Determining relative permeability of porous media

Abstract

A method for determining the relative permeability of a porous medium generates a pore scale output from a pore scale flow simulation using a segmented structure image generated from a 3D image. Based on the boundary conditions of the pore scale flow simulation and the initial relative permeability model, a Darcy scale flow model is generated by simulating fluid flow. The Darcy scale output is compared to the pore scale output to determine a degree of match. The initial relative permeability model is updated and the Darcy scale simulation and reverse modeling steps are repeated until the matching degree falls within a predetermined tolerance range.

Description

Technical Field

[0001] The present disclosure relates to the field of fluid flow in porous media. In particular, the present disclosure relates to determining the relative permeability of porous media. Background Art

[0002] Analysis and imaging techniques can help visualize fluid flow in porous media for a variety of applications including, for example, but not limited to, hydrology, pollutant fluid dynamics, petroleum engineering, carbon capture and sequestration, hydrogen storage, fuel cells, electrolysis, CO 2 Converted to basic chemicals.

[0003] For example, in the face of growing global energy consumption and the need to reduce environmental pressure, enhanced oil recovery (EOR) and carbon capture and sequestration (CCS) allow hydrocarbons to be produced in a sustainable way. With EOR, more hydrocarbons can be obtained from a single well by injecting brine, polymers or surfactants, reducing the need for additional drilling. In CCS, carbon dioxide is reinjected into the reservoir so that it does not enter the atmosphere. The flow and dispersion of fluids in rocks also drives geothermal extraction processes, which are another promising source of renewable energy. At the same time, seasonal storage of underground hydrogen is also important. Understanding the properties of fluid flow plays a key role in the deployment of such technologies.

[0004] Many factors affect fluid flow, including, for example, fluid density and viscosity, interfacial tension, fluid flow rate, surface wettability, and pore geometry. Laboratory testing of these factors is time-consuming and very expensive. In addition, the number of samples that can be processed is relatively limited due to the time and expense required to perform each test.

[0005] In conventional processes, although the goal is to use real fluids such as CO 2 and H 2 , it is common practice to measure relative permeabilities using model fluids, such as decalin as the oil or non-wetting phase or hydrogen to represent the hydrogen phase. The preferred method for measuring relative permeabilities is the steady-state method over other methods such as the unsteady-state method, which provides a larger accessible saturation range, is generally easier to understand, and is more resistant to displacement instabilities. However, the steady-state method requires a large amount of pore volume to be injected at different fractional flow rates. For CO 2 and H 2 This can pose significant technical challenges for experimental setup and fluid handling, and may also experience dissolved CO 2 or H 2 Furthermore, measuring capillary pressure and saturation of reactive fluids and gases is challenging, for this reason these measurements are usually performed using dual sample specialist methods.

[0006] Alternatively, an option is to consider non-steady-state methods, which are more suitable for target fluid treatment and usually operate with significantly reduced injection volumes than steady-state methods. Under specific operating conditions, non-steady-state methods allow for simultaneous determination of relative permeability and capillary pressure-saturation function under relevant operating conditions. Traditional interpretation methods for non-steady-state experiments are usually based on analytical models. Although these methods are significantly improved to capture the impact of capillary endpoint effects at the same time, they are usually significantly simplified, and in many cases, these methods do not integrate all measured quantities such as longitudinal profile saturation distribution diagrams, which can be obtained by in-situ X-ray saturation monitoring.

[0007] Information needs to be provided more quickly to enable more timely decisions. Digital rock physics is a technology developed to provide faster, more extensive and cheaper analysis of porous media. Digital rock physics uses digital images of formation rocks to simulate pore-scale rock physical properties and predict the properties of the complex. The main focus of digital rock technology is to accurately simulate the behavior of fluid flow within the pore space in order to predict the rock physical properties of the sample, such as permeability.

[0008] Pore-scale flow simulations based on an understanding of the dynamics of fluid displacement at the pore scale can facilitate the calculation of Darcy-scale flow parameters used as input in reservoir simulations, for example.

[0009] One type of fluid flow simulator is based on the lattice Boltzmann method (LBM), which simulates the collision and flow of microscopic particles and then estimates macroscopic pressure gradients and velocities, from which the permeability of porous materials can be estimated (Chen et al., "Lattice Boltzmann method for fluid flows" Fluid Mechanics Yearbook (Annu Rev Fluid Mech) 30:329-64; 1998). For example, Alpak et al. (“Prediction of fluid topology and relative permeability in imbibition in sandstone rock by direct numerical simulation” Advances in Water Resources122:49-59; 2018 and “Direct simulation of pore-scale two-phase visco-capillary flow on large digital rock images using a phase-field lattice Boltzmann method on general-purpose graphics processing units” Computational Geoscience Geosciences 23:849-880; 2019) described an energy-based LBM (eLBM), which is a two-phase flow simulation system consisting of the following three modules: a forced discharge simulation module, a forced imbibition simulation module, and a steady-state relative permeability calculation module.

