SYSTEMS AND METHODS FOR DETERMINING UNSTEADY-STATE TWO-PHASE RELATIVE PERMEABILITY

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. provisional patent application No.

62/367,749, filed on July 28, 2016, and entitled "SYSTEMS AN D M ETHODS FOR DETERM IN ING

U NSTEADY-STATE TWO-PHASE RELATIVE PERM EABI LITY," the disclosure of which is expressly incorporated herein by reference in its entirety.

STATEMENT REGARDING FEDERALLY FUNDED RESEARCH

[0002] This invention was made with government support under Grant no. DE-SC0001114 awarded by the Department of Energy. The government has certain rights in the invention.

BACKGROUND

[0003] Relative permeability is a key parameter of multiphase flow in porous media. It has been studied in both the petroleum literature [Burdine, 1953; Johnson et al., 1959] and the hydrology literature (also known as unsaturated conductivity) [Irmay, 1954; Gardner and Miklich, 1962] since 1950s. In particular, the two-phase water-oil relative permeabilities and three-phase water-gas-oil relative permeabilities are critical to predict oil production scenarios that use water flooding strategy [Johnson et al., 1959; Land, 1968; Stone, 1973] and water alternating gas flooding strategy [Oak, 1990; Blunt, 2000]. Recently, the growing research interests in C02 geological storage have motivated both experimental measurements of C02-brine relative permeabilities [Chen and DiCarlo, n.d.; Bachu and Bennion, 2007; Krevor et al., 2012; Akbarabadi and Piri, 2013; Pini and Benson, 2013; Chen et al., 2014, n.d.] and modeling studies on C02 geological storage that use C02 relative permeability as a key input [Nordbotten et al., 2005; Kovscek and Cakici, 2005; Juanes et al., 2006; Doughty, 2007; Celia and Nordbotten, 2009] . In terms of experimentally determining relative permeability, the J BN method is a well-known and fast method that obtains data across a broad range of saturations [Welge, 1952; Johnson et al., 1959]. [0004] Relative permeability in porous media is simply a measure of the reduction of permeability to a certain phase when that phase is not at complete saturation. It is used in the Darcy Buckingham equation as:

[0005]

[0006] where Q, is the phase, ; volumetric flow rate, is the phase ; relative permeability, K is the permeability, A is the cross section area, ΔΡ' is the phase ; pressure drop, μ, is the phase ; viscosity, and . is the length of porous media. To measure relative permeability, local measurements of saturation [Si), flow rate, and pressure drop of phase / are needed. In terms of corefloods, obtaining global (i.e., core integrated) measurements of S„ Q„ and ΔΡ' are typically straightforward; unfortunately the global values can vary significantly from the local values. The key to any relative permeability measurement method is designing the method to obtain local values of the three quantities.

[0007] During steady-state methods for determining relative permeabilities, equilibrated two phase fluids are injected into a core until the measured overall pressure drop and the overall saturation do not change with time ^3"25 . The phase flow rates, the saturation, a nd the pressure drop are constants along the core when steady state is reached. Thus, for steady-state methods, the global measurements are equivalent to the local measurements and the relative permeability equation can be used directly. Potentially there are some complicating capillary effects that occur at the inlet and outlet of the core; these effects can be remedied by measuring pressure drops and saturations in the center of the core ^18,26 . This simple equivalence between global a nd local values makes steady-state methods the gold-standard of relative permeability measurements. Since at each steady state only two data points are obtained (the relative permeability of each phase), the process must be repeated for each flux ratio to obtain a full relative permeability curve. This ma kes the steady-state methods very time consuming because steady state is achieved very slowly at the end points and expensive.

[0008] To speed up the measurement process, unsteady-state methods have been developed, particularly in determining oil-brine relative permeabilities ^27"33 . Many of these unsteady- state methods obtain the local saturations and phase flow rates from their global measurements by solving the continuity equation along with Darcy Buckingham equation either analytically or numerically. In the well-known Johnson, Bossier, and Naumann (JBN)method ^ ³⁰ · ³² · ³³ , one phase (invading phase) is injected into a core saturated with another phase (defending phase), and the overall pressure drop and outlet flux (effluent) are measured versus time. By using fractional flow theory

[Buckley and Leverett, 1942] and a mathematical inversion, relative permeability data to both phases are obtained at the outlet as the defending phase saturation decreases. Again, the global measurements of pressure drop, phase flow rate, and saturation (through mass balance) are taken, but now there are large differences between the global and local values both in time and space. The local values are obtained by assuming a 1-dimensional Buckley Leverett ^27,28 type displacement, and from mathematical inversions both the relative permeabilities and the saturation at the outlet are calculated. This procedure is done versus time, allowing many different saturations and relative permeabilities to be obtained from one displacement.

[0009] There are other unsteady-state methods, all of which use slightly different ways of obtaining local values of the three key quantities. Hagoort et al. ³¹ extended the JBN method to oil displacement by air in a centrifuge. The expression for oil relative permeability at the outlet is simplified from the original JBN expression to the oil production rate due to high mobility of air and low capillary pressure. DiCarlo et al. ^1,2 and Kianinejad et al. ^34,35 conducted gravity drainages and measured local saturations and fluxes in-situ using CT scanning. The local pressure gradient was taken to be the gravitational gradient, which was shown to be a good assumption for the center portion of the column. Schembre et al. ³⁶ and Berg et al. ³⁷ did two-phase displacement experiments and measured saturation profiles, pressure drop and effluent versus time. Schembre et al. input a prior model of relative permeability and a capillary pressure curve to numerically solve the coupled Darcy Buckingham equation and continuity equation, and eventually Schembre et al. obtained a posterior relative permeability model that best matched simulated results with measurements

SUMMARY [0010] An example unsteady-state method for determining relative permeability is described herein. The method can include injecting at least one fluid into a core to obtain a plurality of fixed two-phase fractional flow rates, and obtaining a plurality of local measurements at each of the fixed two-phase fractional flow rates. The local measurements can include a respective local saturation profile of a fluid along the core and respective local pressure data. The method can also include calculating a respective local phase flux using the respective local saturation profile at each of the fixed two-phase fractional flow rates, and calculating the relative permeability using local saturation profiles, local pressure data, and local phase fluxes.

[0011] In some implementations, at least two fluids can be co-injected into the core.

[0012] Additionally, the local measurements can be obtained under unsteady-state conditions.

[0013] Alternatively or additionally, the local measurements can be local measurement values at a plurality of locations along the core.

[0014] Alternatively or additionally, the respective local phase flux can be calculated using fractional flow theory. In some implementations, the respective local phase flux can be calculated by integrating on a local saturation profile with respect to spatial changes in saturation at the same time. In other implementations, the respective local phase flux can be calculated by integrating on a local saturation profile with respect to temporal changes in saturation at the same location. Optionally, the integration can be performed using a finite difference approximation.

[0015] Alternatively or additionally, the Darcy Buckingham equation can optionally be used to calculate the relative permeability using local saturation profiles, local pressure data, and local phase fluxes. Optionally, relative permeability can be calculated in a plurality of sections of the core using the Darcy Buckingham equation. Optionally, relative permeability is not calculated in an outlet section of the core.

[0016] Alternatively or additionally, an extended Johnson, Bossier, and Naumann (JBN) method can optionally be used to calculate the relative permeability using local saturation profiles, local pressure data, and local phase fluxes. Optionally, relative permeability can be calculated at a plurality of pressure sensor locations along the core using the extended JBN method. Optionally, relative permeability is not calculated in an outlet of the core.

[0017] Alternatively or additionally, the respective local saturation profile can be obtained using a nondestructive testing (NDT) technique. The NDT technique can optionally be computed tomography (CT) imaging.

[0018] Alternatively or additionally, the respective local pressure data can be obtained using a plurality of pressure sensors. Optionally, the pressure sensors can be spaced apart along the core.

[0019] Alternatively or additionally, the at least one fluid can be gas, oil, or water.

[0020] Alternatively or additionally, the core can be permeable rock.

[0021] An example system for determining relative permeability from unsteady-state saturation profiles is also described herein. The system can include at least one pressure source configured to inject at least one fluid into a core to obtain a plurality of fixed two-phase fractional flow rates, a nondestructive test (N DT) device configured to measure a local saturation profile of a fluid along the core, a plurality of pressure sensors arranged along the core, and a processor and a memory in operative communication with the processor. The processor can be configured to receive a plurality of local measurements at each of the fixed two-phase fractional flow rates, where the local measurements include a respective local saturation profile measured by the N DT device and respective local pressure data measured by the pressure sensors. The processor can be further configured to calculate a respective local phase flux using the respective local saturation profile at each of the fixed two-phase fractional flow rates, and calculate the relative permeability using local saturation profiles, local pressure data, and local phase fluxes.

[0022] An example non-transitory computer readable medium is also described herein. The computer readable medium can have computer-executable instructions stored thereon for determining relative permeability from unsteady-state saturation profiles. When executed by a processor, the computer readable medium can cause the processor to receive a respective local saturation profile of a fluid along a core at each of a plurality of fixed two-phase fractional flow rates, receive respective local pressure data at each of the fixed two-phase fractional flow rates, calculate a respective local phase flux using the respective local saturation profile at each of the fixed two-phase fractional flow rates, and calculate the relative permeability using local saturation profiles, local pressure data, and local phase fluxes.

[0023] Other systems, methods, features and/or advantages will be or may become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features and/or advantages be included within this description and be protected by the accompanying claims.

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] The components in the drawings are not necessarily to scale relative to each other. Like reference numerals designate corresponding parts throughout the several views.

[0025] FIG. 1 is a block diagram of an example system for determining relative permeability from unsteady-state saturation profiles according to implementations described herein.

[0026] FIG. 2A is an example computing device.

[0027] FIG. 2B is a flow chart illustrating example operations for determining relative permeability using unsteady-state saturation profiles.

[0028] FIG. 3 is a diagram illustrating the example core flooding apparatus used in the experiments described herein.

[0029] FIG. 4A is a graph that illustrates water saturation profiles recorded at continuous injection times. FIG. 4B is a graph that illustrates the pressure drops of the upstream four sections of the core measured versus time. Data plotted in both FIGS. 4A and 4B were obtained during the fractional flow step of/ _w =0.1 in Exp5 described below.

[0030] FIG. 5A is a graph that illustrates the local water fractional flow versus space and time calculated by integrating spatial saturation difference based on fractional flow theory (e.g., the

Fractional Flow Method) as described herein. FIG. 5B is a graph that illustrates the local water fractional flow versus space and time calculated by integrating temporal saturation difference on space (e.g., the

Mass Conservation Method) as described herein. Both FIGS. 5A and 5B were calculated for the fractional flow step of / _w =0.1 in Exp5. FIG. 5C is a graph illustrating the unsteady-state brine relative permeability data obtained using local phase flux calculated using both the Fractional Flow Method and the Mass Conservation method, respectively. FIG. 5D is a graph illustrating the unsteady-state CO2 relative permeability data obtained using local phase fluxes calculated using the Fractional Flow Method and the Mass Conservation method, respectively.

[0031] FIGS. 6A and 6B compare the steady state (FIG. 6A) and the unsteady-state (FIG. 6B - local phase flux calculated using the Fractional Flow Method) CO2 and brine relative permeabilities of all five drainage experiments described below.

[0032] FIG. 7 A is a graph that shows brine relative permeability data obtained from the Mass Conservation Method were roughly between 30% more and 20% less of those obtained from the Fractional Flow Method (average relative difference is 25%). FIG. 7B is a graph that shows CO2 relative permeability data obtained from the Mass Conservation Method were within 2% of those obtained from the Fractional Flow Method (average relative difference is 2%).

[0033] FIG. 8A is a graph that illustrates water saturation profile recorded at each injection time. FIG. 8B is a graph that illustrates pressure drop of the five individual sections (i.e., seel, sec2, sec3, sec4, sec5) of the core measured versus the total injected pore volume. FIG. 8C is a graph that illustrates water fractional flow profile calculated at each injection time using the Fractional Flow Method described herein. FIG. 8D is a graph that illustrates water fractional flow profile calculated at each injection time using the Mass Conservation Method described herein.

[0034] FIG. 9A is a graph that illustrates l/t _d plotted versus ΔΡ/ΐ for the overall core

("overall"), section 1 ("seel"), the combination of sections 1 and 2 ("secsl2"), the combination of sections 1, 2 and 3 ("secsl23"), and the combination of sections 1, 2, 3 and 4 ("secsl234") and their cubic equation fits (dashed lines). FIGS. 9B and 9C are graphs that illustrate the resulting brine and CO2 relative permeabilities determined at the plurality of pressure taps using the extended JBN method and those determined only at the core outlet using the J BN method. FIG. 9B uses local water fractional flow

(f _w ) calculated with the Fractional Flow Method. FIG. 9C uses local water fractional flow {f _w ) calculated with the Mass Conservation Method. FIG. 9D is a graph that illustrates steady state relative permeability data compared with those obtained with the extended JBN method using local water fractional flow (f _w ) calculated with the Fractional Flow Method.

