Patent Publication Number: US-2023152483-A1

Title: Nuclear magnetic resonance based archie parameter determination

Description:
BACKGROUND 
     This disclosure generally relates to exploration and production of hydrocarbons in subsurface formations, and more particularly, to formation evaluation that includes determination of an Archie parameter (e.g., cementation exponent) that is based on nuclear magnetic resonance (NMR). 
     There are a number of approaches to perform formation evaluation of subsurface formations as part of hydrocarbon recovery operations from such formations. One approach includes determining an effective conductivity of a formation using Archie&#39;s law. Conventional approaches use electrical methods to measure an Archie cementation exponent, m, of core samples in a laboratory that have been brought to the surface. However, obtaining core samples from each drilled well to study in the laboratory can be expensive. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments of the disclosure may be better understood by referencing the accompanying drawings. 
         FIG.  1 A  depicts an example logging while drilling (LWD) system, according to some embodiments. 
         FIG.  1 B  depicts an example wireline system, according to some embodiments. 
         FIG.  2    depicts a flowchart of example operations for determining and using the Archie cementation exponent based on an NMR response from NMR logging, according to some embodiments. 
         FIG.  3    depicts an example Diffusion coefficient-transverse relaxation time (D−T 2 ) map generated based on the NMR response, according to some embodiments. 
         FIG.  4    depicts a flowchart of example operations for determining and using the Archie cementation exponent based on an NMR response using laser scanning confocal microscopy (LSCM), according to some embodiments. 
         FIG.  5    depicts a flowchart of example operations for determining and using the Archie cementation exponent based on an NMR response using micro-Computed Topography (u-CT) to determine pore size distribution of the formation, according to some embodiments. 
         FIG.  6    depicts an example graph of showing that the roughness corrected BET surface relaxivity and the μ-CT surface relaxivity provide similar results, according to some embodiments. 
         FIG.  7    depicts an example computer, according to some embodiments. 
     
    
    
     DESCRIPTION 
     The description that follows includes example systems, methods, techniques, and program flows that embody aspects of the disclosure. However, it is understood that this disclosure may be practiced without these specific details. For instance, well-known instruction instances, protocols, structures and techniques have not been shown in detail in order not to obfuscate the description. 
     Example embodiments relate to formation evaluation that can include determining an electrical conductivity of a subsurface formation based on the porosity of the rock, water saturation, and conductivity of the formation. For example, a model known as Archie&#39;s equation (also known as Archie&#39;s relation or Archie&#39;s equation) can be used to make this determination. 
     Rock resistivity of formations can be used to estimate hydrocarbon reserves in such formation. The rock resistivity can be used in conjunction with other measurements to estimate the porosity and the water saturation of the formation. For brine-saturated rocks, the solid matrix can be insulating and the fluid conductive. A model known as Archie&#39;s equation (also known as Archie&#39;s relation or Archie&#39;s equation) can provide an empirical correlation between resistivity and porosity of the formation, as set forth below in Equation (1). One parameter of Archie&#39;s equation is a cementation exponent (m). Example embodiments can determine the cementation exponent from Nuclear Magnetic Resonance (NMR) response data. For example, the cementation exponent can be determined from diffusion coefficient and transverse relaxation time (D−T2) data. 
     In some embodiments, the surface relaxivity, p, of the formation is also used as part of determining the cementation exponent. The surface relaxivity for a specific rock type can be available from previous studies, where is calibrated for a rock type. The surface relaxivity for a specific rock type can be measured in the laboratory on core plug samples or can be measured in the laboratory on drill cuttings. Thus, in contrast to conventional approaches, example embodiments can determine cementation exponent (m) from NMR response data. Also, if not readily known, the surface relaxivity can be measured from drill cuttings, instead of using core plugs. 
     Also, example embodiments can be applied in various configurations and applications. For example, some embodiments can be applied to NMR diffusion maps from log measurements. Thus, example embodiments can allow for monitoring the changes in the cementation exponent directly from logs. The correct values of the cementation exponent can be used in various types of formation evaluation. For example, these values can be used for interpretation of water saturation, porosity, etc. from resistivity logs. Additionally, some embodiments can be used in core analysis, in the laboratory, etc. 
     Example Systems 
     Two example systems to determine the Archie cementation exponent using NMR responses are now described.  FIG.  1 A  depicts an example logging while drilling (LWD) system, according to some embodiments. A drilling platform  102  supports a derrick  104  having a traveling block  106  for raising and lowering a drill string  108 . A kelly  110  supports the drill string  108  as it is lowered through a rotary table  112 . A drill bit  114  is driven by a downhole motor and/or rotation of the drill string  108 . As the drill bit  114  rotates, it creates a wellbore  116  that passes through various formations  118 . A pump  120  circulates drilling fluid through a feed pipe  122  to the kelly  110 , downhole through the interior of the drill string  108 , through orifices in the drill bit  114 , back to the surface via the annulus around the drill string  108 , and into a retention pit  124 . The drilling fluid transports cuttings from the borehole into the retention pit  124  and aids in maintaining the borehole integrity. 
     An NMR logging tool  126  can be integrated into the bottom-hole assembly near the drill bit  114 . As the drill bit  114  extends the wellbore  116  through the formations  118 , the bottom-hole assembly collects NMR measurements relating to spin relaxation time (e.g., T1, T2, etc.) distributions, as well as various other formation properties and information regarding tool orientation and various other drilling conditions. The NMR logging tool  126  may take the form of a drill collar (i.e., a thick-walled tubular that provides weight and rigidity to aid the drilling process). The NMR logging tool  126  can also include one or more navigational packages for determining the position, inclination angle, horizontal angle, and rotational angle of the tool. Such navigational packages can include, for example, accelerometers, magnetometers, and/or sensors. 
     For purposes of communication, a downhole telemetry sub  128  can be included in the bottom-hole assembly to transfer measurement data to a surface receiver  130  and to receive commands from the surface. Mud pulse telemetry is one common telemetry technique for transferring tool measurements to surface receivers and receiving commands from the surface, but other telemetry techniques can also be used. In some embodiments, the telemetry sub  128  can store logging data for later retrieval at the surface when the logging assembly is recovered. 
     At the surface, the surface receiver  130  can receive the uplink signal from the downhole telemetry sub  128  and can communicate the signal to a data acquisition module  132 . The data acquisition module  132  can include one or more processors, storage mediums, input devices, output devices, software, etc. The data acquisition module  132  can collect, store, and/or process the data received from the NMR logging tool  126  to determine characteristics (e.g., porosity, pore size distribution, permeability, hydrocarbon saturation, etc.) of the formations  118  (as further described herein). For example, the Archie cementation exponent using NMR diffusion can be determined and then used to determine various formation characteristics. 
     At various times during the drilling process, the drill string  108  may be removed from the borehole as shown in  FIG.  1 B . In particular,  FIG.  1 B  depicts an example wireline system, according to some embodiments. 
     Once the drill string has been removed, logging operations can be conducted using a wireline logging tool  134  (i.e., a sensing instrument sonde suspended by a cable  142  having conductors for transporting power to the tool and telemetry from the tool to the surface). The wireline logging tool  134  may have pads and/or centralizing springs to maintain the tool near the central axis of the borehole or to bias the tool towards the borehole wall as the tool is moved downhole or uphole. The wireline logging tool  34  can also include one or more navigational packages for determining the position, inclination angle, horizontal angle, and rotational angle of the tool. Such navigational packages can include, for example, accelerometers, magnetometers, and/or sensors. In some embodiments, a surface measurement system (not shown) can be used to determine the depth of the wireline logging tool  134 . 
     As explained further below, the wireline logging tool  134  can include an NMR logging instrument that collects NMR measurements associated with the formations  118  within the wellbore  116 . A logging facility  144  includes a computer, such as those described with reference to  FIG.  7   , for collecting, storing, and/or processing the measurements gathered by the wireline logging tool  134  (e.g., to determine characteristics such as porosity, pore size distribution, permeability, and/or hydrocarbon saturation of the formations  118 ). 
     Although  FIGS.  1 A and  1 B  depict specific borehole configurations, it should be understood by those skilled in the art that the present disclosure is equally well suited for use in wellbores having other orientations including vertical wellbores, horizontal wellbores, slanted wellbores, multilateral wellbores and the like. Also, even though  FIGS.  1 A and  1 B  depict an onshore operation, it should be understood by those skilled in the art that the present disclosure is equally well suited for use in offshore operations. Moreover, it should be understood by those skilled in the art that the present disclosure is not limited to the environments depicted in  FIGS.  1 A and  1 B , and can also be used, for example, in other well operations such as non-conductive production tubing operations, jointed tubing operations, coiled tubing operations, combinations thereof, and the like. 
     Cementation Exponent (m) 
     Rock resistivity for a formation can be used to estimate or determine the amount of hydrocarbons in the formation. The rock resistivity can used in conjunction with other measurements to estimate the porosity and the water saturation of the formation. For brine-saturated rocks, the solid matrix is insulating, and the fluid is conductive. The empirical correlation between resistivity and porosity is known as the Archie&#39;s relation and is set forth in Equation (1) below: 
     
