Patent Publication Number: US-2023152252-A1

Title: Method for measuring the spatial water permeability profile of porous media by using non-destructive nuclear magnetic resonance technique

Description:
BACKGROUND 
     Permeability is one of the most important properties of porous media. Permeability information of various materials is desired in geology. For example, the permeability of subsurface formations is crucial in predicting the potential of a reservoir (a hydrocarbon-bearing formation) to product hydrocarbons, like oil and gas. The reservoir configuration may vary in their structures. Some reservoirs, such as carbonate and sandstones, are highly heterogeneous in their pore structures, making predictions about permeability difficult. The continuous permeability profile information of these complicated reservoir configurations is becoming increasingly important for estimating the amount of hydrocarbons stored within. 
     Currently, the most developed technique used to measure and predict the spatial permeability profile of a reservoir is the probe permeameter technique. In this technique, the decrease in pressure on specific locations on a rock sample are measured with a probe to derive gas permeability. This technique is accepted as relatively robust for identifying general heterogeneity of rock samples, including quantifying thin beds, highly permeable beds, and permeability barriers. 
     SUMMARY 
     This Summary is provided to introduce a selection of concepts that are further described in the Detailed Description. This Summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter. 
     In one aspect, embodiments of the present disclosure relate to methods that include saturating the rock sample with an aqueous solution, performing T 2  NMR (nuclear magnetic resonance) on the saturated rock sample to detect spatial NMR data along a core sample axis, desaturating the rock sample, performing T 2  NMR on the desaturated rock sample to detect spatial NMR data along the core sample axis, determining the spatial cutoff data for the saturated and desaturated rock sample along a core sample axis, and analyzing the spatial NMR data to derive spatial permeability. 
     In another aspect, embodiments of the present disclosure relate to methods that include saturating the rock sample with an aqueous solution, performing T 2  NMR on the saturated rock sample to detect spatial NMR data along a core sample axis, performing T 2  NMR on the saturated rock sample to detect spatial NMR data along a second core sample axis, desaturating the rock sample, performing T 2  NMR on the desaturated rock sample to detect spatial NMR data along the core sample axis, performing T 2  NMR on the desaturated rock sample to detect spatial NMR data along a second core sample axis, determining the spatial cutoff data for the saturated and desaturated rock sample along a core sample axis, determining the spatial cutoff data for the saturated and desaturated rock sample along a second core sample axis, and analyzing the spatial NMR data to derive spatial permeability. 
     Other aspects and advantages will be apparent from the following Detailed Description and the appended Claims. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG.  1 A-C  illustrates varying porosity and permeabilities of the subsurface formations or rocks. 
         FIG.  2    illustrates a workflow according to one or more embodiments. 
         FIG.  3    illustrates the shape and structure of a rock sample according to one or more embodiments. 
         FIG.  4    illustrate a drilling site and an NMR facility according to one or more embodiments. 
         FIG.  5 A  illustrates a general step of the workflow according to one or more embodiments. 
         FIG.  5 B  illustrates a general step of the workflow according to one or more embodiments. 
         FIG.  5 C  illustrates a general step of the workflow according to one or more embodiments. 
         FIG.  5 D  illustrates a general step of the workflow to one or more embodiments. 
         FIG.  6    illustrates a graph of T 2  NMR distribution that may be used to calculate T 2  cutoff values according to one or more embodiments. 
         FIG.  7    illustrates a workflow according to one or more embodiments. 
         FIG.  8 A  illustrates detecting spatial permeability profiles along a first sample axis according to one or more embodiments. 
         FIG.  8 B  illustrates determining spatial permeability profiles along a first sample axis according to one or more embodiments. 
         FIG.  8 C  illustrates detecting spatial permeability profiles along a second sample axis according to one or more embodiments. 
