CN107944620A - A kind of Non-linear of individual well steady state productivity - Google Patents

A kind of Non-linear of individual well steady state productivity Download PDF

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CN107944620A
CN107944620A CN201711164546.0A CN201711164546A CN107944620A CN 107944620 A CN107944620 A CN 107944620A CN 201711164546 A CN201711164546 A CN 201711164546A CN 107944620 A CN107944620 A CN 107944620A
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聂仁仕
欧进晶
何希亮
邓祺
黄静
何彦均
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Southwest Petroleum University
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Abstract

The invention belongs to Petroleum finance technical field, more particularly to a kind of Non-linear of individual well steady state productivity.The present invention is mainly to overcome shortcoming of the prior art, the Non-linear of a kind of precision higher for surveying stable productivity of proposition, more accurately individual well steady state productivity, obtains the bottom pressure p of individual wellw, supply pressure pe, oil compressibility Co, drainage radius re, skin factor S, oil volume factor Bo, viscosity of crude μ, core intersection h, reservoir permeability k；Yield is obtained further according to the above-mentioned individual well stable productivity formula for getting parameter and nonlinear radial flow system.It is the precision higher of present invention prediction stable productivity, more accurate, establish theoretical foundation for researching and analysing for oil reservoir steady seepage production capacity.

Description

A kind of Non-linear of individual well steady state productivity
Technical field
The invention belongs to Petroleum finance technical field, more particularly to a kind of nonlinear prediction of individual well steady state productivity Method.
Background technology
The linear prediction method of linear seepage theory is usually used when carrying out the prediction of stable productivity, using linear pre- more The main reason for survey method is due to for convenience of calculation, in reservoir engineering design and percolation equationk are derived and calculated, passes through vacation If small oil compressibility or small barometric gradient, the elastic liquid density of pressure function is unfolded with Maclaurin series, Ignore higher order term, occur without nonlinear terms in derived seepage flow continuity equation based on this, so as to ignore non-linear to oozing The influence of stream.Carry out capability forecasting there are large error using linear forecasting technology, especially to Deep tight reservoir compact oil reservoir etc. When nontraditional reservoir carries out AOF calculation, the barometric gradient of very little may cause significant capability forecasting error.At present, to deep layer The exploitation of hypotonic densification nontraditional reservoir is increasingly taken seriously, therefore has with non-linear PRODUCTION FORECASTING METHODS prediction stable productivity There is good popularizing application prospect.
The content of the invention
The present invention mainly overcomes shortcoming of the prior art, proposes a kind of nonlinear prediction of individual well steady state productivity Method, the precision higher, more accurate of method prediction stable productivity.
The present invention solves the technical solution that above-mentioned technical problem is provided：A kind of nonlinear prediction of individual well steady state productivity Method, including：
Obtain the bottom pressure p of individual wellw, supply pressure pe, oil compressibility Co, drainage radius re, skin factor S, original Oil volume coefficient Bo, viscosity of crude μ, core intersection h, reservoir permeability k；
Obtain considering effective well radius r of epidermis further according to following formulawe：
rwe=ree-S
In formula：reFor drainage radius, its unit is m；rweTo consider effective well radius of epidermis, its unit is m；S is mono- to be Skin factor, no factor；
Finally produced according to the above-mentioned individual well stable productivity formula for getting parameter and nonlinear radial flow system Amount, the individual well stable productivity formula of the nonlinear radial flow system are：
In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit For MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider the effective of epidermis Well radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
It is further that the individual well stable productivity formula of the nonlinear radial flow system is obtained by following steps Arrive：
Step S100, obtained by the principle of mass conservation when monophasic fluid passes through porous media under rectangular coordinate system Flow continuity equation：
In formula：ρ is elastic liquid density, its unit is kg/m3；φ be rock porosity, zero dimension；uxFor x directions Percolation flow velocity, its unit are m/s；uyFor the percolation flow velocity in y directions, its unit is m/s；uzFor the percolation flow velocity in z directions, its list Position is m/s；