       
         
           
             
               
                 
                   
                     R 
                     0 
                   
                   = 
                   
                     
                       F 
                       ⁢ 
                       
                         R 
                         w 
                       
                     
                     ≈ 
                     
                       
                         1 
                         
                           ϕ 
                           m 
                         
                       
                       ⁢ 
                       
                         R 
                         w 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where Ro is the resistivity of the fully brine saturated rock, R w  is the resistivity of brine, F is the formation factor, ϕ is the porosity and m is Archie cementation exponent. For example for sandstone, it was determined that the Archie cementation exponent is m≈2. However, the Archie cementation exponent needs to be determined for each particular field or formation being evaluated. The cementation exponent is closely connected to the rock properties which are not universal. 
     The cementation exponent can be calculated by Equation (2): 
     
       
         
           
             
               
                 
                   m 
                   = 
                   
                     
                       log 
                       ⁢ 
                       
                         
                           R 
                           w 
                         
                         
                           R 
                           0 
                         
                       
                     
                     
                       log 
                       ⁢ 
                       ϕ 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The variations in m can be large enough to affect the interpretation of water saturation from resistivity logs which can result in using values of m which are not correct in the Archie&#39;s equation. To illustrate, water saturation (S w   n ) can be calculated using Equation (3): 
     
       
         
           
             
               
                 