         FIG.  8 D  illustrates determining spatial permeability profiles along a second sample axis according to one or more embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     The previously-described probe permeameter technique has several limitations. Some of the limitations are due to the technique itself. For example, it is difficult to make measurements if the rock samples have a rough or uneven surface. A tight seal is required between the probe and the rock sample surface. Also, this technique provides inaccurate measurements for rock samples with high pore connectivity and heterogeneity because measurements are made from one spot that is dependent of the size of the probe aperture. This technique is also incapable of specifically measuring water permeability. 
     The method uses nuclear magnetic resonance (NMR) to measure the spatial water permeability along a porous rock sample. The water permeability profile is obtained with a simple and robust workflow using NMR measurements, analysis, and subsequent predictions. Also, the NMR technique is nondestructive because NMR utilized electromagnetism to measures the fluids filled inside the pores of a rock sample. The shape, surface, or composition of a rock sample are not affected by and do not affect the accuracy of measurements. 
     Various illustrative embodiments of the disclosed subject matter are described. In the interest of clarity, not all features of an actual implementation are described in this specification. It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions must be made to achieve the specific goals of the developers, such as compliance with system-related and business-related constraints, which will vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming but may be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure. 
     The present subject matter will now be described with reference to the attached figures. Various structures, systems, and devices are schematically depicted in the Drawings for purposes of explanation only and to not obscure the present disclosure with details that are well known to those skilled in the art. Nevertheless, the attached Drawings are included to describe and explain illustrative examples of the present Detailed Disclosure. The words and phrases used should be understood and interpreted to have a meaning consistent with the understanding of those words and phrases by those skilled in the relevant art. No special definition of a term or phrase, that is, a definition that is different from the ordinary and customary meaning as understood by those skilled in the art, is intended to be implied by consistent usage of the term or phrase. To the extent that a term or phrase is intended to have a special meaning, that is, a meaning other than that understood by skilled artisans, such a special definition will be expressly set forth in the specification in a definitional manner that directly and unequivocally provides the special definition for the term or phrase. With reference to the attached figures, various illustrative embodiments of the systems, devices and method disclosed will now be described in more detail. 
       FIG.  1 A-C  illustrates varying porosity and permeabilities of the subsurface formations or rocks. Porosity and permeability are related properties of reservoir material; however, the differences between these properties are useful in understanding and predicting the ability of a reservoir to produce oil or gas. Generally, porosity is a measure of the void spaces inside materials. More specifically, the porosity of reservoir material is a measure of the ability of the reservoir material to hold a fluid. Mathematically, porosity may be defined as the open space in a rock divided by the total rock volume. Permeability is a measure of the ability of a reservoir material to transmit fluids. Here, permeability is the measure of the ability of a porous rock to transmit fluids through a reservoir material. The reservoir material may have varying degrees of porosity and permeability. 
       FIG.  1 A  is a diagram illustrating a rock  20  from a possible subsurface formation. The rock  20  is composed of many rock grains  11  that fit together. Each rock grain  11  is made of minerals and chemical compounds that may have different shapes and sizes. The rock grains  11  in the rock  20  are tightly joined together such that there are no voids between adjacent rock grains  11 . The absence of voids between adjacent rock grains  11  prevents fluids from being released by the rock  20  as well as from flowing between the rock grains  11  and the rock  20 . A subsurface formation composed of rocks without voids will have relatively low porosity and little permeability. 
       FIG.  1 B  is a diagram illustrating a rock  40  from a different possible subsurface formation. The rock  40  is composed of many rock grains  11  that fit together. The rock grains  11  in the rock  40  are joined; however, there are also voids between adjacent rock grains  11 . The voids between the rock grain  11  are known as pores  13 . Rocks with many pores are described to have high porosity. Even though the rock  40  has high porosity, the permeability may be low if the pores  13  are not inter-connected. If the pores are not connected, fluid may not be able to flow though the rock. 
       FIG.  1 C  is a diagram illustrating a rock  60  from another different possible subsurface formation. The rock  60  is composed of many rock grains  11  that fit together. The rock grains  11  in the rock  60  are joined, but there are also pores  13  between adjacent rock grains  11 . Further, the pores  13  between rock grains  11  are connected to other pores  13  by channels  15 . For example, a channel  15 A fluidly connects a pore  13 A to an adjacent pore  13 B. The presence of channels connecting pores allows fluid to flow though the rock, making the rock permeable. The presence of many channels connecting the pores may make a rock permeable. Fluids, such as hydrocarbons, may be able to flow though the rock and the corresponding formation for retrieval. 