Step S200, the equation of motion is obtained by Darcy's law：
In formula：μ is viscosity of crude, its unit is mPas；K is reservoir permeability, its unit is μm2；U is seepage flow speed Degree, its unit is m/s；▽ p are barometric gradient, its unit is MPa/m；
Step S300, by considering that the isothermal flow event of single-phase micro- compressible liquid obtains state equation：
In formula：φ be formation condition under porosity, zero dimension；φ0For rock porosity under normal pressure, zero dimension；ρ is bullet Property fluid density, kg/m3；ρ0For fluid density under normal pressure；P is strata pressure, its unit is MPa； peFor supply pressure, its list Position is MPa；CoFor oil compressibility, its unit is MPa-1；CfFor rock compressibility, its unit is MPa-1；
Step S400, nonlinear stability seepage flow side is obtained by above-mentioned flowing continuity equation, the equation of motion, state equation Journey；
Step S500, the steady seepage nonlinear mathematical model for examining single-phase radial fluid flow is established；
Step S600, the individual well stable productivity formula of nonlinear radial flow system is obtained：
In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit For MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider the effective of epidermis Well radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
It is further that the detailed process of the step S400 is：
Step S401, the permeability of individual well is constant and isotropism, can be obtained：
In formula：P is strata pressure, its unit is MPa；X, y, z is three coordinate length, its unit is m；CoFor crude oil pressure Contracting coefficient, its unit are MPa-1；
Step S402, have to steady seepageNonlinear stability seepage flow side can be then obtained under cylinder areal coordinate Journey：
In formula：R is that its unit is m with a distance from well；CoFor oil compressibility, its unit is MPa-1；P is laminated for ground Power, its unit are MPa.
It is further that the step S500 detailed processes are：
Step S501, the boundary condition of nonlinear stability percolation equationk is obtained：
Internal boundary condition：
Outer Boundary Conditions：
Step S502, the ground crude output calculating formula at the borehole wall is obtained：
In formula：P is strata pressure, its unit is MPa；H is core intersection, its unit is m；R is its list with a distance from well Position is m；rweTo consider effective well radius of epidermis, its unit is m；Q is yield, its unit is m3/d；K is reservoir permeability, μ m2；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；
Step S503, substitution of variable is made
Obtain following relation：
Joining solution by above equation can obtain：
Above formula derivation can be obtained：
In formula：P is strata pressure, its unit is MPa；X is x-axis coordinate length, its unit is m；CoCompressed for crude oil and be Number, its unit is MPa-1；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；reFor supply half Footpath, its unit are m；rweTo consider effective well radius of epidermis, its unit is m；
Step S504, can be obtained by joining solution above formula：
In formula：X is x-axis coordinate length, its unit is m；rweTo consider effective well radius of epidermis, its unit is m；peFor Supply pressure, its unit are MPa；pwFor bottom pressure, its unit is MPa；reFor drainage radius, its unit is m；CoFor crude oil The compressed coefficient, its unit are MPa-1；
Step S505, substitution of variable is made
Extrapolate following formula：
It is so as to obtain the steady seepage nonlinear mathematical model of radial fluid flow：
In formula：pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；reFor drainage radius, its Unit is m；rweTo consider effective well radius of epidermis, its unit is m；CoFor oil compressibility, MPa-1；X grows for x-axis coordinate Degree, its unit is m.
Beneficial effects of the present invention are：It is the precision higher of present invention prediction stable productivity, more accurate, it is oil reservoir steady seepage Theoretical foundation has been established in researching and analysing for production capacity.
Brief description of the drawings
Fig. 1 is that nonlinear method predicts contrast curve with linear approach in the embodiment of the present invention 1；
Fig. 2 is that nonlinear method predicts contrast curve with linear approach in the embodiment of the present invention 2；
Fig. 3 is that nonlinear method predicts contrast curve with linear approach in the embodiment of the present invention 3；
Fig. 4 is that nonlinear method predicts contrast curve with linear approach in the embodiment of the present invention 4；
Fig. 5 is that nonlinear method predicts contrast curve with linear approach in the embodiment of the present invention 5；
Fig. 6 is that nonlinear method predicts contrast curve with linear approach in the embodiment of the present invention 6.