[0010] There is a need to improve the speed and accuracy of porous media parameters while reducing the time and overhead of computational resources. Summary of the invention

[0011] According to one aspect of the present invention, a method for determining the relative permeability of a porous medium is provided, the method comprising the following steps: (a) providing a 3D image for a porous medium sample; (b) generating a segmented structure image from the 3D image to identify pore space and solid material; (c) simulating fluid flow on the segmented structure image using a pore-scale flow simulation to produce a pore-scale output; (d) selecting an initial relative permeability model; (e) using the initial relative permeability model, based on the boundary conditions of the pore-scale flow simulation, generating a Darcy-scale flow model by simulating fluid flow to generate a Darcy-scale output; and (f) comparing the Darcy-scale output with the pore-scale output to determine a match; and (g) updating the initial relative permeability model and repeating steps (e) to (g) until the match falls within a predetermined tolerance range. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The method of the present invention may be better understood by reference to the following detailed description of the preferred embodiments and the accompanying drawings referred to therein, in which:

[0013] Figure 1 is a schematic diagram showing one embodiment of the method of the present invention;

[0014] Figures 2 to 5 is a graphical illustration of pore-scale output in an example of the present invention;

[0015] Figure 6 and Figure 7 is a graphical illustration of an initial relative permeability model in an example of the present invention compared to a matched relative permeability model after inverse modeling;

[0016] Figures 8 to 10 Compare the match between the pore-scale output and the Darcy-scale output in the examples of the present invention; and

[0017] Fig.11 is a schematic diagram of the computational domain discussed in the examples of the present invention. DETAILED DESCRIPTION

[0018] The present invention provides a method for determining the relative permeability of a porous medium from a 3D image of the porous medium. The 3D image is segmented to generate a segmented structural image. A pore-scale flow simulation is performed on the segmented structural image to generate a pore-scale output. A Darcy flow model is then generated based on the boundary conditions of the pore-scale flow simulation using an initial relative permeability model. The generated Darcy-scale output is compared with the pore-scale output to determine a match. The initial relative permeability model is updated. The Darcy-scale simulation and inverse modeling are repeated until the match is within a predetermined tolerance.

[0019] Using segmented structural images of rocks in 3 dimensions at pore-scale resolution, petrophysical properties of rock samples are measured using digital rock techniques. Typically, pore-scale simulations of 3D images of porous media require high-performance computing resources with huge computational costs.

[0020] For reservoir simulation, the Darcy scale concept is dedicated to relevant flow parameters such as relative permeability and capillary pressure-saturation function. In traditional workflows, these are determined in time-consuming and expensive laboratory experiments, i.e., so-called "core flood" experiments, in which fluid flows through cylindrical rock samples obtained by flow field, and relative permeability is obtained by interpreting the experimental production curves, pressure drops, and saturation profiles of saturation profiles. In addition, Berg et al. (Transmission 140:27-57 for "Sensitivity and Uncertainty Analysis of Parameterizations for Multiphase Flow Models in Porous Media" in Porous Media; 2021) describe an auxiliary history matching workflow in which experimental core flood data are matched with numerical solutions of the two-phase Darcy equation to extract the relative permeability function.

[0021] According to the present invention, the Darcy scale flow model is interpreted for pore scale flow simulation. The inventors surprisingly found that the method of the present invention can accurately and timely predict relative permeability with low requirements on computing resources.

[0022] By "pore scale" is meant the length dimension at which the individual pores of a porous material are resolved. For porous rocks, the pore dimension typically requires, for example, but not limited to, a resolution of 3D segmented images of 1 to several microns. For example, the resolution limits of pore-scale simulations are discussed in Saxena et al., "Effect of image segmentation & voxel size on micro-CT computed effective transport & elastic properties," Marine Petroleum Geology 86: 972-990; September 2017), which is incorporated herein by reference. For porous media in general, the length dimension depends on the actual porous media and, therefore, can be significantly larger or smaller than 1 micron.

[0023] "Darcy scale" means the size at which a porous medium is described by continuum mechanics methods and corresponding parameters such as porosity and permeability. For single-phase flow properties, this is typically the case when the length dimension is equal to or greater than the characterizing unit volume (e.g., J. Bear, Fluid Dynamics in Porous Media, Dover, 1988.), and for porous gravel rocks, is often between 2 mm and 4 mm. For porous media in general, the length dimension depends on the actual porous medium and, therefore, can be significantly greater or less than 2 mm to 4 mm.

[0024] "Relative permeability model" means the relative permeability and capillary pressure-saturation functions and parameterizations thereof in functional form, such as (but not limited to) the Corey or LET model for relative permeability and the Skjavran model for capillary pressure, including parameter values ​​used in these models, or tabulated values ​​with respective interpolations as required for numerical calculations. Additionally or alternatively, the relative permeability model may include a saturation endpoint, a capillary endpoint, and combinations thereof.