       FIG.  2    illustrates a workflow. The steps may include: Providing a rock sample  101 ; saturating the rock sample with water  103 ; detecting spatial NMR data from the saturated rock sample by performing T 2  NMR measurements  105 ; desaturating the rock sample  107 ; detecting spatial NMR data from the desaturated rock sample by performing T 2  NMR measurements  109 ; determining the spatial cut-off data for both the saturated and desaturated samples  110 ; and analyzing the spatial NMR cut-off data to determine spatial permeability of the rock sample  111 . 
       FIG.  3    illustrates the shape and structure of a rock sample. A rock sample  301  may be provided during step  101  of method  200 . For example, the sample may be obtained during a drilling operation. A rock sample may be composed of many rock grains that fit and are cemented together through natural cementation processes. In doing so, a portion of the rock sample is porous, that is, it is comprised of voids defined by the surrounding rock grains, and a portion of the sample that is non-porous, better known as the rock matrix material comprising the rock grains and cementation material. The rock sample  301  may be carbonate or sandstone; however, those having skill in the art will appreciate that several different types of formations and formation rocks may be use in this workflow. 
     A person of ordinary skill in the art may envision that the provided rock sample may take any geometric or non-geometric form. As shown in  FIG.  3   , the rock sample  301  may be in a cylindrical configuration. Rock sample  301  is well-defined, having both a sample length  6000  and a sample diameter  6500 . Further, in one or more embodiments, the rock sample may be a cutting rock sample. Cutting rock samples, better known in the industry as simply “cuttings”, are fragments of a formation that are the resultant of drilling operation, such as by drilling by a drill bit, acting against the subsurface. 
     As also shown in  FIG.  3   , one or more scanning axis may be envisioned in relation to the configuration of the rock sample, which may be useful in detecting properties of the rock sample. For example, a first axis  7000  may be an imaginary line that is in alignment with the sample length  6000 , that is, running parallel with in this case with the height dimension of the cylindrical configuration of the rock sample  301 . Similarly, a second axis  7500  may be an imaginary line that is in alignment with the sample diameter  6500 , that is, running parallel with the width dimension of the cylindrical configuration of the rock sample. In one or more embodiments, where two or more axis are utilized, two or more axes may be aligned perpendicular with one another, such as in the case of rock sample  301  of  FIG.  3   , where first axis  7000  is perpendicular with second axis  7500 . 
       FIG.  4    depicts an example of a drilling site  310  and an NMR facility  321 . In some instances, providing a rock sample may include obtaining it from a formation. First a rock sample  301  may be obtained from a subsurface  307 . Interest may exist in determining the permeability of a particular formation  303 , such as a reservoir. The formation  303  may have complex and heterogenous porosity as are found in certain carbonate formations; therefore, analysis may need to be performed at the surface  305 . A rock sample may be removed from the formation  303  during a drilling operation, transported though a wellbore  314  and collected at the surface  305 . Once the rock sample is collected at the surface  305 , the rock sample  301  may be relocated to an NMR facility  321 . 
     In one or more embodiments, the rock sample may be washed and cleaned to remove impurities, such as mud and oil. In one or more embodiments, a solvent may be used to remove the impurities contained on the surface and in the pores of the rock sample. In one or more embodiments, a device, such as a centrifuge, may also be used to remove the impurities as well as the solvent. Impurities and solvents are removed because they may interfere with detecting NMR spectral data. Those skilled in the art will appreciate that various additional techniques that may be used to wash and clean the rock sample. 
     In one or more embodiments, an additional processing step may include modifying the shape and size of the rock sample. The resultant size and shape of the modified rock sample may depend on downstream apparatus limitations, such as the centrifuges and NMR equipment at an NMR facility, such as shown in  FIG.  4   . 
