CN105808913A - Shale gas yield decline analysis method - Google Patents

Abstract

The invention discloses a shale gas yield decline analysis method and belongs to the technical field of shale gas exploitation. The method comprises the steps of detecting a diffusion coefficient, a gas pressure and the shale gas density of shale gas in a target shale reservoir, and determining the quality of free gas and desorbed gas flowing out of fractures according to the parameters; and according to the quality of the free gas and the desorbed gas, the detected initial concentration of the shale gas before exploitation, the detected fracture interval in the target shale reservoir, the detected wellhead gas yield in a preset cycle and a duration from the beginning of shale gas exploitation to the cyclic end time for detecting the wellhead gas yield in the preset cycle, determining the total shale gas exploitation time when the wellhead gas yield is zero, and according to a preset target exploitation time, determining the wellhead gas yield in the preset cycle corresponding to the preset target exploitation time. With the adoption of the method, the accuracy of yield decline analysis can be improved.

Description

A kind of shale gas Production Decline Analysis method

Technical field

The present invention relates to shale gas production technique field, particularly to a kind of shale gas Production Decline Analysis method.

Background technology

Development along with shale gas production technique, people are day by day frequent to the activity of shale gas, the Production Decline Analysis of shale gas is also increasingly paid attention to, it is desirable to, by the change of production of shale gas being analyzed and studying, instruct people that shale gas is more reasonably exploited.

In practice, people, when carrying out Production Decline Analysis, can be summarized some empirical equations according to conventional natural gas pool reservoir, by these empirical equations, the yield of natural gas pool reservoir are analyzed.When shale gas is carried out Production Decline Analysis, people adopt these empirical equations that the yield of shale gas is analyzed, it is possible to the time according to exploitation shale gas, it was predicted that gas production, can also gas production time on the estimation, calculate the tolerance that can exploit shale gas in the gas production time estimated.

In the process realizing the present invention, inventor have found that prior art at least there is problems in that

The adsorptivity of most natural gas pool reservoir is relatively low, and the adsorptivity of shale gas reservoir reservoir is very strong, therefore, by the empirical equation summarized, shale gas carried out Production Decline Analysis, the adsorptivity of shale gas cannot be considered, thus causing that the accuracy of Production Decline Analysis is relatively low.

Summary of the invention

In order to solve problem of the prior art, embodiments provide a kind of shale gas Production Decline Analysis method.Described technical scheme is as follows:

First aspect, it is provided that a kind of shale gas Production Decline Analysis method, described method includes:

Detect the diffusion coefficient of shale gas in target shale reservoir, gas pressure and shale gas density；

Determine the quality of the free gas flowed out in crack according to described diffusion coefficient, and determine the quality of desorption gas according to described gas pressure and described shale gas density；

Detect the initial concentration before shale gas in described target shale reservoir is exploited, the fracture interval in described target shale reservoir and the well head gas production in predetermined period；

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting the duration of exploiting shale gas to the period end time of the well head gas production detected in described predetermined period, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production detected in described predetermined period and default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

Optionally, the diffusion coefficient of shale gas in described detection target shale reservoir, gas pressure and shale gas density, including:

Detect the maximum adsorption ability of the diffusion coefficient of shale gas in target shale reservoir, gas pressure, shale gas density and described target shale reservoir；

The described quality determining the free gas flowed out in crack according to described diffusion coefficient, determines the quality of desorption gas according to described gas pressure and described shale gas density, including:

Determine the quality of the free gas flowed out in crack according to described diffusion coefficient, determine the quality of desorption gas according to the maximum adsorption ability of described gas pressure, described shale gas density and described target shale reservoir.

Optionally, initial concentration before the exploitation of the described quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from the gas production starting to exploit the shale gas duration to the period end time of the well head gas production detected in described predetermined period and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production detected in described predetermined period and default target recovery time, determine the well head gas production in the described predetermined period that described default target recovery time is corresponding, including:

The detection length in crack, height and width；

According to the length in described crack, height and width, it is determined that the filtrational resistance of shale gas；

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

Optionally, initial concentration before the exploitation of the described quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and default target recovery time, determine the well head gas production in the described predetermined period that described default target recovery time is corresponding, including:

The diameter of detection temperature within pit shaft, the vertical length of the oil pipe in described pit shaft and described oil pipe；

Obtain the coefficient of friction resistance of shale gas, the relative density of shale gas and wellbore gas deviation factor；

Temperature within described pit shaft, the vertical length of the oil pipe in described pit shaft, the diameter of described oil pipe, the described coefficient of friction resistance, the relative density of described shale gas and described wellbore gas deviation factor, it is determined that the bottom pressure of described pit shaft；

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting the gas production of exploiting shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

Optionally, initial concentration before the exploitation of the described quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and default target recovery time, determine the well head gas production in the described predetermined period that described default target recovery time is corresponding, including:

Fracture interval in quality according to described free gas, the quality of described desorption gas, the initial concentration of described shale gas, described shale reservoir, the well head gas production in described predetermined period, from starting to exploit the gas production in the shale gas duration to the period end time of the well head gas production detected in described predetermined period and crack, it is determined that

In coefficient, the gas production according to crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Fracture interval in quality according to described free gas, the quality of described desorption gas, the initial concentration of described shale gas, described shale reservoir, the well head gas production in described predetermined period, from starting to exploit the gas production in the shale gas duration to the period end time of the well head gas production detected in described predetermined period and crack, it is determined that

In coefficient, according to default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

The technical scheme that the embodiment of the present invention provides has the benefit that

In the embodiment of the present invention, detect the diffusion coefficient of shale gas in target shale reservoir, gas pressure and shale gas density；Determine the quality of the free gas flowed out in crack according to described diffusion coefficient, and determine the quality of desorption gas according to described gas pressure and described shale gas density；Detect the initial concentration before shale gas in described target shale reservoir is exploited, the fracture interval in described target shale reservoir and the well head gas production in predetermined period；Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting the duration of exploiting shale gas to the period end time of the well head gas production detected in described predetermined period, it is determined that total recovery time of shale gas when well head gas production is zero；Or, quality according to described free gas, the quality of described desorption gas, initial concentration before the exploitation of described shale gas, fracture interval in described target shale reservoir, well head gas production in described predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production detected in described predetermined period and the target recovery time preset, determine the well head gas production in the predetermined period that described default target recovery time is corresponding, so, the quality of adsorbed gas can be conciliate according to the quality of free gas, shale gas is carried out Production Decline Analysis, therefore, it is considered that the adsorptivity of shale gas, such that it is able to improve the accuracy of Production Decline Analysis.

Accompanying drawing explanation

In order to be illustrated more clearly that the technical scheme in the embodiment of the present invention, below the accompanying drawing used required during embodiment is described is briefly described, apparently, accompanying drawing in the following describes is only some embodiments of the present invention, for those of ordinary skill in the art, under the premise not paying creative work, it is also possible to obtain other accompanying drawing according to these accompanying drawings.

Fig. 1 is a kind of shale gas Production Decline Analysis method flow diagram that the embodiment of the present invention provides；

Fig. 2 is the shale gas diffusion schematic diagram that the embodiment of the present invention provides；

Fig. 3 is that the adsorbed gas that the embodiment of the present invention provides resolves characteristic curve schematic diagram；

Fig. 4 is that a kind of imfinitesimal method that the embodiment of the present invention provides sets up model schematic；

Fig. 5 is the target shale reservoir that the embodiment of the present invention provides is the schematic diagram of ideal model.

Detailed description of the invention

For making the object, technical solutions and advantages of the present invention clearly, below in conjunction with accompanying drawing, embodiment of the present invention is described further in detail.

Embodiment one

The handling process of the method embodiments provides a kind of shale gas Production Decline Analysis method, as it is shown in figure 1, can comprise the following steps that

Step 101, detects the diffusion coefficient of shale gas in target shale reservoir, gas pressure and shale gas density.

