JP4831407B2 - Estimation method of permeability of porous objects using renormalization - Google Patents

Description

  The present invention estimates the effective permeability of a heterogeneous porous object, for example, a formation as a whole, containing voids of various sizes from the void size distribution data using a technique called renormalization. It relates to the invention.

Underground fluids such as groundwater and petroleum exist in the voids of the porous formation, and when a well is dug and sucked into the formation, the fluid is driven by a pressure gradient and flows inside the void and moves toward the well. Permeability of porous formation (unit: m ² ) is a quantity that determines the flow rate per unit area when a certain pressure gradient is applied, and is an economically important hydraulic characteristic that defines the production rate of wells. It is. The pore size data of the porous formation is an amount necessary for calculating the permeability, and is measured by a nuclear magnetic resonance (NMR) method, a mercury porosimeter indoor analysis method of the collected core, or the like.

In this case, the problem is a significant non-uniformity in the pore size of natural rocks (see, for example, FIG. 3). FIG. 3 is a two-dimensional X-ray CT image of a natural Chichibu porous sandstone, showing a cross-sectional view of a cylindrical sample having a diameter of 4 mm, with the black portions being voids and the gray and white portions being rock-forming minerals. It can be seen that gaps of various sizes are mixed in only 4 mm.
In general, a gap of several nanometers to a few millimeters is mixed in a rock sample of several grams that looks uniform at first glance. According to the Kalman-Cozeny model (cylindrical pipe model), the permeability is the square of the void size (see, for example, Non-Patent Document 1). Therefore, for example, the permeability calculated from the 10 nm gap size is different from that calculated from the 10 mm gap size by 1 million times, so there is a trillion times increase in the permeability. In order to correctly evaluate the permeability that governs the production rate of wells, macroscopically, for example, several centimeters to several hundred meters in size from porous rocks with microscopically wide pore sizes as shown in FIG. There is a need for a method to determine the penetration rate of the effective (effective) formation.
Y. Nakashima and T. Yamaguchi (2004) X-ray CT for Geomaterials; Soils, Concrete, Rocks. (Edited by Otani, J. and Obara, Y.) AA Balkema Publishers, pp. 103-110

As one of the conventional technologies frequently used, as shown in FIG. 10, the nuclear magnetic resonance sensor is lowered into the well and the wall of the well is continuously scanned, so that the proton profile of the porous formation filled with water or oil is obtained. There is a geophysical exploration method that obtains relaxation time distribution data (equivalent to void size distribution data) in-situ and then estimates the permeability of the formation (eg by K.-J. Dunn, DJ Bergman, GA Latorraca, Nuclear Magnetic Resonance Petrophysical and Logging Applications, Pergamon, Amsterdam, 2002).
FIG. 10 is a schematic diagram of NMR logging, in which a permanent magnet that precesses protons and a coil for exciting and detecting spin are mounted in an NMR sonde. Sensitivity area 1.9cm deep from the surface of the porous wall of the porous formation filled with groundwater or oil while pushing the sonde against the pit wall and pulling up the wire (hatched area, outline is 1.3cm in diameter and 12cm in length) The proton transverse relaxation time of the interstitial fluid is scanned.

In this so-called nuclear magnetic resonance logging, the macroscopic permeability k of the formation is estimated by the following equation (1) or (2).
[Schlumberger-Doll-Research method]
φ is the porosity of the porous rock, that is, the volume fraction of the void (0 ≦ φ ≦ 1). T _2LM is the logarithmic mean value of the proton transverse relaxation time (T2) distribution. If this logarithmic average value is in ms units, k in equation (1) is in millidalcy units.
[Timur-Coates method]
FFV is called Free Fluid Volume (Free Fluid Volume), which is the volume fraction of large voids. BFV is called the bound fluid volume fraction (Bound Fluid Volume). There is a relationship of φ = FFV + BFV. In terms of NMR relaxation time, FFV is the volume fraction of voids with a T2 greater than a certain threshold, and BFV is the volume fraction of voids with a T2 smaller than a certain threshold. In this specification, in equation (2), the threshold value of FFV and BFV (cut-off value of lateral relaxation time) is set to a commonly used value of 33 ms. Note that k in the equation (2) is in millidalcy units.