     As provided in the  FIG.  2    workflow  200 , after the rock sample  301  has been provided and, if necessary, cleaned, the rock sample is saturated in a water solution  103 . In one or more embodiments, brine may be used to saturate the rock sample instead of water. In one or more embodiments, the sample is placed in a water filled container and saturated under vacuum. To verify that a rock sample is fully saturated, the rock sample may be weighed multiple times until no change in the weight of the rock sample occurs. Even though water saturation is described in the workflow  200 , a person having skill in the art will appreciate that the rock sample  301  may be saturated with other fluids if permeability data of other fluids is desired instead of water. 
     At an NMR facility  321  of  FIG.  4   , several properties of the rock sample may be analyzed. The NMR facility  321  may be equipped with various tools and apparatus allowing additional processing and data collection and analysis of the rock sample  301 , including those apparatus required to carry out the steps described in the workflow  200 . Some of the apparatus housed in the NMR facility may include centrifuges  325  used to desaturate the rock samples, NMR equipment  323  used to obtain NMR data, and processors  327  to analyze the results. Those having skill in the art would appreciate that additional tools, apparatus, equipment, and chemical solutions may be required to complete the steps in workflow  200 . 
     As provided in the  FIG.  2    workflow  200 , after saturating the rock sample with water, spatial NMR spectroscopy of the rock sample is performed  105 . The primary functioning parts of an NMR system may include a magnet, gradient coils, RF (radio frequency) equipment, and computer processors. The magnet is used to create an external magnetic field that penetrates throughout the rock sample. The RF equipment may be used to transmit the RF pulse that induces the atoms to emit a signal, receive the emitted signal and amplify it so it can be manipulated by the computer processors. 
     In the NMR technique, a rock sample may be positioned in a magnetic field. In the presence of a magnetic field, B 0 , the magnetic nuclear spins of protons (hydrogens) align in the direction of the magnetic field. The applied magnetic field may be uniform or there may be a magnetic field gradient along a certain direction. The alignment of the magnetic nuclear spins may be perturbed by a weak oscillating magnetic field pulse. The weak oscillating magnetic field pulse that may be a radio-frequency pulse. The RF pulse, B 1 , induces movement in the transverse (x-y) plane. After the RF pulse, the magnetic nuclear spins of the protons realign in the direction of the magnetic field, B 0 , and the movement induced by the RF pulse in the transverse plane diminishes. Measurements may be made of the decaying signals of the nuclear spin of protons at various times in response to the magnetic field. The signal decaying time of the protons to completely dephase in the transverse plane may be referred to as the T 2  transverse relaxation time, and the amplitude of the signals may be detected. The amplitudes of the signal decaying time are used by the computer processors to generate an image. 
     Typically, the area to be imaged is scanned by a sequence of measurement cycles where the gradient varies according to the particular position measurement method being used. The resulting set of received NMR signals is digitized and processed to reconstruct the image using one of many well-known reconstruction techniques known to those having skill in the art. The measurements of amplitude versus time and the derived image may be used to determine the T 2  spatial distribution of the rock sample. The T 2  spatial distributions obtained from the NMR technique may be used to accurately predict various properties of the rock sample, including the porosity, pore-size, pore-fluid properties, and permeability of the rock sample and the corresponding geological formation. 
     In one or more embodiments, the method used to measure spatially-resolved T 2  (T 2  mapping) is Spin Echo Single Point Imaging (SE-SPI). Many conventional methods used for T 2  mapping may be problematic either due to local gradient distortions or short T 2 . In SE-SPI, spatial encoding precedes T 2  relaxation. The magnetization is phase encoded during the first pulse interval and then readout through multiple refocusing. To preserve the introduced phase shift upon refocusing, an XY-16 phase cycle is applied and a hard (full-excitation) 90-degree pulse is used. 
     Each of  FIGS.  5 A-D  illustrate a general step in method  200 ; however, a person having skill in the art will appreciate that certain steps may be modified or rearranged. 