DETAILED DESCRIPTION

[0035] Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art. Methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present disclosure. As used in the specification, and in the appended claims, the singular forms "a," "an," "the" include plural referents unless the context clearly dictates otherwise. The term "comprising" and variations thereof as used herein is used synonymously with the term "including" and variations thereof and are open, non-limiting terms. The terms "optional" or "optionally" used herein mean that the subsequently described feature, event or circumstance may or may not occur, and that the description includes instances where said feature, event or circumstance occurs and instances where it does not. Ranges may be expressed herein as from "about" one particular value, and/or to "about" another particular value. When such a range is expressed, an aspect includes from the one particular value and/or to the other particular value. Similarly, when values are expressed as approximations, by use of the antecedent "about," it will be understood that the particular value forms another aspect. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint. While implementations will be described for determining relative permeability from unsteady-state saturation profiles based local core

measurement using the Darcy Buckingham equation or an extended JBN method, it will become evident to those skilled in the art that the implementations are not limited thereto.

[0036] Systems, methods, and computer readable medium for unsteady-state unsteady state determination of two-phase relative permeability by obtaining local values of three key parameters (i.e., saturation, pressure drop, and phase flux) versus time during a displacement are described herein. As used herein, local values of saturation, pressure drop, and phase flux constitute measured or derived values of these parameters in space and time. Local values of these parameters may vary with respect to space and/or time during unsteady-state conditions. It should be understood that local values are different than global values of these same parameters (e.g., values for the entire core under steady-state conditions), which are used in conventional techniques to determine relative permeability. By using global values, conventional techniques for calculating relative permeability may be plagued by saturation variations within the core, for example (but not limited to) heterogeneity and/or capillary end effects. In some implementations, the three key parameters can be substituted to two-phase Darcy Buckingham Equation to directly determine relative permeability. In other implementations, the three key parameters can be used with an extended JBN method to determine relative permeability to two phases at a plurality of pressure taps (and not just at the core outlet as is possible using the conventional JBN method). Although example implementations are described using the two-phase Darcy Buckingham Equation and the extended JBN method, this disclosure contemplates that other techniques may be used to determine relative permeability based on local values of saturation, pressure drop, and phase flux.

[0037] Referring now to FIG. 1, an example system for determining relative permeability from unsteady-state saturation profiles is shown. The system includes a core 100. The core 100 can optionally be permeable rock (e.g., non-sandpack). For example, the core 100 can optionally be a sample obtained from the field, for example, from a formation that contains a desirable fluid such as oil and/or gas. It should be understood that the core 100 can be analyzed, for example in a laboratory environment, to obtain information (e.g., relative permeability) about the formation. This information can then be used during operations to extract the desirable fluid from the formation. The core 100 defines an entry end 100A and an exit end 100B. Although the core 100 is vertically oriented in the examples, this disclosure contemplates that the core 100 (or portions thereof) can optionally be oriented horizontally and/or at angles between horizontal and vertical.

[0038] The system can include at least one pressure source 110 configured to inject at least one fluid into the core 100 to obtain a plurality of fixed two-phase fractional flow rates. The fluid(s) can be selected from gas, water, or oil. The system can be used to perform primary drainage experiments, where fluid(s) is injected into an entry end 100A of the core 100 and displacements occurs along the core 100, and the effluents leave at the exit end 100B of the core 100. Using the pressure source 110, a single fractional flow step can be performed, where a single phase is injected (e.g., 100% gas injection and no water injection) into the core 100. In other implementations, the system can include a plurality of pressure sources 110 and 110η configured to co-inject at least two fluids into the core 100. The fluids can be different phases (e.g., a gas such as CO2 and water). Using the pressure sources 110 and 110η, a plurality of fractional flow steps can be performed, wherein two phases (e.g., a gas such as C02 and water) are co-injected at each of the fractional flow steps into the core 100. As described below, during each subsequent fractional flow step, the water fractional flow is lower than the water fractional flow during the previous fractional flow step. During the final fractional flow step, only one of the two phases is injected (e.g., 100% gas injection and no water injection). Optionally, the number of fractional flow steps can be five. It should be understood, however, that five fractional flow steps is provided only as an example and that more or fewer than five fractional flow steps can be performed.

[0039] Optionally, the pressure source can be a pump and/or an accumulator, for example. Pumps and accumulators are well known in the art and are therefore not described in further detail below. The pump and/or accumulator can be configured to inject the fluid into the core 100 at the desired constant flow rate. As shown in FIG. 1, fluid(s) can be injected into the entry end 100A of the core 100 using the pressure source 110 and/or 110η. Alternatively or additionally, when the injected fluid is gas (e.g., CO2), the pressure source can optionally include a pressurized reservoir, a gas pressure regulator, and a mass flow controller, for example. A mass flow controller uses a feedback mechanism to keep the gas flow at a certain rate. Mass flow controllers are well known in the art and are therefore not described in further detail below.

[0040] The system can also include a nondestructive test (NDT) device 120 configured to measure a local saturation profile of a fluid along the core 100 in time and space. In some

implementations, the local saturation profile is measured along an entire length of the core 100, e.g., from the entry end 100A to the exit end 100B. Alternatively, in some implementations, the local saturation profile is measured along a portion of the core 100. For example, as shown by the dotted arrows in FIG. 1, the core 100 can be configured to move relative to the NDT device 120. For example, the core 100 can be arranged in a positioning system that is configured to move relative to the NDT device 120 (e.g., up and down). In other words, the positioning system can be configured to move up and down such that the core 100 can be imaged by the NDT device 120. In some implementations, the N DT device 120 is a CT imaging device. Although a CT imaging device is provided as an example herein, it should be understood that the NDT device 120 can be any device configured to measure local saturation profiles along the core 100, including but not limited to, devices using gamma ray, neutron probes, and/or other electrical measurement technologies.

[0041] The system can also include a plurality of pressure sensors 140a, 140b,... 140n configured to measure local pressure data. As shown in FIG. 1, the pressure sensors 140a, 140b,... 140n are arranged along the core 100, e.g., inserted into pressure taps drilled into the core 100. The pressure sensors 140a, 140b,... 140n can be spaced apart along a lengthwise extent of the core 100 as shown in FIG. 1. Optionally, the pressure sensors can be differential pressure transducers. It should be understood that a differential pressure transducer is only provided as an example pressure sensor and that other types of pressure sensors can be used. As shown in Fig. 1, the system includes six pressure sensors 140a, 140b,... 140n. These pressure sensors can be used to measure the overall pressure drop of the core 100 (e.g., using pressure sensors 140a and 140n), as well as the respective pressure drops of five individual sections of the core 100, continuously. It should be understood that more or less than six pressure sensors as shown in FIG. 1 can be used with the implementations described herein.

[0042] The system can also include a computing device 130. Optionally, the computing device 130 can include one or more of the components of the example computing device of FIG. 2A

(e.g., a processor and a memory operatively coupled to the processor). As shown in FIG. 1, the NDT device 120 and the computing device 130 can be communicatively connected via a communication link.

Additionally, the pressure sensors 140a, 140b,... 140n and the computing device 130 can be communicatively connected via a communication link. This disclosure contemplates a communication link is any suitable communication link. For example, a communication link may be implemented by any medium that facilitates data exchange between the network elements including, but not limited to, wired, wireless and optical links. Example communication links include, but are not limited to, a LAN, a WAN, a MAN, Ethernet, the Internet, or any other wired or wireless link such as WiFi, WiMax, 3G or 4G. Alternatively or additionally, the NDT device 120 and the computing device 130 and/or the pressure sensors 140a, 140b,... 140n and the computing device 130 can be communicatively connected via a network. The N DT device 120 and the computing device 130 and/or the pressure sensors 140a, 140b,... 140n and the computing device 130 can be coupled to the network through one or more

communication links. This disclosure contemplates that the network is any suitable communication network. The network can include a local area network (LAN), a wireless local area network (WLAN), a wide area network (WAN), a metropolitan area network (MAN), a virtual private network (VPN), etc., including portions or combinations of any of the above networks.

[0043] As described below, the computing device 130 can be configured to receive a plurality of local measurements at each of the fixed two-phase fractional flow rates, where the local measurements include a respective local saturation profile measured by the NDT device 120 and respective local pressure data measured by the pressure sensors 140a, 140b,... 140n. The computing device 130 can be further configured to calculate a respective local phase flux using the respective local saturation profile at each of the fixed two-phase fractional flow rates, and calculate the relative permeability using local saturation profiles, local pressure data, and local phase fluxes.

[0044] Referring to FIG. 2A, an example computing device 200 upon which embodiments of the invention may be implemented is illustrated. It should be understood that the example computing device 200 is only one example of a suitable computing environment upon which embodiments of the invention may be implemented. Optionally, the computing device 200 can be a well-known computing system including, but not limited to, personal computers, servers, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, network personal computers (PCs), minicomputers, mainframe computers, embedded systems, and/or distributed computing

environments including a plurality of any of the above systems or devices. Distributed computing environments enable remote computing devices, which are connected to a communication network or other data transmission medium, to perform various tasks. In the distributed computing environment, the program modules, applications, and other data may be stored on local and/or remote computer storage media.

[0045] In its most basic configuration, computing device 200 typically includes at least one processing unit 206 and system memory 204. Depending on the exact configuration and type of computing device, system memory 204 may be volatile (such as random access memory (RAM)), nonvolatile (such as read-only memory (ROM), flash memory, etc.), or some combination of the two. This most basic configuration is illustrated in FIG. 2A by dashed line 202. The processing unit 206 may be a standard programmable processor that performs arithmetic and logic operations necessary for operation of the computing device 200. The computing device 200 may also include a bus or other communication mechanism for communicating information among various components of the computing device 200.

[0046] Computing device 200 may have additional features/functionality. For example, computing device 200 may include additional storage such as removable storage 208 and nonremovable storage 210 including, but not limited to, magnetic or optical disks or tapes. Computing device 200 may also contain network connection(s) 216 that allow the device to communicate with other devices. Computing device 200 may also have input device(s) 214 such as a keyboard, mouse, touch screen, etc. Output device(s) 212 such as a display, speakers, printer, etc. may also be included. The additional devices may be connected to the bus in order to facilitate communication of data among the components of the computing device 200. All these devices are well known in the art and need not be discussed at length here.

[0047] The processing unit 206 may be configured to execute program code encoded in tangible, computer-readable media. Tangible, computer-readable media refers to any media that is capable of providing data that causes the computing device 200 (i.e., a machine) to operate in a particular fashion. Various computer-readable media may be utilized to provide instructions to the processing unit 206 for execution. Example tangible, computer-readable media may include, but is not limited to, volatile media, non-volatile media, removable media and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. System memory 204, removable storage 208, and non-removable storage 210 are all examples of tangible, computer storage media. Example tangible, computer-readable recording media include, but are not limited to, an integrated circuit (e.g., field-programmable gate array or application-specific IC), a hard disk, an optical disk, a magneto-optical disk, a floppy disk, a magnetic tape, a holographic storage medium, a solid-state device, RAM, ROM, electrically erasable program read-only memory (EEPROM), flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices.

[0048] In an example implementation, the processing unit 206 may execute program code stored in the system memory 204. For example, the bus may carry data to the system memory 204, from which the processing unit 206 receives and executes instructions. The data received by the system memory 204 may optionally be stored on the removable storage 208 or the non-removable storage 210 before or after execution by the processing unit 206.

[0049] It should be understood that the various techniques described herein may be implemented in connection with hardware or software or, where appropriate, with a combination thereof. Thus, the methods and apparatuses of the presently disclosed subject matter, or certain aspects or portions thereof, may take the form of program code (i.e., instructions) embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, or any other machine-readable storage medium wherein, when the program code is loaded into and executed by a machine, such as a computing device, the machine becomes an apparatus for practicing the presently disclosed subject matter. In the case of program code execution on programmable computers, the computing device generally includes a processor, a storage medium readable by the processor (including volatile and non-volatile memory and/or storage elements), at least one input device, and at least one output device. One or more programs may implement or utilize the processes described in connection with the presently disclosed subject matter, e.g., through the use of an application programming interface (API), reusable controls, or the like. Such programs may be implemented in a high level procedural or object-oriented programming language to communicate with a computer system. However, the program(s) can be implemented in assembly or machine language, if desired. In any case, the language may be a compiled or interpreted language and it may be combined with hardware implementations.