                   
                     S 
                     w 
                     n 
                   
                   = 
                   
                     
                       a 
                       
                         ϕ 
                         m 
                       
                     
                     ⁢ 
                     
                       
                         R 
                         w 
                       
                       
                         R 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where n is the saturation exponent, a≈1 a constant, and R t  is the formation resistivity. The variations in m are large enough to affect the interpretation of water saturation from resistivity logs. Thus, the interpretation of water saturation can be incorrect if the value of the cementation exponent, m, which are not correct in the Archie&#39;s equation. 
     Example embodiments can include a new approach for accurately estimating or determining m. In some embodiments, this approach can include estimating m from an NMR response and surface relaxivity. For example, m can be estimated or determined from D−T 2  maps generated from an NMR response. For instance, fully brine saturated cores can be used for estimating the cementation exponent, m, by using a truncated singular value decomposition (SVD) with a regularization parameter. 
     In some embodiments, at short diffusion times, the diffusion coefficient D(t) can deviate from the bulk molecular diffusion coefficient of the fluid, D 0 , by a term proportional to the surface-to-volume ratio, S/V p . This is known as restricted diffusion. At long diffusion times, when the diffusion length is large compared with the heterogeneity length-scale of the pore space, the spins can probe the connectivity of the pore space and the diffusion coefficient can approach the diffusion coefficient in the tortuosity limit given by Equation (4): 
         τ=Fϕ=ϕ   1-m   (4)
 
     As the diffusion coefficient D(t) tends to approach D ∞ , the diffusion coefficient in the tortuosity limit is given by Equation (5): 
     
       
         
           
             
               
                 
                   
                     
                       D 
                       ⁡ 
                       ( 
                       t 
                       ) 
                     
                     → 
                     
                       D 
                       ∞ 
                     
                   
                   = 
                   
                     
                       D 
                       o 
                     
                     τ 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     An expression for D(t) that can interpolate between the short-time behavior and the long-time asymptotic value of 1/τ. The time-dependent D(t) for short times and long times can be connected using the Padé approximation. The D−T 2  based cementation exponent, m, based on D−T2 can then be determined based on Equation (6): 
     
       
         
           
             
               
                 
                   
                     
                       D 
                       ⁡ 
                       ( 
                       
                         T 
                         
                           2 
                           ⁢ 
                           S 
                         
                       
                       ) 
                     
                     
                       D 
                       0 
                     
                   
                   = 
                   
                     1 
                     - 
                     
                       γ 
                       ⁢ 
                       
                         
                           α 
                           ⁢ 
                           
                             L 
                             D 
                           
                         
                         
                           
                             α 
                             ⁢ 
                             
                               L 
                               D 
                             
                           
                           + 
                           γ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     where Equations (7)-(9) define the following variables of Equation (6): 
     
       
         
           
             
               
                 
                   γ 
                   = 
                   
                     
                       1 
                       - 
                       
                         
                           D 
                           ∞ 
                         
                         
                           D 
                           0 
                         
                       
                     
                     = 
                     
                       
                         1 
                         - 
                         
                           1 
                           τ 
                         
                       
                       = 
                       
                         1 
                         - 
                         
                           ϕ 
                           
                             m 
                             - 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     L 
                     D 
                   
                   = 
                   
                     
                       
                         D 
                         0 
                       
                       ⁢ 
                       
                         t 
                         D 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       4 
                       
                         9 
                         ⁢ 
                         
                           π 
                         
                       
                     
                     ⁢ 
                     
                       1 
                       
                         
                           T 
                           
                             2 
                             ⁢ 
                             S 
                           
                         
                         ⁢ 
                         
                           ρ 
                           
                             D 
                             ⁢ 
                             
                               T 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where D 0  is the bulk diffusion coefficient of the fluid, D ∞  is D(t) in the tortuosity limit, and t D  the diffusion time, i.e., the time the molecules are allowed to diffuse. The cementation exponent, m, can then be estimated from the Padé D−T 2  phenomenological fit to restricted-diffusion versus T2 (Equation 6), when the surface relaxivity and porosity are known. In some embodiments, this determination can be made based on a BET (Brunauer, Emmett, and Teller) surface area measurement using gas (such as nitrogen) as the adsorbate. However, when the pore grain surface is rough, the specific surface area (surface area per unit sample weight) of rough surfaces would be larger than that of smooth surfaces and therefore the surface relaxivity value from BET, ρ BET , can be underestimated. 
     In a first example, in a brine saturated rock with ϕ=23.4 p.u., D 0 =2.6×10 −9  m 2 /s at 30° C., the diffusion time t D =50 ms and surface relaxivity, ρ=29.84 μm/s, the Padé D−T 2  fit line that gives the best fit to the D−T 2  distribution (i.e., that goes through the peak of the distribution) is for m=2 (Error! Reference source not found.). 
     Surface Relaxivity 
     Different examples of measuring surface relaxivity of a formation are now described. Depending on the method, the surface relaxivity can vary by orders of magnitude. One example for determining NMR surface relaxivity is based on the classical BET (Brunauer, Emmett, and Teller) surface area measurement using nitrogen gas as the adsorbate. 
     However, when the pore grain surface is rough, the specific surface area (surface area per unit sample weight) of rough surfaces would be larger than that of smooth surfaces and therefore the surface relaxivity value from BET, ρ BET , is underestimated. The ρ BET  value is mineralogy and surface chemistry dependent, it. Also, the ρ BET  value parameterizes the strength of the interaction between fluid nuclear spins and the smooth pore wall. To extract correct data regarding hydrocarbons from the data, the effect of surface roughness should also be considered. 
     In some implementations, a roughness corrected surface relaxivity is the surface relaxivity from BET surface measurement multiplied by a factor that depends on the surface roughness, R, given by Equation (10): 
       ρ LSCM =ρ BET (1+2 R )   (10)
 
     wherein R is the surface roughness, which can be calculated based on the ratio of rough surfaces to that of smooth surfaces, given by Equation (11): 
         R=S   true   /S   geometric    (11)
 
     In some embodiments, R can be measured from a laser Confocal Scanning Microscope (LSCM). 
     A second example for determining surface relaxivity is based on gas adsorption and BET. For example, surface relaxivity can be determined using the Brunauer, Emmett, and Teller (BET) surface area measurement using nitrogen gas as the adsorbate. For instance, a BET relation for surface area measurements can be given by Equation (12): 
     