       FIG.  5 A  illustrates a general step of the workflow.  FIG.  5 A  is a diagram of detected NMR measurements of the T 2  NMR distributions of a saturated rock sample that are spatially arranged according to their sequential position along a first core sample axis, such as received as the result of performing step  105  of method  200 . The T 2  NMR distribution of a saturated rock sample may be obtained by acquiring T 2  NMR scan measurements along a first core sample axis of the rock sample in a desired orientation. In  FIG.  5 A , four T 2  NMR distribution measurements are detected along a first core sample axis of a rock sample. The four T 2  NMR distribution measurements are represented as four T 2  NMR distribution curves  203 ,  205 ,  207 , and  209 , respectively, and are sequentially arranged based on their positions along a first core sample axis of a rock sample. A particular T 2  NMR distribution may be similar or different from other detected T 2  NMR distributions from a rock sample. Also, the detected T 2  NMR distribution curves may have different shapes and may have one or more peaks. For example, the detected T 2  NMR distribution curve  207  only has one peak whereas the detected T 2  NMR distribution curve  209  has two identifiable peaks. This may indicate that a first position along a first core sample axis for the rock sample, as shown by the detected T 2  NMR distribution curve  207 , has a different porosity and pore structure with respect to a second position along a first core sample axis for the rock sample, as shown by the detected T 2  NMR distribution curve  209 . The detected data from multiple T 2  NMR distribution curves may be utilized to determine the heterogeneity, complexity, porosity, and the pore structure along a first core sample axis of the rock sample. 
     The NMR technique allows T 2  NMR distributions to be detected for any desired orientation—it is not required to detect the T 2  NMR distributions along a normal plane to an axis as provided in the workflow. Therefore, the permeability profile may be obtained along any positional direction of a rock sample, such as by using acute or obtuse angles relative to the axis from which the scans are being performed. 
     In one or more embodiments, the measurements generated by the NMR technique may be logged by NMR logging systems and apparatus. Logged NMR measurements may be illustrated as a T 2  signal amplitude versus time to determine a T 2  spatial distribution as a function of their T 2  times (forthcoming). In one or more embodiments, identified pore systems through this process may be saved and used later for subsequent NMR logging of the same or different formations having the same rock type. 
     As provided in workflow  200  as shown in  FIG.  2   , the water is removed from the rock sample  301  by desaturation  107 . In one or more embodiments, a technique used is to desaturate the rock sample  301  is to spin the rock sample  301  in a centrifuge unit  325 . This causes a certain amount of fluid to drain from the rock sample  301  due to centrifugal forces. Generally, spinning the rock sample  301  in one direction produces a saturation distribution along the core of the rock sample  301 . To achieve uniform saturation distribution. The rock sample  301  may be spun in the centrifuge unit  325  at low speeds. In one or more embodiments, the rock sample  301  is spun for a second time in the opposite direction in the centrifugal unit  325 . The second spin may be performed at the same speed that was utilized the first time. Spinning the rock sample in the opposite direction the second time ensures that a uniform saturation distribution is obtained. 
     In one or more embodiments, the porous plate technique may be used to desaturate the rock sample. In such embodiments, a water wet porous plate is placed between the rock sample  301  and the downstream end of a rock sample holder. A desaturation pressure may be applied. Also, to ensure capillary continuity, water-saturated filter paper may be placed between the porous-plate and the rock sample  301 . Subsequently, a pressurized gas may be applied to the rock sample until no more water production is measured in a fluid receiver placed downstream. In this technique, only water is produced at the downstream end due to water-wet nature of the porous-plate. 
     As provided in the  FIG.  2    workflow  200 , after water is removed from the rock sample  301  by desaturation  107  NMR spectroscopy of the rock sample is performed  109  for a second time. The spatial T 2  NMR distribution of the desaturated rock sample  301  is detected by T 2  mapping NMR pulse sequence. The spatial T 2  measurements of the desaturated rock sample may be performed with the same or substantially the same vertical resolution that was used the first time to obtain spatial T 2  measurements for the saturated rock sample  301 . 
