Abstract:
Values for porosity and permeability of core samples in a borehole are estimated by generating radial waves with an acoustic source in fluid around the core sample, and measuring pressure in the fluid. Moreover, the acoustic source operates at frequency close to a resonant frequency of the core sample. After the acoustic source no longer operates at the resonant frequency, pressure in the fluid attenuates over time. The pressure attenuation is recorded by the pressure measurements, along with the pressure in the fluid at the first harmonic (spectral component). The pressure attenuation and spectral component each are dependent on porosity and permeability of the core sample. Thus values for the porosity and permeability are determined based on the arithmetic relationships between pressure attenuation and the spectral component and porosity and permeability.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of Invention 
         [0002]    The present disclosure relates in general to a method of estimating parameters of a subterranean formation. More specifically, the present disclosure relates to a method of estimating porosity and/or permeability of a subterranean formation by measuring a spectral component of a resonant frequency of a sample of the formation. 
         [0003]    2. Description of Prior Art 
         [0004]    A hydrocarbon production rate and overall yield in a subterranean formation can be estimated based on the porosity and permeability of the strata making up the formation. Sometimes a coring tool is used to extract rock samples from within the formation and bring the samples to surface where properties of the sample are measured. Characteristics of the formation, such as porosity and permeability, can be estimated based on values of the measured properties. 
         [0005]    However, while the core sample is being pulled to the surface, the hydrostatic pressure decreases gradually allowing gas to come out of solution inside the core. During its escape from the core, the expanding gas drives the formation fluids expelling oil and/or water out of pores. spaces, and cracks in the core. Standard core analysis thus cannot obtain accurate information about the in situ condition of the core and any entrained fluids, and thus is generally not used to properly quantify fluid parameters. 
       SUMMARY OF THE INVENTION 
       [0006]    Disclosed herein is a method of evaluating porosity and permeability of subterranean rock formation by analyzing a core sample downhole. In an example, a method of estimating characteristics of a core sample in a borehole includes measuring values of pressure in a fluid surrounding the core sample that fluctuate in response to radial waves that propagate through the fluid and the core sample, identifying an attenuation of pressure in the fluid over time based on the measured pressure; identifying a pressure in the fluid at the frequency of a resonant peak, and estimating at least one of porosity or permeability of the core sample based on the values of pressure and pressure attenuation. The radial waves can be at a frequency which is at about a resonant frequency of the core sample. In this example, the step of identifying an attenuation of pressure in the fluid can take place after radial waves at the resonant frequency of the core sample no longer propagate through the fluid. The method can further include providing an acoustic source for generating the radial waves. Optionally, the acoustic source operates at about the first resonant frequency of the core sample. The resonant peak may optionally be at the first resonant peak. In an alternative, the core sample is obtained with a coring tool, and wherein the step of measuring values of pressure takes place in the coring tool. 
         [0007]    Also disclosed herein is a method of estimating characteristics of a core sample in a borehole that includes obtaining the core sample with a coring tool in the borehole and immersing the core sample with a liquid, radial waves are generated in the liquid with a transmitter that operates at a resonant frequency of the core sample, an attenuation of peaks of pressure in the fluid over time are identified, where pressure peaks are from the radial waves that come from the core sample. The method further includes identifying a pressure in the fluid at the frequency of a resonant peak that is from at least one of the radial waves and estimating at least one of porosity or permeability of the core sample. The step of estimating at least one of porosity or permeability of the core sample can be based on the identifying the pressure attenuation or pressure peak. The method may further optionally include terminating generating radial waves at the resonant frequency prior to measuring the pressure attenuation. In one example, the resonant frequency of the core sample is estimated. 
         [0008]    Further disclosed herein is a system for estimating characteristics of a core sample in a borehole and which is made up of a coring bit having an axial bore in which the core sample is selectively disposed, a tubular transducer that circumscribes the axial bore and selectively oscillates at about the resonant frequency of the core sample, an acoustic receiver disposed in the axial bore and immersed in liquid that contacts the core sample, and that selectively receives radial waves generated by the transducer that reflect from the core sample. The system may further include a processor for estimating a value for at least one of a porosity or permeability of the core sample based on a measurement of an attenuation of pressure of the radial waves received by the receiver and a pressure measured in the fluid at a first resonant frequency of the core sample. The system can further have springs for suspending the receiver in the fluid. Optionally included with the system is a housing on which the coring bit is attached and which defines a coring tool; in this example a drill string can be attached to an end of the coring tool for rotating the coring bit. The liquid can be disposed in the axial bore when the coring bit is at surface. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0009]    Some of the features and benefits of the present invention having been stated, others will become apparent as the description proceeds when taken in conjunction with the accompanying drawings, in which: 
           [0010]      FIG. 1  is a side partial sectional view of an example of a coring tool deployed in a wellbore and in accordance with the present invention. 