[0025] To date, those skilled in the art have clearly distinguished the effects of pore-scale simulations and Darcy flow simulations based on the length scale used by each. The length scale at which this separation occurs is called the Characterization Element Volume (REV). When used for traditional Darcy-scale inverse modeling, when the computational domain of digital rock direct flow simulations is divided into grid blocks (e.g., 50 grid blocks), the length scale of each grid block is less than the REV. Accordingly, it is generally believed that Darcy-scale physics is not valid at the pore scale. Accordingly, although the entire computational domain of pore-scale direct flow simulations can be REV, a single grid block is generally understood to be much smaller than REV and, therefore, is not suitable for the two-phase Darcy equations used for inverse modeling.

[0026] A better understanding of the flow and distribution of fluids through porous media can have implications for hydrology, pollutant fluid dynamics, petroleum engineering, carbon capture and sequestration, hydrogen storage, fuel cells, electrolysis, CO 2 Make better decisions when switching to base chemicals.

[0027] For many modeling studies, e.g., for underground storage of carbon dioxide and hydrogen, it is important to have a consistent set of relative permeability and capillary pressure-saturation functions. Advantageously, both functions are determined in a single experiment using the same porous medium and fluid sample. However, experimental measurements are often challenging. In order to assess the availability and development of individual reservoirs, e.g., in underground formations of rock, e.g., in depleted oil and gas fields, or in saline aquifers, numerical models are used. The demand for such models ranges from estimating storage capacity and plume migration to assessing risks such as potential leakage or displacement stability.

[0028] Relative permeability models are highly dependent on the wetting properties of the porous media and the fluids that flow into the pore space of the rock, directly affecting parameters such as the capture of certain fluids. For example, supercritical CO 2 It may not be a completely non-wetting fluid towards the hydrophilic medium, but may have different wetting behavior relative to other non-wetting fluids, such as n-decane. 2 , H 2 Have different wetting behaviors.

[0029] The inventors surprisingly found that a cost-effective and time-efficient unsteady-state type of pore-scale flow simulation can be used to simultaneously derive a relative permeability model within acceptable uncertainty. By using Darcy-scale flow simulation, comparing the Darcy-scale output with the pore-scale output, and adjusting the relative permeability model in situ, the accuracy of the predictions is improved compared to traditional methods.

[0030] The present invention generally explains the capillary endpoint effect and the capillary effect, which can have a large impact on explaining the key parameters caused, such as residual oil saturation. In traditional methods, steady-state simulations are usually used to try to capture the capillary endpoint effect. Steady-state simulations are computationally expensive due to the long convergence time. According to the present invention, inverse modeling for unsteady-state pore-scale simulations is faster and the computational cost is greatly reduced. According to the method of the present invention, unsteady-state pore-scale simulations can be used to reliably extract relative permeability models.

[0031] Reference Figure 1 , one embodiment of the method 10 of the present invention involves providing a 3D image 12 of a porous media sample and generating a segmented structural image of the sample at step 14 .

[0032] 3D images

[0033] The 3D image 12 is obtained using pore-scale imaging techniques. The 3D image 12 may be obtained by, for example, but not limited to, scanning electron microscopy (SEM), X-ray computed tomography, acoustic microscopy, magnetic resonance imaging, etc. X-ray computed tomography includes, but is not limited to, X-ray micro-computed tomography (micro-CT) and X-ray nano-computed tomography (nano-CT). Most preferably, the 3D image 12 is obtained by micro-CT to provide a sufficient field of view of the porous medium to avoid the total porosity of the side pores distorting the generated image, as well as to reduce scanning time and computational requirements (which may be required for higher resolution tomography (micro-CT)).

[0034] The imaging 3D image 12 obtained by the pore-scale imaging technique consists of a plurality of voxels, wherein the volume defined by each voxel represents the maximum resolution of the image. The resolution of the 3D image 12 should be selected to provide a voxel size at which the dominant pore throats for fluid flow in the porous media are adequately resolved and provide a sufficient field of view to represent the entire medium for which the fluid transport properties of the entire medium will be analyzed.

[0035] The resolution of the 3D image 12 can be selected based on the sample size, the relative average pore size, the time required for imaging, and the computing power required to store and perform further computational activities on the imaging data. For example, the pore scale resolution for a micro-CT image can be from, for example, 0.1 μm per voxel to 3 Up to 30 μm per voxel 3 For sandstone, micro-CT images are preferably taken in the range of 1 μm per voxel. 3 Up to 25μm 3 More preferably, 2.5 μm per voxel 3 Up to 15μm 3 For carbonate rocks, the resolution of micro-CT images is preferably in the range of 0.5 μm per voxel. 3 Up to 20μm 3 More preferably, 1 μm per voxel 3 Up to 10μm 3 For shales, the resolution of micro-CT (or nano-CT) images is preferably in the range of 0.1 μm per voxel. 3 Up to 10μm 3 More preferably, 0.5 μm per voxel 3 Up to 5μm 3 The resolution is generated.

