Calculation method of gas pressure in porous media considering the influence of gas pressure difference

Abstract

The present invention discloses a method for calculating the gas pressure in a porous medium under the influence of gas pressure difference. The method starts from the fact that the pressure distribution curve of the gas in the porous medium is nonlinear. The area of ​​a figure enclosed by the pore fluid pressure and its flow distance is first determined. Then, the integral mean value theorem is used to convert the so-called figure into a rectangle, where the length of the rectangle is the flow distance and the width is the average pressure of the gas. In the calculation process, the compressible characteristics of the pore fluid are fully considered, and the error caused by simplifying the gas into an incompressible fluid in the prior art is overcome, thereby providing a basis for accurately evaluating the permeability-pore fluid pressure relationship and reservoir development.

Description

Technical Field

[0001] The invention relates to measuring the relationship between the permeability of a porous medium and the pressure change of a pore fluid, and in particular to determining the average pressure of a fluid when a fluid pressure gradient exists in the porous medium. Background Art

[0002] According to the compressibility of fluids, fluids are generally divided into incompressible fluids and compressible fluids, among which incompressible fluids are liquids, while compressible fluids are gases. Understanding the movement of fluids in porous media is the basis for implementing groundwater extraction, oil and gas resource development, and carbon sequestration projects. In the 19th century, French engineer Henry Darcy observed that the water output was proportional to the hydraulic gradient and the water-passing cross-sectional area during one-dimensional water seepage in a sand column, and used the proportionality coefficient (i.e., permeability) as a key parameter for evaluating the permeability of geotechnical media, namely Darcy's law. Without changing the size of the sand column, the permeability of the sand column is only related to the water pressure at both ends. Therefore, accurately measuring the relationship between permeability and pore fluid pressure is the key to evaluating the permeability of porous media.

[0003] Due to the need for oil and gas resource development, later generations extended the calculation method of water (liquid) permeability in rock and soil media to the gas seepage process based on the basic principle of Darcy's law. Compared with the compressibility of liquids, the compressibility of gases is significantly affected by their pressure. The core analysis method [SY-T 5336-2006] takes into account the influence of gas compressibility on permeability, and can reduce the nonlinear change of gas pressure along the core by reducing the pressure difference at both ends of the core. It is worth noting that, limited by the performance of the seepage instrument, most experiments cannot control the downstream pressure of the core, and can only drain the seepage fluid, that is, the downstream pressure is atmospheric pressure (about 0.1MPa). When the difference between the upstream and downstream pressures is large, the effect of gas compressibility on the nonlinear distribution characteristics of gas pressure along the core becomes more obvious. At present, most methods refer to the Darcy seepage experimental method and simplify the average pressure of gas in porous media to the average value of the sum of upstream and downstream pressures. In fact, this method simplifies the gas into an incompressible fluid, which causes a large error in the accurate evaluation of the permeability-pore fluid pressure relationship of porous media. In view of this, it is necessary to provide a calculation method for pore fluid pressure considering the influence of fluid compressibility.

[0004] Chinese patent application No. 201811376090.9 discloses a pressure-corrected pore gas permeability calculation method, which is designed to obtain more accurate pore gas permeability results, calibrate the existing average pressure, and eliminate the effect of temperature on permeability. This method confirms that pore pressure changes have an effect on the permeability of porous materials. Summary of the invention

[0005] In order to accurately evaluate the relationship between porous medium permeability and pore fluid pressure, the present invention provides a method for calculating the gas pressure in porous media under the influence of gas pressure difference. This method can determine the average pressure of the pore fluid (i.e., gas) in the porous medium under any pressure difference condition, and reduce the calculation error caused by the nonlinearity of the gas pressure gradient.