       
         
           
             
               
                 
                   
                     1 
                     
                       W 
                       ⁡ 
                       ( 
                       
                         
                           ( 
                           
                             
                               P 
                               0 
                             
                             P 
                           
                           ) 
                         
                         - 
                         1 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           W 
                           m 
                         
                         ⁢ 
                         C 
                       
                     
                     + 
                     
                       
                         
                           C 
                           - 
                           1 
                         
                         
                           
                             W 
                             m 
                           
                           ⁢ 
                           C 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           P 
                           
                             P 
                             0 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where W is the weight of gas adsorbed, P/P 0  is the relative pressure, W m  is the weight of adsorbate as monolayer, and C is the BET constant. The specific surface area (S g ) can be given (in units of m 2 /g) by Equation (13): 
     
       
         
           
             
               
                 
                   
                     S 
                     g 
                   
                   = 
                   
                     
                       
                         W 
                         m 
                       
                       ⁢ 
                       
                         N 
                         A 
                       
                       ⁢ 
                       
                         A 
                         
                           C 
                           ⁢ 
                           S 
                         
                       
                     
                     
                       
                         M 
                         g 
                       
                       ⁢ 
                       M 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     where N A  is the Avogadro&#39;s number (6.0323×10 23 ), M is the molecular weight of adsorbate, A cs  is the adsorbate cross sectional area (16.2 Å 2  for N 2 ), and M g  is the sample weight. The surface to pore-volume 
     
       
         
           
             ( 
             
               S 
               
                 V 
                 p 
               
             
             ) 
           
         
       
     
     is then given by Equation (14): 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       V 
                       p 
                     
                   
                   = 
                   
                     
                       
                         
                           1 
                           - 
                           ϕ 
                         
                         ϕ 
                       
                       ⁢ 
                       
                         S 
                         
                           V 
                           g 
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           ϕ 
                         
                         ϕ 
                       
                       ⁢ 
                       
                         ρ 
                         g 
                       
                       ⁢ 
                       
                         S 
                         g 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     where ρ g  is the grain density. 
     The surface relaxivity (ρ BET ) for T 2  can then calculated from Equation (1) using Equation (15): 
     
       
         
           
             
               
                 
                   
                     ρ 
                     BET 
                   
                   = 
                   
                     
                       1 
                       
                         T 
                         
                           2 
                           , 
                           LM 
                         
                       
                     
                     ⁢ 
                     
                       
                         V 
                         p 
                       
                       S 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     where T 2LM  is the log-mean of the T 2  distribution. Alternatively or in addition, the mean of the distribution in rate, R 2M , (also given by the initial slope of the T 2  decay) can be used in Equation (15). In some embodiments, T 2LM  is used since the analysis of the D−T 2  measurements below are also done on a log scale. In some embodiments, T 2peak  is used. The surface relaxivity can then be corrected for surface roughness using Equation (10) shown above. 
     A third example for determining surface relaxivity is based on μ-CT (pore size distribution). For example, the surface relaxivity based on μ-CT can be determined using Equation (16): 
     
       
         
           
             
               
                 