Embodiment
Further description is done to the present invention with reference to embodiment and attached drawing.
A kind of Non-linear of individual well steady state productivity of the present invention, comprises the following steps：
Obtain the bottom pressure p of individual wellw, supply pressure pe, oil compressibility Co, drainage radius re, skin factor S, original Oil volume coefficient Bo, viscosity of crude μ, core intersection h, reservoir permeability k；Wherein supply pressure peFor being strata pressure, production Pressure differential deltap P is supply pressure peWith bottom pressure pwDifference, bottom pressure p can be tried to achieve by above-mentioned relationw,
Obtain considering effective well radius r of epidermis further according to following formulawe：
rwe=ree-S
In formula：reFor drainage radius, its unit is m；rweTo consider effective well radius of epidermis, its unit is m；S is mono- to be Skin factor, no factor；
Finally produced according to the above-mentioned individual well stable productivity formula for getting parameter and nonlinear radial flow system Amount, the individual well stable productivity formula of the nonlinear radial flow system are：
In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit For MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider the effective of epidermis Well radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
The individual well stable productivity formula of wherein described nonlinear radial flow system is obtained by following steps：
Step S100, obtained by the principle of mass conservation when monophasic fluid passes through porous media under rectangular coordinate system Flow continuity equation：
In formula：ρ is elastic liquid density, its unit is kg/m3；φ be rock porosity, zero dimension；uxFor x directions Percolation flow velocity, its unit are m/s；uyFor the percolation flow velocity in y directions, its unit is m/s；uzFor the percolation flow velocity in z directions, its list Position is m/s；
Step S200, the equation of motion is obtained by Darcy's law：
In formula：μ is viscosity of crude, its unit is mPas；K is reservoir permeability, its unit is μm2；U is seepage flow speed Degree, its unit is m/s；▽ p are barometric gradient, its unit is MPa/m；
Step S300, by considering that the isothermal flow event of single-phase micro- compressible liquid obtains state equation：
In formula：φ be formation condition under porosity, zero dimension；φ0For rock porosity under normal pressure, zero dimension；ρ is bullet Property fluid density, kg/m3；ρ0For fluid density under normal pressure；P is strata pressure, its unit is MPa； peFor supply pressure, its list Position is MPa；CoFor oil compressibility, its unit is MPa-1；CfFor rock compressibility, its unit is MPa-1；
Step S400, nonlinear stability seepage flow side is obtained by above-mentioned flowing continuity equation, the equation of motion, state equation Journey；
Step S500, the steady seepage nonlinear mathematical model for examining single-phase radial fluid flow is established；
Step S600, the individual well stable productivity formula of nonlinear radial flow system is obtained：
In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit For MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider the effective of epidermis Well radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
Wherein, the detailed process of the step S400 is：
Step S401, the permeability of individual well is constant and isotropism, can be obtained：
In formula：P is strata pressure, its unit is MPa；X, y, z is three coordinate length, its unit is m； CoFor crude oil The compressed coefficient, its unit are MPa-1；
Step S402, have to steady seepageNonlinear stability seepage flow side can be then obtained under cylinder areal coordinate Journey：
In formula：R is that its unit is m with a distance from well；CoFor oil compressibility, its unit is MPa-1；P is laminated for ground Power, its unit are MPa.
