Studies of rock stress sensitivity are mainly focused on experimental and data processing methods, and the mechanism cannot be adequately explained using specific pore shape models. This study, based on a random pore network simulation, explains the rock stress sensitivity mechanism for the first time. Based on the network model theory, the hydraulic conductivity equation, the dimensionless radius equation, and the effective stress equation for partially saturated rock are used to generate a three-dimensional random pore network model based on the QT platform. The simulation results show that the influence of the effective stress on the dimensionless radius becomes more significant as the aspect ratio decreases, and the relationship between dimensionless radius and effective stress can be effectively interpreted through different combinations of pore shapes. Moreover, the mechanism behind permeability stress sensitivity can be explained by establishing the relationship between permeability and effective stress.
Rock stress sensitivity refers to the changes in rock petrophysical parameters caused by effective stress, including porosity stress sensitivity [
Although some researchers have previously built theoretical models based on specific pore shapes, such as circular [
Pore network models include pore space-based imaging network models [
Percolation theory is central to the study of randomized pore network models, which is mainly used to describe random structures and connectivity. It is used to study the fluid distribution and flow in a random and disorderly medium, as in randomized pore network models using statistical methods. The randomized pore network model employed in this study is designed as follows: the lines in the model symbolize throats of a certain volume and flow resistance with different distribution modes such as a uniform distribution and the intersections of the lines symbolize the throats without volume and flow resistance that only function as connections; therefore, the calculation of percolation parameters should be mainly focused on the linearity calculation without considering the node. Such a technique, based on the linear distribution of the network, studies the impact of the microdistribution on the macroproperties of the porous media. Since the percolation theory complies with the probability theory and the statistical accuracy depends on the sample size, the samples should meet the requirements for a reliable statistical result, and therefore the network simulation based on percolation theory requires that the three-dimensional network model has nodes of at least 20 × 20 × 20 [
The network simulation is based on the similarity principle between water and electricity. By analyzing the flow in the porous medium, the network structure of the circuit can be used to conduct a simulation analysis. The current in the circuit follows Ohm’s law:
By analyzing the network circuit, we assume that the circular cross section pipes filled with conduction fluid are made of resistors:
Assuming that flow is laminar, the simplest cylindrical pipe is taken as an example and the Poiseuille equation is used:
The circuit network and fluid pore network are all made up of pipes. According to (
Based on this, the Poiseuille equation can be simplified to Ohm’s law:
It can be seen from the similarity principle between water and electricity above that the fluid in the porous network is similar to the current in the circuit network; therefore, we can conduct the fluid flow simulation in the pore network model based on the current in the circuit network and use the analytical method for the current to analyze the simulated fluid model.
Kirchhoff’s circuit laws are divided into (1) Kirchhoff’s current law, whereby the current inflows of any node in the circuit are equal to the current outflows, and (2) Kirchhoff voltage law, whereby the voltage in any circuit loop shall be zero after completing the loop in the direction of current flow. For the pore network model, Kirchhoff’s current law is adopted to build the equation set of the model.
This study is focused on the quasi-static, two-phase flow, random pore network model. The fluid flow in the model is fully controlled by the capillary pressure, assuming that the fluid is incompressible and the influence of a viscous force is ignored. There are several assumptions reflected in the physical model. The oil-water or gas-water interface is relatively static; namely, the fluid distribution of the corresponding pore throat unit will change once displacement occurs. This assumption is the same as the low-speed percolation of most porous media. Furthermore, in the network simulation we generally assume that the displacement process is instantly completed and then balanced; namely, the nonwetting fluid pressure (
Throats with displacement contain a type of flowing fluid whereas the wetting fluid at the corner is taken as a static and nonflowing fluid; its pressure value remains at 0, the flow inside the pipes can be deemed as single-phase flow, and the fluid pressure (
This theory was proposed by Bishop [
When wetting-phase saturation values are at the extreme endpoints (0 and 1), the product of wetting-phase saturation and capillary pressure (
This expression is the effective stress equation for two-phase fluids and is identical to (
The key equation of this method involves the hydraulic conductivity at the throats and the dimensionless radius equation. The flow equation for a hyperbolic triangle throat [
Because there is no dimensionless radius equation for hyperbolic triangle throats in the literature, here we present a simple deduction based on the formula from Gangi [
Moreover, it is known from another Gangi formula that [
In a similar manner, the hydraulic conductivity equation and dimensionless radius equations for circular throats, oval throats, conical throats, hyperbolic triangle throats, and star throats can be deduced, shown in Table
Key equation.
Diagrammatic drawing | Hydraulic conductivity equation | Dimensionless radius equation | |
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Circular throat |
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Oval throat |
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Conical throat |
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Hyperbolic triangle throat |
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Star throat |
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In these equations,
The dimensionless radius equations for hyperbolic triangle throats and star-shaped throats are similar; only the coefficient and index are different. Therefore, their conclusions are similar, which is illustrated by Figure
Quadrilateral-triangles stress transformation.