[0036] Where the target porous medium is rock, the rock sample can be obtained from the formation, and the fluid transmission properties for the formation are important. As an example, the rock can be sandstone, carbonate, shale, and combinations thereof from hydrocarbon-bearing formations. Alternatively, the rock can be from an underground formation considered for carbon sequestration. The rock can be obtained by conventional means for obtaining rock samples from the formation. In a preferred embodiment, a core sample of the rock is obtained by coring a portion of the formation from a well in the formation (e.g., the entire core or a sidewall core). Alternatively, the rock sample can be obtained from drill cuttings generated when drilling a wellbore in the formation, preferably undisturbed drill cuttings. The rock can be obtained from the same wellbore as the electrical property measurement. Alternatively, the rock can be obtained from another wellbore in the same field as the wellbore in which the electrical property measurement is produced.

[0037] Alternatively, for fuel cells, electrolysis, and CO2 conversion to base chemicals, examples of porous media include, but are not limited to, ceramics and membranes.

[0038] Accordingly, the porous medium may be selected from rock, ceramic, membrane, and combinations thereof.

[0039] The porous media sample should be of sufficient size to obtain a 3D image 12 of sufficient volume at the size at which the image is to be generated. In particular, the sample should be of sufficient size so that volume features of the sample dominate over edge features of the sample at the size or field of view at which the image will be generated.

[0040] In a preferred embodiment, the 3D image 12 can be pre-processed to reduce noise and image artifacts. Noise can be filtered out of the acquired image by filtering using a local mean filter to reduce noise. Imaging artifacts that dominate at the outer edges of the acquired image can be reduced by processing the image while excluding the outer edges of the image.

[0041] segmentation

[0042] The 3D image 12 is segmented 14 to identify pore spaces and solid materials.

[0043] In one embodiment of the present invention, voxels of the 3D image 12 are segmented into voxels representing pore spaces in a porous medium or solid materials in a porous medium, thereby producing a binary image in which pore voxels have a value of zero and solid material voxels have a value of 1 (or vice versa). The 3D image 12 can be a grayscale image, and processing the voxels of the image to segment the image into voxels representing pore spaces or solid materials may be affected by assigning voxels to pore spaces or solid materials based on a threshold, wherein voxels with image intensities above the threshold can be assigned values ​​representing pores (or solid materials) and voxels with image intensities below the threshold can be assigned values ​​representing solid materials (or pores). The threshold selection method described in Otsu ("A Threshold Selection Method from Gray-level Histogram") can be used. IEEE Transactions on Systems, Man and Cybernetics (IEEE Trans.SMC) The threshold is calculated using the Otsu method described in 9:62-66; 1979) or other threshold calculation algorithms known in the art.

[0044] Segmentation algorithms are known to those skilled in the art. Preferably, the segmentation method is selected to identify pore space from a solid matrix. Examples of segmentation methods are described in Otsu (“A Threshold Selection Method from Gray-level Histogram”). IEEE Transactions on Systems, Man, and Cybernetics Trans.SMC)》 9:62-66; 1979); Andra et al. (“Digital Rock Physics Benchmarks-Part II: Computing Effective Properties” "count Computers and Geosciences 50:33-43;2013); Saxena et al. (“Effect of Image Segmentation & Voxel Size on Micro-CT Computed Effective Transport & Elastic Properties” Marine and Petroleum Geology 86:972-990; 2019); and Chuang et al. (“Fuzzy C-Means Clustering with Spatial Information for Image Segmentation” Computerized Medical Imaging and Graphics (Comput.Med.Imaging Graph.)》 30:9-15; 2006). Those skilled in the art will understand the desired choice of segmentation. Segmentation using a segmentation algorithm is preferably performed automatically using a data processing system.

[0045] In a preferred embodiment, the 3D image 12 is segmented in step 14 using a watershed segmentation algorithm (Beucher et al., "The morphological approach to segmentation: The watershed transformation", E.R. Dougherty (ed.), Math. Morphol. Image Process., Marcel Dekker Inc., New York, 1993: pp. 433-481) to segment the 3D image.

[0046] In another embodiment of the present invention, the 3D image 12 is segmented at step 14 using a multi-phase segmentation technique to correct for partial porosity and / or porous material.

[0047] Pore-Scale Flow Simulation

[0048] According to the present invention, the segmented structure image from step 14 is subjected to fluid flow simulation using pore scale flow simulation 16 .

[0049] Suitable types of pore-scale flow simulations 16 are well known to those skilled in the art and include, but are not limited to, direct flow simulations (which operate directly on segmented pore-scale images to dynamically solve the flow equations with both viscous and capillary forces acting simultaneously), quasi-static methods (which also operate directly on the image but with capillary dominance), pore network modeling (quasi-static and dynamic), machine learning-based methods (which estimate pore-scale flow fields and pressure gradients), and combinations and hybrids thereof. Preferably, the pore-scale flow simulation 16 is a direct flow simulation.