Wherein, the step S500 detailed processes are：
Step S501, the boundary condition of nonlinear stability percolation equationk is obtained：
Internal boundary condition：
Outer Boundary Conditions：
Step S502, the ground crude output calculating formula at the borehole wall is obtained：
In formula：P is strata pressure, its unit is MPa；H is core intersection, its unit is m；R is its list with a distance from well Position is m；rweTo consider effective well radius of epidermis, its unit is m；Q is yield, its unit is m3/d；K is reservoir permeability, μ m2；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；
Step S503, substitution of variable is made
Obtain following relation：
Joining solution by above equation can obtain：
Above formula derivation can be obtained：
In formula：P is strata pressure, its unit is MPa；X is x-axis coordinate length, its unit is m；CoCompressed for crude oil and be Number, its unit is MPa-1；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；reFor supply half Footpath, its unit are m；rweTo consider effective well radius of epidermis, its unit is m；
Step S504, can be obtained by joining solution above formula：
In formula：X is x-axis coordinate length, its unit is m；rweTo consider effective well radius of epidermis, its unit is m；peFor Supply pressure, its unit are MPa；pwFor bottom pressure, its unit is MPa；reFor drainage radius, its unit is m；CoFor crude oil The compressed coefficient, its unit are MPa-1；
Step S505, substitution of variable is made
Extrapolate following formula：
It is so as to obtain the steady seepage nonlinear mathematical model of radial fluid flow：
In formula：pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；reFor drainage radius, its Unit is m；rweTo consider effective well radius of epidermis, its unit is m；CoFor oil compressibility, MPa-1；X grows for x-axis coordinate Degree, its unit is m.
The individual well stable productivity calculation formula of wherein linear radial flow system：
In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit For MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider the effective of epidermis Well radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
Embodiment 1
Obtain oil deposit parameter：Reservoir depth D=3839.5m, strata pressure 45.09MPa, pressure coefficient α=1.46, oil reservoir Permeability k=5mD, oil volume factor Bo=1.08, oil compressibility Co=0.005 (MPa-1), viscosity of crude μ= 10mPas, core intersection h=10m, drainage radius re=500m.
S is calculated respectively1For -3.7, S2For -3, you can try to achieve and consider that effective well radius of epidermis is respectively 0.1m, 1.39m, The individual well stable productivity calculation formula of linear radial flow system and the list of nonlinear radial flow system are respectively adopted again Well stable productivity formula calculates single well yield, and for its production index curve to such as Fig. 1, result of calculation such as table 1, considers secondary pressure Increase during power gradients affect with conventional model crude output and absolute deviation with the reduction of skin factor, and consider secondary pressure Yield is higher than conventional model crude output during gradients affect.
Pass through comparative analysis, it can be seen that with the expansion of producing pressure differential, oil well output dramatically increases, linear approach prediction production Amount is higher than nonlinear method forecast production, and as the expansion of producing pressure differential, deviation are also increasing.Predicted when using nonlinear method When, consider that quadratic pressure gradient influences with the reduction of skin factor to increase with conventional model crude output and absolute deviation, and Consider that yield is higher than conventional model crude output when quadratic pressure gradient influences, Non-linear is more accurate.
Nonlinear method is contrasted with linear approach under 1 skin factor of different coefficient of table
Embodiment 2
Obtain oil deposit parameter, k=0.05 μm of reservoir permeability2, core intersection h=10m, oil volume factor Bo=1.2, Viscosity of crude μ=5mPas, drainage radius re=300m, considers effective well radius r of epidermiswe=0.1m, skin factor S= 1.25.C is calculated respectivelyo2For 0.003 (MPa-1) and Co2For 0.005 (MPa-1) non-linear hour method and linear approach prediction result, it is given birth to Indicative curve is produced as shown in Fig. 2, result of calculation is as shown in table 2.
It is 0.003 (MPa to oil compressibility-1), linear approach forecast production least absolute deviation is 0.43m3/ d, it is minimum Relative deviation is 1.50%；Nonlinear method forecast production least absolute deviation is 0.21m3/ d, minimum relative deviation are 0.75%. It is 0.005 (MPa to oil compressibility-1), linear approach forecast production least absolute deviation is 0.43m3/ d, minimum relative deviation For 1.50%；Nonlinear method forecast production least absolute deviation is 0.21m3/ d, minimum relative deviation are 0.75%.By right Than analysis, when using nonlinear method prediction, crude output increases with the increase of the compressed coefficient of oil, and error is also increasingly Greatly, Non-linear is more accurate.