The point common to both of the above formulas is from the distribution of void sizes widely distributed over several digits, the total area of the distribution (void ratio φ) and the average value (logarithmic average value or FFV / BFV ratio) It is the point that the penetration rate of the formation is estimated using only these two statistics. Since this method requires a small amount of calculation, it is useful if you want to know the permeability quickly from the proton transverse relaxation time distribution. However, the shape of the size distribution cannot be determined uniquely using only two statistics. FIG. 11 shows two hypothetical gap size distributions with diagonal lines and black. The diagonal line is bimodal and the distance between the two peaks is narrow. Black is also bimodal, but the peak interval is wide. Both the diagonal and black distributions have the same total area (φ) and logarithmic mean (T _2LM ), but the shape of the distribution is clearly different. Therefore, there is no guarantee that the macroscopic penetration rate will show the same value.
In this way, equations (1) and (2) using only two statistics may be used for special formations such as sandstone and limestone that store crude oil, but any shape size distribution (lateral relaxation time) If it can be applied to (distribution), it is wrong. There is no guarantee that it can be applied to all natural formations with various void structure characteristics other than oil reservoirs. In fact, although it is about hypothetical data, there are already reports of cases where the true penetration value cannot be calculated correctly using the geometric mean (synonymous with logarithmic mean) (Transport in Porous Media, 4, 37- 58, 1989).

  The present invention applies a renormalization theory used in statistical physics to a void structure model, and a system of heterogeneous porous objects, such as formations, including voids of various sizes as shown in FIG. The total effective permeability (also called “permeability coefficient”) is estimated. By using all the data points of the void size distribution, only the total area and the average value are used. The purpose is to make it possible to estimate the penetration rate with higher accuracy than the above method.

  “Renormalization” was originally a method for formulating the concept of coarse-graining in the statistical mechanics of many-body systems and deriving macroscopic properties, and was especially successful when applied to critical phenomena. The first application of renormalization to the calculation of macroscopic permeability in groundwater science was P.R. King (Transport in Porous Media, 4, 37-58, 1989).

FIG. 1 shows the basic concept of “renormalization”. Although the present invention is a three-dimensional renormalization, a two-dimensional case is shown for easy understanding.
Suppose now that there is a porous rock that is a non-uniform porous object consisting of 4 × 4 cells. This is the original rock texture. The 16 cells have different void sizes. As a result, the permeation rate also takes a different value (K1 _A , K1 _B ,,,,,, K1 _P ) for each cell according to a certain distribution K1. In the first step of renormalization, four cells are virtually considered as subsystems, and the total penetration rate (so-called combined resistance of the electric network) of the four systems is strictly calculated using Kirchhoff and Ohm's law. It is to solve. If the operation is repeated four times, a penetration rate combination of four subsystems is obtained. As a result, as a penetration index, obtain K2 (component _{K2 A, K2 B, K2 C} , K2 D).
In the second step of renormalization, assuming an upper 2 × 2 super subsystem, the overall penetration rate K3, which integrates four subsystems (components are K2 _A , K2 _B , K2 _C , K2 _D ), Strictly using Kirchhoff and Ohm's law. The estimation of the permeability of the entire system by such coarse-grained cell is a renormalization operation. Since the actual rock is composed of an infinite number of cells instead of 4x4, the renormalization operation is usually carried out many times, and the columns K2, K3, K4, etc. of the obtained permeability distribution are sufficiently small in standard deviation. Is confirmed to converge to the distribution, and the average value converged is regarded as the effective penetration rate of the entire system.