Step 102, determines the quality of the free gas flowed out in crack, and determines the quality of desorption gas according to gas pressure and shale gas density according to diffusion coefficient.

Step 103, detects the initial concentration before shale gas in target shale reservoir is exploited, the fracture interval in target shale reservoir and the well head gas production in predetermined period；

Step 104, according to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit the shale gas duration to the period end time of the well head gas production in detection predetermined period, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

According to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production in detection predetermined period and default target recovery time, it is determined that the well head gas production in the predetermined period that default target recovery time is corresponding.

nullIn the embodiment of the present invention，The diffusion coefficient of the shale gas in detection target shale reservoir、Gas pressure and shale gas density，The quality of the free gas flowed out in crack is determined according to diffusion coefficient，And the quality of desorption gas is determined according to gas pressure and shale gas density，Detect the initial concentration before shale gas exploitation in target shale reservoir、Fracture interval in target shale reservoir and the well head gas production in predetermined period，Quality according to free gas、The quality of desorption gas、Initial concentration before shale gas exploitation、Fracture interval in target shale reservoir、Well head gas production in predetermined period、From starting to exploit the shale gas duration to the period end time of the well head gas production in detection predetermined period，Determine total recovery time of shale gas when well head gas production is zero，Or，Quality according to free gas、The quality of desorption gas、Initial concentration before shale gas exploitation、Fracture interval in target shale reservoir、Well head gas production in predetermined period、From starting to exploit shale gas to the duration of the period end time of the well head gas production in detection predetermined period and the target recovery time preset，Determine the well head gas production in the predetermined period that default target recovery time is corresponding，So，The quality of adsorbed gas can be conciliate according to the quality of free gas，Shale gas is carried out Production Decline Analysis，Therefore，It is considered that the adsorptivity of shale gas，Such that it is able to improve the accuracy of Production Decline Analysis.

Embodiment two

Embodiments provide a kind of shale gas Production Decline Analysis method.

Below in conjunction with detailed description of the invention, the handling process shown in Fig. 1 being described in detail, content can be such that

Step 101, detects the diffusion coefficient of shale gas in target shale reservoir, gas pressure and shale gas density.

In force, operator can install detecting instrument in target shale reservoir, detects the diffusion coefficient of shale gas, gas pressure and shale gas density by detecting instrument.

Optionally, other parameters in target shale reservoir can also be detected, accordingly, the processing procedure of step 101 can be such that the maximum adsorption ability of the diffusion coefficient of shale gas in detection target shale reservoir, gas pressure, shale gas density and target shale reservoir.

In force, operator can install detecting instrument in target shale reservoir, is detected the maximum adsorption ability of the diffusion coefficient of shale gas, gas pressure, shale gas density and target shale reservoir by detecting instrument, in order to follow-up use.

Optionally, other parameters in target shale reservoir can also be detected, accordingly, the processing procedure of step 101 can be such that the density of the diffusion coefficient of shale gas in detection target shale reservoir, gas pressure, shale gas density, the maximum adsorption ability of target shale reservoir and target shale reservoir.

In force, operator can install detecting instrument in target shale reservoir, the density of the diffusion coefficient of shale gas, gas pressure, shale gas density, the maximum adsorption ability of target shale reservoir and target shale reservoir is detected, in order to follow-up use by detecting instrument.

Step 102, determines the quality of the free gas flowed out in crack, and determines the quality of desorption gas according to gas pressure and shale gas density according to diffusion coefficient.

In force, after the diffusion coefficient of shale gas, gas pressure and shale gas density being detected, it is possible to determine the quality of the free gas flowed out in crack according to the diffusion coefficient of shale gas, determine the quality of desorption gas according to gas pressure and shale gas density.

Optionally, when the maximum adsorption ability of the diffusion coefficient of shale gas in above-mentioned detection target shale reservoir, gas pressure, shale gas density and target shale reservoir, accordingly, the processing procedure of step 102 can be such that the quality determining the free gas flowed out in crack according to diffusion coefficient, determines the quality of desorption gas according to the maximum adsorption ability of gas pressure, shale gas density and target shale reservoir.

In force, after the diffusion coefficient of shale gas, gas pressure and shale gas density being detected, the quality of the free gas flowed out in crack can be determined according to the diffusion coefficient of shale gas, determine the quality of desorption gas according to the maximum adsorption ability of gas pressure and shale gas density and target shale reservoir.

Optionally, when the density of the diffusion coefficient of shale gas in above-mentioned detection target shale reservoir, gas pressure, shale gas density, the maximum adsorption ability of target shale reservoir and target shale reservoir, accordingly, the processing procedure of step 102 can be such that the quality determining the free gas flowed out in crack according to diffusion coefficient, determines the quality of desorption gas according to gas pressure, shale gas density, the maximum adsorption ability of target shale reservoir and the density of target shale reservoir.

In force, after the diffusion coefficient of shale gas, gas pressure and shale gas density being detected, the quality of the free gas flowed out in crack can be determined according to the diffusion coefficient of shale gas, determine the quality of desorption gas according to the density of the maximum adsorption ability of gas pressure, shale gas density and target shale reservoir and target shale reservoir.

Specifically, target shale reservoir, in recovery process, can be carried out pressure break by operator, forms many cracks, and in shale gas, shale gas then can be diffused in crack from target shale reservoir, as shown in Figure 2.Owing to there is pressure differential between crack and horizontal wellbore, shale gas in crack can flow in horizontal wellbore from crack, the shale gas flowed out in many cracks converges in horizontal wellbore, so can form the pressure differential of horizontal wellbore and well head, then shale gas can flow to well head from horizontal wellbore, thus realizing exploitation shale gas.When shale gas is carried out Production Decline Analysis, it is possible to detect in target shale reservoir, the gas pressure in each crack, thus obtaining the gas production in each crack.Hereinafter shale gas is carried out in the analysis process of production decline, it is possible to illustrate for a certain crack.

Shale gas desorbing diffusion process must comply with the principle of mass conservation, is also called the principle of continuity.The process of foundation is to take a small cell cube in the earth formation, and as shown in Figure 4, the gaseous mass flowing out an infinitesimal area within a certain period of time should be equal to time period concentration difference variable quality and desorption makings amount sum.Desorbing-diffusional flow mathematical model refers to gas concentration diffusion problem in shale reservoir, and gas concentration is defined as unit volume gaseous mass.With function C (x, y, z, t) in representation space object O position (x, y, z) and the concentration of moment t, the quality of institute's diffusate in its representation unit volume.

Utilizing integration method to set up desorbing-diffusion equation of continuity, object flows through the quality dM of an infinitely small face ds and the object concentration directional derivative along curved surface ds normal direction in infinitely small period dt along normal direction nIt is directly proportional, it may be assumed that

$\mathrm{dM}=-D(d,y,z)\frac{\∂C}{\∂n}\mathrm{dSdt}---(1-1-1)$

In formula, (x, y, z) at point, (x, y, z) diffusion coefficient at place, unit is m to object to D²/ s or cm²/s.Diffusion coefficient represents the physical quantity of gas diffusibility, refers to along dispersal direction, when unit time per unit Concentraton gradient, perpendicular through the quality of the spread something of unit are；

Gas concentration in C shale reservoir, g/cm³；

Represent the directional derivative of unit exterior normal direction n, dimensionless unit along Γ.

In formula (1-1-1), diffusion coefficient D (x, y, z) should take on the occasion of, the appearance of negative sign is owing to concentration always flows to, from the side that concentration is high, the side that concentration is low, and therefore, dM should be withContrary sign.