                   
                     ρ 
                     
                       C 
                       ⁢ 
                       T 
                     
                   
                   = 
                   
                     d 
                     
                       6 
                       ⁢ 
                       
                         T 
                         
                           2 
                           , 
                           LM 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     where d is the mean pore size from μ-CT 
     In some embodiments, to extract correct hydrocarbon information of the subsurface formation from the NMR response, the effect of surface roughness is considered. Therefore, ρ LSCM  or ρ CT  can be used to determine m (instead of BET, which has values that can underestimated). 
     Example Operations 
     Example operations for determining and using the Archie cementation exponent using NMR responses are now described.  FIG.  2    depicts a flowchart of example operations for determining and using the Archie cementation exponent based on an NMR response from NMR logging data, according to some embodiments.  FIG.  2    depicts a flowchart  200  can be performed by software, firmware, hardware or a combination thereof. Such operations are described with reference to the systems of  FIGS.  1 A- 1 B . However, such operations can be performed by other systems or components. For example, at least some of the operations of the flowchart  200  are described as being performed by a computer at a surface of the wellbore. In some embodiments, one or more of these operations can be performed by a computer downhole in the wellbore. For example, the NMR logging tool  126  or the wireline logging tool  134  can include a computer to perform one or more of these operations. The operations of the flowchart  200  start at block  202 . 
     At block  202 , an NMR logging tool is positioned in a wellbore formed in a subsurface formation. For example, with reference to  FIGS.  1 A- 1 B , the NMR logging tool  126  or the wireline logging tool  134  can be positioned in the wellbore  116  where NMR logging is to be performed. 
     At block  204 , an NMR response is detected by the NMR logging tool and from the subsurface formation. The NMR response can be derived from emission of a radio frequency (RF) magnetic field into the subsurface formation. For example, with reference to  FIGS.  1 A- 1 B , the NMR logging tool  126  or the wireline logging tool  134  can emit the RF magnetic field into the subsurface formation. 
     At block  206 , a Diffusion coefficient-transverse relaxation time (D−T 2 ) map is generated based on the NMR response. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can generate the D−T 2  map. To illustrate,  FIG.  3    depicts an example diffusion coefficient-transverse relaxation time (D−T 2 ) map generated based on the NMR response, according to some embodiments. A D−T 2  map  300  is an example map for a subsurface formation sample having a surface relaxivity (ρ) of 29.84 micrometers (μm)/second (s) and is saturated with brine. The D−T 2  map  300  includes an x-axis  302  that is the transverse relaxation time (T 2 ). The D−T 2  map  300  includes a y-axis  304  (the diffusion coefficient (D)/the bulk diffusion coefficient (D 0 )). The sample porosity is ϕ=23.4 p.u. The bulk diffusion coefficient of brine is D 0 =2.6×10 −9  m 2 /s at 30° C. The diffusion time t D =50 ms. A line  306  is the D/D 0  of bulk water−10 0 . A line  308  is the T2 of bulk water−10 0.5 . The D−T 2  map  300  also includes a Padé D−T 2  fit line  310  that passes through a peak  312 . Selection of the Padé D−T 2  fit  310  is further described below in the description of the operation at block  210 . Returning to  FIG.  2   , operations of the flowchart  200  continue at block  208 . 
     At block  208 , NMR surface relaxivity of the subsurface formation is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. Different examples of determining surface relaxivity of the subsurface formation are described above. 
     At block  210 , a Padé D−T 2  phenomenological fit to restricted-diffusion versus T2 is determined in the D−T 2  map that is nearest a peak of distribution of the NMR response. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. With reference to  FIG.  3   , the Padé D−T 2  fit is the line  310  that intersects the peak of distribution ( 312 ). In some embodiments, the Padé D−T 2  fit is created based on Equation (6) set forth above. In Equation (6), all the terms can be known except the cementation exponent, m. Thus, cementation exponent, m, can be set for Equation (6) such that the line intersects or is nearest the peak  312 . The peak  312  can be defined as the maximum intensity of the signal amplitude of the NMR response. The Padé D−T 2  fit line  310  that passes through the peak  312  can be selected because it is considered the best fit to the D−T 2  distribution (i.e., that goes through a peak  312  of the distribution). In this example, the value of m for the best fit line  310  is 2.02 μm/s. So the value of m can be 2. 
     At block  212 , the Archie cementation exponent (m) for the Padé D−T 2  fit line in the D−T 2  map and the surface relaxivity is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. As described above, Equation (6) can used to determine the Archie cementation exponent (m) for the Padé D−T 2  fit such that the fit line intersects or is nearest the peak of the distribution. 
     At block  214 , formation evaluation based on the Archie cementation exponent is performed. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make perform the formation evaluation. For instance, the computer can determine porosity, permeability, saturation, etc. based on Archie&#39;s law and the Archie cementation exponent. 
     At block  216 , a wellbore operation is adjusted based on the formation evaluation. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this operation. For instance, drilling operations (such as direction, speed of the drill bit, weight on bit, etc.) can be adjusted based on the formation evaluation. Other wellbore operations can also be adjusted. For example, post drilling operations (such as production-related operations) can be adjusted based on the formation evaluation. For example, different parameters of a fracking operation can be adjusted based on the formation evaluation. Operations of the flowchart  200  are complete. 
     A second example of operations for determining and using the Archie cementation exponent based on an NMR response is now described. In particular,  FIG.  4    depicts a flowchart of example operations for determining and using the Archie cementation exponent based on an NMR response using laser scanning confocal microscopy (LSCM), according to some embodiments.  FIG.  4    depicts a flowchart  400  can be performed by software, firmware, hardware or a combination thereof. Such operations are described with reference to the systems of  FIGS.  1 A- 1 B . However, such operations can be performed by other systems or components. For example, at least some of the operations of the flowchart  400  are described as being performed by a computer at a surface of the wellbore. In some embodiments, one or more of these operations can be performed by a computer downhole in the wellbore. For example, the NMR logging tool  126  or the wireline logging tool  134  can include a computer to perform one or more of these operations. The operations of the flowchart  400  start at block  402 . 
     At block  402 , an NMR logging tool is positioned in a wellbore formed in a subsurface formation. For example, with reference to  FIGS.  1 A- 1 B , the NMR logging tool  126  or the wireline logging tool  134  can be positioned in the wellbore  116  where NMR logging is to be performed. 
