Method and system for determining permeability of a porous medium

Abstract

A method for determining permeability of a porous medium is described which comprises the steps: a) obtaining a three-dimensional picture of the porous medium by an imaging system, b) dividing the three-dimensional picture into a number n of two-dimensional parallel slices, wherein n is an integer of 2 or more, c) identifying one or more pores in a first outermost slice (n.sub.1) using a grid which defines image pixels, d) identifying one or more pores in a second slice (n.sub.2) directly neighboring the first outermost slice (n.sub.1) using the same grid which defines image pixels as for the first outermost slice (n.sub.1), and e) labelling the one or more pores in the second slice (n.sub.2) as connected if at least one of its neighbours in the first outermost slice (n.sub.1) is a pore to give a number of connected pores as a connectivity result. Also described is a system comprising means for carrying out such a method.

Claims

1. A method for determining permeability of a porous medium, comprising the steps: a) obtaining a three-dimensional picture of the porous medium by an imaging system, b) dividing the three-dimensional picture into a number n of two-dimensional parallel slices, wherein n is an integer of 2 or more, c) identifying one or more pores in a first outermost slice (n.sub.1) using a grid which defines image pixels of the outermost slice (n.sub.1), d) identifying one or more pores in a second slice (n.sub.2) directly neighboring the first outermost slice (n.sub.1) using the same grid, which defines image pixels of the second slice (n.sub.2), as for the first outermost slice (n.sub.1), e) labelling the one or more pores in the second slice (n.sub.2) as connected to the one or more pores in the first outermost slice (n.sub.1) if the one or more pores in the second slice (n.sub.2) neighbors the one or more pores in the first outermost slice (n.sub.1) to give a number of connected pores as a connectivity result.

2. The method of claim 1 wherein steps c) to e) are a first iteration and wherein these steps are iterated to subsequent slices (n.sub.3, n.sub.4, . . . ) until a last slice (n.sub.last) is reached, wherein in each iteration the connectivity result in the previous iteration is regarded as a first slice, to give a number of connected pores as a final connectivity result.

3. The method of claim 1, wherein the method is repeated with respect to steps c) to e) in the opposite direction, i.e. starting from the last slice (n.sub.2 or n.sub.last), wherein after the repetition an average number of connected pores is computed as an average connectivity result.

4. The method according to claim 3 wherein a permeability connectivity index (PCI) is computed which is defined as the average number of connected pores divided by the total number of image pixels.

5. The method according to claim 1 wherein the grid defines squared image pixels.

6. The method according to claim 5 wherein each image pixel has nine neighboring image pixels in the previous and/or the subsequent slice.

7. The method according to claim 1 wherein the porous medium is a rock.

8. System comprising means for carrying out the method according to claim 1.

9. The system according to claim 8 wherein said means comprise an imaging system and a computer.

10. The system according to claim 8 wherein said means comprise an imaging system which is a computed tomography scanner.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present invention will be described with reference to the accompanying drawings of which:

(2) FIG. 1 is an example used to illustrate the operation of the developed pore connectivity index; (a) first slice, e.g. n.sub.1, (b) second slice, e.g. n.sub.2. Pores are labelled by the index 1.

(3) FIG. 2 is another example used to illustrate the operation of the developed pore connectivity index; (a) first slice, e.g. n.sub.1, (b) second slice, e.g. n.sub.2. Pores are labelled by the index 1.

DETAILED DESCRIPTION OF THE INVENTION

(4) According to the present invention, there is provided a method for determining permeability of a porous medium, comprising the steps:

(5) a) obtaining a three-dimensional picture of the porous medium by an imaging system,

(6) b) dividing the three-dimensional picture into a number n of two-dimensional parallel slices, wherein n is an integer of 2 or more,

(7) c) identifying one or more pores in a first outermost slice (n.sub.1) using a grid which defines image pixels,

(8) d) identifying one or more pores in a second slice (n.sub.2) directly neighboring the first outermost slice (n.sub.1) while using the same grid which defines image pixels as used for the first outermost slice (n.sub.1),

(9) e) labelling the one or more pores in the second slice (n.sub.2) as connected if at least one of its neighbours in the first outermost slice (n.sub.1) is a pore to give a number of connected pores as a connectivity result.

(10) In the inventive method, the image is preferably an electronical image so that it can be easily processed by e.g. a computer. Further, the two-dimensional parallel slices of the produced image are indexed as n.sub.1 and n.sub.2 for the first two slices. In case three slices are obtained in step b), the slices would be indexed as n.sub.1, n.sub.2 and n.sub.3. The same logic applies in case n is an integer of 4 or more. The final or last slice of a set of slices as produced in step b) is indexed as n.sub.last. So in case in total e.g. 100 slices are created, n.sub.last would equal n.sub.100. The entire stack of slices form the three-dimensional picture obtained in step a), and its outermost layers are slices n.sub.1 and n.sub.last. Further, in the sense of this invention slice n.sub.1 has one direct neighbor, namely slice n.sub.2, while slice n.sub.2 itself typically has two direct neighbours, namely slices n.sub.1 and n.sub.3—provided n.sub.3 exists; otherwise n.sub.2 is equal to n.sub.last and thus also has only one direct neighbour, namely n.sub.1. The slices n.sub.1 and n.sub.last are the outermost slices of the stack of n slices generated in step b).