Based on FIG. 2, a specific method for calculating the penetration rate by renormalization will be described.
Taking the 4 × 4 two-dimensional original system of FIG. 1 as an example, a renormalization operation is performed every 2 × 2 = 4 cells. For example, in the case of four cells having permeability of K1 _A , K1 _B , K1 _E , K1 _F , each cell has a resistance value 1 / K1 _A , 1 / K1 _B , 1 / K1 _E , 1 / K1 _F It is regarded as an element having an electrical resistance value of. Note that the resistance value is an index of difficulty in flowing, and the permeability is an index of ease of flowing, so they are in a reciprocal relationship with each other. Further, for enabling connections between cells, for example cells of K1 _A is subdivided into four elements in the resistance 1 / 2K1 _A, and assembled as shown in FIG. 2 (b), 4 types of electrical resistance To form an electric circuit network consisting of a total of 16 resistive elements. As shown in FIG. 2C, considering that a current is virtually passed through this circuit, the combined resistance value of the entire circuit, 1 / K2 _A , is solved exactly using Kirchhoff and Ohm's law. as a result,
Get. K2 _A is a permeability value when 2 × 2 cells having permeability values of K1 _A , K1 _B , K1 _E , and K1 _F are combined into one cell (coarse-grained) by a renormalization operation.

In order to achieve the above-described ability to estimate the permeability with high accuracy, in the method for estimating the permeability of a non-uniform porous object containing voids of various sizes using the renormalization according to the present invention, it becomes an estimation target. Measure the distribution data of the porosity φ and the void size d of the porous object, select a lot of data randomly from the distribution of the void size d, calculate the permeability k from the following equation (4), and calculate the original permeability A distribution K1 is created, and the original permeability distribution K1 is gradually converged to a small standard deviation distribution by a renormalization operation, and the average value is used as the permeability of the entire porous object to be estimated. .

By adopting the above-mentioned means, the present invention has the following excellent effects.
(1) In the conventional nuclear magnetic resonance method, which extracts only two statistics (area and average value) from the distribution of pore sizes widely distributed over several digits, oil storage rocks (sandstone, limestone, etc.) The present invention is applicable to all of the natural porous objects having various void structures, as compared to that which can only be used for a specific formation.
(2) Compared to the conventional nuclear magnetic resonance method, it is possible to obtain a value that is much closer to the true permeability value (actually measured value).

  The best mode for carrying out the method for estimating the permeability of a porous object using renormalization according to the present invention will be described below with reference to the drawings based on the embodiments.

  In order to use the renormalization theory, a void structure model (pipe model) for renormalization was first created as shown in FIG. The porous cell 1 is a cube with a side length of L, which corresponds to each of the 4 × 4 original cells in FIG. Each cell 1 is embedded with a group of straight pipes 2 that run orthogonal to three directions as shown in this figure. The number of pipes 2 running in each direction is the same. The fluid flows through the pipe 2 at a low Reynolds number. The diameter d of the pipe 2 is different for each cell 1, but takes a common value in one cell. The porosity φ of the cell 1 is common to all the cells regardless of d. That is, assuming a sufficiently large L value model as a cell, the value φ (a value converged by the central limit theorem) having a common porosity in all cells can be taken.

Since equation (4) does not depend on L, there is no arbitrary cell size L arbitrary. The width of the distribution of the gap size data is taken into the value of d. That is, the d value of each cell takes a value sampled at random from the measured gap size data, and a large system in which the cells are randomly stacked in three dimensions as shown in FIG. 5 is virtually considered. FIG. 5 is a conceptual diagram of the three-dimensional random porous object 3 (in this case, composed of 50 × 50 × 50 cells). Each cell 1 is made of the porous cube of FIG. 4 and has an isotropic and random permeability distribution with no spatial correlation as a whole. For a system with an infinite number of pixels, the ease of flow when a fluid is passed through the system, that is, the effective penetration rate of the entire system is estimated by the renormalization method of FIG.