Nonlinear method is contrasted with linear approach under the 2 different crude oils compressed coefficient of table
Embodiment 3
Obtain oil deposit parameter：Reservoir depth D=3839.5m, strata pressure 45.09MPa, pressure coefficient α=1.46, oil reservoir Thickness h=10m, oil volume factor Bo=1.08, viscosity of crude μ=10mPas, drainage radius re=500m, considers epidermis Effective well radius rwe=0.1m, skin factor S=-0.37, oil compressibility Co=0.005 (MPa-1)。
Reservoir permeability k is calculated respectively1=5mD, k2During=50mD, its production index curve is as shown in figure 3, result of calculation It is as shown in table 3 below.By analysis it can be seen that in identical permeability, with the increase of producing pressure differential, oil during big producing pressure differential Well production dramatically increases, and yield is higher than the yield of conventional model when consideration quadratic pressure gradient influences, and with producing pressure differential Expand, deviation is also increasing.Under identical producing pressure differential, crude output increases with the increase of permeability, and into line Sexual intercourse is as shown in figure 3, crude output and the absolute deviation of conventional model yield when consideration quadratic pressure gradient influences also are got over Come bigger.So when producing pressure differential is larger, relative deviation is larger, and the influence of quadratic pressure gradient item is also bigger.
Nonlinear method is contrasted with linear approach under the different permeabilities of table 3
Embodiment 4
Obtain oil deposit parameter：Reservoir depth D=3839.5m, strata pressure 45.09MPa, pressure coefficient α=1.46, oil reservoir Permeability k=5mD, core intersection h=10m, oil volume factor Bo=1.08, drainage radius re=500m, considers epidermis Effective well radius rwe=0.1m, skin factor S=-0.37, oil compressibility Co=0.005 (MPa-1)。
μ is calculated respectively1=26mPas and μ2Nonlinear prediction and linear prediction under=10mPas different viscosities, such as Shown in table 4.It can be seen that consider secondary pressure as the increase of producing pressure differential, oil well output dramatically increase in big producing pressure differential Yield is higher than the yield of conventional model during power gradients affect, and as the expansion of producing pressure differential, deviation are also increasing.Crude oil produces Amount and absolute deviation reduce with the increase of viscosity of crude.
Nonlinear method is contrasted with linear approach under 4 different viscosities of table
Embodiment 5
If oil deposit parameter：Reservoir depth D=3839.5m, strata pressure 45.09MPa, pressure coefficient α=1.46, oil reservoir ooze Saturating rate k=5mD, oil volume factor Bo=1.08, consider effective well radius r of epidermiswe=0.1m, S=-0.37, oil reservoir are thick Spend h=10m, oil compressibility Co=0.005 (MPa-1), viscosity of crude μ=10mPas.
R is calculated respectivelye1=500m, re2During=150m, production index curve is as shown in figure 5, result of calculation such as table 5 below institute Show.Consider that quadratic pressure gradient influences, conventional model crude output and absolute deviation reduce with the increase of drainage radius.
Nonlinear method is contrasted with linear approach under the different drainage radius of table 5
Embodiment 6
Obtain and hide parameter：Reservoir depth D=3839.5m, strata pressure 45.09MPa, pressure coefficient α=1.46, oil reservoir ooze Saturating rate k=5mD, oil volume factor Bo=1.08, drainage radius re=500m, considers effective well radius r of epidermiswe= 0.1m, skin factor S=-0.37, oil compressibility Co=0.005 (MPa-1), viscosity of crude μ=10mPas.
H is calculated respectively1=10m, h2During=2m, production index curve is as shown in fig. 6, result of calculation is as shown in table 6 below.Can To find out in high pressure as the increase of producing pressure differential, oil well output dramatically increase, yield when quadratic pressure gradient influences is considered Higher than conventional model yield, and as the expansion of producing pressure differential, absolute deviation are also increasing.Crude output and absolute deviation with The increase of core intersection and increase, and it is linear.