[0006] To achieve the above object, the technical solution adopted by the present invention is:

[0007] A method for calculating the gas pressure in a porous medium under the influence of gas pressure difference, wherein the gas pressure is used to evaluate the relationship between the gas permeability in the porous medium and the fluid pressure under a certain pressure difference condition measured by a steady-state method, characterized in that the specific method is:

[0008] Step 1: Determine the steady-state seepage parameters of the porous medium

[0009] The permeability test of a cylindrical porous medium with a length of L and a cross-sectional area of ​​A was carried out using the steady-state test method. The test parameters were the upstream pressure p(x=0,t)=p up , downstream pressure p(x=L,t)=p down , the seepage flow is Q, the fluid viscosity is μ, the fluid density is ρ, and the cross-sectional area of ​​the porous medium is A;

[0010] Step 2: Determine the average pressure of the fluid in the porous medium

[0011] Under steady-state seepage conditions, the gas pressure distribution in porous media is only related to the upstream and downstream pressures and the flow length. Therefore, the pressure distribution function of gas in porous media is shown in formula (a):

[0012]

[0013] in:

[0014] x is the flow distance of the injected gas;

[0015] L is the length of the sample; t is the steady-state flow time;

[0016] p(x,t) is the gas pressure at the flow distance x at time t;

[0017] p(0,t) is the upstream pressure;

[0018] p(L,t) is the downstream pressure;

[0019] The area S enclosed by the gas pressure and its flow distance com It can be expressed as formula (b)

[0020]

[0021] Using the mean value theorem of integral, we can convert the area S com The equivalent area is a rectangle with the same length as the sample, i.e. L, and width as W. Then formula (b) is changed to formula (c)

[0022]

[0023] The width W of the rectangle is the average pressure p of the gas in the sample. eq ,

[0024]

[0025] The pressure p calculated by formula d eq As the pressure difference p 上游 -p 下游 The average pressure of the gas under the conditions. As the average gas pressure, formula d reduces the error caused by the nonlinearity of the gas pressure gradient and can provide an accurate pore gas pressure value for evaluating the gas phase permeability.

[0026] In order to facilitate application, the present invention introduces a gas pressure correction coefficient λ to correct the average value p of the sum of the upstream pressure and the downstream pressure. mean ,Right now:

[0027]

[0028] Formula (e) can be rewritten as

[0029]

[0030] The λ is obtained by the following method:

[0031] First, using formula (f), we can get different pressure differences p 上游 -p 下游 The gas pressure correction coefficient λ corresponding to the conditions is then calculated based on the upstream pressure p 上游 As the horizontal axis, the correction coefficient λ is used as the vertical axis to plot different downstream pressures p 下游 In this way, when evaluating the relationship between the gas permeability and gas pressure of porous media with known length and cross-sectional area, it is only necessary to know the upstream pressure and downstream pressure. 上游 -Correction coefficient λ curve can be used to obtain the correction coefficient λ value corresponding to the pressure condition, and then according to formula (e), the average pressure of the gas in the porous medium can be accurately evaluated.

[0032] The pressure conditions described herein refer to different combinations of upstream pressure and downstream pressure of the sample during steady-state testing.

[0033] The advantages of the present invention are:

[0034] 1. The present invention starts from the fact that the pressure distribution curve of gas in porous media is nonlinear. The area of ​​the figure enclosed by the pore fluid pressure and its flow distance is first determined, and then the integral mean value theorem is used to convert the so-called figure into a rectangle. The length of the rectangle is the flow distance, and the width is the average pressure of the gas. The compressible characteristics of the pore fluid are fully considered in the calculation process, which provides a basis for accurately evaluating the permeability-pore fluid pressure relationship.

[0035] 2. In order to facilitate the calculation of the average pressure of the gas in the porous medium, the present invention proposes to use a correction coefficient to correct the average pressure value calculated by the common method, and draw the corresponding correction coefficient curve under various upstream pressure and downstream pressure combinations. In practical applications, first, the corresponding correction coefficient curve can be found according to the downstream pressure of the porous medium, and then the value of the correction coefficient can be determined according to the upstream pressure. Then, the correction coefficient is multiplied by the average pressure of the incompressible fluid to obtain the average pressure value of the compressible fluid, which provides a simple method for accurately evaluating the permeability-pore fluid pressure relationship. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 It is a schematic diagram of the pressure distribution curve of compressible and incompressible fluids in porous media;

[0037] Figure 2 It is a graph showing the relationship between the average pressure of compressible fluid and the average pressure of incompressible fluid;

[0038] Figure 3 It is the effect of pore gas compressibility on the permeability-average pressure curve (taking coal as an example);

[0039] Figure 4 is the pressure correction coefficient diagram under different gas injection pressure conditions;

[0040] Figure 5 It is a schematic diagram of testing pore gas permeability using the steady-state pressure difference method. DETAILED DESCRIPTION

[0041] The technical solution of the present invention is described in detail below in conjunction with specific examples and the derivation process of the formula of the present invention.