Appointing in object O and take a Γ, its area encompassed is designated as Ω, by (1-1-1) formula from moment t₁To t₂The all-mass flowing out this closed surface is:

$M{=\∫}_{t_1}^{t_2}\left[\underset{\mathrm{\Γ}}{\∫\∫}D(x,y,z)\frac{\∂C}{\∂n}\mathrm{dS}\right]\mathrm{dt}---(1-1-2)$

The quality flowed out includes two parts, and a part causes owing to interior of articles pressure differential causes that concentration difference changes, at interval (t₁,t₂) in object concentration from C (x, y, z, t₁) it is reduced to C (x, y, z, t₂), the gaseous mass that it should reduce is:

$\underset{\mathrm{\Ω}}{\∫\∫\∫}[C(x,y,z,t_1)-C(x,y,z,t_2)]\mathrm{dxdydz}---(1-1-3)$

Owing to shale has adsorbed gas desorption properties (Fig. 4), the quality that then another part flows out is owing to shale causes concentration to reduce in pressure reduction, cause shale adsorbed gas desorbing, as it is shown on figure 3, resolve characteristic curve for shale gas adsorbed gas when temperature is 30 DEG C.

At interval (t₁,t₂) in object absorption tolerance by Q (x, y, z, t₁) it is reduced to Q (x, y, z, t₂), absorption tolerance, by Langmuir adsorption isotherm model, is shown in formula (1-1-4), adopts downstream rights method gas downstream container pressure to determine, then the shale gas weight of desorbing is formula (1-1-5):

$Q=\frac{V_{L}p}{p_L+p}---(1-1-4)$

In formula, q adsorbs tolerance, cm³/g；

P gas pressure, MPa；

V_LLangmuir volume, represents maximum adsorption ability；

p_LLangmuir pressure, the pressure corresponding to the half of Langmuir volume.

$\underset{\mathrm{\Ω}}{\∫\∫\∫}{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2[Q(x,y,z,t_1)-q(x,y,z,t_2)]\mathrm{dxdydz}---(1-1-5)$

In formula, Q (x, y, z, t) shale absorption tolerance, cm³/g；

ρ₁For sample rate, g/cm³；

ρ₂For gas density, g/cm³。

Step 103, detects the initial concentration before shale gas in target shale reservoir is exploited, the fracture interval in target shale reservoir and the well head gas production in predetermined period.

In force, the initial concentration before shale gas exploitation can be detected in target shale reservoir, then target shale reservoir can be carried out frac treatment, the fracture interval of detection target shale reservoir, then shale gas exploitation is carried out, and can record in predetermined period, the well head gas production of shale gas.

Step 104, according to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from the gas production starting to exploit the shale gas duration to the period end time of the well head gas production in detection predetermined period and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, according to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production in detection predetermined period and default target recovery time, it is determined that the gas production of default target recovery time internal fissure.

Optionally, it is possible to first determine the filtrational resistance of shale gas, then carry out volume analysis, accordingly, the processing procedure of step 104 can be such that the detection length in crack, height and width；According to the length in crack, height and width, it is determined that the filtrational resistance of shale gas；According to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production in detection predetermined period, the filtrational resistance of shale gas and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, according to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production in detection predetermined period, the filtrational resistance of shale gas and default target recovery time, it is determined that the well head gas production in the predetermined period that default target recovery time is corresponding.

In force, target shale reservoir can be carried out pressure break by fracturing technique by operator in target shale reservoir, in the process of pressure break, after operator can determine pressure break according to producing needs, the quantity in crack in target shale reservoir, operator can also detect the length in crack, height and width by detecting instrument, in order to carries out further data process.

Optionally, it may be determined that the bottom pressure of pit shaft, then carrying out Production Decline Analysis, corresponding processing procedure can be such that detection temperature within pit shaft；Temperature within pit shaft, it is determined that the bottom pressure of pit shaft；According to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production in detection predetermined period, the filtrational resistance of shale gas, bottom pressure and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, according to the initial concentration before the quality of free gas, the quality of desorption gas, shale gas exploitation, the fracture interval in target shale reservoir, the well head gas production in predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production in detection predetermined period, the filtrational resistance of shale gas, bottom pressure and default target recovery time, it is determined that the well head gas production in the predetermined period that default target recovery time is corresponding.

In force, operator can place temperature detector being used for exploiting in the pit shaft of shale gas, in order to detection temperature within pit shaft, the temperature within pit shaft, it is determined that the bottom pressure of pit shaft, in order to carry out subsequent analysis.

Optionally, it is possible to other parameters within pit shaft are detected, corresponding processing procedure can be such that the diameter of detection temperature within pit shaft, the vertical length of the oil pipe in pit shaft and oil pipe.

In force, operator can also place temperature detector being used for exploiting in the pit shaft of shale gas, to detect the temperature within pit shaft, it is also possible to by the diameter of the vertical length of the oil pipe in distance detector detection pit shaft and oil pipe, in order to carry out further data process.

Optionally, when parameter within above-mentioned detection pit shaft, accordingly, it is determined that the processing procedure of the bottom pressure of pit shaft can be such that the diameter of the temperature within pit shaft, the vertical length of the oil pipe in pit shaft and oil pipe, it is determined that states the bottom pressure of pit shaft.

In force, after operator detect the temperature within pit shaft, the vertical length of the oil pipe in pit shaft and the diameter of oil pipe, it is possible to the data detected are substituted into relevant formula and carries out mathematical calculation, to obtain the bottom pressure of pit shaft.

Optionally, it is also possible to obtain the relevant parameter of shale gas, to determine the bottom pressure of pit shaft, corresponding processing procedure can be such that and obtains the coefficient of friction resistance of shale gas, the relative density of shale gas and wellbore gas deviation factor；Temperature within pit shaft, the vertical length of the oil pipe in pit shaft, the diameter of oil pipe, the coefficient of friction resistance, the relative density of shale gas and wellbore gas deviation factor, it is determined that the bottom pressure of pit shaft.

In force, the coefficient of friction resistance of shale gas, the relative density of shale gas and wellbore gas deviation factor can obtain by the experimental analysis in room by experiment, can also be obtained by the empirical data in exploiting, operator can based on the parameter got and the data detected, it is determined that the bottom pressure of pit shaft.

Optionally, the gas production equation in shale gas crack can be determined according to the above-mentioned parameter detected, and then carry out volume analysis, corresponding processing procedure can be such that according to the fracture interval in the quality of free gas, the quality of desorption gas, the initial concentration of shale gas, shale reservoir, well head gas production in predetermined period, from the gas production starting to exploit the shale gas duration to the period end time of the well head gas production in detection predetermined period and crack, it is determined that

In coefficient, gas production according to crack, determine total recovery time of shale gas when well head gas production is zero, or, according to the fracture interval in the quality of free gas, the quality of desorption gas, the initial concentration of shale gas, shale reservoir, well head gas production in predetermined period, from the gas production starting to exploit the shale gas duration to the period end time of the well head gas production in detection predetermined period and crack, it is determined that

In coefficient, according to default target recovery time, it is determined that the well head gas production in the predetermined period that default target recovery time is corresponding.