[0050] It should be appreciated that the logical operations described herein with respect to the various figures may be implemented (1) as a sequence of computer implemented acts or program modules (i.e., software) running on a computing device (e.g., the computing device described in FIG. 2A), (2) as interconnected machine logic circuits or circuit modules (i.e., hardware) within the computing device and/or (3) a combination of software and hardware of the computing device. Thus, the logical operations discussed herein are not limited to any specific combination of hardware and software. The implementation is a matter of choice dependent on the performance and other requirements of the computing device. Accordingly, the logical operations described herein are referred to variously as operations, structural devices, acts, or modules. These operations, structural devices, acts and modules may be implemented in software, in firmware, in special purpose digital logic, and any combination thereof. It should also be appreciated that more or fewer operations may be performed than shown in the figures and described herein. These operations may also be performed in a different order than those described herein.

[0051] Referring now to FIG. 2B, a flow chart illustrating example operations for determining relative permeability using unsteady-state saturation profiles. At 220, a fluid is injected into a core (e.g., a permeable rock core such as core 100 of FIG. 1). The fluid can be injected into the core using a pressure source (e.g., pressure source 110 and/or 110η of FIG. 1). Fluid(s) can be selected from a gas, water, and/or oil. As described herein, fluid(s) can be injected into the core to obtain a plurality of fixed two-phase fractional flow rates. As used herein, a phase fractional flow is a single phase flux divided by the total flux of multiple phases. For example, water fractional flow is the water flux divided by the total flux (e.g., water and gas). Example fixed two-phase fractional flows are described below such as the following flow rate of brine to flow rate of CO2 (qbrine:qCC>2) - 50:50

(fractional flow step 1), 10:90 (fractional flow step 2), 1:99 (fractional flow step 3), 0.1:99.9 (fractional flow step 4), and 0:100 (fractional flow step 5). It should be understood that the fluids, phases, number of steps, and/or flow rates are provided only as examples and that other fluids, phases, number of steps, and/or flow rates can be used in accordance with this disclosure. In some implementations, a single fractional flow step can be performed, where a single phase is injected (e.g., 100% gas injection and no water injection) into the core. In other implementations, a plurality of fractional flow steps can be performed, where two phases (e.g., a gas such as C02 and water) are co-injected at each of the fractional flow steps into the core. Steady-state flow conditions can be achieved (e.g., initial conditions can be established) between fractional flow steps. During each subsequent fractional flow step, the water fractional flow is lower than the water fractional flow during the previous fractional flow step.

[0052] At 222, a plurality of local measurements are obtained at each of the fixed two-phase fractional flow rates. The local measurements are obtained under unsteady-state conditions as opposed to during steady-state conditions, when phase flow rates, saturation, and pressure drop are constants throughout the core. As discussed herein, local measurements include local values for pressure drop, saturation and phase flux. It should be understood that local values are different than global values for these same parameters. One advantage gained by using local measurements is that capillary end effects can be experimentally avoided, e.g., by not calculating relative permeability for the core outlet and/or outlet section of the core. This may not be possible using conventional steady-state and unsteady-state techniques. Local saturation profiles of a fluid along the core are measured using an N DT device (e.g., NDT device 120 of FIG. 1), and local pressure drops are obtained using a plurality of pressure sensors (e.g., pressure sensors 140a, 140b,...140n of FIG. 1). The plurality of pressure sensors can be spaced apart along the core. As described herein, this facilitates obtaining respective pressure drops between the core inlet and each respective pressure sensor, e.g., respective pressure drops for a plurality of sections of the core. Using the six pressure sensors shown in the example of FIG. 1 (i.e., core inlet, core outlet, and four pressure taps along the lengthwise extent), the overall pressure drop between core inlet and outlet, as well as respective pressure drops over five core sections can be obtained. It should be understood that the number of pressure sensors is not intended to be limited by this example.

[0053] At 224, a respective local phase flux is calculated using the respective local saturation profile at each of the fixed two-phase fractional flow rates. The local phase flux can be calculated using fractional flow theory. As described below, two example techniques can be used to calculate local phase fluxes. In some implementations, the local phase flux can be calculated by integrating on a local saturation profile with respect to spatial changes in saturation. This is referred to herein as a Fractional Flow Method and uses fractional flow theory. In other implementations, the local phase flux can be calculated by integrating on a local saturation profile with respect to temporal changes in saturation at the same location. This is referred to herein as a Mass Conservation Method and is based on mass conservation. Optionally, the integration can be performed using a finite difference approximation.

[0054] With the local measurements, the relative permeability is calculated using local saturation profiles, local pressure data, and local phase fluxes at 226. As described below, two example techniques can be used to calculate two-phase relative permeability. In some implementations, the Darcy Buckingham equation can optionally be used to calculate two-phase relative permeability. As described below, the Darcy Buckingham equations facilitates direct calculation of relative permeability in a plurality of sections of the core. The plurality of sections of the core are those sections created by the plurality of pressure sensors along the lengthwise extent of the core. Additionally, it is possible to avoid capillary end effects experimentally by determining relative permeabilities only for upstream sections of the core and not determining relative permeabilities for capillary dominated downstream sections of the core. For example, relative permeability for the outlet section of the core cannot be calculated. In other implementations, an extended JBN method can be used to calculate two-phase relative permeability. Unlike the conventional JBN method which is capable only of calculating two- phase relative permeability at the outlet of the core, the extended JBN method is capable of calculating two-phase relative permeability at a plurality of locations along the core. In particular, the extended JBN method allows calculation of relative permeability at the location of each of the plurality of pressure sensors. In other words, the JBN method is extended into the core. Additionally, it is possible to avoid capillary end effects experimentally by not calculating relative permeability at the outlet of the core.

[0055] CALCULATING RELATIVE PERMEABILITY WITH THE TWO-PHASE DARCY

BUCKINGHAM EQUATION [0056] An unsteady-state unsteady state method for measuring two-phase relative permeability by obtaining local values of the three key parameters (i.e., saturation, pressure drop, and phase flux) versus time during a displacement is presented below. These three parameters can be substituted into two-phase Darcy Buckingham Equation to directly determine relative permeability. To obtain the first two local values (saturation and pressure drop), a medical X-ray Computed Tomography (CT) scanner, which continuously measures saturation in time and space, and six differential pressure transducers, which continuously measure the overall pressure drop and the respective pressure drops of five individual sections (divided by four pressure taps on the core), are used. The local phase flux at a certain injection time is obtained by integrating on the saturation profile measured at this particular time based on a fractional flow analysis.

[0057] Five CCh-brine primary drainage experiments were conducted in a 60.8 cm long and 116 mD Berea sandstone core at 20 °C and 1500 psi to illustrate this unsteady-state method. In return, hundreds of unsteady-state CO2 and brine relative permeability data points were obtained. The data points were consistent with steady-state relative permeability data from the same experiments. Due to the large amount of relative permeability data obtained by the unsteady-state method described herein, the uncertainties of the exponents in the Corey-type fits decrease by roughly 90% compared with the steady-state method. The unsteady-state method using the Fractional Flow Method was also compared with a previously published unsteady-state method based on the Mass Conservation Method by DiCarlo et al. ^1,2 . The consistency between the relative permeability data obtained by the two unsteady-state methods indicates that the unsteady-state method using the Fractional Flow Method is a sound alternative for determining relative permeability.

[0058] An unsteady-state method is described herein that is based on steady-state experiments - two phases are simultaneously injected into the core at a certain water fractional flow

(which is the water flux divided by the total flux), local saturation is measured non-destructively using

CT scanning, and pressure drops are measured through pressure taps to avoid the capillary end effect.

Unlike the steady-state method, local measurements are obtained at all times at each new water fractional flow and not just at steady state. Local saturations are obtained from repeated CT scans, local pressure gradients are obtained from multiple pressure taps with continuously recorded pressure drops, and local fluxes are derived from saturation profile along the core versus time and a fractional flow analysis. From these local measurements at each water fractional flow, many points on a relative permeability curve are obtained. This is in contrast to the single point on a relative permeability curve obtained using the steady-state method. Compared with standard u nsteady-state methods, such as the J BN method, the unsteady-state method described herein experimentally avoids the capillary end effect by only using the upstream four sections of the core.

[0059] In the example described below, the unsteady method is used to obtain CCh-brine relative permeability. CO2 relative permeability has been a key parameter in modeling multiphase flow scenarios such as enhanced oil recovery using CO ₂ as an agent ^38_4 °, and CO ₂ geological storage in deep saline aquifers ^41"43 . The unsteady-state method is detailed below, and results are compared to steady- state data and those obtained using other unsteady-state methods.

[0060] Obtaining Local Phase Flux From Saturation Data

[0061] Any unsteady-state method requires the local flux. Two different techniques that use the measured saturation profiles with time to find the local flux during the unsteady portion of the flow, which is the part that occurs during each fractional flow step before steady state is reached, are presented below. Note, each drainage experiment consists of a few fractional flow steps, during which two phase fluids are injected at a fixed fractional flow.

[0062] According to the first technique, a method that uses the fractional flow theory and integrates spatial saturation difference is presented for the first time. This is referred to herein as the Fractional Flow Method. According to the second technique, a previously published method that integrates the temporal saturation difference on space ^{1 2} ' ³⁴ ' ³⁵ is presented. This is referred to herein as the Mass Conservation Method.

[0063] Both techniques have the same initial condition and boundary condition. The initial condition is that the core starts with the uniform saturation achieved at the earlier fractional flow step:

[0064]

S _w (x, t— 0)— S _w (x, steady state of previous fractional flow step) . Eqn (2) [0065] In Equation 2, S _w is the water saturation, t is the time from the onset of a new imposed fractional flow, and x is the distance from the inlet of core (unit: cm). The boundary condition is the application of a new fractional flow:

[0066] f _w (x = 0, t > 0) = f _w (imposed at the pumps, t > 0) . Eqn. (3)

[0067] In Equation 3, / _w is water fractional flow. The first technique uses the fractional flow theory ^27,28 to obtain the local water fractional flow as a function of space and time. By assuming incompressible fluids and no diffusive transport, the continuity equation is:

[0068] dt dx Eqn. (4)

[0069] In Equation 4, φ is porosity and u is the total Darcy velocity of the two phases (unit: cm/min). If gravitational and capillary forces are neglected, then the fractional flow is only a function of the saturation. Thus, Equation 4 is hyperbolic, and can be solved by introducing a new variable using the similarity transform: ξ =— . Then Equation 4 becomes the following equation:

t

[0070] ξ ^dS _w , dS _w _ _Q

u άξ dS _w άξ _{Eqn (5)}

[0071] Equation 5 has two solutions: a) a trivial one is dS _w /di, = 0, this occurs ahead of the front where the change in boundary condition has not yet propagated, and b) [0072]

[0073] Since the variables on the right hand side are easily found, Equation 6 gives the derivative of water fractional flow with respect to water saturation at any position and time. According to the Fractional Flow Method, to obtain the local fractional flow at a particular injection time, this derivative is numerically integrated with respect to the spatial changes in saturation at the same time; this integration is done by finite difference approximation shown in Equation 7. It should be noted that when applying the boundary condition Equation 3, the most upstream slice of the core measured by CT is set to have the imposed f _w by pumps. [0074]

[0075] The Mass Conservation Method of calculating local phase fluxes was previously published and used by DiCarlo et al. ^1,2 and Kianinejad et al. ^34,35 , which is based on mass conservation. In the gravity drainage experiments, the phase fractional flow was calculated by integrating the temporal phase saturation difference on space, and the integration was approximated with finite difference shown by Equation 8.

[0076] (* _> 0 = L (0, t) + -^- j ^* (S _w (x ₀ , t - At) - S _w (x ₀ , t)) dx _{

uAt

^ f _w (x = i, t) = f _w (x = 0, t) + ^-^∑{S _w (x ₀ , t - At) - S _w (x ₀ , t)) .

uAt ,

¾ ⁼¹ Eqn. (8)

[0077] The local flux obtained above along with the measured pressure drop of each section of the core can be used to find the relative permeabilities with the inversion of Equation 1, and the associated water saturation was measured locally. [0078] Materials and Methods [0079] Rock and Fluids

[0080] The rock sample is a 60.8-cm long and 7.14-cm diameter cylindrical Berea sandstone core. It has a uniform porosity of 17.6% along the axial direction. The experiments were conducted at

20 °C and 1500 psi with 2 wt% NaCI brine and CO ₂ . To avoid mass transfer during the experiment, brine and CO ₂ were equilibrated with each other and stored separately in two vertical piston-accumulators; details on making sure phase equilibration are in Chen et al., "Measurements of C02-Brine Relative Permeability in Berea Sandstone Using Pressure Taps and a Long Core. Greenh. Gases Sci. Technol." ²⁶ . The viscosities of non-equilibrated brine, equilibrated brine and equilibrated CO2 are 1.040 cp ⁴⁴ , 1.081 cp ⁴⁵ ' ⁴⁶ , and 0.087 cp ⁴⁷ ' ⁴⁸ .