[0050] Direct flow simulations include, for example, but are not limited to, finite difference methods, finite element methods, finite volume methods, and lattice Boltzmann methods. Several methods have been developed for simulating single-phase and two-phase flows at the molecular, pore, and other mesoporous scales. These methods include lattice gas and lattice Boltzmann method models, Monte Carlo models, molecular dynamics, smoothed particle fluid dynamics, dissipative particle dynamics, and Euler computational fluid dynamics. The latter class of techniques includes interface tracking methods, fluid volume methods, level set methods, and phase field methods.

[0051] In a preferred embodiment, the fluid flow is simulated using an LBM simulator.Examples of LBM simulators include, but are not limited to, energy-based LBM (eLBM) simulators and multiple relaxation time (MRT) LBM simulators.

[0052] According to the present invention, numerical modeling is used to simulate fluid flow at a pore discrete scale. Preferably, the fluid flow is multiphase, such as a two-phase flow. More preferably, the multiphase fluid flow is performed using at least two immiscible fluid phases. Most preferably, the multiphase fluid flow includes a wetting fluid and a non-wetting fluid. Preferably, the pore scale flow simulation 16 is performed on a continuum fluid dynamics scale using the Navier-Stokes flow equations for two-phase flow.

[0053] In the pore-scale flow simulation 16, the fluid dynamics flow equations are solved directly on the complex pore space in the segmented image from step 14. By performing the pore-scale flow simulation 16 directly on the segmented image, the re-partitioning uncertainty is avoided. Unlike traditional pore network modeling techniques and morphological modeling methods, capillary forces and viscous forces act simultaneously in the pore-scale flow simulation 16. Accordingly, capillary and viscosity-dominated flows can be rigorously captured by the pore-scale flow simulation 16. In addition, the pore-scale flow simulation enables the description of a wide range of flow patterns and the simulation of a wide range of pore-scale dynamics, such as cooperative and / or non-local displacement processes.

[0054] In traditional methods, pore-scale flow simulations are performed using a steady-state approach, where the wetted and non-wetted phases flow at monotonically increasing (intake) or decreasing (exhaust) fractional flow rates f w Simultaneous injection, where in each split, steady state is achieved at constant pressure and constant average saturation. This may require hundreds of pore volumes of injected liquid. This is time consuming and computationally expensive.

[0055] According to the present invention, pore-scale flow simulation 16 is performed using a non-problematic approach, where the amount of pore volume injection required is orders of magnitude smaller. Conventional techniques avoid non-steady-state approaches because relative permeability predictions experience capillary endpoint effects and high uncertainty. In addition, those skilled in the art understand that non-unique solutions are possible.

[0056] Boundary conditions for pore-scale flow simulations include, but are not limited to, conditions regarding fluid type, fluid viscosity, interfacial tension, flow rate, fluid ratio, pressure, temperature, and combinations thereof.

[0057] According to the present invention, a pore-scale flow simulation 16 is performed on the segmented structural image to determine a pore-scale output. The pore-scale output includes, but is not limited to, fluid distribution, fluid pressure distribution, total pressure drop of the fluid phase in the simulation domain, fluid production curves and / or spatial gradients and / or other characteristics related to fluid distribution and pressure, such as flow rate. In step 18, a Darcy-scale flow model is generated by simulating fluid flow based on boundary conditions of the pore-scale flow distribution and an initial relative permeability model. Preferably, the Darcy-scale flow simulation is performed for a plurality of predetermined fluid flow rates.

[0058] Darcy-scale flow model

[0059] A Darcy scale flow model is generated at step 18. The Darcy scale flow model may be 1D, 2D or 3D. In a preferred embodiment, the Darcy scale flow model is 1D.

[0060] The governing equations for one-dimensional two-phase flow in homogeneous porous media are formulated in the absence of gravity to relate the Darcy velocity v (which is the flow velocity q divided by the cross-sectional area A) to the pressure gradient for 1-dimensional flow in the x-direction The volume flux v of the fluid phase a a Given by Equation 1, where a = w for the wetting fluid phase and a = n for the non-wetting fluid phase:

[0061]

[0062] where k r，α is the relative permeability of phase a, K is the absolute permeability of the porous medium, μ α is the viscosity of phase a, and p α It's the pressure of phase a.

[0063] The continuity equation (2) expresses the conservation of mass, where the saturation varies with time, t, and is related to the divergence of the flow:

[0064]

[0065] Where S a is the saturation of phase a, and is the time derivative.

[0066] Assumptions may include, for example, but not limited to:

[0067] -For the flow of two incompressible fluids, the constant total flux of the wetting and non-wetting fluid phases, v T =v w +v n ;

[0068] - Saturation of wetting and non-wetting phases S w+S n =1 total; and

[0069] -Relative permeability and capillary pressure are functions of saturation only.