By the principle of mass conservation, the gaseous mass flowing out an infinitesimal area within a certain period of time should be equal to time period concentration difference variable quality and desorption makings amount sum, make formula:

$\begin{array}{c}{\∫}_{t_1}^{t_2}\left[\underset{\mathrm{\Γ}}{\∫\∫}D(x,y,z)\frac{\∂C}{\∂n}\mathrm{dS}\right]\mathrm{dt}=\underset{\mathrm{\Ω}}{\∫\∫\∫}[C(x,y,z,t_1)-C(x,y,z,t_2)]\mathrm{dxdydz}\\ +\underset{\mathrm{\Ω}}{\∫\∫\∫}{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2[Q(x,y,z,t_1)-Q(x,y,z,t_2)]\mathrm{dxdydz}\end{array}---(1-1-6)$

Assume function C (x, y, z, t) about variable, (x, y, z) have Second Order Continuous partial derivative, having the continuous partial derivative of single order about variable t, utilize green theorem that the right-hand member of equation (1-1-6) is converted, Green's theorem is as follows:

If Closed domain is made up of piecewise smooth curve L, if function P (x, y) and Q (x y) has the continuous partial derivative of single order on D, then has

Assume function C (x, y, z, t) about variable (x, y, z) have Second Order Continuous partial derivative, has the continuous partial derivative of single order about variable t, utilizes green theorem, it is possible to conversion formula (1-1-6) left end item is:

$\begin{array}{c}{\∫}_{t_1}^{t_2}\left[\underset{\mathrm{\Γ}}{\∫\∫}D(x,y,z)\frac{\∂C}{\∂n}\mathrm{dS}\right]\mathrm{dt}\\ ={\∫}_{t_1}^{t_2}\underset{\mathrm{\Ω}}{\∫\∫\∫}[\frac{\∂}{\∂x}\left(D\frac{\∂C}{\∂x}\right)+\frac{\∂}{\∂y}\left(D\frac{\∂C}{\∂y}\right)+\frac{\∂}{\∂y}\left(D\frac{\∂C}{\∂z}\right)]\mathrm{dxdydz}\end{array}---(1-1-8)$

Assume function Q (x, y, z, t) have the continuous partial derivative of single order about variable t, then conversion formula (1-1-6) right-hand vector is:

$\begin{array}{c}\underset{\mathrm{\Ω}}{\∫\∫\∫}[C(\mathrm{dx},y,z,t_1)-C(x,y,z,t_2)]\mathrm{dxdydz}+\\ \underset{\mathrm{\Ω}}{\∫\∫\∫}{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2[Q(x,y,z,t_2)-Q(x,y,z,t_1)]\mathrm{dxdydz}\\ =\underset{\mathrm{\Ω}}{\∫\∫\∫}\left[{\∫}_{t_1}^{t_2}\frac{\∂C}{\∂t}\mathrm{dt}\right]\mathrm{dxdydz}+\underset{\mathrm{\Ω}}{\∫\∫\∫}\left[{\∫}_{t_1}^{t_2}{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2\frac{\∂Q}{\∂t}\mathrm{dt}\right]\mathrm{dxdydz}\end{array}---(1-1-9)$

Then can obtain the reformulations of formula (1-1-6):

$\begin{array}{c}{\∫}_{t_1}^{t_2}\underset{\mathrm{\Ω}}{\∫\∫\∫}[\frac{\∂}{\∂x}\left(D\frac{\∂C}{\∂x}\right)+\frac{\∂}{\∂y}\left(D\frac{\∂C}{\∂y}\right)+\frac{\∂}{\∂z}\left(D\frac{\∂C}{\∂z}\right)]\mathrm{dxdydz}\\ =\underset{\mathrm{\Ω}}{\∫\∫\∫}[{\∫}_{t_1}^{t_2}\frac{\∂C}{\∂t}\mathrm{dt}+{\∫}_{t_1}^{t_2}{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2\frac{\∂Q}{\∂t}\mathrm{dt}]\mathrm{dxdydz}\end{array}---(1-1-10)$

Integrate Sequence in switch type (1-1-10), it is possible to obtain:

$\begin{array}{c}{\∫}_{t_1}^{t_2}\underset{\mathrm{\Ω}}{\∫\∫\∫}[\frac{\∂}{\∂x}\left(D\frac{\∂C}{\∂x}\right)+\frac{\∂}{\∂y}\left(D\frac{\∂C}{\∂y}\right)+\frac{\∂}{\∂z}\left(D\frac{\∂C}{\∂z}\right)]\mathrm{dxdydz}\\ =\underset{\mathrm{\Ω}}{\∫\∫\∫}[\frac{\∂C}{\∂t}+{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2\frac{\∂Q}{\∂t}]\mathrm{dxdydz}\end{array}---(1-1-11)$

Portable (1-1-11) right-hand vector, to left end, arranges:

${\∫}_{t_1}^{t_2}\underset{\mathrm{\Ω}}{\∫\∫\∫}\left[\begin{array}{c}\frac{\∂}{\∂x}\left(D\frac{\∂C}{\∂x}\right)+\frac{\∂}{\∂y}\left(D\frac{\∂C}{\∂y}\right)\\ +\frac{\∂}{\∂z}\left(D\frac{\∂C}{\∂z}\right)-\frac{\∂N}{\∂t}-{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2\frac{\∂Q}{\∂t}\end{array}\right]\mathrm{dxdydzdt}=0---(1-1-12)$

Due to t₁、t₂Be all arbitrary with Ω, then formula (1-1-12) becomes:

$\frac{\∂}{\∂x}\left(D\frac{\∂C}{\∂x}\right)+\frac{\∂}{\∂y}\left(D\frac{\∂C}{\∂y}\right)+\frac{\∂}{\∂z}\left(D\frac{\∂C}{\∂z}\right)=\frac{\∂C}{\∂t}+{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2\frac{\∂Q}{\∂t}---(1-1-13)$

Consider that D is constant, then obtaining desorbing-diffusing qualities conservation equation is:

$D\frac{{\∂}^{2}C}{{\∂x}^2}+D\frac{{\∂}^{2}C}{{\∂y}^2}+D\frac{{\∂}^{2}C}{{\∂z}^2}=\frac{\∂C}{\∂t}+{\mathrm{\ρ}}_1{\mathrm{\ρ}}_2\frac{\∂Q}{\∂t}---(1-1-14)$

As seen from Figure 2, gas at shale reservoir constantly to desorbing and diffusion in artificial fracturing crack, this desorbing is one-dimensional with diffusion, by staged fracturing man-made fracture, reservoir is divided into multiple diffuser, the half of the spacing that distance is man-made fracture of diffusion, it is the big distance of semo-infinite for first and last diffuser, can simplified style (1-1-14) be:

$D\frac{{\&PartialD;}^{2}C}{{\&PartialD;x}^2}=\frac{\&PartialD;C}{\&PartialD;t}+{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2\frac{\&PartialD;Q}{\&PartialD;t}(t\&GreaterEqual;\mathrm{0,0}<x<L)---(1-1-15)$

In formula, L refers to the half of fracture interval, is that semo-infinite is big for first last diffuser, m.

Discounting for the impact of desorption, then formula (1-1-15) is reduced to:

$D\frac{{\&PartialD;}^{2}C}{{\&PartialD;x}^2}=\frac{\&PartialD;C}{\&PartialD;t}(t\&GreaterEqual;\mathrm{0,0}<x<L)---(1-1-16)$

Equation (1-1-16) is that through type (1-1-15) simplifies and the diffusion model being left out desorbing that obtains, is incorrect for this model of shale gas.

OrderThen formula (1-1-15) becomes:

$D\frac{{\&PartialD;}^{2}C}{{\&PartialD;x}^2}-\frac{\&PartialD;C}{\&PartialD;t}=f(x,t)(t\&GreaterEqual;\mathrm{0,0}<x<L)---(1-1-17)$

Formula (1-1-17) is exactly shale reservoir desorbing-diffusing qualities conservation equation.

If it is known that the object concentration at borderline concentration profile and object initial time, the concentration in just can determine object completely moment afterwards, therefore the diffusion the most natural demarcation problem of math equation be exactly to initial condition and boundary condition under seek the solution of problem.