[0081] Core Flood Experiment Setup

[0082] FIG. 3 is a diagram illustrating the example core flooding apparatus used in the experiments described herein. The three-layer-wrapped (heat-shrink tubing, aluminum foil, heat-shrink tubing) core sample was placed in a vertical alumin um core holder with a confining pressure of 2000 psi.

Four pressure taps on the core divided it, from upstream to downstream, into five individual sections: section 1, section 2, section 3, section 4 and section 5. The wrapping was drilled through at taps located at 15.24, 25.40, 35.56, and 45.72 cm away from the inlet. The overall pressure drop (AP _t0 ) and the pressure drops of five sections (from upstream to downstream named as ΔΡα, ΔΡη, ΔΡ23, ΔΡ34, and ΔΡ40) were continuously monitored using six differential pressure transducers. During experiments, the piston

Accumulator C (vertically placed, FIG. 3) received the two-phase effluents, which then pushed the piston to discharge pure water on the bottom through a backpressure regulator (BPR) set at 1500 psi. The use of the outlet accumulator prevented CO2 decompression through the BPR and minimized pressure fluctuations during the experiment.

[0083] Determination of Saturation with X-ray Computed Tomography

[0084] The method for determining relative permeability using the Darcy Buckingham

Equation is predicated on obtaining accurate spatia l saturations versus time. To achieve that, the core holder was vertically mounted onto a vertical positioning system (VPS) that can move up and down through a medical X-ray CT scanner. During the two-phase flow, when the total flow rate was 2 ml/min,

60 1-cm thick slices of CT images along the core axis were obtained, which took 6 minutes and 9 seconds. The imaging frequency was every 0.05 PV and the index between consecutive slice centers was

1 cm. When the total flow rate was 4 ml/min, 30 1-cm thick slices of CT images along the core axis were obtained, which took 3 minutes and 5 seconds. The imaging frequency was still every 0.05 PV but the index between consecutive slice centers was 2 cm. When the total flow rate was 8 ml/min and 16 ml/min, 30 slices of CT scans along the core axis were obtained, but the scanning frequency was 0.10 PV and 0.20 PV, respectively. The local saturations data are missing at the pressure taps where X-rays were blocked by the stainless steel fitting. The accuracy/error of CT-measured slice-wise water saturation is 0.4% (absolute saturation).

[0085] Although the saturation measurement along the core axis took a certain amount of time, it is assumed that the saturation profile was taken as a snapshot when using it for calculating local water fractional flows. As a consequence, this approximation brings a 3% relative error to the calculated water fractional flow using the method described herein, and this uncertainty propagates to the relative permeability.

[0086] C0 ₂ -Brine Two-Phase Primary Drainage

[0087] To determine two-phase relative permeabilities, five primary drainage experiments were conducted by simultaneously injecting (e.g., co-injecting) equilibrated-brine (hereafter referred as brine) and equilibrated-CCh (hereafter referred as CO ₂ ) into a brine-saturated core. Each experimental step included: (i) applying a water fractional flow {f _w ) that was lower than the water fractional flow {f _w ) of the previous step, and (ii) measuring the saturation versus space and time and the local pressure drops during the unsteady-state flow that occurred. It should be understood that the measurements of step (ii) were taken under unsteady-state conditions. Once steady state flow was reached for a step, another lower/ _w was applied, and step (ii) was repeated. Five primary drainage experiments were performed and their names and specific flow rates are listed in Table 1.

Table 1. Injection rates during primary drainage experiments qbrine:qCC>2 100:0 50:50 10:90 1:99 0.1:99.9 0:100

qbrine(ml/min) 2.000 1.000 0.200 0.040 0.008 0

[0088] Results

[0089] Single-Phase Brine Relative Permeability

[0090] Five primary drainage experiments (Exp5, Exp6, Exp7d, Exp8d, and Exp9d) were conducted. Before the beginning of each experiment, the single-phase absolute permeability to non- equilibrated brine was measured and reported in Chen et al., "Measurements of C02-Brine Relative Permeability in Berea Sandstone Using Pressure Taps and a Long Core. Greenh. Gases Sci. Technol." ²⁶ '. Table 2 shows that the brine permeability of the entire core and each individual section changed within 10% throughout all the five primary drainages.

Table 2. Single phase brine permeability (k _w ) whole section lsection2setion3section4section5

length(cm) 60.80 15.24 10.16 10.16 10.16 15.08

Exp5 115.98 95.17 128.27 138.78 143.12 110.87

Exp6 104.76 80.45 119.98 126.66 126.91 104.50

Exp7d 114.36 91.82 128.50 135.03 141.15 110.02

k _w (md) Exp8d 117.63 101.55 134.22 138.24 135.15 116.04

Exp9d 120.56 101.67 135.86 147.46 132.13 113.66

avg(md) 114.66 94.13 129.37 137.24 135.69 111.01

std/avg (%) 5.22 9.28 4.82 5.46 4.87 3.92

[0091] Saturation and Pressure Drop Data versus Space and Time

[0092] As an example, the fractional flow step of f _w = 0.1 (i.e., qbrine:qCC = 10:90) during Exp5 is taken to show how relative permeability is calculated with the new unsteady state method. FIG. 4A is a graph illustrating the saturation profile development versus time at the fractional flow step of / _w =0.1 during Exp5. The initial condition of the core was after a steady-state flood conditions were achieved at / _w =0.5. With the application of the lower water fractional flow of/ _w = 0.1, the water saturation decreased from 0.90 to about 0.75 as the front moved through the core from left to right as shown in FIG. 4A.

[0093] FIG. 4B is a graph that illustrates the measured pressure drops of the upstream four sections of the core versus total injected volume at the fractional flow step of / _w =0.1 in Exp5. Note that the pressure drop in section 1 was higher than the other sections, as it is longer (see Table 2). Section 5 is also longer (see Table 2) but not shown, as it is not used in calculating relative permeability. The pressure drops of the core's upstream four sections monotonically decreased with time until reaching stable values. By comparing FIGS. 4A and 4B, it was found that the evolutions of saturation and pressure drop of every section of the core were synchronized. For instance, for the entrance section of the core (section 1), between 0.05 PV and 0.20 PV, both the water saturation and the pressure drop decreased with time; since 0.25 PV, they both approximately reached steady-state values.

[0094] Local Phase Flux and C0 ₂ -Brine Relative Permeability

[0095] FIG. 5A is a graph that illustrates the local water fractional flow versus space and time calculated by integrating spatial saturation difference based on fractional flow theory (e.g., the Fractional Flow Method) as described herein. FIG. 5B is a graph that illustrates the local water fractional flow versus space and time calculated by integrating temporal saturation difference on space (e.g., the Mass Conservation Method) as described herein. Both FIGS. 5A and 5B were calculated for the fractional flow step of / _w =0.1 in Exp5. The details of each of the Fractional Flow Method and the Mass Conservation Method are described above with regard to Equations (2)-(8). In general, both FIGS. 5A and 5B show the calculated water fractional flow started at 0.1 at the inlet where the flow was controlled; progressing into the column the water fractional flow rose quickly in space to the fractional flow of the previous step (0.5). At later times, 0.55-0.60 PV, the imposed fractional flow of 0.1 was seen throughout the column. FIG. 5C is a graph illustrating the unsteady-state brine relative permeability data obtained using local phase flux calculated using both the Fractional Flow Method and the Mass Conservation method, respectively. FIG. 5D is a graph illustrating the unsteady-state CO2 relative permeability data obtained using local phase fluxes calculated using the Fractional Flow Method and the Mass Conservation method, respectively. In FIGS. 5C and 5D, the steady-state data at/ _w =0.1 and / _w =0.5 are also shown.

[0096] As a comparison, the differences between the slice-wise water fractional flows calculated with the Fractional Flow Method and the Mass Conservation Method are: (1) the water fractional flow profile at every injection time calculated with the Fractional Flow Method was more scattered than that calculated with the Mass Conservation Method; (2) the water fractional flow profile calculated with the Fractional Flow Method monotonically decreased with increasing time, while the Mass Conservation Method had some water fractional flows that increased with increasing time (see 0.20 PV and 0.25 PV). The first difference is because the Fractional Flow Method integrates on spatial saturation difference that has more scatter due to saturation heterogeneity. The second difference is because: (1) in the Fractional Flow Method, the integrand, df _w /dS _w =cpx/ut, monotonically decreases with time and hence the integral at the same space also monotonically decreases with time; (2) in the Mass Conservation Method, the integrand, cpAS _w /uAt, does not monotonically decrease with time, but depends on the measured temporal saturation difference. For instance, in the downstream portion of the core, the temporal water saturation difference between 0.20 and 0.25 PV was greater than that between 0.15 and 0.20 PV; hence the water fractional flows at 0.25 PV was higher than those at 0.20 PV when using the Mass Conservation Method.

[0097] Since the local pressure drop is measured over each section, the / _w is averaged over each section. This causes some of the scatter in f _w obtained by the Fractional Flow Method to be smoothed out. Comparing the section-wise f _w of these two methods, it is found that they are consistent with an average relative difference (defined as the absolute difference in fractional flow between the two methods divided by the fractional flow obtained from the Fractional Flow Method) of 20%.

[0098] Using the section-wise fractional flow of both methods and the pressure drop of each section, the unsteady-state CO2 and brine relative permeability can be directly calculated with the simple inversion of Equation 1. In Equation 1, Qcoi and Qbrine =Q *fw where Q _t are the total flow rates of CO2 and brine (Table 1); are the viscosity of equilibrated CO2 and brine, which are 0.087 cp and 1.081 cp; . and /Care the length and single phase brine permeability of the entire core or a certain section, (Table 2); A is cross section a rea of the core, which is 40.08 cm ² . The pressure drop ΔΡ was the measured pressure drop of each of the upstream four sections at each injection time; it has been previously shown that this is the pressure drop of both phases ΔΡ' in the center of the core and the pressure drop of the CO2 phase ΔΡ ^∞2 in the entrance section (section 1) ²⁶ .

[0099] FIGS. 5C and 5D show the unsteady-state brine and CO2 relative permeability data obtained during the unsteady-state flow portion of / _w =0.1 step with both methods along with the steady state data. For both methods, there are dozens of unsteady-state relative permeability data points (30 for brine and 40 for CO2) that fill the saturation gap between the steady state data of consecutive steps.

[00100] By comparing the relative permeabilities obtained with the two methods, the following was found that: (1) in terms of brine relative permeability, the average relative difference between these two methods is 20%, which reflects the 20% average relative difference between the water fractional flows calculated with these two methods; and (2) in terms of CO2 relative permeability, the average relative difference between these two methods is 6% which is because at this step, local unsteady-state f _w is around 0.1. Thus, the average relative difference in CO2 fractional flow between these two methods is one order of magnitude less than the average relative difference in water fractional flow.

[00101] Using both the Fractional Flow Method and the Mass Conservation Method, the local water fractional flows were calculated. Then, the unsteady-state CO2 and brine relative permeability data were obtained from Exp5, Exp6 and Exp7d (5 steps primary drainage) at water fractional flow steps of f _w = 0.5, 0.1 and 0.01, and from Exp8d and Exp9d (1 step primary drainage at/ _w = 0). Note that some fractional flow steps and sections do not have any data. This is because sometimes the measured saturation changes within the experiment were too small to apply the Fractional Flow Method of calculating unsteady-state local water fractional flow profiles. There were other instances of pressure tap failures. Still, through 5 experiments, 388 CO2 relative permeability data points and 266 brine relative permeability data points were determined with both the Fractional Flow Method and the Mass Conservation Method. The uncertainty in water saturation is 0.4 % (absolute saturation) due to measurement error. For both the Fractional Flow Method and the Mass Conservation Method, the uncertainty of CO2 relative permeability obtained in the center three sections of the core is 3% of its own value, and the uncertainty of CO2 relative permea bility obtained in the entrance section of the core is 6% of its own value. The uncertainty of brine relative permeability obtained in the center three sections of the core is 4% of its own value for the Fractional Flow Method, and 8% of its own value for the Mass Conservation Method.

[00102] Comparison Between Steady-State and Unsteady-State Relative Permeability [00103] FIGS. 6A and 6B compares the steady state (FIG. 6A) and the unsteady-state

(FIG. 6B - local phase flux calculated using the Fractional Flow Method) CO ₂ and brine relative permeabilities of all five drainages. The dashed line (Equation 9) and the solid line (Equation 10) in FIGS. 6A and 6B the Corey-type models that fit to the steady-state ( FIG. 6A) and the unsteady-state (FIG. 6B) CO ₂ and brine relative permeabilities, respectively.