[0070] The system of equations is closed by relating the pressure difference between the wetting and non-wetting phases to the capillary pressure in equation (3):

[0071] p w -p n =p c (3)

[0072] For phase α, the fluidity is defined as The split flow is defined by equation (4):

[0073]

[0074] Equations (1) to (4) are combined in Equation 5, which describes the saturation S w The spatiotemporal evolution of (x, t).

[0075]

[0076] In the process of the present invention, equation (5) is numerically solved and the formation of the resulting curve can be calculated by integrating the longitudinal profile saturation distribution map over the calculation domain in x.

[0077]

[0078] The reduced or movable saturation is defined in equation (7), where S w，c is the irreducible wetting phase saturation, and S n，r is the irreducible non-wetting phase saturation. In the case of a hydrocarbon-bearing formation, for example, the wettable phase may be connate water and the non-wetting phase may be residual oil.

[0079]

[0080] Relative permeabilities and capillary pressures are parameterized using appropriate models. For example, Corey (“The interrelation between gas and oil relative permeabilities”) Monthly production 19:1:38-41; 1954) expressed the relative permeability of the wetting and non-wetting phases as the reduced saturation S as described by equation (8): red A simple power law of:

[0081]

[0082] in and is the endpoint relative permeability of the wetting and non-wetting phases at the irreducible saturation of the respective other phases, and n w and n n is the power law "Corry" exponent. The Corry exponent defines the curve of the relative permeability-saturation relationship.

[0083] Alternatively, the LET model (Lomeland et al., “A new versatile relative permeability correlation” International Symposium of the Society of Core Analysts, Toronto, Canada, August 21-25, 2005, paper SCA2005-032), provides more degrees of freedom than the Corey model to describe the relative permeabilities of the wetting and non-wetting phases according to equation (9):

[0084]

[0085] The parameters and Define k r (S w ) shape. In one embodiment of the present invention, the relative permeability is parameterized using the LET model, but the E and T parameters are kept fixed.

[0086] For example, the capillary pressure-saturation function p c (S w ) can be expressed using, for example, the model from Skjavlan et al. (“Capillary pressure correlation for mixed-wet reservoirs” SPE India Oil and Gas Conference and Exhibition, February 17-19, 1998, New Delhi, India, SPE 39497), as shown in equation (10), where S n =1-S w 、c w 、c n 、a w and a n It is an adjustable parameter.

[0087]

[0088] Equation (5) is solved numerically using an explicit difference scheme using time-stepping control for two-phase incompressible flow with capillary pressure and gravity. In one embodiment of the invention, this is implemented as native Python code and computationally intensive components are compiled using the Python Numba package using a just-in-time compiler.

[0089] Other two-phase flow simulators known to those skilled in the art with capillary pressure can be used. As an example, these can be combined with the inverse modeling framework in Python via a wrapper.

[0090] In step 18, a Darcy scale flow model is generated by simulating fluid flow based on boundary conditions of the pore scale flow distribution and an initial relative permeability model. The initial relative permeability model can be randomly selected using, for example, but not limited to, a best guess approach, using analytical methods such as (but not limited to) the JBN method (Johnson et al. "Calculation of Relative Permeability from Displacement Experiments", Petroleum Transactions AIME (1959)), directly inverting the two-phase Darcy equation on the subdomain with a finite saturation gradient to estimate the initial relative permeability-saturation function and the pressure difference between the fluid phases to estimate the initial capillary pressure-saturation function and combinations thereof. Other methods are known to those skilled in the art, such as, but not limited to, methods that take into account the effect of capillary pressure on relative permeability compared to the JBN method that assumes the capillary pressure is zero.

[0091] Regardless of the method used, the more accurate the initial relative permeability model, the faster the convergence during inverse modeling. For example, the inventors found that using the JBN method to estimate the initial relative permeability from the fluid phase pressure in the simulation domain, the inverse modeling converged in 20-100 iterations.

[0092] The Darcy scale flow simulation may be performed using flow rates and other conditions consistent with the boundary conditions used in the pore scale simulation.Preferably, the Darcy scale flow simulation is performed for a plurality of predetermined fluid flow rates.

[0093] The Darcy scale output 20 is compared to the pore scale output to determine the degree of match based on, for example (but not limited to), the generation of curves for one or more of fluid phase, fluid distribution (converted to a longitudinal saturation profile for comparison with the Darcy scale simulation), fluid phase pressure, and phase flow velocity, pressure drop over domain.

[0094] Reverse Modeling

[0095] The Darcy scale output 20 is then subjected to inverse modeling, which is an iterative inversion technique, in step 22. An objective function is constructed to measure the difference between the Darcy scale output 20 and the pore scale output from step 16. The relative permeability model is iteratively updated until the objective function finds a minimum.

[0096] For example, by adjusting the relative permeability values ​​and / or capillary pressure values ​​in the initial relative permeability model, the Darcy scale output 20 can be compared and matched within predetermined tolerances to the simulation-generated curves, pressure drop, and fluid profile saturation distribution maps from the pore scale output.