[0042] The present invention takes into account the nonlinear effect of gas pressure difference on the change of fluid pressure in porous media, and is mainly used to determine the relationship between gas permeability and pore fluid pressure in porous media. The gas permeability is measured by a steady-state method under a certain pressure difference condition. Figure 5 The permeability test is performed on a cylindrical porous material with a length of L and a cross-sectional area of ​​A. The test parameters are the upstream pressure p 上游 , downstream pressure p 下游, the seepage flow is Q, the fluid viscosity is μ, the fluid density is ρ, and the cross-sectional area of ​​the porous medium is A. The purpose of the present invention is to accurately obtain the average pressure of the pore fluid in the porous material under a certain pressure difference condition based on the compressible characteristics of the pore fluid. The specific steps are as follows:

[0043] Step 1: Determine the average pressure of the gas in the porous medium

[0044] 1.1: Determine the pressure distribution function of a gas in a porous medium

[0045] According to the law of conservation of mass, the continuity equation of the fluid is as follows:

[0046]

[0047] Where φ is the porosity of the porous medium and ρ is the density of the fluid. represents the cumulative amount of pore gas in the sample, represents the fluid mass gradient, and s represents the mass source per unit volume in the porous medium. When the fluid seepage is in a stable state, s = 0, and formula (1) is simplified to:

[0048]

[0049] For a one-dimensional percolation process, Equation 2 is simplified to:

[0050]

[0051] According to Darcy's law, Substituting into the above formula (3), u x is the flow rate, k is the permeability, μ is the fluid viscosity, is the fluid pressure gradient along the x direction. Formula (3) is rewritten as Formula (4):

[0052]

[0053] For compressible fluids, the gas density ρ can be replaced by the ideal gas state equation ρ = pM / zRT, and combined with formula (4) to obtain formula (5):

[0054]

[0055] Where M is the molecular weight of the gas, z is the gas compressibility factor, R is the Proctor constant, T is the temperature, and p is the pore fluid pressure at a certain position x.

[0056] For steady-state seepage conditions, the permeability k, porosity φ and pore gas viscosity μ are constants. Considering the ideal pore gas condition (i.e. z = 1), equation (5) is transformed into:

[0057]

[0058] Formula (6) is rewritten as formula (7):

[0059]

[0060] When the fluid is in steady state flow, the gas pressure p at any position x does not change with time t, that is,

[0061]

[0062] Therefore, formula (7) becomes:

[0063]

[0064] The pressure function is obtained by integrating equation (9) twice:

[0065] p 2 (x) = Ax + B (10)

[0066] x is the flow distance of the compressible fluid;

[0067] Using the following boundary conditions, the integration constants A and B can be determined:

[0068] When x = 0, it indicates the upstream pressure boundary: p(x = 0, t) = p 上游

[0069] When x=L, it indicates the downstream pressure boundary: p(x=L,t)=p 下游

[0070] Substituting the upstream boundary and downstream boundary conditions into equation (10), we obtain:

[0071]

[0072]

[0073] Formula (10) can be expressed as:

[0074]

[0075] The pressure distribution function of the compressible fluid is shown in equation (14), and its distribution curve is shown in Figure 1 :

[0076]

[0077] According to formula (14), under steady-state seepage conditions, the pore fluid pressure is only related to the upstream and downstream pressures and the flow length.

[0078] The area enclosed by the pore fluid pressure and its flow distance can be expressed as (15)

[0079]

[0080] Using the mean value theorem of integral, we can convert the area S com It is equivalent to a rectangular area with a length of L (that is, the sample length) and a width of W, then formula (15) is changed to formula (16)

[0081]

[0082] The width W of the rectangle is the average pressure p of the compressible fluid. eq Formula (17):

[0083]

[0084] Thus, the pressure p calculated by formula (17) is eq It can represent the average pressure value of the pore fluid in the porous medium as a whole.