1, initial condition

The given gas initial concentration in 0 moment is distributed, and initial condition equation is:

C (x, 0)=C₀(0 < x < L) (1-1-18)

In formula, C₀For gas concentration, g/cm in initial time shale reservoir³。

2, boundary condition

Research shale reservoir desorbing-diffusion model boundary condition must utilize diffusion experiment law (Newton's law) in physics: from rock sample flow out quality of natural gas to and concentration difference be directly proportional:

DM=σ (C₁-C₂)dSdt(1-1-19)

In formula, σ flow coefficient, dimensionless unit；

C₁Concentration before diffusion, g/cm³；

C₂Concentration after diffusion, g/cm³。

Investigating the quality flowing through shale reservoir surface, from rock sample interior side, it should be determined by Fourier law, and from the Diffusion Law of object Yu medium contact surface, it should be determined by Newton's law, therefore sets up relational expression:

$-D\frac{\&PartialD;C}{\&PartialD;x}\mathrm{dSdt}=\mathrm{\&sigma;}(C_1-C_2)\mathrm{dSdt}---(1-1-20)$

Simplification can obtain:

$-D\frac{\&PartialD;C}{\&PartialD;x}=\mathrm{\&sigma;}(C_1-C_2)---(1-1-21)$

This arbitrary boundary conditions can be write as

$-D\frac{\&PartialD;C}{\&PartialD;x}{|}_{x=0}=0---(1-1-22)$

$-D\frac{\&PartialD;N}{\&PartialD;x}{|}_{x=L}=\mathrm{\&sigma;}(C_1-C_2)---(1-1-23)$

This kind of boundary condition that formula (1-1-22) and formula (1-1-23) form is called the third boundary condition of diffusion equation.

Comprehensive quality conservation equation (1-1--17), definite condition equation (1-1-18), (1-1-22) and (1-1-23) constitute cumulated volume structure equation group (1-1-24) of shale reservoir gas desorbing-diffusion mathematical model:

$\left\{\begin{array}{c}D\frac{{\&PartialD;}^{2}C}{{\&PartialD;x}^2}-\frac{\&PartialD;C}{\&PartialD;t}=f(x,t)(t\&GreaterEqual;\mathrm{0,0}<x<L)\\ C(x,0)=C_0(0<x<L)\\ -D\frac{\&PartialD;C}{\&PartialD;x}{|}_{x=0}=0,-D\frac{\&PartialD;N}{\&PartialD;x}{|}_{x=L}=\mathrm{\&sigma;}(C_1-C_2)(t>0)\end{array}\right.---(1-1-24)$

In formula, (x, t) represents desorption item to f, it is possible to regard a gaseous diffusion source as, along with the generation of Gas desorption continuously provides free gas.

By constitutive equation it can be seen that desorbing-diffusion mathematical model is the nonhomogeneous equation of the two mediation numbers of a mediation number about time t and one-dimensional space x, wherein, initial condition and boundary condition are all nonhomogeneous, are typical case's parabolic equations nonhomogeneous.

The analytic solutions of the equation that the separation of variable asks can be adopted for equation (1-1-24).

For the typical nonhomogeneous desorbing diffusion equation of equation (1-1-24) formula one, wherein definite condition and equation of continuity are all nonhomogeneous, first inhomogeneous boundary condition are carried out homogeneous partial differential, order

C (x, t)=V (x, t)+W (x, t) (1-1-25)

Select suitable W (x, t) so that W (x, t) meets boundary condition:

$-D\frac{\&PartialD;W}{\&PartialD;x}{|}_{x=0}=0---(1-1-26)$

$-D\frac{\&PartialD;W}{\&PartialD;x}{|}_{x=L}=\mathrm{\&sigma;}(C_1-C_2)---(1-1-27)$

Generally take the minimum form meeting boundary condition:

W (x, t)=A (t) x²(1-1-28)

Substitute into boundary condition (1-1-26) and (1-1-27) can draw:

$A\left(t\right)=\frac{\mathrm{\&sigma;}(C_2-C_1)}{2\mathrm{DL}}---(1-1-29)$

Thus, make replacement C (x, t)=V (x, t)+A (t) x², then

$\left\{\begin{array}{c}D\frac{{\&PartialD;}^{2}V}{{\&PartialD;x}^2}-\frac{\&PartialD;V}{\&PartialD;t}=f(x,t)+A^{\&prime;}\left(t\right)x^2-2A\left(t\right)\\ V(x,0)=C_0-A\left(0\right)x^2\\ \frac{\&PartialD;V(0,t)}{\&PartialD;t}=0,\frac{\&PartialD;V(L,t)}{\&PartialD;t}=0\end{array}\right.---(1-1-30)$

Make f (x, t)+A ' (t) x²-2A (t)=f₁(x, t),Then can obtain new unknown function V (x, equation t) meet homogeneous boundary condition:

For equation (4-1-31), available principle of stacking, above-mentioned initial boundary problems can be decomposed into following two initial boundary problems:

$\left\{\begin{array}{c}D\frac{{\&PartialD;}^2{V}_1}{{\&PartialD;x}^2}-\frac{{\&PartialD;V}_1}{\&PartialD;t}=f_1(x,t)\\ V_1(x,0)=0\\ \frac{{\&PartialD;V}_1(0,t)}{\&PartialD;t}=0,\frac{{\&PartialD;V}_1(L,t)}{\&PartialD;t}=0\end{array}\right.---(1-1-32)$

It is also clear that there is the establishment of equation:

V (x, t)=V₁(x,t)+V₂(x,t)(1-1-34)

It is homogeneous equation for equation (1-1-33), adopts the separation of variable to solve, shown in the process that concrete method for solving is following.

By separation of variable solving equation group: order

V₂(x, t)=X (x) T (t) (1-1-35)

Here X (x) and T (t) represent respectively only with x about and only relevant with t function, they are substituted in equation (8), obtain:

XT=DXT (1-1-36)

Namely

$\frac{T}{\mathrm{DT}}=\frac{X}{x}---(1-1-37)$

This equation only has both sides just to set up when being equal to constant.Making this constant is-λ², λ is greater than the constant of 0, then have:

T+λ²DT=0 (1-1-38)

X+λ²X=0 (1-1-39)

For formula (1-1-38), the general solution is had to be

For formula (1-1-39), the general solution is had to be

X (x)=Bcos (λ x)+Csin (λ x) (1-1-41)

(x, general solution t) is then to obtain N

V₂(x, t)=Ae^-Dλt[Bcos(λx)+Csin(λx)](1-1-42)

First according to the Boundary Condition for Solving equation (1-1-41) in equation (1-1-33):

Owing to T (t) is not equal to 0, then obtain

$\frac{\&PartialD;X}{\&PartialD;x}{|}_{x=0}=0---(1-1-45)$

$\frac{\&PartialD;x}{\&PartialD;x}{|}_{x=L}=0---(1-1-46)$

Try to achieve according to (1-1-41) formula

X (x) '=-B λ sin (λ x)+C λ cos (λ x) (1-1-47)

Boundary condition (1-1-45) is substituted into (1-1-47) formula

X (0) '=-B λ sin (0)+C λ cos (0)=C λ=0

Because λ > 0, then obtain C=0.

Then equation (1-1-41) and (1-1-47) become

X (x)=Bcos (λ x) (1-1-48)

X (x) '=-B λ sin (λ x) (1-1-49)

(1-1-46) formula is substituted into (1-1-49) formula

X (x) '=-B λ sin (λ L)=0

Sin (λ L)=0 (1-1-50)

Namely

Equation (30) is the normal solution of a transcendental equation, there are infinite many centimeters of eigenvalue λ_k(k=1,2 ...).