[0097] Inverse modeling 22 is performed using a two-phase flow simulator with capillary action, whose numerical solutions of the two-phase Darcy equations are matched to the simulation data, using gradient-based optimization algorithms such as the Levenberg-Marquardt method or Bayesian methods such as the Markov Chain Monte Carlo theory.

[0098] Preferably, the Levenberg-Marquardt algorithm can be used to perform the least squares fit, where the data y i The sum of the squared differences between is used to minimize the objective function based on equation (11):

[0099]

[0100] where y i is the data point at the parameter value, and x i and f(x i ) are the respective values ​​calculated by the model. 2 is the sum of squared misfit between the model and the data, given by uncertainty ∈ i Normalization, which can be, for example, the standard deviation in the simulation data. In this case, the data includes the generated curve Q(t), the pressure drop Δp and the longitudinal saturation distribution diagram S w (x, t).

[0101] According to the objective function of equation (12), the generated data Q for the water phase and the oil phase are i , pressure drop △p and saturation S of the reference data set (index ref) w and the weighting factor w used to generate the data, pressure drop and longitudinal saturation distribution diagram Q 、w p and w S Merge separately.

[0102]

[0103] One advantage of inverse modeling is that relative permeability can be objectively obtained. By directly matching the Darcy scale output 20 to the pore scale output within a predetermined tolerance, the method produces less overall uncertainty by iteratively updating the relative permeability model, rather than manually matching, for example, each flow distribution profile in a steady-state experiment and subsequently fitting the relative permeability model.

[0104] Example

[0105] The following non-limiting example of one embodiment of the method of the present invention described herein is for illustrative purposes only.

[0106] Subvolumes of cylindrical core samples were measured using a voxel size of 1.51 um to resolve pore-scale features of the sandstone. The samples were well sorted from fine to medium grained. The 3D image was segmented and values ​​for porosity (19%) and absolute permeability (470 mD) were obtained.

[0107] The existing lattice Boltzmann method is used to perform pore-scale simulations of unsteady two-phase flow. Water totaling nearly two pore volumes is injected at a rate of 0.1 ml / min. The water and oil phases are assumed to have equal densities of 1 g / cm 3 and equal viscosity 1 cP. Table 1 summarizes the boundary conditions for the pore-scale flow simulations.

[0108] Table 1

[0109] Figures 2 to 5 is a graphical illustration of the pore-scale output for this example. Specifically, Figure 2 The total liquid production (solid line), water production (dashed line) and oil production (dashed line) are shown. Figure 3 Represents the average water saturation.

[0110] Figure 4 The fluid pressure drop over the computational domain as a function of the injected pore volume (PV) is shown, where the pressure drops for water and oil are represented by dashed and dotted lines, respectively. Figure 4 Also shown is the capillary pressure at the outlet as a function of the injected PV.

[0111] Figure 5 A longitudinal profile saturation profile is shown along the injection (z) direction, which is the direction in which the pressure gradient is applied. The different lines represent the longitudinal profile saturation profiles listed in Table 2 as a function of time.

[0112] Table 2

[0113] Time (seconds) Line Type 0.03 Solid Line 0.13 Dot 0.25 Square Point 0.38 Dash 0.50 Dash Dot 0.63 Em dash 0.76 Long dash dot

[0114] As shown in Table 1, the dimensions of the pore-scale simulation domain are 0.98 mm × 0.98 mm × 1.96 mm. For typical sandstones with porosities and permeabilities from Table 1, REV may be in the range of 2 mm to 4 mm. Therefore, in the 1D characteristic attributes in the flow direction (z), it is expected that the reflection of the discrete pore-grain structure will not be completely averaged at this scale. This largely explains the variation in pore scale. Figures 2 to 5 The visible saturation S w (z), pressure p w,o (z) and capillary pressure p z (S w ) significant changes.

[0115] This variation explains why those skilled in the art generally understand that, since the computational domain size is close to the REV size, pore-scale variations thus imply that the pore-scale output is still pore-scale. The inventors recognized that pore-scale variations actually affect Darcy-scale properties. And therefore, the inventors surprisingly discovered a method for Darcy-scale physics that captures pore-scale variations, even though the pore-scale variations are not visible in the Darcy-scale computational domain.

[0116] The ground truth dataset is generated starting with the definition of the relative permeability and capillary pressure-saturation function sets. The Corey and Skjavlan models are used for the relative permeability and capillary pressure relationships, respectively. The respective production curves, pressure drops, and longitudinal saturation profiles are then numerically computed by solving the Darcy-scale flow model. The Darcy-scale output is then matched to the pore-scale output by inverse modeling. The Darcy-scale output and inverse modeling match in Figures 6 to 10 Described in diagram form.