Nonlinear method is contrasted with linear approach under the different core intersections of table 6
Embodiment 7
Quasi- Ge Er Basin In Westerns north area deep layer abnormal pressure LOW PERMEABILITY RESERVOIR well parameter：The well is fractured well, after pressure Man-made fracture half long 81.52m, skin factor S=-6.38, reservoir depth 3922m, producing pressure differential Δ P=21MPa, yield q= 55.22m3/ d, strata pressure 55.72MPa, pressure coefficient α=1.45, porosity φ=9.8%, reservoir permeability k= 3.75mD, core intersection h=12m, oil volume factor Bo=1.22, viscosity of crude μ=3.66mPas, drainage radius re= 400m, considers effective well radius r of epidermiswe=0.069m, oil compressibility Co=0.005 (MPa-1), oil density 0.83g/cm3。
Survey yield q=53.90m3/ d, carries out capability forecasting to comparative analysis such as table 3, linear prediction yield and actual production yield Relative deviation be 7.43%, consider that the non-linear yield relative deviation surveyed of Quadratic Pressure Gradient is 2.40%, therefore nonlinear prediction is accurate Exactness is higher than linear prediction, and Nonlinear Predictor is more suitable for the exploitation prediction of Deep tight reservoir densification nontraditional reservoir.
Table 7 is without considering yield and the actual production variance analysis with considering quadratic pressure gradient influence
The above, not makees the present invention limitation in any form, although the present invention is taken off by above-described embodiment Dew, but the present invention is not limited to, any person skilled in the art, is not departing from technical solution of the present invention scope It is interior, when the technology contents using the disclosure above make a little equivalent embodiments changed or be modified to equivalent variations, as long as being not The content of disengaging technical solution of the present invention, any simple modification made according to technical spirit of the invention to above example, Equivalent variations and modification, in the range of still falling within technical solution of the present invention.

Claims (4)

  1. A kind of 1. Non-linear of individual well steady state productivity, it is characterised in that including：
    Obtain the bottom pressure p of individual wellw, supply pressure pe, oil compressibility Co, drainage radius re, skin factor S, crude oil body Product coefficient Bo, viscosity of crude μ, core intersection h, reservoir permeability k；
    Obtain considering effective well radius r of epidermis further according to following formulawe：
    rwe=ree-S
    In formula：reFor drainage radius, its unit is m；rweTo consider effective well radius of epidermis, its unit is m；S mono- is epidermis The factor, no factor；
    Yield is finally obtained according to the above-mentioned individual well stable productivity formula for getting parameter and nonlinear radial flow system, institute The individual well stable productivity formula for stating nonlinear radial flow system is：
    <mrow> <mi>q</mi> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mi>h</mi> </mrow> <mrow> <mn>1.842</mn> <mo>&amp;times;</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <msub> <mi>B</mi> <mi>o</mi> </msub> <mi>&amp;mu;</mi> </mrow> </mfrac> <mfrac> <mn>1</mn> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>o</mi> </msub> </mfrac> <mo>&amp;lsqb;</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>e</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> </mrow>
    In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider effective well of epidermis Radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
  2. 2. the Non-linear of a kind of individual well steady state productivity according to claim 1, it is characterised in that described non-thread The individual well stable productivity formula of the radial flow system of property is obtained by following steps：
    Step S100, the flowing when monophasic fluid passes through porous media under rectangular coordinate system is obtained by the principle of mass conservation Continuity equation：
    <mrow> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;rho;u</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>y</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;rho;u</mi> <mi>y</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>z</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&amp;rho;u</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <mi>t</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mi>&amp;rho;</mi> <mi>&amp;phi;</mi> <mo>)</mo> </mrow> </mrow>