Then corresponding a series of intrinsic functions are

X_k(x)-B_kcos(λ_kx)(1-1-51)

(1-1-53)

Owing to equation (1-1-32) and boundary condition are all homogeneous, therefore the solution of available principle of stacking structure progression form

Wherein, G_k=A_k×B_k。

For seeking coefficient G_k, can obtain according to formula (10) initial condition

For determining coefficient G_k, must first prove intrinsic function system { X_k}={ cos (λ_kX) } upper orthogonal in [0, L], if intrinsic function X_nAnd X_mCorrespond respectively to different λ_nAnd λ_m, then (19) formula is substituted into

$X_n^{\&prime;\&prime;}+{\mathrm{\&lambda;}}_n^2{X}_n=0---(1-1-56)$

$X_m^{\&prime;\&prime;}+{\mathrm{\&lambda;}}_m^2{X}_m=0---(1-1-57)$

Use X_mAnd X_nIt is multiplied by formula (35) and (36) respectively, obtains

$X_m{X}_n^{\&prime;\&prime;}+X_m{\mathrm{\&lambda;}}_n^2{X}_n---(1-1-58)$

$X_n{X}_m^{\&prime;\&prime;}+X_n{\mathrm{\&lambda;}}_m^2{X}_m---(1-1-59)$

Formula (1-1-58) and (1-1-59) subtract each other and in [0, L] upper integral, obtain

${\&Integral;}_0^L(X_m{X}_n^{\&prime;\&prime;}-X_n{X}_m^{\&prime;\&prime;})\mathrm{dx}+({\mathrm{\&lambda;}}_n-{\mathrm{\&lambda;}}_m){\&Integral;}_0^L{X}_N{X}_m\mathrm{dx}=0$

$\begin{array}{c}({\mathrm{\&lambda;}}_n-{\mathrm{\&lambda;}}_m){\&Integral;}_0^L{X}_N{X}_m\mathrm{dx}={\&Integral;}_0^L(X_n{X}_m^{\&prime;\&prime;}-X_m{X}_n^{\&prime;\&prime;})\mathrm{dx}=(X_n{X}_m^{\&prime;}-X_m{X}_n^{\&prime;}){|}_0^L\\ =X_n\left(L\right)X_m^{\&prime;}\left(L\right)-X_m\left(L\right)X_n^{\&prime;}\left(L\right)-X_n\left(0\right)X_m^{\&prime;}\left(0\right)+X_m\left(0\right)X_n^{\&prime;}\left(0\right)+\frac{\mathrm{\&sigma;}}{D}{X}_n\left(L\right)X_m\left(L\right)\\ -\frac{\mathrm{\&sigma;}}{D}{X}_n\left(L\right)X_m\left(L\right)=X_n\left(L\right)[X_m^{\&prime;}\left(L\right)+\frac{\mathrm{\&sigma;}}{D}{X}_m\left(L\right)]-X_m\left(L\right)[X_n^{\&prime;}\left(L\right)+\frac{\mathrm{\&sigma;}}{D}{X}_n\left(L\right)]\\ -X_n\left(0\right)X_m^{\&prime;}\left(0\right)+X_m\left(0\right)X_n^{\&prime;}\left(0\right)\\ \end{array}$

Due to X_mAnd X_nMeet boundary condition, then boundary condition formula substituted into (1-1-58) and obtain:

$({\mathrm{\&lambda;}}_n-{\mathrm{\&lambda;}}_m){\&Integral;}_0^L{X}_N{X}_m\mathrm{dx}=0---(1-1-60)$

Due to λ_n≠λ_m

Therefore obtain intrinsic function cording and have orthogonality:

${\&Integral;}_0^L{X}_N{X}_m\mathrm{dx}={\&Integral;}_0^L\cos \left({\mathrm{\&lambda;}}_{n}x\right)\cos \left({\mathrm{\&lambda;}}_{m}x\right)\mathrm{dx}=0---(1-1-61)$

Order

$\begin{array}{c}{M}_k={\&Integral;}_0^L{\cos }^2\left({\mathrm{\&lambda;}}_{k}x\right)\mathrm{dx}={\&Integral;}_0^L\frac{1+\cos \left(2{\mathrm{\&lambda;}}_{k}x\right)}{2}\mathrm{dx}={\&Integral;}_0^L(\frac{1}{2}+\frac{\cos \left(2{\mathrm{\&lambda;}}_{k}x\right)}{2})\mathrm{dx}=\frac{L}{2}+\frac{\sin \left({2\mathrm{\&lambda;}}_{k}L\right)}{{4\mathrm{\&lambda;}}_k}\\ =\frac{L}{2}+\frac{1}{{4\mathrm{\&lambda;}}_k}\frac{2\tan \left({\mathrm{\&lambda;}}_{k}L\right)}{1+{\tan }^2\left({\mathrm{\&lambda;}}_{k}L\right)}=\frac{L}{2}+\frac{\mathrm{\&sigma;}}{2({\mathrm{\&lambda;}}_k^2+{\mathrm{\&sigma;}}^2)}\end{array}$

It is multiplied by cos (λ on (1-1-53) formula both sides simultaneously_kX), then it is integrated, utilizes orthogonality to obtain

${\&Integral;}_0^L{N}_k(x,t)\cos \left({\mathrm{\&lambda;}}_{k}x\right)\mathrm{dx}={\&Integral;}_0^L{G}_k{e}^{-D{{\mathrm{\&lambda;}}_k}^{2}t}{\cos }^2\left({\mathrm{\&lambda;}}_{k}x\right)\mathrm{dx}$

${\&Integral;}_0^L{N}_k(x,0)\cos \left({\mathrm{\&lambda;}}_{k}x\right)\mathrm{dx}=G_k{\&Integral;}_0^L{\cos }^2\left({\mathrm{\&lambda;}}_{k}x\right)\mathrm{dx}=G_k{M}_k$

Then obtain

Formula (1-1-62) is substituted into formula (1-1-55) and obtains shale gas CONCENTRATION DISTRIBUTION expression formula

It is nonhomogeneous equation for equation (1-1-32), by the solution V of equation (1-1-32)₁(x, t), function f₁(x,t)、Launch by the intrinsic function of equation (1-1-33):

$V_1(x,t)=\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{M}_k\left(t\right)\sin \frac{\mathrm{k\&pi;}}{L}x---(1-1-64)$

$f_1(x,t)=\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{f}_k\left(t\right)\sin \frac{\mathrm{k\&pi;}}{L}x---(1-1-65)$

Wherein, $f_k\left(t\right)=\frac{2}{L}\underset{0}{\overset{L}{\&Integral;}}{f}_1(x,t)\sin \frac{\mathrm{k\&pi;}}{L}\mathrm{xdx},$

By equation V₁(x, t) substitutes in equation (1-1-33):

By initial condition:

Comparison equation (1-1-67), equation (1-1-66) and (1-1-68) coefficient obtain:

If M_k(t)、f_kT the Laplace transform of () exists, respectively M_k(p) and F_k(P), formula (1-1-69) and formula (1-1-70) both sides are taken Laplace transform:

L(M′_k(t)]=pM_k(p)-M_k(0)(1-1-71)

Then can obtain the solution of equation (1-1-32):

The solution of equation (1-1-24) may finally be obtained:

Furthermore it is also possible to set up shale man-made fracture seepage experiment.

In crack, the pressure size of gas directly affects crack and flows to the gas flow of horizontal wellbore, also directly affects shale gas in shale substrate and flows to fluid ability in crack.Shale gas flowing in crack belongs to Darcy Flow, meets the application conditions of equivalent flow resistance method, calculates the pressure of gas in crack according to flow, bottom pressure, equivalent flow resistance, and equivalent flow resistance is shown in following formula:

$q=\frac{p_f-p_{\mathrm{wf}}}{R_u+R_n}---(1-2-1)$

In formula, q is the gas flow flowing to pit shaft from crack, m³/d；p_fFor bottom pressure ofgas well, MPa；R_uFor parallel filtrational resistance, it is called extrernal resistance；R_nFor Radial Flow Through Porous Media resistance, it is called internal resistance.

It practice, formula (1-2-1) divide into two sections the flowing in crack to pit shaft, one section is the way flow of the imaginary fluid drain flowing near wellbore from crack, and another section is the planar radial seepage flow flowing to pit shaft near wellbore, sees shown in figure.