A key part of the network model simulation is building operation expressions, the basic idea of which is based on Kirchhoff’s law, nodes, and line conductivity. Noble and Daniels [
For a simple network, the coefficient matrix can be obtained through a structural analysis; however, the simulation method of the model is based on statistical theory, which dictates that the number of nodes and lines in the network model needs to be large enough to ensure the reliability of the simulation. As a result, when the number of nodes and connections is large enough, the model cannot be solved by simple algebra, and the Cholesky decomposition is required in order to solve the coefficient matrix
The principal of the iterative method, which gradually approaches the real solution, is to take the assumed value as the solution and perform continuous iterations until it meets the convergence condition and obtains the solution of the equation. For the convergent system set, the deviation obtained from each displacement will decrease and its solution becomes closer to the real solution. The iteration method can also automatically adjust the occasional calculation error that occurs in the iteration. The method includes simple iteration and super-relaxed iteration and, in the following section, the simplest two-dimensional square network model is used as an example to introduce the solution process.
The basis of the simple iteration, also known as successive iteration, is to construct the fixed-point equation in order to obtain the approximate solution. The simple iteration method is solved via the following steps.
One node cell of square network model.
This equation can be used for every node in the network and, in this way, we obtain the system of linear equations whose quantities are identical to the node quantities:
To conduct the iteration solution process, (
As the convergence rate of the simple iteration is low, we use the super-relaxed iteration, and the program obtains the adjacent node voltage values from the last step calculation during the node calculation. For example, when we calculate the points
The method is called Gauss-Seidel iteration, and it replaces the new values obtained from the previous last step in order to speed up the convergence rates. The increment can be written as
To speed up convergence and introduce the relaxing factor
It can be seen from the above equation that the radius of any throat in the network is between the maximum throat radius and minimum throat radius. It should be particularly noted that the hydraulic radii (i.e., twice the volume to surface ratio of the pores,
(Log) uniform distribution.
Log uniform distribution | Uniform distribution |
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Value of
Log uniform distribution | Uniform distribution | ||||
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0.05 | 43.4563 | 36.5437 | 0.05 | 43.3557 | 36.4448 |
0.30 | 59.2411 | 20.7589 | 0.30 | 55.7657 | 17.6288 |
0.55 | 70.1570 | 9.84301 | 0.55 | 59.9655 | 1.45480 |
0.80 | 76.0414 | 3.95856 | |||
1.05 | 78.6472 | 1.35281 |
Based on the theory of the network model and the key programming points, the C++ language is used to compile random network model procedures on the QT platform. Regarding the grid size (
When running the program, the first output is a 3D cube random network model, whose size is set at 100 × 100 × 100 (Figure
Program output parameters.
Output parameters | Value |
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Pore volume |
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Pore surface area | 135504 |
Hydraulic radius | 39.7605 |
Numbers of tube bundle | 2970000 |
Total input electric current | 0.109619 |
Total output electric current | 0.0851105 |
Average coordination number | 5.9374 |
3D pore network model.
In order to analyze the influence of the aspect ratio on the calculation result, the program is set according to a specific throat, and the change in dimensionless radius due to effective stress is studied by setting different aspect ratios. See Figures
Effect of aspect ratio on oval pores.
Effect of aspect ratio on taper pores.
Effect of aspect ratio on triangular pores.
Effect of aspect ratio on star-shaped pores.
It is observed from Figures
Star-shaped throat at different aspect ratios.
The relationship described above is between the dimensionless radius and the effective stress in a specific throat and does not consider the range in aspect ratio values or the combination of the aspect ratio and the proportion. Therefore, taking the star-shaped pore as an example, the plan and operation results shown in Figure
Star-shaped [1.0 (60%) + 0.1 (20%) + 0.05 (10%) + 0.01 (10%)].
In addition to the different aspect ratios and proportion combinations for one specific pore shape, other relationships exist between dimensionless radius and effective stress with combinations of two-, three-, or four-pore throat types. Due to limited space, the combination of three different throat types is taken as an example, and the relationship between dimensionless radius and effective stress is effectively explained by a specific combination plan (Figure
Combination of three pores types. 20% cone [1.0 (60%) + 0.1 (20%) + 0.005 (10%) + 0.001 (10%)] + 30% triangle [1.0 (60%) + 0.1 (20%) + 0.005 (10%) + 0.001 (10%)] + 50% star [1.0 (60%) + 0.1 (20%) + 0.005 (10%) + 0.001 (10%)].
The analysis above indicates that the pore structure of a rock changes with the degree of effective stress, which induces changes in the rock physical properties. Therefore, a crucial factor in studies of rock stress sensitivity, particularly permeability stress sensitivity, is understanding the response of the pore radius to effective stress; namely,
The relationship between the pore radius derived from the model and the dimensionless radius
Comparison diagram of the program calculation curve and experiment data. 50% oval [0.01 (60%) + 0.005 (30%) + 0.0012 (10%)] + 50% star [0.1 (27%) + 0.005 (36%) + 0.0012 (37%)].
The effective stress equation for partially saturated rock is derived and verified, and the core operation expressions of the network model are obtained using iteration methods. The network model simulation shows that a circular throat is the special case for an ellipse, and the smaller the aspect ratio, the greater the effect of stress on the dimensionless radius. The relationship between the experimental dimensionless radius and the effective stress can be explained through different pore shape combinations based on the network model. The permeability stress sensitivity is effectively explained using the network simulation.
The authors declare that they have no competing interests.