       FIG.  5 B  illustrates a general step of the workflow.  FIG.  5 B  is a diagram of detected NMR measurements of T 2  NMR distributions of a desaturated rock sample that are spatially arranged according to their sequential position along a first core sample axis, such as in step  109  of method  200 . The T 2  NMR distribution of the desaturated rock sample may be obtained by acquiring T 2  NMR measurements along the first core sample axis of the rock sample. In one or more embodiments, the same orientation and positions that were used for the saturated rock sample are used for the desaturated rock sample when making T 2  measurements along the first core sample axis. In  FIG.  5 B , four T 2  NMR distribution measurements are detected along the first core sample axis of the rock sample in the same first core sample axis positions that were scanned in step  105 . The four T 2  NMR distribution measurement positions are represented as four detected T 2  NMR distribution curves  223 ,  225 ,  227 , and  229 , respectively, and are sequentially arranged based on their positions along the first core sample axis of a rock sample. The detected four T 2  NMR distribution curves  223 ,  225 ,  227 ,  229  for the desaturated rock sample may be analyzed with the four corresponding detected T 2  NMR distribution curves  203 ,  205 ,  207 ,  209  from  FIG.  5 A  (step  105 ) for the saturated rock sample. The detected T 2  NMR distribution curves may have different shapes and may have one or more peaks or may have different detected curves in the saturated versus the desaturated state. For example, the T 2  NMR distribution curve  203  at a particular position has three peaks when the rock sample is saturated but only two peaks for the T 2  NMR distribution curve  223  of the desaturated sample at the same position along the first core sample axis. Also, the T 2  NMR distribution curves for the saturated and the corresponding desaturated rock sample positions may vary by sizes. For example, the T 2  NMR distribution curve  203  for a saturated rock sample is larger than the corresponding T 2  NMR distribution curve  223  for the desaturated sample at the same position along the first core sample axis. 
     Once T 2  NMR distributions have been detected by NMR spectroscopy of the saturated and desaturated rock sample  105 ,  109  the generated data may be analyzed. An analysis preformed for pore structure evaluation may be used to derive NMR T 2  cutoff values for the T 2  NMR distributions.  FIG.  6    shows a simple graph of a T 2  NMR distribution that plots amplitude over time  610 . The amplitude distribution may have one or more higher peaks and one or more lower peaks. The higher amplitude peaks correspond to larger pores and the lower amplitude peaks correspond to smaller pores. The smaller pores are less likely to contain free fluid and therefore less likely to produce oil and gas. The larger pores are more likely to contain free fluid (mobile fluid) and therefore more likely to produce oil and gas. Referring to  FIG.  6   , the T 2  cutoff  613  in a T 2  NMR distribution  615  is the T 2  value that divides the smaller pores from the larger pores. Thus, the T 2  NMR distribution of the curve above the T 2  cutoff is a measure of the free fluid in the rock  620 . The T 2  NMR distribution of the curve below the T 2  cutoff is a measure of the bound fluid  630 . Bound fluid may be made up of clay bound fluid and the capillary bound fluid. In one or more embodiments, reference T 2  cutoff values may be used along with the detected core data to derive T 2  cutoff values. For example, reference value for sandstones is approximately 33 ms (milliseconds) and approximately 90 ms to 100 ms for carbonates. 
       FIG.  5 C  illustrates a general step of the workflow.  FIG.  5 C  shows a series of analyses to determine spatial T 2  NMR permeability. This image is in respect to step  110  of method  200 . The determined T 2  cutoff values for each position along the core sample axis of the rock sample may be plotted in relative position to give spatial information about the porosity and pore structure along the core sample axis of the rock sample. In  FIG.  5 C , four determined T 2  cutoff values  243 ,  245 ,  247 , and  249 , respectfully, are plotted in a graph displaying positional T 2  cutoff values  202  along the first core sample axis. Each T 2  cutoff value is determined from data obtained from the detected saturated and desaturated T 2  NMR distribution curves at the same position along the core sample axis of the rock sample. For example, T 2  cutoff value  249  is determined from the corresponding detected saturated rock sample T 2  NMR distribution curve  209  from  FIG.  5 A  and detected desaturated rock sample T 2  NMR distribution curve  229  from  FIG.  5 B  at the same position along the core sample axis. The T 2  cutoff values in the rock samples along the core sample axis may differ significantly. For example, the T 2  cutoff values for position  245  and  247  along the core sample axis vary significantly, showing possible variations in porosity and pore structure in the rock sample. Rock samples from heterogenous formations may have varying T 2  values, which may possibly indicate complex pore structures and heterogeneity along the core sample axis of the rock sample and the formation from which the rock was obtained. 