[00104]

f V,

l - 5„ ^

l - 5„

Eqn. (9)

[00105]

[00106] In Equation 9 and Equation 10, only the exponents, n _g and n _w , are the fitting parameters. The irreducible water saturation {S _wr ) is the average measured S _wr of the five primary drainages, with an error being the standard deviation; so the result is S _wr = (26 ± 2) %. Using the least square method, the exponents that fit to the steady-state relative permeability data are: n _g = 1.8 ± 0.1 and n _w = 5.2 ± 0.3 (given in Chen et al., "Measurements of C02-Brine Relative Permeability in Berea Sandstone Using Pressure Taps and a Long Core. Greenh. Gases Sci. Technol." ²⁶ ); while the exponents that fit to the unsteady-state data are: n _g = 1.78 ± 0.01 and n _w = 4.64 ± 0.04. Because of the larger amount of data points in the unsteady state method compared with the steady state method, the uncertainty in the fitting exponents, n _g and n _w , decrease by roughly 90%, leading to better constrained relative permeability models.

[00107] In general, the steady state and the unsteady state CO ₂ and brine relative permeabilities are consistent. In particular, the Corey-type exponents, n _g and n _w , in both methods are different by 1% and 11%, respectively. The relative large difference in n _w reflects the uncertainty in the calculated local water fractional flow with the Fractional Flow Method described herein. The smaller relative difference in n _g is due to the local CO ₂ fractional flow (f _g ) being close to 1 in the primary drainages. Hence, the uncertainty in f _g is one order (or two orders) of magnitude smaller than the uncertainty in / _w . One limitation of the unsteady-state method compared with steady state methods is that values cannot be obtained at very low water saturations. The reason is at the final steps of the primary drainage (very low imposed f _w ), water saturation changes within the experimental duration were too small to apply the unsteady-state method to calculate local unsteady-state water fractional flows.

[00108] Comparison Between the Fractional Flow Method and the Mass Conservation

Method

[00109] The Fractional Flow Method and the Mass Conservation Method use slightly different ways to obtain local water fractional flows: one integrates spatial saturation difference, and the latter integrates temporal saturation difference on space ^{1 2} ' ³⁴ ' ³⁵ . If the actual flow exactly follows the fractional flow theory, then these two methods should give identical results. However, this is not the case as FIG. 5C shows that the brine relative permeabilities calculated by these two methods are different. By simple eye test, the Fractional Flow Method is better because the brine relative permeabilities of the Fractional Flow Method are less scattered.

[00110] FIGS. 7 A and 7B compare the brine and CO2 relative permeability data calculated with the Fractional Flow Method and the Mass Conservation Method for all five experiments. The solid lines in FIG. 7A are y=x, y=1.3x and y=0.8x, respectively. The solid line in FIG. 7B is y=x. FIG. 7A is a graph that shows brine relative permeability data obtained from the Mass Conservation Method were roughly between 30% more and 20% less of those obtained from the Fractional Flow Method (average relative difference is 25%). FIG. 7B is a graph that shows CO2 relative permeability data obtained from the Mass Conservation Method were within 2% of those obtained from the Fractional Flow Method (average relative difference is 2%). As discussed above, the relative difference in the brine relative permeability data between the two methods only results from the same amount of relative difference in the section-wise water fractional flows between the two methods. And since CO2 fractional flow is order/orders of magnitude higher than water fractional flow in the primary drainages, this 25% relative difference in water fractional flow becomes order/orders of magnitude smaller for CO ₂ fractional flow.

[00111] Again, the difference in water fractional flow calculated by the two methods is because the actual flow did not exactly follow the fractional flow theory. This could be because the large viscosity ratio of the C0 ₂ -brine system makes capillary forces not negligible. By choosing a different fluid pair with a smaller viscosity ratio, the difference between f _w obtained by the two methods may turn out smaller.

[00112] An unsteady-state method of determining two-phase relative permeability by using the local pressure drop, local saturation, and local phase fluxes during the unsteady-state flow portion of conventional steady-state primary drainage experiments is described herein. The former two parameters (local pressure drop and local saturation) are experimentally measured, and the last (local phase flux) is obtained by integrating spatial saturation difference based on the fractional flow theory. In particular, the Fractional Flow Method was used to determine C0 ₂ -brine relative permeability from five primary drainage experiments in a Berea sandstone core. It was found that that the unsteady-state and steady-state C0 ₂ -brine relative permeability data are consistent. Compared with the steady state method, the large amount of relative permeability data obtained by the new unsteady-state method decrease the uncertainties of the exponents in the Corey-type fits by roughly 90%. The Mass

Conservation Method by DiCarlo et al. ^1,2 was also used to calculate local phase fluxes based on mass balance. The Mass Conservation Method integrates temporal saturation difference on space. The comparison between the Fractional Flow Method and the Mass Conservation Method in terms of section-wise phase fluxes and relative permeabilities proves that the Fractional Flow Method is rigorous in finding the local phase fluxes and unsteady-state relative permeabilities.

[00113] The Uncertainty of Water Fractional Flow and Relative Permeability Calculated with the Fractional Flow Method and the Mass Conservation Method

[00114] To estimate the uncertainty in water fractional flow due to neglecting saturation measurement time, Exp5 is taken as an example, and , the time offsets of the saturation profiles are corrected at the water fractional steps of/ _w =0.5, 0.1 and 0.01. When the total flow rate was 2 ml/min, 60 X-ray CT scans were taken along the core axial direction to measure slice-wise saturations, which took 6 minute and 9 seconds. Except for the saturations at x=30 cm, saturation scanned at any other place of the core had a time offset with the planned time. The scanning frequency at total flow rate of 2 ml/min was every 0.05 PV. The first order correction of saturation profile is to assume that saturations at each position of the core were changing linearly with time between consecutive scanning times. Based on this assumption, all saturation profiles were corrected to be exactly at the pla nned times. Using the corrected saturation profiles, the local water fractional flows {f _w ) were calculated again using both the Fractional Flow Method and the Mass Conservation Method.

[00115] It was found that the average relative difference in / _w between using the actual saturation profiles and using the corrected saturation profiles is 3% for the Fractional Flow Method and 7% for the Mass Conservation Method based on 160 f _w data points. Hence, due to neglecting the saturation measurement time, the water fractional flow has a 3% relative error of its own value for the Fractional Flow Method, and a 7% relative error of its own value for the Mass Conservation Method.

[00116] In Chen et al., "Measurements of C02-Brine Relative Permeability in Berea

Sandstone Using Pressure Taps and a Long Core. Greenh. Gases Sci. Technol." ²⁶ , it was shown that due to the measured pressure drop fluctuations and the non-uniform saturation in the entrance section of the core, steady-state CO2 and brine relative permeability determined in the center three sections of the core had a 3% relative error of its own value and steady-state CO2 relative permeability determined in the entrance section of the core had a 6% relative error of its own value. For the unsteady-state relative permeability determined herein, the extra error source is from the local fractiona l flows of water and CO2. As for the uncertainty in the local fractional flow of CO2 (f _g ), since f _g was close to 1 in the primary drainages, its uncertainty is negligible compared with the uncertainty due to pressure fluctuations.

[00117] Applying the uncertainty propagation to the two-phase Darcy's law, the uncertainties in unsteady-state relative permeability data were estimated using Equation 11. First, for both the Fractional Flow Method and the Mass Conservation Method, the uncertainty of CO2 permeability determined in the core's center sections is 3% of its own value, and the uncertainty of CO2 relative permeability determined in the core's entrance section is 6% of its own value. Second, the uncertainty of brine relative permeability determined in the core's center sections is 4% of its own value for the Fractional Flow Method, and 8% of its own value for the Mass Conservation Method.

[00118] Unsteady-State Relative Permeability Data Obtained with the Fractional Flow

Method and the Mass Conservation Method

CQ2-Brine Relative Permeability Calculated with the Fractional Flow Method in Exp5

Time seel sec2 sec3 sec4

(PV) Sw krg Sw krw krg Sw krw krg Sw krw krg

0.30 0.887 0.0507 0.898 0.5014 0.0390 0.903 0.5127 0.0385 0.910 0.4739 0. 0341

0.5 0.35 0.890 0.0509 0.904 0.4887 0.0379 0.907 0.5561 0.0421 0.913 0.4761 0.0351

0.40 0.892 0.0509 0.900 0.4852 0.0383 0.904 0.5479 0.0424 0.907 0.4674 0.0360

0.05 0.810 0.1176 0.875 0.4850 0.0548 0.882 0.5191 0.0448 0.884 0.4849 0398 0.10 0.764 0.1545 0.845 0.3982 0.0751 0.862 0.4462 0.0612 0.874 0.4214 0455 0.15 0.753 0.1793 0.823 0.3122 0.0923 0.851 0.3650 0.0716 0.869 0.3781 0565 0.20 0.748 0.1903 0.803 0.2662 0.1054 0.826 0.2933 0.0861 0.855 0.3550 0694 0.25 0.739 0.2027 0.783 0.2772 0.1332 0.799 0.2655 0.1031 0.831 0.3365 0859

0.1

0.30 0.739 0.2039 0.779 0.2456 0.1291 0.792 0.2375 0.1073 0.820 0.3076 0974 0.40 0.738 0.1874 0.766 0.2534 0.1546 0.768 0.2083 0.1236 0.788 0.2398 1123 0.45 0.733 0.1962 0.765 0.2301 0.1405 0.766 0.2068 0.1233 0.788 0.2369 1130 0.55 0.738 0.1875 0.764 0.2188 0.1417 0.757 0.1930 0.1300 0.772 0.2102 1228 0.60 0.730 0.1859 0.759 0.2451 0.1577 0.756 0.2002 0.1309 0.772 0.2276 1291

0.30 0.644 0.2989 0.726 0.1755 0.1872 0.736 0.1802 0.1502

0.40 0.632 0.3026 0.711 0.1584 0.2213 0.726 0.1498 0.1589

0.45 0.627 0.3153 0.706 0.1440 0.2230 0.721 0.1392 0.1626

0.55 0.622 0.3305 0.698 0.1216 0.2285 0.717 0.1266 0.1721

0.60 0.619 0.3593 0.695 0.1096 0.2255 0.711 0.1170 0.1812

0.65 0.618 0.3328 0.689 0.1151 0.2602 0.705 0.1092 0.1851

0.75 0.613 0.3431 0.682 0.1040 0.2698 0.698 0.0997 0.1936

0.01

0.90 0.600 0.3581 0.662 0.0988 0.3057 0.677 0.0839 0.1989

1.00 0.595 0.3890 0.657 0.0817 0.2703 0.672 0.0840 0.2146

1.10 0.594 0.3954 0.657 0.0806 0.2795 0.670 0.0800 0.2219

1.15 0.595 0.3598 0.652 0.0845 0.3205 0.666 0.0765 0.2263

1.25 0.592 0.3689 0.650 0.0816 0.3161 0.664 0.0740 0.2301

1.40 0.592 0.3678 0.646 0.0786 0.3326 0.658 0.0699 0.2419

1.50 0.590 0.3670 0.640 0.0797 0.3591 0.651 0.0650 0.2460

CO?-Brine Relative Permeability Calculated with the Mass Conservation Method in Exp5

Time seel sec2 sec3 sec4

0.30 0.887 0.0511 0.898 0.4926 0.0397 0.903 0.5098 0.0388 0.910 0.5219 0.0302 0.5 0.35 0.890 0.0517 0.904 0.4568 0.0404 0.907 0.4942 0.0471 0.913 0.4050 0.0409 0.40 0.892 0.0522 0.900 0.4779 0.0389 0.904 0.5485 0.0424 0.907 0.4815 0.0349

0.05 0.810 0.1002 0.875 0.6098 0.0447 0.882 0.6304 0.0359 0.884 0.6205 0.0289 0.10 0.764 0.1467 0.845 0.5086 0.0663 0.862 0.5513 0.0527 0.874 0.4978 0.0394 0.1 0.15 0.753 0.1839 0.823 0.2825 0.0947 0.851 0.3131 0.0757 0.869 0.2950 0.0632 0.20 0.748 0.1945 0.803 0.2543 0.1064 0.826 0.3208 0.0839 0.855 0.3641 0.0687 0.25 0.739 0.2021 0.783 0.3473 0.1276 0.799 0.4041 0.0920 0.831 0.4886 0.0737 0.30 0.739 0.2075 0.779 0.1907 0.1336 0.792 0.1900 0.1111 0.820 0.2298 0.1037