     At block  404 , an NMR response is detected by the NMR logging tool and from the subsurface formation. The NMR response can be derived from emission of a radio frequency (RF) magnetic field into the subsurface formation. For example, with reference to  FIGS.  1 A- 1 B , the NMR logging tool  126  or the wireline logging tool  134  can emit the RF magnetic field into the subsurface formation. 
     At block  406 , a D−T 2  map is generated based on the NMR response. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can generate the D−T 2  map (as described above in reference to  FIG.  3   ). 
     At block  408 , surface area with BET gas adsorption is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this operation. For instance, the surface area with BET gas adsorption can be determined using Equations (12)-(14) described in detail above. 
     At block  410 , ρ BET  is calculated based on the surface area with BET gas adsorption. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this calculation. For instance, ABET can be calculated based on the BET gas adsorption using Equation (15) described in detail above. 
     At block  412 , a surface roughness of the subsurface formation (R) is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. For instance, the surface roughness can be determined based on the type of the rock of the subsurface formation being processed. In some embodiments, the surface roughness can be determined based on the type of rock by accessing such data from a database, library, etc. that includes a surface roughness for given types of rock. 
     At block  414 , a roughness corrected surface relaxivity based on the surface roughness (R) is calculated. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this calculation. For instance, roughness corrected surface relaxivity (ρ LSCM ) can be calculated based on based on the surface roughness (R) using Equation (16) described in detail above. 
     At block  416 , a Padé D−T 2  fit line is determined in the D−T 2  map (based on the roughness corrected surface relaxivity) that is nearest a peak of distribution of the NMR response. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. With reference to  FIG.  3   , the Padé D−T 2  fit is the line  310  that intersects the peak of distribution ( 312 ). In some embodiments, the Padé D−T 2  fit is created based on Equation (6) set forth above. In Equation (6), all the terms can be known except the cementation exponent, m. Thus, cementation exponent, m, can be set for Equation (6) such that the line intersects or is nearest the peak  312 . The peak  312  can be defined as the maximum intensity of the signal amplitude of the NMR response. The Padé D−T 2  fit line  310  that passes through the peak  312  can be selected because it is considered the best fit to the D−T 2  distribution (i.e., that goes through a peak  312  of the distribution). In this example, the value of m for the fit line  310  is 2.02 μm/s. So the value of m can be 2. 
     At block  418 , the Archie cementation exponent (m) for the Padé D−T 2  fit line in the D−T 2  map and the surface relaxivity is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. As described above, Equation (6) can used to determine the Archie cementation exponent (m) for the Padé D−T 2  fit such that the line intersects or is nearest the peak of the distribution. 
     At block  420 , formation evaluation based on the Archie cementation exponent is performed. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make perform the formation evaluation. For instance, the computer can determine porosity, permeability, saturation, etc. based on Archie&#39;s law and the Archie cementation exponent. 
     At block  422 , a wellbore operation is adjusted based on the formation evaluation. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this operation. For instance, drilling operations (such as direction, speed of the drill bit, weight on bit, etc.) can be adjusted based on the formation evaluation. Other wellbore operations can also be adjusted. For example, post drilling operations (such as production-related operations) can be adjusted based on the formation evaluation. For example, different parameters of a fracking operation can be adjusted based on the formation evaluation. Operations of the flowchart  400  are complete. 
     A third example of operations for determining and using the Archie cementation exponent based on an NMR response is now described. In particular,  FIG.  5    depicts a flowchart of example operations for determining and using the Archie cementation exponent based on an NMR response using micro-Computed Topography (μ-CT) to determine pore size distribution of the formation, according to some embodiments.  FIG.  5    depicts a flowchart  500  can be performed by software, firmware, hardware or a combination thereof. Such operations are described with reference to the systems of  FIGS.  1 A- 1 B . However, such operations can be performed by other systems or components. For example, at least some of the operations of the flowchart  500  are described as being performed by a computer at a surface of the wellbore. In some embodiments, one or more of these operations can be performed by a computer downhole in the wellbore. For example, the NMR logging tool  126  or the wireline logging tool  134  can include a computer to perform one or more of these operations. The operations of the flowchart  500  start at block  502 . 
     At block  502 , an NMR logging tool is positioned in a wellbore formed in a subsurface formation. For example, with reference to  FIGS.  1 A- 1 B , the NMR logging tool  126  or the wireline logging tool  134  can be positioned in the wellbore  116  where NMR logging is to be performed. 
     At block  504 , an NMR response is detected by the NMR logging tool and from the subsurface formation. The NMR response can be derived from emission of a radio frequency (RF) magnetic field into the subsurface formation. For example, with reference to  FIGS.  1 A- 1 B , the NMR logging tool  126  or the wireline logging tool  134  can emit the RF magnetic field into the subsurface formation. 
     At block  506 , a D−T 2  map is generated based on the NMR response. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can generate the D−T 2  map (as described above in reference to  FIG.  3   ). 
     At block  508 , pore size distribution is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this operation. As described above, assuming spherical pores, the surface relaxivity can be determined from the from μ-CT. 
     At block  510 , surface relaxivity is calculated from the pore size distribution and the average T 2 . For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this operation. For instance, the surface relaxivity can be calculated from the pore size distribution using Equation (16) described in detail above. 
     At block  512 , a Padé D−T 2  fit line is determined in the D−T 2  map that is nearest a peak of distribution of the NMR response. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. With reference to  FIG.  3   , the Padé D−T 2  fit is the line  310  that intersects the peak of distribution ( 312 ). In some embodiments, the Padé D−T 2  fit line is created based on Equation (6) set forth above. In Equation (6), all the terms can be known except the cementation exponent, m. Thus, cementation exponent, m, can be set for Equation (6) such that the line intersects or is nearest the peak  312 . The peak  312  can be defined as the maximum intensity of the signal amplitude of the NMR response. The Padé D−T 2  fit line  310  that passes through the peak  312  can be selected because it is considered the best fit to the D−T 2  distribution (i.e., that goes through a peak  312  of the distribution). In this example, the value of m for the best fit line  310  is 2.02 μm/s. So the value of m can be 2. 