  By the way, the void structure model submitted by K. Xu, JF Daian, and D. Quenard (Transport in Porous Media, 26, 51-73, 1997) is too realistic to pursue the efficiency of mathematical processing. It is unnatural because a void structure model is used. For example, they assume that the distribution of cell size L is a power, and further make the precondition that the ratio of L and d is constant. However, this is inconsistent with reality, as is clear from recent research on the three-dimensional void structure of real porous rocks (Lindquist et al., Journal of Geophysical Research, 105, 21509-21527, 2000). is there. On the other hand, the patent does not make such an unrealistic assumption, so it can be said that the model is more realistic.

  The procedure for estimating permeability by the renormalization method is as follows.

Procedure (1):
Measure porosity and void size distribution data. For example, the collected core is analyzed with a mercury porosimeter. Alternatively, NMR logging is performed to obtain a transverse relaxation time distribution. However, since the raw data of the NMR logging is a transverse relaxation time distribution and not a size distribution, for example, using a core sample, the proton surface relaxation rate (ρ ₂ ) of rock is measured to measure the transverse relaxation time, T2 , Must be converted to a size distribution (eg, K.-J. Dunn, DJ Bergman, GA Latorraca, Nuclear Magnetic Resonance Petrophysical and Logging Applications, Pergamon, Amsterdam, 2002). Here, since it is assumed the cylindrical pipe shown in FIG. 4, converted by the expression d = 4T2ρ _2.

Procedure (2):
A large number of data are selected at random from the distribution of the gap size d, and the permeability k given by the equation (4) is calculated to create a permeability distribution. This corresponds to the original permeability data set K1 in FIG.

Procedure (3):
A renormalization simulation of a three-dimensional cell is performed with the concept of FIG. For a 2 × 2 × 2 three-dimensional cubic subsystem, the overall permeability of the eight system (the combined resistance of the electrical network) is solved exactly using Kirchhoff and Ohm's law. Detailed calculation method follows Transport in Porous Media, 4, 37-58, 1989). While monitoring on the computer how the original permeability distribution K1 with a wide standard deviation gradually converges to a small standard deviation distribution by the renormalization operation, if it is determined that it has converged to a sufficient accuracy, the renormalization operation is terminated. The average value is regarded as the effective penetration rate of the entire system.

[Experimental example]
For two porous sandstones and one porous lava, the permeability was estimated by the renormalization technique of the present invention from the void size distribution data by a mercury porosimeter. For comparison, the permeability was estimated by the conventional nuclear magnetic resonance method, that is, the equations (1) and (2). The results are shown in Table 1.
[Description of rock samples used in the experiment]
(1) Sandstone produced in Chichibu, Saitama Prefecture (Fig. 3) has a porosity of φ = 0.14. Data such as proton surface relaxation rate (ρ ₂ = 5.6x10 ^-5 m / s) and measured values of permeability are “ Y. Nakashima, T. Nakano, K. Nakamura, K. Uesugi, A. Tsuchiyama, and S. Ikeda (2004) Journal of Contaminant Hydrology, 74, 253-264. The distribution data of the void size d by the mercury porosimeter experiment is shown in FIG. The vertical axis represents the void volume per gram of rock sample. The d in FIG. 6 d = 4T2ρ ₂ results of analysis for the transverse relaxation time distribution converted by the expression, FFV = 0.0155, BFV = 0.1245 , was T _2LM = 2.61 ms.
(2) Andesitic lava from Sumikawa, Akita Prefecture, is derived from the core taken from 57.3m underground by drilling on the premises of the Sumikawa geothermal power plant. The porosity is φ = 0.0286, and the measured value of the core permeability test is 3 × 10 ⁻¹⁹ m ² . FIG. 7 shows distribution data of the gap size d by the mercury porosimeter experiment. The vertical axis represents the void volume per gram of rock sample. The proton surface relaxation rate ρ ₂ required when converting the pore size distribution measured by the mercury porosimeter into the NMR transverse relaxation time is 9.5 × 10 ⁻⁴ m / s. As a result, a result of analysis for the transverse relaxation time distribution converted by the expression to d in FIG. _{7 d = 4T2ρ 2, FFV =} 0, BFV = 0.0286, was T _2LM = 0.0143 ms.