0.40 0.738 0.1895 0.766 0.2440 0.1554 0.768 0.2511 0.1202 0.788 0.3137 0.1064

0.45 0.733 0.1951 0.765 0.2534 0.1387 0.766 0.2306 0.1214 0.788 0.2246 0.1140

0.55 0.738 0.1901 0.764 0.1745 0.1453 0.757 0.1764 0.1313 0.772 0.2061 0.1232

0.60 0.730 0.1830 0.759 0.3233 0.1514 0.756 0.2837 0.1242 0.772 0.2877 0.1243

0.30 0.644 0.2985 0.726 0.1698 0.1876 0.736 0.1590 0.1520

0.40 0.632 0.3042 0.711 0.1564 0.2214 0.726 0.1524 0.1587

0.45 0.627 0.3157 0.706 0.1305 0.2241 0.721 0.1359 0.1629

0.55 0.622 0.3328 0.698 0.0969 0.2305 0.717 0.0957 0.1746

0.60 0.619 0.3612 0.695 0.0746 0.2283 0.711 0.0957 0.1829

0.65 0.618 0.3357 0.689 0.1032 0.2612 0.705 0.1244 0.1839

0.75 0.613 0.3436 0.682 0.0982 0.2703 0.698 0.1033 0.1933

0 ⁰¹ 0.90 0.600 0.3567 0.662 0.1758 0.2996 0.677 0.1722 0.1918

1.00 0.595 0.3888 0.657 0.0986 0.2690 0.672 0.1015 0.2132

1.10 0.594 0.3960 0.657 0.0451 0.2824 0.670 0.0393 0.2252

1.15 0.595 0.3647 0.652 0.0485 0.3234 0.666 0.0838 0.2257

1.25 0.592 0.3678 0.650 0.0779 0.3164 0.664 0.0648 0.2308

1.40 0.592 0.3698 0.646 0.0593 0.3342 0.658 0.0606 0.2427

1.50 0.590 0.3667 0.640 0.0928 0.3581 0.651 0.0986 0.2434

CO?-Brine Relative Permeabilitv Calculated with the Fractional Flow Method in Exp6

Time seel sec2 sec3 sec4

fw (PV) Sw kr Sw krw kr Sw krw kr Sw krw krg

* At fw=0.01, relative permeability data of section 3 are calculated from the total pressure drops of sec 3&4

C02-Brine Relative Permeabilitv Calculated with the Mass Conservation Method in Exp6

Time seel sec2 sec3 sec4 fw (PV) Sw krg Sw krw krg Sw krw krg Sw krw krg

0.5 0.32 0.879 0.0511 0.892 0.5584 0.0479 0.901 0.3458 0.0267 0.911 0.5907 0.0368

0.10 0.754 0.1362 0.825 0.5893 0.0568 0.854 0.3353 0.0238 0.870 0.7003 0.0405

0.20 0.730 0.1906 0.780 0.3490 0.1132 0.806 0.2243 0.0471 0.833 0.4187 0.0638

0.1 0.32 0.728 0.2034 0.766 0.2200 0.1394 0.782 0.1465 0.0717 0.808 0.2133 0.0791

0.41 0.729 0.2050 0.763 0.2029 0.1477 0.773 0.1242 0.0811 0.800 0.1511 0.0836

0.50 0.732 0.2050 0.763 0.1877 0.1503 0.768 0.1162 0.0872 0.787 0.1344 0.0858

1.10 0.586 0.4271 0.665 0.1231 0.2773 0.703 0.1240 0.1877

0.01

1.47 0.560 0.4419 0.628 0.1390 0.2968 0.665 0.1279 0.1574

* At fw=0.01, relative permeability data of section 3 are calculated from the total pressure drops of sec 3&4

CO?-Brine Relative Permeabilitv Calculated with the Fractional Flow Method in Exp7d

Time seel sec2 sec4

fw (PV) Sw krg Sw krw krg Sw krw krg

0.25 0.896 0.0483 0.924 0.4998 0.0353

0.5 0.30 0.900 0.0485 0.921 0.4853 0.0365 0.35 0.899 0.0483 0.919 0.5130 0.0389

0.20 0.745 0.1654 0.796 0.2763 0.1111

0.1 0.25 0.743 0.1773 0.787 0.2422 0.1156

0.30 0.741 0.1778 0.778 0.2310 0.1221 0.35 0.742 0.1786 0.776 0.2167 0.1233

0.40 0.742 0.1784 0.773 0.2179 0.1288

0.45 0.744 0.1792 0.775 0.2135 0.1301

0.50 0.745 0.1791 0.774 0.2087 0.1303

0.55 0.746 0.1800 0.772 0.2086 0.1332

0.60 0.743 0.1810 0.773 0.2122 0.1354

0.65 0.746 0.1806 0.775 0.2110 0.1362

0.70 0.745 0.1804 0.772 0.2076 0.1358

0.75 0.744 0.1801 0.774 0.2062 0.1346

0.80 0.743 0.1808 0.774 0.2083 0.1365

0.85 0.746 0.1815 0.774 0.2055 0.1364

0.90 0.745 0.1804 0.774 0.2052 0.1367

0.95 0.745 0.1806 0.773 0.2044 0.1370

1.00 0.744 0.1818 0.774 0.2015 0.1347

1.10 0.740 0.1813 0.770 0.2038 0.1374

1.20 0.732 0.1811 0.764 0.2049 0.1383

1.30 0.730 0.1813 0.762 0.2031 0.1377

0.45 0.636 0.3223 0.731 0.1472 0.1768

0.50 0.631 0.3293 0.728 0.1384 0.1799

0.55 0.627 0.3334 0.721 0.1246 0.1802

0.60 0.625 0.3398 0.717 0.1164 0.1837

0.65 0.621 0.3378 0.710 0.1092 0.1889

0.70 0.618 0.3405 0.706 0.1034 0.1906

0.75 0.615 0.3465 0.704 0.1007 0.1962

0.80 0.612 0.3435 0.698 0.1004 0.2098

0.85 0.611 0.3444 0.695 0.0974 0.2154

0.90 0.608 0.3464 0.692 0.0947 0.2183

0.95 0.605 0.3535 0.690 0.0931 0.2236

1.00 0.605 0.3626 0.686 0.0867 0.2230

1.10 0.595 0.3657 0.675 0.0826 0.2280

1.20 0.582 0.3496 0.661 0.0786 0.2320

1.30 0.581 0.3590 0.659 0.0742 0.2330

1.40 0.579 0.3626 0.655 0.0721 0.2396

1.50 0.574 0.3621 0.648 0.0695 0.2438

1.60 0.567 0.3494 0.641 0.0670 0.2434

1.70 0.567 0.3671 0.642 0.0649 0.2449

C0 ₂ -Brine Relative Permeability Calculated with the Mass Conservation Method in Exp7d

Time seel sec2 sec4

fw (PV) Sw krg Sw krw krg Sw krw krg

0.25 0.896 0.0490 0.924 0.5296 0.0329

0.5 0.30 0.900 0.0503 0.921 0.4599 0.0385

0.35 0.899 0.0480 0.919 0.5135 0.0389

0.20 0.745 0 1676 0.796 0.2505 0.1132

0.25 0.743 0 1806 0.787 0.2032 0.1188

0.30 0.741 0 1783 0.778 0.2235 0.1228

0.35 0.742 0 1820 0.776 0.1647 0.1275

0.40 0.742 0 1796 0.773 0.1924 0.1309

0.45 0.744 0 1821 0.775 0.1519 0.1351

0.50 0.745 0 1805 0.774 0.1826 0.1324

0.55 0.746 0 1815 0.772 0.1823 0.1353

0.60 0.743 0 1802 0.773 0.2126 0.1354

0.1

0.65 0.746 0 1837 0.775 0.1553 0.1407

0.70 0.745 0 1803 0.772 0.2084 0.1357

0.75 0.744 0 1802 0.774 0.1931 0.1357

0.80 0.743 0 1804 0.774 0.2013 0.1370

0.85 0.746 0 1841 0.774 0.1604 0.1401

0.90 0.745 0 1805 0.774 0.1963 0.1374

0.95 0.745 0 1811 0.773 0.1969 0.1377

1.00 0.744 0 1821 0.774 0.1911 0.1356

1.10 0.740 0 1807 0.770 0.2196 0.1361 1.20 0.732 0.1792 0.764 0.2394 0.1355

1.30 0.730 0.1812 0.762 0.2067 0.1374

0.45 0.636 0.3240 0.731 0. .1140 0. .1795

0.50 0.631 0.3306 0.728 0. .0990 0. .1831

0.55 0.627 0.3353 0.721 0. .0924 0. .1828

0.60 0.625 0.3416 0.717 0. .0790 0. .1867

0.65 0.621 0.3389 0.710 0. .0962 0. .1899

0.70 0.618 0.3410 0.706 0. .0824 0. .1923

0.75 0.615 0.3480 0.704 0. .0693 0. .1987

0.80 0.612 0.3442 0.698 0. .0935 0. .2104

0.85 0.611 0.3460 0.695 0. .0707 0. .2175

0.90 0.608 0.3461 0.692 0. .0816 0. .2194

0.95 0.605 0.3523 0.690 0. .0724 0. .2253

1.00 0.605 0.3660 0.686 0. .0467 0. .2262

1.10 0.595 0.3631 0.675 0. .1331 0. .2240

1.20 0.582 0.3463 0.661 0. .1551 0. .2259

1.30 0.581 0.3598 0.659 0. .0443 0. .2355

1.40 0.579 0.3631 0.655 0. .0590 0. .2407

1.50 0.574 0.3617 0.648 0. .0894 0. .2423

1.60 0.567 0.3468 0.641 0. .1008 0. .2407

1.70 0.567 0.3690 0.642 0. .0283 0. .2478

CC -Brine Relative Permeabilitv Calculated with the Fractional Flow Method in Exp8d

Time seel sec2 sec3 sec4 fw (PV) Sw krg Sw krw krg Sw krw krg Sw krw krg

0.25 0.673 0.2524 0.790 0.1919 0.1132 0.833 0.2237 0.0774 0.874 0.2905 0.0603

0.30 0.660 0.2590 0.774 0.1796 0.1332 0.813 0.1925 0.0889 0.844 0.2405 0.0746

0.35 0.648 0.2642 0.761 0.1549 0.1383 0.797 0.1662 0.0951 0.827 0.2155 0.0841

0.40 0.638 0.2852 0.751 0.1429 0.1472 0.784 0.1542 0.1059 0.814 0.1986 0.0934

0.45 0.630 0.2967 0.741 0.1398 0.1675 0.776 0.1381 0.1087 0.806 0.1803 0.0969

0.50 0.622 0.3097 0.735 0.1279 0.1696 0.770 0.1317 0.1150 0.797 0.1728 0.1066

0.55 0.618 0.3091 0.727 0.1129 0.1690 0.760 0.1272 0.1271 0.789 0.1616 0.1130

0.60 0.611 0.3139 0.722 0.1060 0.1732 0.755 0.1168 0.1282 0.784 0.1556 0.1183

0.65 0.607 0.3166 0.715 0.0956 0.1737 0.749 0.1088 0.1311 0.777 0.1538 0.1296

0.70 0.602 0.3168 0.710 0.0901 0.1780 0.741 0.0988 0.1317 0.770 0.1443 0.1330

0.76 0.594 0.3377 0.700 0.0823 0.1802 0.735 0.0970 0.1371 0.763 0.1361 0.1362

0.80 0.589 0.3360 0.694 0.0779 0.1813 0.723 0.0907 0.1447 0.753 0.1267 0.1377

0.85 0.588 0.3434 0.694 0.0777 0.1911 0.727 0.0915 0.1487 0.756 0.1218 0.1388

0.90 0.585 0.3513 0.690 0.0727 0.1893 0.723 0.0879 0.1525 0.750 0.1127 0.1396

0.95 0.578 0.3535 0.687 0.0732 0.1969 0.720 0.0865 0.1557 0.748 0.1117 0.1434

0 1.00 0.578 0.3673 0.683 0.0696 0.2035 0.715 0.0808 0.1597 0.744 0.1100 0.1518

1.05 0.575 0.3651 0.679 0.0701 0.2167 0.711 0.0771 0.1597 0.739 0.1075 0.1576

1.10 0.572 0.3629 0.676 0.0659 0.2143 0.708 0.0743 0.1616 0.737 0.1047 0.1595

1.15 0.568 0.3757 0.674 0.0656 0.2206 0.706 0.0722 0.1618 0.735 0.1059 0.1681

1.20 0.564 0.3677 0.669 0.0634 0.2213 0.701 0.0704 0.1660 0.731 0.1021 0.1680

1.25 0.563 0.3751 0.667 0.0595 0.2222 0.698 0.0658 0.1657 0.728 0.0974 0.1695

1.30 0.560 0.3735 0.664 0.0574 0.2218 0.696 0.0653 0.1700 0.724 0.0935 0.1706

1.35 0.557 0.3785 0.661 0.0547 0.2203 0.692 0.0633 0.1735 0.722 0.0909 0.1711

1.40 0.554 0.3794 0.660 0.0545 0.2250 0.690 0.0657 0.1851 0.720 0.0884 0.1721

1.45 0.552 0.3786 0.656 0.0525 0.2258 0.689 0.0638 0.1827 0.716 0.0841 0.1713

1.50 0.549 0.3803 0.654 0.0529 0.2332 0.687 0.0660 0.1951 0.715 0.0831 0.1732

1.60 0.541 0.3831 0.643 0.0480 0.2317 0.675 0.0607 0.1970 0.704 0.0781 0.1764

1.70 0.530 0.3831 0.633 0.0458 0.2345 0.666 0.0590 0.2001 0.694 0.0755 0.1810

1.80 0.525 0.3832 0.627 0.0421 0.2285 0.660 0.0565 0.2044 0.689 0.0725 0.1841

1.90 0.520 0.3816 0.624 0.0416 0.2367 0.657 0.0542 0.2051 0.685 0.0679 0.1832

2.00 0.518 0.3817 0.619 0.0393 0.2409 0.650 0.0495 0.2029 0.680 0.0666 0.1893

CO?-Brine Relative Permeabilitv Calculated with the Mass Conservation Method in Exp8d