[0085] In order to illustrate the rationality of the method for calculating the average gas pressure proposed in the present invention, the method is used below to derive the average pressure value of the gas assumed to be an incompressible fluid to verify whether it is consistent with the commonly used method.

[0086] For incompressible fluids, the density ρ is a constant. For steady-state seepage conditions, the porosity φ and permeability k do not change with time t and are also constants, so Equation 4 can be rewritten as:

[0087] Right now

[0088] Using boundary conditions:

[0089] When x = 0, it indicates the upstream pressure boundary: p(x = 0, t) = p 上游

[0090] When x=L, it indicates the downstream pressure boundary: p(x=L,t)=p 下游

[0091] Integrating equation 5 gives the pressure distribution function of the incompressible fluid in the porous medium. The corresponding curve is shown in Figure 1

[0092]

[0093] Where x is the flow distance of the incompressible fluid and L is the length of the sample.

[0094] The area enclosed by the pressure curve of an incompressible fluid and its flow distance can be expressed as

[0095]

[0096] Using the mean value theorem of integral, we can convert the area S im It is equivalent to the area of ​​a rectangle with a length of L and a width of W.

[0097]

[0098] The width W of the rectangle is the average pressure of the incompressible fluid in the sample.

[0099]

[0100] Since the pressure of an incompressible fluid is a linear function of its flow distance x, the average pressure of an incompressible fluid in a porous medium is the upstream p 上游 With downstream pressure p 下游 This shows that the average pressure value of the incompressible fluid in the porous medium obtained by the derivation method of the present invention is completely consistent with the commonly used method, which further shows that in order to accurately evaluate the permeability-pore gas pressure relationship of the porous medium, it is necessary to consider the effect of gas compressibility on the average pressure of the gas in the porous medium.

[0101] Figure 1 The pressure curve distribution diagrams of two types of fluids are drawn. Figure 1 The compressible fluid pressure curve in is drawn according to the formula (14) of the present invention, and the incompressible fluid pressure curve is drawn according to the formula (19). Figure 1 As can be seen from the figure, the pressure of compressible fluid is higher than that of incompressible fluid.

[0102] Figure 2 Taking the core length L as 100 mm, the downstream pressures are set to 0.1 MPa, 0.5 MPa, 1 MPa, 2 MPa, 3 MPa, and the upstream pressure changes from the corresponding downstream pressure to 10 MPa as an example, the compressible fluid pressure is calculated using the formula (17) of the present invention. Figure 2 It can be seen that under the same upstream pressure-downstream pressure combination conditions, compared with the commonly used pore gas method to calculate the incompressible fluid pressure value, as the downstream pressure increases, the compressible fluid pressure value approaches the incompressible fluid pressure.

[0103] In order to prove that fluid compressibility has an effect on the permeability of porous media, the permeability test was carried out using the steady-state seepage method, taking methane pore gas as an example. The relationship between the permeability of coal and the methane pore pressure under the confining pressure conditions of 4MPa, 5MPa, 6MPa and 7MPa was tested. Figure 3 , where the pressure of the compressible pore gas fluid is calculated according to the formula (17) of the present invention, and the pressure of the incompressible pore gas fluid is calculated according to the common method (i.e., formula 22). Figure 3 It can be seen that under the same fluid pressure conditions, the permeability of compressible fluid is less than that of incompressible fluid.

[0104] Furthermore, for ease of use, the present invention introduces a fluid pressure correction coefficient λ to correct the average pressure p calculated in the common method when the pore fluid is simplified to an incompressible fluid. mean After correction, the average pressure value of the compressible fluid is obtained, that is:

[0105] p eq =λp mean (twenty three)

[0106] in

[0107] First, using formula (23), we can get different pressure differences p 上游 -p 下游 The pore gas pressure correction factor λ corresponding to the condition is then calculated based on the upstream pressure p 上游 As the horizontal axis, the correction coefficient λ is used as the vertical axis to plot different downstream pressures p 下游 Correction coefficient curve of pore fluid pressure under certain conditions.