[0117] exist Figure 6 and Figure 7 In , the initial relative permeability model is described by the solid line. Specifically, Figure 6 Shows water k r，w (S w ) initial (solid line) relative permeability and oil k r，o (S w ) as a function of fluid saturation. Figure 7 shows the initial (solid line) capillary pressure p c (S w ), as a function of fluid saturation. Figure 6 and Figure 7 The dashed line in represents the matched relative permeability model as a function of saturation after inverse modeling.

[0118] Figures 8 to 10 The match between the pore-scale output (solid line) and the Darcy-scale output (dashed line) is shown. Figure 8shows the match for the fluid pressure drop as a function of time, while Fig. 9 The match is shown for oil production as a function of time.

[0119] Fig.10 The matching of the longitudinal saturation profiles for the pore-scale output and the Darcy-scale output along the injection (z) direction is shown. Fig.10 The Darcy scale output is a smoother set of curves. The different lines represent the time-varying longitudinal saturation distribution plots listed in Table 3.

[0120] Table 3

[0121] Time (seconds) Line Type 0.08 Solid Line 0.21 Dot 0.34 Square Point 0.47 Dash 0.62 Dash Dot 0.75 Em dash 0.99 Long dash dot 1.22 Long dash dot

[0122] The numerical flow model and relative permeability and capillary pressure parameterizations are numerically solved using 1D native Python code accelerated by the Numba just-in-time compiler.

[0123] The numerical solver is a 1D explicit difference scheme with time stepping control. It treats two-phase incompressible flows with capillary and gravity forces in unidirectional flow, i.e., without convective self-imbibition.

[0124] Fig.11 is a schematic representation of the computational domain. The 1D linear grid has n x grid blocks (where n x =50). At the inlet, the water phase flows at a flow rate of n Injection. For the grid blocks in the computational domain, define the flow parameters (porosity φ, permeability K, relative permeability k r，α (S w ) and capillary pressure p c (S w ) simulation function), and the initial conditions used for the simulation (S w，i ). Apply a constant flow boundary condition at the inlet. Apply a constant pressure at the outlet. In addition, apply a capillary pressure p c =0 boundary condition.

[0125] Although embodiments have been described with reference to various implementations and utilization modes, it should be understood that these embodiments are exemplary and the scope of the subject matter of the present invention is not limited thereto. Many variations, modifications, additions and improvements are possible. Various combinations of the technology provided herein can be used.

Claims

1. A method for determining the relative permeability of a porous medium, the method comprising the following steps: (a) Provide 3D images for porous media samples; (b) generating a segmented structural image from the 3D image to identify pore space and solid material; (c) simulating fluid flow on the segmented structure image using a pore-scale flow simulation to generate a pore-scale output; (d) Selecting an initial relative permeability model; (e) generating a Darcy scale flow model by simulating fluid flow based on the boundary conditions of the pore scale flow simulation using the initial relative permeability model to generate a Darcy scale output; as well as (f) comparing the Darcy scale output to the pore scale output to determine a match; and (g) updating the initial relative permeability model and repeating steps (e) to (g) until the matching degree falls within a predetermined tolerance range.

2. The method according to claim 1, wherein the pore-scale fluid flow simulation of step (c) is performed on multiphase fluid flow.

3. The method of claim 2, wherein the multiphase fluid flow is performed using at least two immiscible fluid phases. The method of claim 2 , wherein the multiphase fluid flow comprises a wetting fluid and a non-wetting fluid.

5. The method of claim 1, wherein the pore-scale fluid flow simulation is 1D. The method of claim 1 , wherein the pore-scale fluid flow simulation is based on a lattice Boltzmann method.

7. The method of claim 1, wherein the pore-scale output comprises one or more of fluid distribution, fluid pressure drop, fluid production curve, gradient, and combinations thereof.

8. The method of claim 1, wherein the boundary conditions of the pore-scale flow simulation include conditions related to fluid type, flow rate, pressure, temperature, viscosity, and combinations thereof.

9. The method of claim 1, wherein the Darcy-scale flow model is generated by simulating fluid flow for a plurality of fluid flow rates.

10. The method of claim 1, wherein the porous medium is selected from the group consisting of: rock, ceramic, membrane, and combinations thereof.

11. The method of claim 1, wherein the segmented structural image is generated by segmenting the 3D image into voxels representing pore spaces in the porous medium and voxels representing solid materials in the porous medium.

12. The method according to claim 1, wherein the segmented structure image is segmented by a multi-phase segmentation technique.

13. The method of claim 1, wherein the 3D image of the rock is obtained by X-ray computed tomography.

14. The method of claim 1, wherein the relative permeability model includes saturation endpoint, capillary endpoint effect, and combinations thereof.

15. The method of claim 1, further comprising the step of using the updated relative permeability for making decisions regarding recovery of hydrocarbons from the subterranean formation.

16. The method of claim 1, further comprising the step of using the updated relative permeability for making decisions regarding carbon capture and sequestration in the subsurface formation.

17. The method of claim 1, further comprising the step of using the updated relative permeability for making decisions regarding geothermal extraction processes.

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