Time seel sec2 sec3 sec4 fw (PV) Sw krg Sw krw krg Sw krw krg Sw krw krg

0.25 I 0.673 0.2533 | 0.790 0.1683 0.1151 | 0.833 0.2387 0.0762 | 0.874 0.3376 0.0565 0.30 0.660 0.2572 0.774 0.1709 0.1340 0.813 0.1927 0.0889 0.844 0.2748 0.0718

0.35 0.648 0.2630 0.761 0.1564 0.1382 0.797 0.1779 0.0942 0.827 0.2356 0.0825

0.40 0.638 0.2845 0.751 0.1318 0.1481 0.784 0.1492 0.1063 0.814 0.1963 0.0936

0.45 0.630 0.2960 0.741 0.1197 0.1691 0.776 0.1265 0.1097 0.806 0.1566 0.0988

0.50 0.622 0.3086 0.735 0.1168 0.1705 0.770 0.1146 0.1164 0.797 0.1508 0.1084

0.55 0.618 0.3106 0.727 0.0762 0.1719 0.760 0.1080 0.1286 0.789 0.1413 0.1147

0.60 0.611 0.3123 0.722 0.0922 0.1743 0.755 0.0993 0.1296 0.784 0.1208 0.1212

0.65 0.607 0.3169 0.715 0.0837 0.1747 0.749 0.1018 0.1317 0.777 0.1400 0.1307

0.70 0.602 0.3163 0.710 0.0727 0.1794 0.741 0.0918 0.1322 0.770 0.1400 0.1334

0.76 0.594 0.3356 0.700 0.1108 0.1779 0.735 0.1279 0.1347 0.763 0.1602 0.1342

0.80 0.589 0.3347 0.694 0.1058 0.1791 0.723 0.1526 0.1398 0.753 0.2346 0.1290

0.85 0.588 0.3436 0.694 0.0260 0.1953 0.727 0.0206 0.1544 0.756 0.0196 0.1471

0.90 0.585 0.3511 0.690 0.0688 0.1896 0.723 0.0777 0.1534 0.750 0.1071 0.1400

0.95 0.578 0.3502 0.687 0.0902 0.1956 0.720 0.0960 0.1550 0.748 0.1063 0.1438

1.00 0.578 0.3681 0.683 0.0412 0.2058 0.715 0.0579 0.1616 0.744 0.0873 0.1536

1.05 0.575 0.3649 0.679 0.0685 0.2169 0.711 0.0802 0.1594 0.739 0.1078 0.1576

1.10 0.572 0.3617 0.676 0.0592 0.2148 0.708 0.0685 0.1621 0.737 0.0864 0.1610

1.15 0.568 0.3753 0.674 0.0596 0.2211 0.706 0.0637 0.1624 0.735 0.0774 0.1704

1.20 0.564 0.3643 0.669 0.0951 0.2188 0.701 0.0980 0.1638 0.731 0.1373 0.1652

1.25 0.563 0.3750 0.667 0.0404 0.2238 0.698 0.0487 0.1671 0.728 0.0717 0.1716

1.30 0.560 0.3726 0.664 0.0541 0.2221 0.696 0.0627 0.1702 0.724 0.0908 0.1708

1.35 0.557 0.3777 0.661 0.0607 0.2198 0.692 0.0731 0.1728 0.722 0.0914 0.1711

1.40 0.554 0.3783 0.660 0.0494 0.2254 0.690 0.0542 0.1861 0.720 0.0707 0.1735

1.45 0.552 0.3772 0.656 0.0538 0.2257 0.689 0.0658 0.1826 0.716 0.0763 0.1720

1.50 0.549 0.3788 0.654 0.0554 0.2330 0.687 0.0612 0.1955 0.715 0.0636 0.1748

1.60 0.541 0.3806 0.643 0.0962 0.2279 0.675 0.1298 0.1915 0.704 0.1629 0.1695

1.70 0.530 0.3791 0.633 0.1050 0.2297 0.666 0.1296 0.1945 0.694 0.1527 0.1748

1.80 0.525 0.3824 0.627 0.0546 0.2275 0.660 0.0748 0.2030 0.689 0.0911 0.1826

1.90 0.520 0.3809 0.624 0.0429 0.2367 0.657 0.0484 0.2056 0.685 0.0573 0.1841

2.00 0.518 0.3819 0.619 0.0328 0.2415 0.650 0.0558 0.2024 0.680 0.0757 0.1886

CC -Brine Relative Permeabilitv Calculated with the Fractional Flow Method in Exp9d

Time seel sec2 sec3 sec4 fw (PV) Sw krg Sw krw krg Sw krw krg Sw krw krg

0.25 0.677 0.2356 0.752 0.1395 0.1302 0.791 0.1942 0.0933 0.842 0.2847 0.0656

0.30 0.663 0.2572 0.737 0.1296 0.1484 0.768 0.1581 0.1043 0.809 0.2286 0.0813

0.35 0.651 0.2721 0.726 0.1164 0.1535 0.756 0.1429 0.1126 0.795 0.2137 0.0940

0.41 0.641 0.2956 0.715 0.1069 0.1674 0.746 0.1280 0.1204 0.783 0.1986 0.1076

0.45 0.637 0.3120 0.707 0.0955 0.1758 0.737 0.1171 0.1295 0.772 0.1752 0.1127

0.50 0.630 0.3257 0.702 0.0912 0.1864 0.729 0.1071 0.1346 0.767 0.1643 0.1172

0.624 0.3259 0.698 0.0874 0.1896 0.724 0.1019 0.1394 0.761 0.1576 0.1235 o °- ⁵⁵

0.60 0.615 0.3274 0.690 0.0858 0.2008 0.719 0.1013 0.1473 0.756 0.1520 0.1287

0.65 0.612 0.3366 0.685 0.0799 0.2069 0.713 0.0932 0.1508 0.749 0.1426 0.1349

0.70 0.608 0.3483 0.679 0.0736 0.2142 0.707 0.0891 0.1583 0.744 0.1342 0.1380

0.75 0.603 0.3552 0.676 0.0713 0.2161 0.703 0.0867 0.1652 0.741 0.1307 0.1428

0.80 0.603 0.3608 0.672 0.0646 0.2192 0.698 0.0858 0.1839 0.737 0.1174 0.1396

0.95 0.588 0.3734 0.659 0.0612 0.2405 0.687 0.0779 0.1893 0.725 0.1096 0.1544

1.00 0.583 0.3845 0.650 0.0558 0.2459 0.678 0.0691 0.1848 0.716 0.1115 0.1690

CO?-Brine Relative Permeabilitv Calculated with the Mass Conservation Method in Exp9d

Time seel sec2 sec3 sec4 fw (PV) Sw krg Sw krw krg Sw krw krg Sw krw krg

0.25 0.677 0.2332 0.752 0.2033 0.1251 0.791 0.2798 0.0865 0.842 0.3914 0.0570

0.30 0.663 0.2539 0.737 0.1960 0.1431 0.768 0.2335 0.0982 0.809 0.3112 0.0747

0.35 0.651 0.2668 0.726 0.1697 0.1492 0.756 0.1868 0.1091 0.795 0.2283 0.0928

0 0.41 0.641 0.2941 0.715 0.1310 0.1654 0.746 0.1370 0.1197 0.783 0.1831 0.1089

0.45 0.637 0.3161 0.707 0.0867 0.1765 0.737 0.1307 0.1284 0.772 0.1811 0.1123

0.50 0.630 0.3233 0.702 0.1020 0.1855 0.729 0.1196 0.1336 0.767 0.1408 0.1191

0.55 0.624 0.3231 0.698 0.1043 0.1882 0.724 0.1084 0.1389 0.761 0.1281 0.1259 0.60 0.615 0.3232 0.690 0.1491 0.1957 0.719 0.1505 0.1434 0.756 0.1703 0.1273

0.65 0.612 0.3362 0.685 0.0761 0.2072 0.713 0.0897 0.1511 0.749 0.1243 0.1364

0.70 0.608 0.3498 0.679 0.0746 0.2141 0.707 0.0998 0.1574 0.744 0.1179 0.1393

0.75 0.603 0.3522 0.676 0.0847 0.2150 0.703 0.0894 0.1650 0.741 0.1040 0.1450

0.80 0.603 0.3651 0.672 0.0224 0.2226 0.698 0.0517 0.1867 0.737 0.0670 0.1436

0.95 0.588 0.3712 0.659 0.1001 0.2374 0.687 0.1114 0.1866 0.725 0.1180 0.1537

1.00 0.583 0.3830 0.650 0.1330 0.2397 0.678 0.1797 0.1759 0.716 0.2332 0.1592

[00119] CALCULATING RELATIVE PERMEABILITY USING AN EXTENDED JBN METHOD

[00120] An extension of the J BN method, which can be used to determine relative permeability using local measurements, is presented herein. Similar as described above, the extended

J BN method requires the local pressure drop, local saturation, and local phase fluxes to be measured or derived during the unsteady-state flow portion of conventional primary drainage experiments. In particular, by obtaining ( 1) section-wise pressure drop measurements between the core inlet, four pressure taps on the core, and the core outlet (i.e., local pressure drops), (2) local saturation measurements, and (3) local phase fluxes, relative permeability to both phases at each pressure tap of the core can be determined using the extended J BN method. In other words, with these local measurements and a mathematical inversion, the extended J BN method can obtain relative permeability data to both phases at every tap location of the core and not just at the core outlet. It should be understood that unlike the conventional J BN method which determines relative permeability to both phase only at the core outlet, the extended J BN method can be used to determine relative permeability at a plurality of locations along the core, i.e., at each pressure senor (or pressure tap).

Although this requires measuring in situ saturation and a plurality of pressure taps in the core (e.g., additional measurements), it has the advantage of avoiding the capillary end effect. As described below, the J BN extension is shown using one data set where CO2 invades a brine-filled core. From this, it is found that the advantages of the extended J BN method over the regular JBN method are: (1) four times more data are obtained, and (2) the extended J BN method data is closer to measured steady-state data than the regular JBN method (e.g., more accurate) because the capillary end effect is experimentally avoided.

[00121] Methods

[00122] Overview of the JBN Method [00123] As introduced, the J BN method is a popular and fast unsteady-state method of determining relative permeability across a wide range of saturations [Welge, 1952; Johnson et al., 1959]. One invading phase is injected into a core fully saturated with a defending phase. During the displacement, both the overall pressure drop and the effluent phase ratio are measured versus time. By applying the fractional flow theory [Buckley and Leverett, 1942] (continuity equation) to the Darcy Buckingham equation and a mathematical inversion, it is possible to obtain the relative permeabilities to both the defending phase (d) and the invading phase (i) at the core outlet. Two main assumptions are stable displacements and incompressible fluids, which can be satisfied by maintaining high flow rate and high experimental pressure. The J BN method also assu mes no capillary or gravitational forces.

[00124] With these assumptions and inversion, Equations 12-14 are used to find the defending phase and the invading phase relative permeabilities at the core outlet (k _r di and kra), and the defending phase saturation at the core outlet (Sd?). Note, the subscript '2' means the outlet of the core. In Equation 12, fdi is the defending phase fractional flow (defined as the phase flux divided by the total flux) at the core outlet, L is the length of core, Q _t is the total flow rate, is the defending phase viscosity, td is the injected pore volume, K is the single phase permeability, A is the cross section area of core, and ΔΡ is the measured overall pressure drop. In Equation 13, μ, is the invading phase viscosity. In Equation 14, Sdavg is the core-average defending phase saturation.