The simulation results for Chichibu sandstone are shown in Fig. 9 (a) to (e). The vertical axis of the histogram is the number of penetration data. The horizontal axis is the permeability value (unit: 10 ^-15 m ² ). The total number of data points in each histogram of K2, K3, K4, and K8 is 7000. The arithmetic mean and standard deviation of the histogram are also written (unit is 10 ^-15 m ² ). It can be seen that the standard deviation of the very wide original permeability distribution (K1) decreases as the renormalization operation is repeated. In this case, notice that becomes sufficiently small standard deviation is performed seven times renormalization, and with a mean ^{1.32x10 -15 m 2 ≒ 1x10 -15 m} 2 of the distribution K8 and the estimated value of the penetration rate.

[Consideration of experimental results]
Table 1 summarizes the comparison of estimation results for the above three rock samples by the renormalization method of the present invention and the conventional NMR method (SDR method of equation (2) and Timur-Coates method of equation (3)). As is clear from Table 1, the estimated values of the present invention are much more true for sandstones from Chichibu, Saitama and andesitic lava from Sumikawa, Akita than the values estimated by the conventional nuclear magnetic resonance method (actual measurements). Was close to the permeability value). Comparing the NMR method with the renormalization method, the SDR method barely gave the same estimation value as the renormalization method for the sandstones from Bellaire, but in other cases, the estimation accuracy was extremely low. The estimated accuracy of Chichibu sandstone differs by an order of magnitude from the true value (actual measurement). However, natural strata have inherently uncertainties of more than 10 orders of magnitude from about 10 ^-21 m ² to about 10 ^-10 m ² . Therefore, it can be said that it is an acceptable error that the estimation of the Chichibu sandstone by the renormalization method was able to suppress the deviation from the true value to only one digit. Thus, the effectiveness of the patent was confirmed by an example using three samples.

It is a conceptual diagram for demonstrating "renormalization" employ | adopted in this invention. It is a figure which shows the calculation method of the specific penetration rate by "renormalization." This is a 2D X-ray CT image of natural Chichibu porous sandstone. It is the perspective view which showed the pipe model for carrying-in. It is a conceptual diagram of the three-dimensional random porous object comprised from 50x50x50 of each cell which consists of a porous cube of FIG. It is a figure which shows the distribution data of the space | gap size d by the mercury porosimeter experiment of the Chichibu sandstone. It is a figure which shows the distribution data of the space | gap size d by the mercury porosimeter experiment of Sumikawa lava. (a) Distribution data of pore size d by mercury porosimeter experiment of Berea sandstone, (b) Distribution data of proton transverse relaxation time by NMR experiment of Berea sandstone (using 2.2MHz NMR logging sonde) FIG. It is a figure which shows the renormalization simulation result of the Chichibu sandstone. It is a schematic diagram of NMR logging. It is a figure which shows two virtual space | gap size distribution.

Explanation of symbols

1 cell 2 pipe 3 3D random porous object

Claims (1)

In the method for estimating the permeability of a non-uniform porous object containing voids of various sizes, distribution data of the porosity φ and the void size d of the porous object to be estimated are measured, and the distribution of the void size d Because of renormalization, a large number of data are selected at random from each other, and two straight pipes with a diameter d are embedded perpendicularly to the two opposite faces of three pairs of cubes each having a length of L. of voids model below it does not depend on the obtained L by using the (4) to calculate the permeability k from equation to create original permeability distribution K1, the original permeability distribution K1, the porous the quality object is converged on the distribution of progressively smaller standard deviation by performing the renormalization operation assumes that piled up randomly in three dimensions the cube, the porous body all the average value of the estimated target Porous estimation method of permeability Objects by renormalization, characterized in that the penetration rate.