    In formula：ρ is elastic liquid density, its unit is kg/m3；φ be rock porosity, zero dimension；uxFor the seepage flow in x directions Speed, its unit are m/s；uyFor the percolation flow velocity in y directions, its unit is m/s；uzFor the percolation flow velocity in z directions, its unit is m/s；
    Step S200, the equation of motion is obtained by Darcy's law：
    <mrow> <mi>u</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mi>k</mi> <mi>&amp;mu;</mi> </mfrac> <mo>&amp;dtri;</mo> <mi>p</mi> </mrow>
    In formula：μ is viscosity of crude, its unit is mPas；K is reservoir permeability, its unit is μm2；U is percolation flow velocity, its list Position is m/s；▽ p are barometric gradient, its unit is MPa/m；
    Step S300, by considering that the isothermal flow event of single-phase micro- compressible liquid obtains state equation：
    <mrow> <mi>&amp;phi;</mi> <mo>=</mo> <msub> <mi>&amp;phi;</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>-</mo> <msub> <mi>p</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> </mrow>
    <mrow> <mi>&amp;rho;</mi> <mo>=</mo> <msub> <mi>&amp;rho;</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>C</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>-</mo> <msub> <mi>p</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </msup> </mrow>
    In formula：φ be formation condition under porosity, zero dimension；φ0For rock porosity under normal pressure, zero dimension；ρ is elastic liquid Volume density, kg/m3；ρ0For fluid density under normal pressure；P is strata pressure, its unit is MPa；peFor supply pressure, its unit is MPa；CoFor oil compressibility, its unit is MPa-1；CfFor rock compressibility, its unit is MPa-1；
    Step S400, nonlinear stability percolation equationk is obtained by above-mentioned flowing continuity equation, the equation of motion, state equation；
    Step S500, the steady seepage nonlinear mathematical model for examining single-phase radial fluid flow is established；
    Step S600, the individual well stable productivity formula of nonlinear radial flow system is obtained：
    <mrow> <mi>q</mi> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mi>h</mi> </mrow> <mrow> <mn>1.842</mn> <mo>&amp;times;</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <msub> <mi>B</mi> <mi>o</mi> </msub> <mi>&amp;mu;</mi> </mrow> </mfrac> <mfrac> <mn>1</mn> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>o</mi> </msub> </mfrac> <mo>&amp;lsqb;</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>e</mi> </msub> <mo>-</mo> <msub> <mi>p</mi> <mi>w</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> </mrow>
    In formula：Q is yield, its unit is m3/d；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；CoFor oil compressibility, its unit is MPa-1；reFor drainage radius, its unit is m；rweTo consider effective well of epidermis Radius, its unit are m；BoFor oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；H is thick for oil reservoir Degree, its unit is m；K is reservoir permeability, its unit is μm2。
  3. A kind of 3. Non-linear of individual well steady state productivity according to claim 2, it is characterised in that the step The detailed process of S400 is：
    Step S401, the permeability of individual well is constant and isotropism, can be obtained：
    <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> <mo>+</mo> <msub> <mi>C</mi> <mi>o</mi> </msub> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>y</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>p</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>z</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> <mo>=</mo> <mn>0</mn> </mrow>
    In formula：P is strata pressure, its unit is MPa；X, y, z is three coordinate length, its unit is m；CoCompressed for crude oil and be Number, its unit is MPa-1；
    Step S402, have to steady seepageNonlinear stability percolation equationk can be then obtained under cylinder areal coordinate：
    <mrow> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mi>r</mi> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mi>o</mi> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow>
    In formula：R is that its unit is m with a distance from well；CoFor oil compressibility, its unit is MPa-1；P is strata pressure, its Unit is MPa.