     Spatial permeability of T 2  NMR distributions is calculated by NMR permeability models to produce permeability profile along the rock sample. The permeability of the rock sample  301  cannot be measured directly but can be derived by NMR data. In one or more embodiments, spatial permeability may be derived from permeability models that may utilize the porosity data obtained from NMR T 2  relaxation times to predict permeability. 
     In one or more embodiments, the permeability estimation model used to derive spatial permeability for NMT T 2  NMR distribution is the Timur-Coates model. 
     In one or more embodiments, the permeability estimation model used to derive spatial permeability is the geometric mean of relaxation time model (T2 gm). 
       FIG.  5 D  illustrates a general step of the workflow.  FIG.  5 D  shows a series of analyses to determine spatial T 2  NMR permeability. This image is in respect to step  111  of method  200 . The data collected from T 2  NMR may be used to predict the permeability of a rock sample. For example, the relaxation times, T 2  NMR distributions, T 2  cutoff values, and other NMR data previously determined may be used to determine permeability by using permeability models along the core sample axis for the rock sample. In  FIG.  5 D , four permeability prediction values  263 ,  265 ,  267 , and  269 , respectively, are plotted in a graph displaying positional permeabilities  204  to give a permeability profile along the core sample axis for the rock sample. Each particular permeability point is determined from data obtained from the T 2  cutoff, the saturated T 2  NMR distribution curve, and the desaturated T 2  NMR distribution curve at the same position along the core sample axis of the rock sample. For example, a permeability point  269  is determined from a corresponding T 2  cutoff value  249  from  FIG.  5 C , the corresponding saturated T 2  NMR distribution curve  209  From  FIG.  5 A , and the desaturated rock sample  229  from  FIG.  5 B  at the same position along the core sample axis of the rock sample. The permeability estimates along the length of a rock sample may vary significantly. For example, the sequential permeability predictions for position  265  and  267  along the core sample axis of the rock sample vary significantly, showing possible variations in the permeability across the rock sample along the core sample axis. Multiple permeability points may be mapped according to their positions to give the spatial permeability of the rock sample and the corresponding formation. As an example, in carbonate formations, the heterogeneity and complexity of pore systems may cause large variations in permeability of the rock, making accurate and detailed spatial predictions and modeling more important. 
       FIG.  7    illustrates a second workflow. The steps may include: Providing a rock sample  801 ; saturating the rock sample with water  803 ; detecting spatial NMR data from the saturated rock sample by performing T 2  NMR measurements along a first core sample axis  805 ; detecting spatial NMR data from the saturated rock sample by performing T 2  NMR measurements along a second core sample axis  807 ; desaturating the rock sample  809 ; detecting spatial NMR data from the desaturated rock sample by performing T 2  NMR measurements along a first core sample axis  811 ; detecting spatial NMR data from the desaturated rock sample by performing T 2  NMR measurements along a second core sample axis  813 ; determining the spatial cut-off data both the saturated and desaturated samples along a first core sample axis  815 ; determining the spatial cut-off data both the saturated and desaturated samples along a second core sample axis  817 ; and analyzing the spatial NMR cut-off data to determine spatial permeability of the rock sample  819 . 
     Each of  FIGS.  8 A-D  illustrate a general step in method  800 ; however, a person having skill in the art will appreciate that certain steps may be modified or rearranged. Method  800  utilizes multiple directional NMR scans to obtain NMR data in different positional directions along a plurality of axis of the rock sample to ultimately derive multiple different permeability profiles of a single sample according to one or more embodiments. 