(11) It is preferable according to the present invention that in the inventive method steps c) to e) are seen as a first iteration. By step e) this iteration gives a connectivity result. In a subsequent step and provided n is larger than 2 this first connectivity result is used a first slice and steps c) to e) are reiterated using this newly generated first slice and the subsequent slice. For example, in case the image is broken up into three slices and after having carried out steps a) to e) for the first time, the result for slices n.sub.1 and n.sub.2 is used as a new first slice, which could be labelled e.g. n.sub.1′. This new first slice n.sub.1′ is then used in a repeated step c) and e) and slice n.sub.3 is used in repeated step d) and e). So after having carried out the inventive method once, steps c) to e) are preferably iterated to subsequent slices (n.sub.3, n.sub.4, . . . ) depending on the number of slices until the last slice (n.sub.last) is reached. This means that in each iteration the connectivity result of the previous iteration is regarded as a first slice, to give a number of connected pores as a final connectivity result.

(12) Following the afore-said protocol pores penetrating through the imaged porous medium basically perpendicular to the plane of the generated parallel and two-dimensional images can be identified. It is preferable for an enhanced accuracy of the measurement to redo the entire exercise, this time reversing the order of the slices, i.e. using n.sub.last as the first slice and using n.sub.1 as the last slice. Thereafter, the resulting connectivities can be averaged. It is therefore particularly preferable in this invention that the method is repeated with respect to steps c) to e) in the opposite direction, i.e. starting from the last slice (n.sub.2 or n.sub.last), wherein after the repetition an average number of connected pores is computed as an average connectivity result.

(13) It is furthermore preferable according to the present invention to calculate a permeability connectivity index (PCI) to obtain a measure for the permeability of the analyzed porous medium. Accordingly, it is preferable that in the inventive method a permeability connectivity index (PCI) is computed which is defined as the average number of connected pores divided by the total number of image pixels.

(14) In a simple, analytically complete and hence particularly preferable method according to the present invention the grid defines squared image pixels, i.e. two-dimensional rectangular pixels of equal length and width are defined by the grid. While other shapes such as hexagons are also contemplated the computing done with image pixels having the shape of a square is more efficient in reducing computational time and memory. In this case the image pixels are preferably arranged such that each image pixel has nine neighbouring image pixels in the previous and/or the subsequent slice.

(15) When the porous medium analyzed by the method of the present invention is actually a rock, the present method is particularly useful for the oil and gas industry.

(16) In order to achieve one or more of the mentioned objects, the present invention furthermore provides a system comprising means for carrying out the method of the present invention as described herein. Preferably, the means comprise an imaging system and a computer. It is also preferable that the means comprise an imaging system which is a computed tomography scanner, especially preferably in combination with a further computer.

(17) A particularly preferred embodiment of the invention is now described. According to this particularly preferred embodiment of the invention, the inventive method is a method for determining the permeability of a rock using a combination of image translation and logical operations. In this particular embodiment the developed method depends on tracking the connectivity of pores within a rock sample. The method first assumes that pores in the first slice are filled with fluids. Next it finds the pores filled with fluids in the second slice. A pore in the second sliced is labelled to be filled with fluid if it is connected to a pore in the first slice. A pore is labelled connected if one of its nine neighbours in the previous slice are pores. FIG. 1 and FIG. 2 show examples that illustrate the connectivity between two slices. These figures show two slices of an image of size 3 by 3.

(18) In FIG. 1 only the centre pixel in the first slice is a pore. In the second slice, all pixels indicate the presence of pores. According to the connectivity definition as used in the present invention all the pores in the second image are connected to the first slice and will allow fluid to pass through them. If the first image did not contain pores then no pores in the second slice would be labelled connected.

(19) In the second example, shown in FIG. 2, only the upper left corner element of the first slice is labelled as a pore. The output of the corresponding method step will indicate the presence of 4 connected pores in the upper left corner of the image. The remaining pores are not connected.

(20) The preceding connectivity computation is iterated to subsequent slices. In each iteration, the connectivity result in the previous iteration is regarded as a first slice. Once the last slice is reached the number of connected pores is used as a measure for the permeability of the rock sample. To reach a more reliable estimate the previous operation is repeated in the opposite direction (i.e. starting from the last slice).

(21) Finally, a permeability connectivity index (PCI) is computed which is defined as the average of the number of connected pores found in the two computed directions divided by the total number of image pixels.

(22) The experiments underlying the present invention established a linear relationship between PCI values and rock permeability index. Computing PCI values for a sample of rocks with varying permeability values is preferable as a pre-processing step to get the parameters calibrating the relationship between PCI and rock permeability index. This calibration process allows to address variations between different scanners and the use of different imaging resolutions. The invention therefore preferably includes a calibration step to be performed on e.g. a computed tomography (CT) scanner from which data is acquired.

(23) Computation of pore connectivity between slices can be performed using binary logical operations. The simplicity of the proposed method and the use of binary logical operations make the developed permeability index an attractive alternative to other methods frequently used in practice.

(24) The invention delivers a reliable estimate for rock permeability computation with significant reductions in computational time.

(25) Although an illustrative embodiment of the invention has been shown and described, it is to be understood that various modifications and substitutions may be made by those skilled in the art without departing from the novel spirit and scope of the invention.

FURTHER REFERENCES

(26) Journal

(27) Sun H F, Vega S, Tao G. Analysis of Heterogeneity and Permeability Anisotropy in Carbonate Rock Samples Using Digital Rock Physics. Journal of Petroleum Science and Engineering, 2017, 156: 419-429. (IF 1.873) Sun H F, Tao G, Vega S, Al-Suwaidi A. Simulation of Gas Flow in Organic-Rich Mudrocks Using Digital Rock Physics. Journal of Natural Gas Science and Engineering, 2017, 41: 17-29. (IF 2.718)
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Others
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