  4. 4. the Non-linear of a kind of individual well steady state productivity according to Claims 2 or 3, it is characterised in that described Step S500 detailed processes are：
    Step S501, the boundary condition of nonlinear stability percolation equationk is obtained：
    Internal boundary condition：
    Outer Boundary Conditions：
    Step S502, the ground crude output calculating formula at the borehole wall is obtained：
    <mrow> <mi>q</mi> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mi>h</mi> </mrow> <mrow> <mn>1.842</mn> <mo>&amp;times;</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <msub> <mi>B</mi> <mi>o</mi> </msub> <mi>&amp;mu;</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mi>r</mi> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <msub> <mo>|</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </msub> </mrow>
    In formula：P is strata pressure, its unit is MPa；H is core intersection, its unit is m；R is that its unit is with a distance from well m；rweTo consider effective well radius of epidermis, its unit is m；Q is yield, its unit is m3/d；K is reservoir permeability, μm2；Bo For oil volume factor, zero dimension；μ is viscosity of crude, its unit is mPas；
    Step S503, substitution of variable is made
    <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>o</mi> </msub> </mfrac> <mi>l</mi> <mi>n</mi> <mi> </mi> <mi>x</mi> </mrow>
    Obtain following relation：
    <mrow> <mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> <mi>x</mi> </mrow> <mrow> <msup> <mi>dr</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow>
    <mrow> <mi>x</mi> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msup> </mrow>
    <mrow> <mi>x</mi> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>O</mi> </msub> <msub> <mi>p</mi> <mi>e</mi> </msub> </mrow> </msup> </mrow>
    Joining solution by above equation can obtain：
    <mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>e</mi> </msub> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mi>l</mi> <mi>n</mi> <mi> </mi> <mi>r</mi> <mo>+</mo> <mfrac> <mrow> <msubsup> <mi>e</mi> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msubsup> <mi>ln</mi> <mi> </mi> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>-</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <msub> <mi>p</mi> <mi>e</mi> </msub> </mrow> </msup> <mi>ln</mi> <mi> </mi> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow>
    Above formula derivation can be obtained：
    <mrow> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>e</mi> </msub> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow>
    In formula：P is strata pressure, its unit is MPa；X is x-axis coordinate length, its unit is m；CoFor oil compressibility, its Unit is MPa-1；pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；reFor drainage radius, its list Position is m；rweTo consider effective well radius of epidermis, its unit is m；
    Step S504, can be obtained by joining solution above formula：
    <mrow> <mo>(</mo> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>)</mo> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>e</mi> </msub> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mfrac> </mrow>
    In formula：X is x-axis coordinate length, its unit is m；rweTo consider effective well radius of epidermis, its unit is m；peFor supply Pressure, its unit are MPa；pwFor bottom pressure, its unit is MPa；reFor drainage radius, its unit is m；CoCompressed for crude oil Coefficient, its unit are MPa-1；
    Step S505, substitution of variable is made
    <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>o</mi> </msub> </mfrac> <mi>l</mi> <mi>n</mi> <mi> </mi> <mi>x</mi> </mrow>
    Extrapolate following formula：
    <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>o</mi> </msub> </mfrac> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow>
    <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>o</mi> </msub> </mfrac> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mrow> </msub> </mrow>
    It is so as to obtain the steady seepage nonlinear mathematical model of radial fluid flow：
    <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mi>&amp;rho;</mi> </msub> </mfrac> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msup> </mfrac> <mfrac> <mrow> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>e</mi> </msub> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>p</mi> <mi>w</mi> </msub> </mrow> </msup> </mrow> <mrow> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>e</mi> </msub> <mo>/</mo> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow> <mi>w</mi> <mi>e</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> </mrow>
    In formula：pwFor bottom pressure, its unit is MPa；peFor supply pressure, its unit is MPa；reFor drainage radius, its unit For m；rweTo consider effective well radius of epidermis, its unit is m；CoFor oil compressibility, MPa-1；X is x-axis coordinate length, Its unit is m.
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CN109281663B (en) * 2018-09-26 2021-09-24 中国海洋石油集团有限公司 Method for determining permeability of coal bed gas reservoir
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CN109555516A (en) * 2018-10-09 2019-04-02 西南石油大学 A kind of calculation method of radial compound multi-region oil reservoir radius of investigation
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CN110608036A (en) * 2019-07-24 2019-12-24 王新海 Method for calculating total skin coefficient of multilayer oil reservoir
CN113338904A (en) * 2021-06-08 2021-09-03 西南石油大学 Offshore oil well productivity evaluation method

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