     At block  514 , the Archie cementation exponent (m) for the Padé D−T 2  fit line in the D−T 2  map and the surface relaxivity is determined. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make this determination. As described above, Equation (6) can used to determine the Archie cementation exponent (m) for the Padé D−T 2  fit such that the fit line intersects or is nearest the peak of the distribution. 
     At block  516 , formation evaluation based on the Archie cementation exponent is performed. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can make perform the formation evaluation. For instance, the computer can determine porosity, permeability, saturation, etc. based on Archie&#39;s law and the Archie cementation exponent. 
     At block  518 , a wellbore operation is adjusted based on the formation evaluation. For example, with reference to  FIGS.  1 A- 1 B , a computer at the surface of the wellbore  116  can perform this operation. For instance, drilling operations (such as direction, speed of the drill bit, weight on bit, etc.) can be adjusted based on the formation evaluation. Other wellbore operations can also be adjusted. For example, post drilling operations (such as production-related operations) can be adjusted based on the formation evaluation. For example, different parameters of a fracking operation can be adjusted based on the formation evaluation. Operations of the flowchart  500  are complete. 
     As described, there are multiple approaches for determining surface relaxivity. The three approaches described herein provide similar results. To illustrate,  FIG.  6    depicts an example graph of showing that the roughness corrected BET surface relaxivity and the p-CT surface relaxivity provide similar results, according to some embodiments. A graph  600  of  FIG.  6    includes a y-axis  602  that is the pore size distribution and an x-axis  604  that is the pore body diameter. 
     A line  606  is the surface relaxivity using BET (ρ BET ). Lines  608  and  610  perform different corrections of ρ BET  with similar results. The line  608  is the surface relaxivity that includes a correctness for roughness (ρ LSCM ) that is used in the operations of  FIG.  5   . A line  610  is the surface relaxivity that is determined based on p-CT that is used in the operations of  FIG.  4   . 
     The flowcharts herein are annotated with a series of numbers. These numbers represent stages of operations. Although these stages are ordered for this example, the stages illustrate one example to aid in understanding this disclosure and should not be used to limit the claims. Subject matter falling within the scope of the claims can vary with respect to the order and some of the operations. The flowcharts are provided to aid in understanding the illustrations and are not to be used to limit scope of the claims. The flowcharts depict example operations that can vary within the scope of the claims. Additional operations may be performed; fewer operations may be performed; the operations may be performed in parallel; and the operations may be performed in a different order. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by program code. The program code may be provided to a processor of a general purpose computer, special purpose computer, or other programmable machine or apparatus. 
     As will be appreciated, aspects of the disclosure may be embodied as a system, method or program code/instructions stored in one or more machine-readable media. Accordingly, aspects may take the form of hardware, software (including firmware, resident software, micro-code, etc.), or a combination of software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” or “system.” The functionality presented as individual modules/units in the example illustrations can be organized differently in accordance with any one of platform (operating system and/or hardware), application ecosystem, interfaces, programmer preferences, programming language, administrator preferences, etc. 
     Any combination of one or more machine-readable medium(s) may be utilized. The machine-readable medium may be a machine-readable signal medium or a machine-readable storage medium. A machine-readable storage medium may be, for example, but not limited to, a system, apparatus, or device, that employs any one of or combination of electronic, magnetic, optical, electromagnetic, infrared, or semiconductor technology to store program code. More specific examples (a non-exhaustive list) of the machine-readable storage medium would include the following: a portable computer diskette, a hard disk, a random-access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a machine-readable storage medium may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device. A machine-readable storage medium is not a machine-readable signal medium. 
     A machine-readable signal medium may include a propagated data signal with machine readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A machine-readable signal medium may be any machine-readable medium that is not a machine-readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device. Program code embodied on a machine-readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber cable, RF, etc., or any suitable combination of the foregoing. 
     Computer program code for carrying out operations for aspects of the disclosure may be written in any combination of one or more programming languages, including an object oriented programming language such as the Java® programming language, C++ or the like; a dynamic programming language such as Python; a scripting language such as Perl programming language or PowerShell script language; and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The program code may execute entirely on a stand-alone machine, may execute in a distributed manner across multiple machines, and may execute on one machine while providing results and or accepting input on another machine. 
     The program code/instructions may also be stored in a machine-readable medium that can direct a machine to function in a particular manner, such that the instructions stored in the machine-readable medium produce an article of manufacture including instructions which implement the function/act specified in the flowchart and/or block diagram block or blocks. 
     Example Computer 
       FIG.  7    depicts an example computer, according to some embodiments. A computer  700  includes a processor  701  (possibly including multiple processors, multiple cores, multiple nodes, and/or implementing multi-threading, etc.). The computer  700  includes a memory  707 . The memory  707  may be system memory (e.g., one or more of cache, SRAM, DRAM, zero capacitor RAM, Twin Transistor RAM, eDRAM, EDO RAM, DDR RAM, EEPROM, NRAM, RRAM, SONOS, PRAM, etc.) or any one or more of the above already described possible realizations of machine-readable media. The computer  700  also includes a bus  703  (e.g., PCI, ISA, PCI-Express, HyperTransport® bus, InfiniBand® bus, NuBus, etc.) and a network interface  705  (e.g., a Fiber Channel interface, an Ethernet interface, an internet small computer system interface, SONET interface, wireless interface, etc.). 
     The computer  700  also includes a signal processor  711 . The signal processor  711  can perform at least of a portion of the operations described herein. Any one of the previously described functionalities may be partially (or entirely) implemented in hardware and/or on the processor  701 . For example, the functionality may be implemented with an application specific integrated circuit, in logic implemented in the processor  701 , in a co-processor on a peripheral device or card, etc. Further, realizations may include fewer or additional components not illustrated in  FIG.  7    (e.g., video cards, audio cards, additional network interfaces, peripheral devices, etc.). The processor  701  and the network interface  705  are coupled to the bus  703 . Although illustrated as being coupled to the bus  703 , the memory  707  may be coupled to the processor  701 . 