[0108] like Figure 4 Taking the core length L of 100mm as an example, the variation curve of the pore gas pressure correction coefficient λ is plotted when the downstream pressure is set to 0.1MPa, 0.5MPa, 1.0MPa, 2.0MPa, and 3.0MPa, and the corresponding upstream pressure is set to 0.1 to 10MPa, 0.5 to 10MPa, 1.0 to 10MPa, 2.0 to 10MPa, and 3.0 to 10MPa. In this way, when evaluating the relationship between the gas permeability of a porous medium of known size and the change of pore fluid, it is only necessary to know the upstream pressure and the downstream pressure. According to the upstream pressure p 上游 -Correction coefficient λ curve can be used to obtain the correction coefficient λ value corresponding to the pressure condition, and then according to formula 24, the average pressure of the pore fluid in the porous medium can be accurately evaluated. Figure 4 It can also be seen that the higher the downstream pressure, the closer the pore fluid pressure correction coefficient λ is to 1, which means that the smaller the pressure difference between upstream and downstream, the weaker the influence of fluid compressibility on the average pressure.

[0109] It should be noted that the fluid described in the present invention is limited to the scope of Newtonian fluid. In addition, the upstream and downstream pressure combination conditions listed in the implementation case are not used as limitations on the present invention, and the specific protection scope recorded in the claims shall prevail.

Claims

1. A method for calculating the gas pressure in a porous medium under the influence of gas pressure difference, wherein the gas pressure is used to evaluate the relationship between the gas permeability in the porous medium and the fluid pressure under a certain pressure difference condition measured by the steady-state method. It is characterized in that The specific method is: Step 1: Determine the steady-state seepage parameters of the porous medium The permeability test of a cylindrical porous medium with a length of L and a cross-sectional area of ​​A was carried out using the steady-state test method. The test parameters were the upstream pressure p(x=0,t)=p 上游 , downstream pressure p(x=L,t)=p 下游 , the seepage flow is Q, the fluid viscosity is μ, the fluid density is ρ, and the cross-sectional area of ​​the porous medium is A; Step 2: Determine the average pressure of the fluid in the porous medium Under steady-state seepage conditions, the gas pressure distribution in the porous medium is only related to the upstream and downstream pressures and the flow length. Therefore, the pressure distribution function of the gas in the porous medium is shown in formula (a): in: x is the flow distance of the injected gas; L is the length of the sample; t is the steady-state flow time; p is the gas pressure at the flow distance x at time t; p 上游 is the upstream pressure; p 下游 is the downstream pressure; The area S enclosed by the gas pressure and its flow distance com It can be expressed as formula (b) Using the mean value theorem of integral, we can convert the area S com It is equivalent to a rectangular area with a length of L and a width of W that is the same as the sample length. Then formula (b) is changed to formula (c) The width W of the rectangle is the average pressure p of the gas in the sample. eq , The pressure p calculated by formula (d) eq As the pressure difference p 上游 -p 下游 Gas pressure in porous media under certain conditions.

2. The method for calculating the gas pressure in a porous medium under the influence of gas pressure difference as claimed in claim 1, It is characterized in that The average value p of the sum of the upstream pressure and the downstream pressure is calculated by using the gas pressure correction coefficient λ mean Correction is performed to obtain the pressure p eq The simplified calculation formula is: Formula (e) can be rewritten as The λ is obtained by the following method: First, using formula (f), we can get different pressure differences p 上游 -p 下游 The gas pressure correction coefficient λ corresponding to the conditions is then calculated based on the upstream pressure p 上游 As the horizontal axis, the correction coefficient λ is used as the vertical axis to plot different downstream pressures p 下游 The correction coefficient curve of gas pressure under the condition of the above mentioned conditions is shown in Figure 1. Thus, when evaluating the relationship between gas permeability and gas pressure of porous media with known length and cross-sectional area, it is only necessary to know the upstream pressure and downstream pressure. 上游 -Correction coefficient λ curve can be used to obtain the correction coefficient λ value corresponding to the pressure condition, and then according to formula (e), the average pressure of the gas in the porous medium can be accurately evaluated; The pressure conditions described herein refer to different combinations of upstream pressure and downstream pressure of the sample during steady-state testing.

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