The shape in pressure-break face determines the size of its filtrational resistance, has two kinds of situations, if the height in seam face is long less than seam, then and filtrational resistance R_uAnd R_nCalculate with following formula respectively:

$R_u=\frac{{\mathrm{\&mu;}}_g(\frac{x_f}{2}-h_f)}{K{h}_f{w}_f}---(1-2-2)$

$R_n=\frac{{\mathrm{\&mu;}}_g}{2\mathrm{\&pi;Kh}}\ln \frac{b}{\mathrm{\&pi;}{h}_f}---(1-2-3)$

In formula, x_fLong for artificial fracture seam, m；h_fFor artificial fracture seam height, m；w_fFor artificial fracture seam width, m.

If the height in seam face is long more than seam, then filtrational resistance R_uAnd R_uCalculate with following formula respectively:

$R_u=\frac{{\mathrm{\&mu;}}_g(h_f-\frac{x_f}{2})}{\mathrm{KBh}}---(1-2-4)$

$R_n=\frac{{\mathrm{\&mu;}}_g}{2\mathrm{\&pi;Kh}}\ln \frac{b}{\mathrm{\&pi;}\frac{x_f}{2}}---(1-2-5)$

Shale gas horizontal well pit shaft flow mathematical model can also be set up.

In gas well or gas reservoir production process, gas is made to flow along with the change of pressure all the time.The bottom pressure of gas well is the power of Wellbore Flow, is the important parameter of gas well liquid loading, and oil pressure and casing pressure material computation that its parameter value measures mainly through well head obtain.

The general expression calculating bottom pressure is:

${\&Integral;}_{p_1}^{p_2}\frac{d^5\mathrm{pZTdp}}{d^5{p}^2+1.324\&times;{10}^{-18}f{\left(q_{\mathrm{sc}}\mathrm{TZ}\right)}^2}=0.03415{\&Integral;}_{H_1}^{H_2}{\mathrm{\&gamma;}}_g\mathrm{dH}---(1-3-1)$

In formula, p pressure, MPa；

q_scGas well head gas production, m³/d；

The f coefficient of friction resistance；

D tubing diameter；

Absolute temperature in T pit shaft, K；

Z wellbore gas deviation factor, dimensionless；

γ_gGas relative density, dimensionless；

The vertical tubing length of H, m.

Bottom pressure when gas well stream pressure is gas well liquid loading.Its pressure loss three part produced by well head stream pressure, gas column gravity and Wellbore Flow forms.In (1) formula, it is considered to temperature in wellbore and Gaseous Z-factor be constant be integrated obtaining bottom pressure be:

$p_{\mathrm{wf}}=\sqrt{p_{\mathrm{tf}}^2+\frac{1.324\&times;{10}^{-18}f{\left(\stackrel{\&OverBar;}{T}\stackrel{\&OverBar;}{Z}\right)}^2{q}_{\mathrm{sc}}^2}{d^5}(e^{2S}-1)}---(1-3-2)$

$S=\frac{0.03415{\mathrm{\&gamma;}}_{g}H}{\stackrel{\&OverBar;}{T}\stackrel{\&OverBar;}{Z}}---(1-3-3)$

In formula, p_wfFor bottom pressure ofgas well, MPa；

p_tfFor gas well mouth pressure, MPa.

Wherein, p_wf、q_sc、H、γ_gJust can obtaining for known quantity or by simple calculating, f utilizes Colebrook method, Jain method or Chen method, is obtained by computer programming calculation.The concrete calculating process of conventional Colebrook method is shown in following formula:

$1/\sqrt{f}=2\mathrm{lt}\frac{d}{e}+1.14-21g(1+9.34\frac{d/e}{\mathrm{Re}\sqrt{f}})---(1-3-4)$

$\mathrm{Re}=1.776\&times;{10}^{-2}\frac{q_{\mathrm{sc}}{\mathrm{\&gamma;}}_g}{d{\mathrm{\&mu;}}_g}---(1-3-5)$

In formula,Ratio for caliber Yu absolute roughness；

Re is Reynolds number；

μ_gFor gas viscosity, mPa s.

According to some gas well basic parameter values, utilize well head pressure and well head gas production data, can send out little in the hope of bottom pressure change by formula (2).Owing to gas is when Bottomhole pressure, gas pressure is parabolically distributed, average pressure in pipeCalculated by following formula and obtain:

$\stackrel{\&OverBar;}{p}=\frac{2}{3}(p_{\mathrm{wf}}-\frac{p_{\mathrm{tf}}^2}{p_{\mathrm{wf}}+p_{\mathrm{tf}}})---(1-3-6)$

So, can according to the well head gas production in many group predetermined period, determine bottom pressure, and then may determine that the gas production in crack, the time of the gas production in crack and this well head gas production of exploitation is brought in expression formula 1-1-76, can pass through to calculate the coefficient obtaining in this expression formula, and then may determine that the variation relation of the time of the gas production in crack and this well head gas production of exploitation, may determine that the gas production in crack, such as 0, then being solved by this expression formula obtains when the gas production in crack is 0, required recovery time, the i.e. shale gas minable time.Alternatively, it is also possible to certain recovery time is brought in this this expression formula, such as 30 years, then can calculate and obtain recovery time when being 30 years, the numerical value of the gas production in crack.

Alternatively, it is also possible to adopt finite difference calculus that equation 1-1-24 is solved, it is possible to forecast production and fracturing parameter sensitivity analysis.

In order to ensure the seriality solved, the initial condition given and boundary condition must are fulfilled for compatibility condition, it may be assumed that

Finite difference calculus is adopted to seek numerical solution, time (t >=0t) and the one-dimensional space (0 < x < L) set up finite difference scheme, first set up the difference scheme of initial boundary problems, rectangular mesh is made in domain (0 < x < L, t >=0).

By on x coordinate axle within the scope of 0 < x < L I decile, wherein I is a positive integer, then step-length isMake to be parallel to the parallel lines of t coordinate axes with this step-length.

In like manner, we make to be parallel to the parallel lines of x-axis with step-length for Δ t, be built such that obtain intersection point () for the node of difference gridding.

WithRepresent respectively N (x, t) and partial derivativePoint () value,Represent not t in the same time_nAdsorbed gas content.

In whole domain (0 < x < L, t >=0), for any node () utilize Taylor expansion formula to obtain:

$N_t^{n+1}+{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2{Q}_L^{n+1}=N_t^n+{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2{Q}_L^n+\mathrm{\&Delta;t}{\left(\frac{\&PartialD;N}{\&PartialD;t}\right)}_t^n+\mathrm{\&Delta;t}{\left(\frac{\&PartialD;Q}{\&PartialD;t}\right)}_t^n+\left(\mathrm{\&Delta;t}\right)---(1-1-78)$

Change equationsFor:

In like manner, it is also possible to utilize Taylor expansion formula to obtain:

Wherein, (Δ t) and (Δ x²) for truncated error.

Formula is substituted in math equation group, it is possible to obtain the difference scheme of math equation:

Note

$\mathrm{\&alpha;}=D\frac{\mathrm{\&Delta;t}}{{\mathrm{\&Delta;x}}^2}---(1-1-84)$

Difference scheme becomes:

$-\mathrm{\&alpha;}{N}_{t-1}^{n+1}+(1+2\mathrm{\&alpha;})N_t^{n+1}+{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2{Q}_L^{n+1}-\mathrm{\&alpha;}{N}_{t+1}^{n+1}=N_t^n+{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2{Q}_L^n---(1-1-85)$

Then according to the boundary value of record and initial value, solve system of linear equations, it is possible to obtain the CONCENTRATION DISTRIBUTION of any time, the flow of different time sections gas can be obtained by integration.

So, can according to shale gas CONCENTRATION DISTRIBUTION at any time in the crack obtained, the gas production of shale gas in crack is determined by integration, calculated by the gas production of shale gas in stitching and obtain bottom pressure (i.e. formula 1-2-1), well head gas production (i.e. formula 1-3-2 and formula 1-3-3) is calculated, such that it is able to the yield of shale gas is predicted according to this bottom hole pressure bomb.It should be noted that, after adopting finite difference calculus that equation 1-1-24 is solved, the result obtained is the gas production of a crack, when carrying out production forecast, assume that target shale reservoir is ideal model, namely the parameter such as the length in each crack, height and width is all identical, (wherein, x as shown in Figure 5_fLong for artificial fracture seam；h_fFor artificial fracture seam height；w_fFor artificial fracture seam width), namely it is believed that the gas production in each crack is identical, therefore, the Number of Fractures of the target shale reservoir that can pre-set, the result of calculating is multiplied by the bar number in crack, obtains total crack gas production, and then obtain total well head gas production.