     While the aspects of the disclosure are described with reference to various implementations and exploitations, it will be understood that these aspects are illustrative and that the scope of the claims is not limited to them. In general, techniques as described herein may be implemented with facilities consistent with any hardware system or hardware systems. Many variations, modifications, additions, and improvements are possible. 
     Example Embodiments 
     Embodiment #1 
     A method comprises determining a Nuclear Magnetic Resonance (NMR) response of a subsurface formation that is based on an NMR response signal that traversed through the subsurface formation and that is result of a magnetic field being emitted into the subsurface formation; determining an Archie cementation exponent for an Archie equation based on the NMR response; and determining a property of the subsurface formation based on the Archie cementation exponent. 
     Embodiment #2 
     The method of Embodiment 1, wherein determining the Archie cementation exponent comprises determining the Archie cementation exponent using an NMR diffusion relaxation distribution map that is generated from the NMR response. 
     Embodiment #3 
     The method of Embodiment 2, wherein determining the Archie cementation exponent using the NMR diffusion relaxation distribution map comprises performing a phenomenological fit to restricted-diffusion versus T2 that is positioned closest to a peak of a distribution of the NMR response. 
     Embodiment #4 
     the method of any one of embodiments 1-3, further comprising determining a surface relaxivity of the subsurface formation. 
     Embodiment #5 
     The method of Embodiment 4, wherein determining the Archie cementation exponent comprises determining the Archie cementation exponent for a Padé D−T 2  fit line and the surface relaxivity of the subsurface formation. 
     Embodiment #6 
     The method of any one of Embodiments 1-5, wherein determining the property of the subsurface formation comprises determining at least one of a conductivity and resistivity of the subsurface formation. 
     Embodiment #7 
     The method of any one of Embodiments 1-6, further comprising adjusting a wellbore operation for a wellbore that is formed in the subsurface formation. 
     Embodiment #8 
     One or more non-transitory machine-readable storage media comprising program code executable by a processor to cause the processor to: determine a Nuclear Magnetic Resonance (NMR) response of a subsurface formation that is based on an NMR response signal that traversed through the subsurface formation and that is result of a magnetic field being emitted into the subsurface formation; determine an Archie cementation exponent for an Archie equation based on the NMR response; and determine a property of the subsurface formation based on the Archie cementation exponent. 
     Embodiment #9 
     The one or more non-transitory machine-readable storage media of Embodiment 8, wherein the program code executable by the processor to cause the processor to determine the Archie cementation exponent comprises program code executable by the processor to cause the processor to determine the Archie cementation exponent using an NMR diffusion relaxation distribution map that is generated from the NMR response. 
     Embodiment #10 
     The one or more non-transitory machine-readable storage media of Embodiment 9, wherein the program code to executable by the processor to cause the processor to determine the Archie cementation exponent using the NMR diffusion relaxation distribution map comprises program code executable by the processor to cause the processor to select a phenomenological fit to restricted-diffusion versus T2 that is positioned closest to a peak of a distribution of the NMR response. 
     Embodiment #11 
     The one or more non-transitory machine-readable storage media of any one of Embodiments 8-10, wherein the program code comprises program code executable by the processor to cause the processor to determine a surface relaxivity of the subsurface formation. 
     Embodiment #12 
     The one or more non-transitory machine-readable storage media of Embodiment 11, wherein the program code to executable by the processor to cause the processor to determine the Archie cementation exponent comprises program code to executable by the processor to cause the processor to determine the Archie cementation exponent for a Padé D−T 2  fit line and the surface relaxivity of the subsurface formation. 
     Embodiment #13 
     The one or more non-transitory machine-readable storage media of any one of Embodiments 8-12, wherein the program code executable by the processor to cause the processor to determine the property of the subsurface formation comprises program code executable by the processor to cause the processor to determine at least one of a conductivity and resistivity of the subsurface formation. 
     Embodiment #14 
     The one or more non-transitory machine-readable storage media of Embodiment 8, wherein the program code comprises program code executable by the processor to cause the processor to adjust a wellbore operation for a wellbore that is formed in the subsurface formation. 
     Embodiment #15 
     A system comprising: a Nuclear Magnetic Resonance (NMR) logging tool to be positioned in a wellbore formed in a subsurface formation, the NMR logging tool to detect an NMR response signal that traversed through the subsurface formation and that is result of a magnetic field being emitted into the subsurface formation; a processor; and a machine-readable storage medium comprising program code executable by the processor to cause the processor to: determine an NMR response of the subsurface formation based on the NMR response signal; determine an Archie cementation exponent for an Archie equation based on the NMR response; and determine a property of the subsurface formation based on the Archie cementation exponent. 
     Embodiment #16 
     The system of Embodiment 15, wherein the program code executable by the processor to cause the processor to determine the Archie cementation exponent comprises program code executable by the processor to cause the processor to determine the Archie cementation exponent using an NMR diffusion relaxation distribution map that is generated from the NMR response. 
     Embodiment #17 
     The system of Embodiment 16, wherein the program code to executable by the processor to cause the processor to determine the Archie cementation exponent using the NMR diffusion relaxation distribution map comprises program code executable by the processor to cause the processor to select a phenomenological fit to restricted-diffusion versus T2 that is positioned closest to a peak of a distribution of the NMR response. 
     Embodiment #18 
     The system of any one of Embodiments 15-17, wherein the program code comprises program code executable by the processor to cause the processor to determine a surface relaxivity of the subsurface formation. 
     Embodiment #19 
     The system of Embodiment 18, wherein the program code to executable by the processor to cause the processor to determine the Archie cementation exponent comprises program code to executable by the processor to cause the processor to determine the Archie cementation exponent for a Padé D−T 2  fit line and the surface relaxivity of the subsurface formation. 
     Embodiment #20 
     The system of any one of Embodiments 15-19, wherein the program code comprises program code executable by the processor to cause the processor to adjust a wellbore operation for the wellbore that is formed in the subsurface formation. 
     As used herein, the term “or” is inclusive unless otherwise explicitly noted. Thus, the phrase “at least one of A, B, or C” is satisfied by any element from the set {A, B, C} or any combination thereof, including multiples of any element.