[00125] dfdl Eqn. ( 13)

S d, l davg ^ JdVd · Eqn. ( 14)

[00126] The Extended JBN Method [00127] Without loss of generality, the JBN method can be extended to determine relative permeability at any position of the core (e.g. a plurality of locations in the core) and not just at the core outlet as is possible using the conventional JBN method. This is possible as along as one has the local measurements of saturation and fractional flow at each of the locations, as wells as the pressure drop measurement between the inlet and each of the locations (which can be obtained using pressure sensors or pressure taps).

[00128] Again, the method includes injecting an invading phase into a core fully saturated with a defending phase. The difference from the regular JBN method is that both the pressure drop and saturation are measured locally. As described above, local pressure drops can be obtained using a plurality of pressure sensors (e.g., differential pressure transducers employed at a plurality of pressure taps along the core). Thus, it is possible to obtain pressure drops of individual sections of the core, not just the overall pressure drop. Local saturation can be obtained using an NDT device such as a CT scanner. Specifically, using the extended JBN method, equation 14 is not used to interpret the outlet saturation from the average saturation; instead, an X-ray Computed Tomography (CT) technique is used to measure the in situ saturation along the core. If the local fractional flow can be found at each pressure tap, then together with the pressure drop and saturation measurements, the regular JBN method can be extended to determine relative permeability data to both phases at each pressure tap using Equation 12 and Equation 13. As described above, local fractional flows can be obtained using either the Fractional Flow Method (e.g., by integrating spatial saturation difference) or the Mass Conservation Method (e.g., by integrating temporal phase saturation difference). The Fractional Flow Method and Mass Conservation Method are described in detail above with regard to Equations 2-8. An advantage of the extended JBN method as compared to the JBN method is that it uses pressure drops away from the capillary dominated exit region, and hence experimentally avoids the capillary end effect that potentially plagues the JBN method data.

[00129] Materials and Methods

[00130] Core Flood Experiment Setup [00131] To illustrate this extended J BN method, a primary d rainage experiment was conducted by just injecting CO2 into a brine-saturated Berea sandstone core (60.8 cm long). During the experiment, differential pressure transducers were used to measure the pressure drops of five individual sections of the core divided by four pressure taps. From the upstream to the downstream, the five sections are named seel, sec2, sec3, sec4 and sec5. An X-ray CT scanner was also used to obtain the in situ saturation profile along the core every 0.05 PV from 0.05 PV to 1.50 PV, and every 0.10 PV from 1.50 PV to 2.00 PV (35 scan series in total). Steady state was not reached at 2.00 PV, but it is enough for method illustration purpose.

[00132] In the experiment, the already known parameters are: the length from the core inlet to each tap and the core outlet L (Table 3), the total flow rate Q _t (2 ml/min), the defending phase (brine) viscosity (1.081 cp), the invading phase (CO2) viscosity μ, (0.087 cp), the injected total pore volume td, the average permeability from the inlet to each tap and the outlet K (Table 3), the cross section area A (40.08 cm ² ), the porosity of the core φ ( 17.61%), and the pore volume PV (429.3 cm ³ ).

Table 3. Length and permeability from the core inlet to each tap and the core outlet

seel sees 12 sees 123 sees 1234 overall

[ (cm) 15.24 25.40 35.56 45.72 60.80

/C (mD) 101.55 112.51 118.83 122.10 120.54

[00133] Results

[00134] FIG. 8A is a graph that illustrates water saturation profile recorded at each injection time. FIG. 8B is a graph that illustrates pressure drop of the five individual sections (i.e., seel, sec2, sec3, sec4, sec5) of the core measured versus the total injected pore volume. FIG. 8C is a graph that illustrates water fractional flow profile calculated at each injection time using the Fractional Flow Method described herein. FIG. 8D is a graph that illustrates water fractional flow profile calculated at each injection time using the Mass Conservation Method described herein.

[00135] For simplicity, FIG. 8A only shows water saturation (S _w ) profiles at a few selected injection times. Before the CO2 injection, the core was uniformly saturated with brine. As CO2 was injected, S _w decreased as the CO2 front moved from the left to the right. Before the breakthrough (0.05, 0.11, 0.15 PVs), roughly the same volume of brine was displaced by the injected CO2. After the breakthrough at 0.20 PV, S _w decreased at a slower rate with time and eventually reached roughly 60% at 2.00 PV. Again, steady state was not reached at 2.00 PV, but it is enough to show until this time for illustration.

[00136] FIG. 8B shows the measured pressure drop of each section of the core versus time. Before the arrival of CO2 front at each section, its pressure drop remained constant because single-phase brine flow was measured between taps. When the front was moving through this section, the pressure drop increased because the CO2 phase occupied the upstream pressure tap of this section and both the viscous pressure drop and the capillary pressure were measured between the taps. Upon the front departure from this section, the pressure drop decreased because both pressure taps of this section were occupied by the CO2 phase and only the viscous pressure drop was measured.

[00137] FIG. 8C shows the local water fractional flow (f _w ) profiles versus time calculated from the saturation profiles using the Fractional Flow Method described herein. Before the

breakthrough (0.05, 0.11, 0.15 PVs),/ _w increased from 0 at the inlet (not shown in log scale) to 1 at the front and stayed at 1 until the outlet. After the breakthrough at 0.20 PV, / _W still increased from the inlet to the outlet, but the value was monotonically decreasing with time and approaching the imposed f _w of 0. At 2.00 PV, the core average /„, is roughly 0.01.

[00138] FIG. 8D shows the/ _w profiles versus time calculated using the Mass

Conservation Method described herein. Besides the similar trend as FIG. 8C, the difference between these two figures is that/ _w in FIG. 8D did not monotonically decrease with time: e.g. f _w at 0.80 PV was higher than / _w at 0.60 PV. This is because the Mass Conservation Method is based on mass balance, and the measured temporal S _w difference did not monotonically decrease with time, e.g. S _w difference between 0.60 PV and 0.80 PV was higher than S _w difference between 0.40 PV and 0.60 PV. Because S _w monotonically decreased with time but/ _w obtained by the Mass Conservation Method did not, so the / in, functionality are scattered.

[00139] FIG. 9A is a graph that illustrates l/t _d plotted versus ΔΡ/ΐ for the overall core

("overall"), section 1 ("seel"), the combination of sections 1 and 2 ("secsl2"), the combination of sections 1, 2 and 3 ("secsl23"), and the combination of sections 1, 2, 3 and 4 ("secsl234") and their cubic equation fits (dashed lines). FIGS. 9B and 9C are graphs that illustrates the resulting brine and CO2 relative permeabilities determined at the plurality of pressure taps using the extended JBN method and those determined only at the core outlet using the J BN method. FIG. 9B uses local water fractional flow (f _w ) calculated with the Fractional Flow Method. FIG. 9C uses local water fractional flow {f _w ) calculated with the Mass Conservation Method. FIG. 9D is a graph that illustrates steady state relative permeability data compared with those obtained with the extended JBN method using local water fractional flow (f _w ) calculated with the Fractional Flow Method.

[00140] The solid line and the dashed line in FIGS. 9B-9D are the Corey-type models

(Equation 15 and Equation 16) that fit to brine and CO2 relative permeability data using the same parameters (S _wr =0.28, n _w =4.66, n _g =1.74). Equation 15 and Equation 16 are the Corey-type models that fit to the wetting phase and the non-wetting phase relative permeability data.

[00141]

' 1-5,

l-5„ Eqn. ( 16)

[00142] FIG. 9A is a graph that illustrates l/t _d plotted versus ΔΡ/t _d for the overall core

("overall"), section 1 ("seel"), the combination of sections 1 and 2 ("secsl2"), the combination of sections 1, 2 and 3 ("secsl23"), and the combination of sections 1, 2, 3 and 4 ("secsl234"). For each plot, a cubic equation was fit to it and the fitting parameters are listed in Table 4. The derivative d(l/t _d )/d(AP/t _d ) at each injection time is calculated by differentiating the respective cubic equation for each section or combination of sections.

Table 4. Fitting Parameters for l/t _d as a function of ΔΡ/t _d , using y=a+bx+cx ² +dx ³

overall sees 1 sees 12 sees 123 sees 1234

a 0.110 0.0399 0.102 0.136 0.144

b 0.222 2.42 1.168 0.713 0.515

c -0.00645 -0.656 -0.119 -0.0469 -0.0264 d 0.000130 0.134 0.00855 0.00212 0.000852 [00143] Using/ _W calculated with both the Fractional Flow Method and the Mass

Conservation Method, the derivative d(l/t _d )/d(AP/t _d ), it is possible to calculate the relative

permeabilities to CO2 and brine at each of the four pressure taps and the core outlet for each injection time after the breakthrough (0.20 PV). In total, 124 pairs of brine and CO2 relative permeability data are obtained with the extended JBN method using the pressure drops between the inlet and the four pressure taps and 31 pairs of brine and CO2 relative permeability data are obtained with the regular JBN method using the overall pressure drop.

[00144] FIG. 9B shows, based on /„ calculated with the Fractional Flow Method, the C0 ₂ and brine relative permeability data obtained with the extended JBN method (squares and circles) compared against the JBN method (diamonds and triangles). The solid line and the dashed line are Corey-type models (Equation 15 and Equation 16, fitting parameters are S _wr =0.28, n _w =4.66, n _g =1.74) that fit to the brine and CO2 relative permeability data obtained with the extended JBN method. It can be seen that, the CO2 and brine relative permeability data obtained at the outlet with the JBN method are roughly 30% lower than the Corey type fits (which fit very well to the data obtained at the taps with the extended JBN method, R ² =0.94 for brine and R ² =0.85 for CO2). This is because for the low viscosity fluid CO2, the capillary pressure (~1 psi estimated from Pentland et al. [Pentland et al., 2011]) is at the same order of magnitude as the viscous pressure drop (~5 psi), so the measured overall pressure drop is affected by the capillary pressure. Unlike the JBN method using the overall pressure drop, the extended JBN method uses the pressure drops between the core inlet and each of the pressure taps on the core. As a result, the extended JBN method avoids the capillary end effect and obtains higher relative permeability data to both phases.

[00145] FIG. 9C shows, based on f _w calculated with the Mass Conservation Method, the

CO2 and brine relative permeability data obtained with the extended JBN method (squares and circles) compared against the JBN method (diamonds and triangles). From the figure, it can be seen that the similar trend that the data obtained by the JBN method are roughly 30% lower than the Corey type fits (same parameters as in FIG. 9B) that fit to the data obtained with the extended JBN method. [00146] The difference between FIG. 9B and FIG. 9C is that the brine relative permeability data in FIG. 9C are more scattered. This can be quantified by the R ² of the brine Corey-type fit, which is 0.94 for Figure 2b and 0.68 for FIG. 9C. The cause has been mentioned above: since f _w calculated with the Fractional Flow Method did not monotonically decrease with time while S _w monotonically decreased with time; putting these together in a relative permeability curve directly causes the scattering in the k _r w-S _w functionality. However, the scatter does not show on the CO2 relative permeability data in FIG. 9C (R ² for the CO ₂ Corey-type fit is 0.83 in FIG. 9C). This is because the fractional flow of CO ₂ {f _g ) is orders of magnitude higher than / _w and hence the scattering of the / _w -S _w functionality has less effect on the/ _g -S _w functionality.

[00147] FIG. 9D compares the unsteady-state CO ₂ and brine relative permeability data obtained with the extended JBN method (squares and circles in FIG. 9B) against the steady-state data obtained in five primary drainage experiments (including this illustration experiment) in the same rock and at the same temperature and pressure. It can be seen that these two datasets are consistent with each other. In particular, the steady-state CO ₂ relative permeability data are within ± 20% of the CO ₂ Corey-type fit; and the steady-state brine relative permeability data are within -40% and +30% of the brine Corey-type fit. This gives us confidence to use this extended JBN method as a timesaving alternative method of determining two-phase relative permeability data.

[00148] By extending the JBN method to the interior of the core, it is possible to quickly obtain two-phase relative permeabilities using an unsteady-state flood. The additional experimental requirements as compared to the JBN method are in situ measurements of the saturation and pressure taps along the core. In cases where the end effect is large (in this case, the end effect is large due to the low viscosity of the invading phase), the relative permeabilities obtained with the extended JBN method match the steady-state measurements, but are considerably higher than those obtained from the regular JBN method. Hence, this JBN extension may be an additional useful relative permeability measurement technique, especially in the case where the invading fluid has a low viscosity, such as a gaseous or supercritical fluid.

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[00198] Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.