In force, can also the numerical value that organize the length in crack, height and width be substituted in above-mentioned formula more, so, different well head gas production can be obtained, the numerical value of the length in crack corresponding to the maximum well head gas production calculated, height and width can be recorded, thus the numerical value of the length of fracture, height and width is optimized, in process of production, seam can be made, to reach the gas production effect of the best according to the length in crack after optimizing, height and width.

nullIn the embodiment of the present invention，The diffusion coefficient of the shale gas in detection target shale reservoir、Gas pressure and shale gas density，The quality of the free gas flowed out in crack is determined according to diffusion coefficient，And the quality of desorption gas is determined according to gas pressure and shale gas density，Detect the initial concentration before shale gas exploitation in target shale reservoir、Fracture interval in target shale reservoir and the well head gas production in predetermined period，Quality according to free gas、The quality of desorption gas、Initial concentration before shale gas exploitation、Fracture interval in target shale reservoir、Well head gas production in predetermined period、From starting to exploit the shale gas duration to the period end time of the well head gas production in detection predetermined period，Determine total recovery time of shale gas when well head gas production is zero，Or，Quality according to free gas、The quality of desorption gas、Initial concentration before shale gas exploitation、Fracture interval in target shale reservoir、Well head gas production in predetermined period、From starting to exploit shale gas to the duration of the period end time of the well head gas production in detection predetermined period and the target recovery time preset，Determine the well head gas production in the predetermined period that default target recovery time is corresponding，So，The quality of adsorbed gas can be conciliate according to the quality of free gas，Shale gas is carried out Production Decline Analysis，Therefore，It is considered that the adsorptivity of shale gas，Such that it is able to improve the accuracy of Production Decline Analysis.

One of ordinary skill in the art will appreciate that all or part of step realizing above-described embodiment can be completed by hardware, can also be completed by the hardware that program carrys out instruction relevant, described program can be stored in a kind of computer-readable recording medium, storage medium mentioned above can be read only memory, disk or CD etc..

The foregoing is only presently preferred embodiments of the present invention, not in order to limit the present invention, all within the spirit and principles in the present invention, any amendment of making, equivalent replacement, improvement etc., should be included within protection scope of the present invention.

Claims (8)

1. a shale gas Production Decline Analysis method, it is characterised in that described method includes:

Detect the diffusion coefficient of shale gas in target shale reservoir, gas pressure and shale gas density；

Determine the quality of the free gas flowed out in crack according to described diffusion coefficient, and determine the quality of desorption gas according to described gas pressure and described shale gas density；

Detect the initial concentration before shale gas in described target shale reservoir is exploited, the fracture interval in described target shale reservoir and the well head gas production in predetermined period；

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting the duration of exploiting shale gas to the period end time of the well head gas production detected in described predetermined period, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production detected in described predetermined period and default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

2. method according to claim 1, it is characterised in that the diffusion coefficient of shale gas in described detection target shale reservoir, gas pressure and shale gas density, including:

Detect the maximum adsorption ability of the diffusion coefficient of shale gas in target shale reservoir, gas pressure, shale gas density and described target shale reservoir；

The described quality determining the free gas flowed out in crack according to described diffusion coefficient, determines the quality of desorption gas according to described gas pressure and described shale gas density, including:

Determine the quality of the free gas flowed out in crack according to described diffusion coefficient, determine the quality of desorption gas according to the maximum adsorption ability of described gas pressure, described shale gas density and described target shale reservoir.

3. method according to claim 2, it is characterised in that the maximum adsorption ability of the diffusion coefficient of shale gas in described detection target shale reservoir, gas pressure, shale gas density and described target shale reservoir, including:

The density of the diffusion coefficient of shale gas in detection target shale reservoir, gas pressure, shale gas density, the maximum adsorption ability of described target shale reservoir and described target shale reservoir；

The described quality determining the free gas flowed out in crack according to described diffusion coefficient, determines the quality of desorption gas according to the maximum adsorption ability of described gas pressure, described shale gas density and described target shale reservoir, including:

Determine the quality of the free gas flowed out in crack according to described diffusion coefficient, determine the quality of desorption gas according to described gas pressure, described shale gas density, the maximum adsorption ability of described target shale reservoir and the density of described target shale reservoir.

4. method according to claim 1, it is characterized in that, initial concentration before the exploitation of the described quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from the gas production starting to exploit the shale gas duration to the period end time of the well head gas production detected in described predetermined period and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of the period end time of the well head gas production detected in described predetermined period and default target recovery time, determine the well head gas production in the described predetermined period that described default target recovery time is corresponding, including:

The detection length in crack, height and width；

According to the length in described crack, height and width, it is determined that the filtrational resistance of shale gas；

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

5. method according to claim 4, it is characterized in that, initial concentration before the exploitation of the described quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas and default target recovery time, determine the well head gas production in the described predetermined period that described default target recovery time is corresponding, including:

Detection temperature within pit shaft；

According to the temperature within described pit shaft, it is determined that the bottom pressure of described pit shaft；

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting the gas production of exploiting shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.

6. method according to claim 5, it is characterised in that the temperature within described detection pit shaft, including:

The diameter of detection temperature within pit shaft, the vertical length of the oil pipe in described pit shaft and described oil pipe；

Described temperature within described pit shaft, it is determined that the bottom pressure of described pit shaft, including:

The diameter of the temperature within described pit shaft, the vertical length of the oil pipe in described pit shaft and described oil pipe, it is determined that state the bottom pressure of pit shaft.

7. method according to claim 6, it is characterised in that the diameter of described temperature within described pit shaft, the vertical length of the oil pipe in described pit shaft and described oil pipe, it is determined that state the bottom pressure of pit shaft, including:

Obtain the coefficient of friction resistance of shale gas, the relative density of shale gas and wellbore gas deviation factor；

Temperature within described pit shaft, the vertical length of the oil pipe in described pit shaft, the diameter of described oil pipe, the described coefficient of friction resistance, the relative density of described shale gas and described wellbore gas deviation factor, it is determined that the bottom pressure of described pit shaft.

8. method according to claim 5, it is characterized in that, initial concentration before the exploitation of the described quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit the shale gas gas production to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or, initial concentration before the exploitation of quality according to described free gas, the quality of described desorption gas, described shale gas, the fracture interval in described target shale reservoir, the well head gas production in described predetermined period, from starting to exploit shale gas to the duration of period end time of well head gas production detected in described predetermined period, the filtrational resistance of described shale gas, described bottom pressure and default target recovery time, determine the well head gas production in the described predetermined period that described default target recovery time is corresponding, including:

Fracture interval in quality according to described free gas, the quality of described desorption gas, the initial concentration of described shale gas, described shale reservoir, the well head gas production in described predetermined period, from starting to exploit the gas production in the shale gas duration to the period end time of the well head gas production detected in described predetermined period and crack, it is determined thatIn coefficient, the gas production according to crack, it is determined that total recovery time of shale gas when well head gas production is zero；Or,

Fracture interval in quality according to described free gas, the quality of described desorption gas, the initial concentration of described shale gas, described shale reservoir, the well head gas production in described predetermined period, from starting to exploit the gas production in the shale gas duration to the period end time of the well head gas production detected in described predetermined period and crack, it is determined thatIn coefficient, according to default target recovery time, it is determined that the well head gas production in the described predetermined period that described default target recovery time is corresponding.