless pseudo pressure solution caused by a j-th gas layer vertical line in the multi-layer gas reservoir, namely a second dimensionless seepage pseudo pressure solution is as follows:
Wherein: the phi sDj;ψsDj after Laplace transformation is the j-th air layer dimensionless s-simulated pressure, and the j-th air layer dimensionless s-simulated pressure is the second dimensionless seepage-simulated pressure defined by adopting the original simulated pressure phi Ij of each layer; /(I) Is transformed by Laplace For the production of the j-th gas formation vertical manifold, m 3/s;qsc is the surface gas production of the gas well, and m 3/s;hwDj is the dimensionless quantity of h wj; h wj is the fracture thickness, i.e., fracture height, m, of the jth gas layer; z mDj is the dimensionless quantity of z mj, dimensionless; z mj is the midpoint coordinate of the jth gas layer pressure gauge section.
5. The well test simulation method according to claim 4, wherein in S5, the dimensionless pseudo-pressure solution containing the to-be-determined dimensionless flow density parameter in the Laplace space, namely, the third dimensionless pseudo-pressure solution is as follows:
Wherein:
Wherein: The PSI sDj;ψsDj after Laplace transformation is the j-th air layer dimensionless s-pseudo pressure, and the third dimensionless seepage-pseudo pressure defined by adopting the original pseudo pressure PSI Ij of each layer is represented here; /(I) Is the undetermined dimensionless flow density of the jth crack in the Laplace space; c 2 is an intermediate variable, dimensionless; xwDj is the dimensionless number of x wj; x wj is the x coordinate, m of any point sink in the j-th gas layer.
6. The well test simulation method according to claim 5, wherein in S6, the pending dimensionless flow density of the ith discrete unit in the Laplace space on the jth fractureThe j-th gas layer dimensionless bottom hole s pseudo pressure/>M x N linear algebraic equations of (a), i.e. the first set of linear algebraic equations, are as follows:
Wherein: j-th gas layer dimensionless bottom-hole s pseudo pressure Also known as the third dimensionless seepage bottom-hole sub-pressure; x Di,j and x Di+1,j are the dimensionless x coordinates of the left and right endpoints of the ith discrete unit of the jth gas laminated slit, respectively; c 3 is an intermediate variable, dimensionless, expressed as:
Wherein: Is Laplace transform form of q fDi,j; q fDi,j is the dimensionless flow density of the fracture discrete units (i, j); /(I) The PSI wsDj;ψwsDj after Laplace transformation is the j-th gas layer dimensionless bottom-hole s pseudo pressure defined by the j-th gas layer original pressure, and the j-th gas layer dimensionless bottom-hole s pseudo pressure is the third dimensionless seepage bottom-hole pseudo pressure; /(I)Is the dimensionless abscissa of the kth node on the jth fracture; k is the node number on each fracture, k=1, 2,3, … N; j is the number of each crack, j=1, 2,3, … M;
In S7, the second linear algebraic equation set is as follows:
wherein: phi IDj is the dimensionless original pseudo pressure of the jth gas layer; is psi wsDj after Laplace transformation;
In the step S8:
The dimensionless flow density normalization equation of the gas well obtained according to the relation between the flow density of the pressure cracks of different discrete units and the yield of the gas well is as follows:
Wherein: is Laplace transform of q fDi,j; q fDi,j is the dimensionless flow density of the fracture discrete units (i, j); Δl fDj is the dimensionless length of the fracture discrete element (i, j).
7. The well test simulation method according to claim 1, wherein in S9:
the well test simulation result comprises a double-logarithmic curve plate of the model in a real time domain, an influence graph of each parameter on the double-logarithmic curve, and analysis of the well test curve according to curve morphology and seepage mechanism, wherein the parameters comprise one or more of opening degree, dimensionless original simulated pressure, fracture length, thickness, stratum coefficient ratio, storage Rong Bi and vertical-horizontal permeability ratio.
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| CN114201932A (en) * | 2021-12-10 | 2022-03-18 | 西南石油大学 | Well testing simulation method for tight reservoir fracturing well under complex condition |
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| CN111927420A (en) * | 2020-08-15 | 2020-11-13 | 西南石油大学 | Method for simulating pressure of asymmetric fractured well with limited diversion for gas reservoir in any shape |
| CN114201932A (en) * | 2021-12-10 | 2022-03-18 | 西南石油大学 | Well testing simulation method for tight reservoir fracturing well under complex condition |
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