       FIG.  8 A  illustrates a general step of the workflow.  FIG.  8 A  illustrates a series of T 2  NMR spatial distribution scans being conducted on a rock sample along a first core sample axis, in this case a length of the cylindrical rock sample  501 . A vertical magnetic field gradient (arrow) is applied to a rock sample  501 . The planes  516 ,  517 ,  518 , and  519 , respectively, depicted passing through the rock sample  501  show the positions where T 2  NMR spatial distribution scans are performed, and data is detected. In  FIG.  8 A , the T 2  NMR scans are performed sequentially along the first core sample axis of the rock sample. For example, plane  516  and plane  517  have a differential in distance along the first core sample axis vertically. 
     The demonstrated actions of  FIG.  8 A  are repeated twice. First, it is done for detecting spatial NMR data from the saturated rock sample by performing T 2  NMR measurements along a first core sample axis  805 . It is also done for detecting spatial NMR data from the desaturated rock sample by performing T 2  NMR measurements along a first core sample axis  811 . In between these two scans, the desaturating the rock sample  809 . 
       FIG.  8 B  illustrates a general step of the workflow.  FIG.  8 B  illustrates a permeability profile derived along the first core sample axis of rock sample  501 . After following the workflow for method  800 , the positional permeabilities are graphed  550  in a similar manner as provided for in  FIG.  5 D  with graph  204 . 
       FIG.  8 C  illustrates a general step of the workflow.  FIG.  8 C  illustrates a series of T 2  NMR spatial distribution scans being conducted on a rock sample along a second core sample axis, in this case a diameter of the cylindrical rock sample  501 . A horizontal magnetic field gradient (arrow) is applied to a rock sample  501 . The planes  536 ,  537 ,  538 , and  539 , respectively, depicted passing through the rock sample  501  show the positions where T 2  NMR spatial distribution scans are performed, and data is detected. In  FIG.  8 C , the T 2  NMR scans are performed sequentially along the second core sample axis of the rock sample. For example, square plane  536  and square plane  537  have a differential in distance along the second core sample axis vertically. 
     The demonstrated actions of  FIG.  8 C  are repeated twice. First, it is done for detecting spatial NMR data from the saturated rock sample by performing T 2  NMR measurements along a second core sample axis  807 . It is also done for detecting spatial NMR data from the desaturated rock sample by performing T 2  NMR measurements along a second core sample axis  813 . In between these two scans, the desaturating the rock sample  809 . 
       FIG.  8 D  illustrates a general step of the workflow.  FIG.  8 D  illustrates a permeability profile derived along the second core sample axis of rock sample  501 . After following the workflow for method  800 , the positional permeabilities are graphed  560  in a similar manner as provided for in  FIG.  5 D  with graph  204 , except in this case the positional permeabilities are graphed with respect to the second core sample axis, in this case horizontally. 
     Unless defined otherwise, all technical and scientific terms used have the same meaning as commonly understood by one of ordinary skill in the art to which these systems, apparatuses, methods, processes, and compositions belong. 
     The singular forms “a,” “an,” and “the” include plural referents, unless the context clearly dictates otherwise. 
     As used here and in the appended claims, the words “comprise,” “has,” and “include” and all grammatical variations thereof are each intended to have an open, non-limiting meaning that does not exclude additional elements or steps. 
     When the word “approximately” or “about” are used, this term may mean that there can be a variance in value of up to ±10%, of up to 5%, of up to 2%, of up to 1%, of up to 0.5%, of up to 0.1%, or up to 0.01%. 
     Ranges may be expressed as from about one particular value to about another particular value, inclusive. When such a range is expressed, it is to be understood that another embodiment is from the one particular value to the other particular value, along with all particular values and combinations thereof within the range. 
     While the disclosure includes a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments may be devised which do not depart from the scope of the present disclosure. Accordingly, the scope should be limited only by the attached claims.