           [0011]      FIG. 2  is a side sectional view of an example of a portion of the coring tool of  FIG. 1  obtaining core samples in a wellbore and in accordance with the present invention. 
           [0012]      FIG. 3  is a side sectional perspective view of an embodiment of acoustically measuring a core sample in the coring tool of  FIG. 2  and in accordance with the present invention. 
           [0013]      FIG. 4  is an example plot of pressure wave attenuation of an acoustic signal adjacent a core sample and in accordance with the present invention. 
           [0014]      FIG. 5  is an example plot of eigenwaves of an acoustic signal adjacent a core sample and in accordance with the present invention. 
       
    
    
       [0015]    While the invention will be described in connection with the preferred embodiments, it will be understood that it is not intended to limit the invention to that embodiment. On the contrary, it is intended to cover all alternatives, modifications, and equivalents, as may be included within the spirit and scope of the invention as defined by the appended claims. 
       DETAILED DESCRIPTION OF INVENTION 
       [0016]    The method and system of the present disclosure will now be described more fully hereinafter with reference to the accompanying drawings in which embodiments are shown. The method and system of the present disclosure may be in many different forms and should not be construed as limited to the illustrated embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey its scope to those skilled in the art. Like numbers refer to like elements throughout. 
         [0017]    It is to be further understood that the scope of the present disclosure is not limited to the exact details of construction, operation, exact materials, or embodiments shown and described, as modifications and equivalents will be apparent to one skilled in the art. In the drawings and specification, there have been disclosed illustrative embodiments and, although specific terms are employed, they are used in a generic and descriptive sense only and not for the purpose of limitation. 
         [0018]    Shown in side sectional view in  FIG. 1  is an example of a coring tool  10  disposed in a borehole  12  that intersects a subterranean formation  14 . The coring tool  10  is deployed on a lower end of a drill string  16 . A rotary table  18  couples with the drill string  16  on surface and rotates the drill string  16  so the coring tool  10  can bore into the formation  14  and obtain core samples from the formation  14 . The rotary table  18  is included as part of a drilling rig  20  on the surface shown having a derrick  21  for supporting the rotary table  18  and drill string  16 . 
         [0019]    Referring now to  FIG. 2 , shown in side sectional view is a detailed portion of the coring tool  10  in the borehole  12  illustrating that a bit  22  is included on a lower end of the coring tool  10  for excavating in the formation  14 . An elongate bore  24  extends axially through the bit  22 , and in which a core sample  26  is received after the bit  22  excavates the rock surrounding the core  26 . Drilling fluid or mud can be circulated downhole through the coring tool  10  within an annulus  28  shown extending axially through the bit  22  and spaced radially outward from the bore  24 . In addition to cooling the bit  22  during excavating, the drilling fluid washes cuttings  30  back uphole that were removed from the formation  14  by the cutting action of the bit  22 . 
         [0020]    Optional bearings  32  are shown in the annulus  28  and between an inner body  34  and outer tubing  35 . Both the inner body  34  and outer tubing  35  are tubular members, where the inner body  34  is coaxially and rotationally set within the outer tubing  35 . The inner body  34  contains the core sample  26 , and the outer tubing  35  defines an outer surface of the annulus  28  and includes cutting elements on its outer surface for cutting through the formation  14 . 
         [0021]    Also disposed in the annulus  28  is a transmitter  36  for generating an acoustic signal; in one example the signal includes a radial wave R ( FIG. 3 ) that contacts, passes through, and reflects from the sample  26 . A receiver  38 , which in one example can be a hydrophone, is illustrated spaced axially away from the transmitter  36 . Examples exist wherein the transmitter  36  and receiver  38  are integrated into a single unit. In an example, the inner body  34  is filled with a selected known fluid before being deployed downhole. Optionally, an elastic liner (not shown) can be provided on an inner surface of the inner body  34  for preventing damage to the sample core  26  and to prevent formation fluids from escaping the sample core  26  or from within the bore  24 . 
         [0022]    Schematically illustrated in side sectional perspective view in  FIG. 3  are radial acoustic waves R being generated in fluid F that surrounds the core sample  26 , and where reflections of the radial waves R from the core sample  26  are recorded by the receiver  38 . As illustrated in the example of  FIG. 3 , the transmitter  36  is a tubular member that is selectively oscillated to generate the radial acoustic waves R in the fluid F. Example materials for the transmitter  36  include piezoelectric and/or magnetostrictive materials, in an example the transmitter  36  is oscillated by applying electricity to the transmitter  36 . 
         [0023]    Also shown in  FIG. 3  is that the hydrophone  38  is suspended in the annulus  28  and supported by springs  42 . In a non-limiting example, the transmitter  36  is operated at the resonant frequency of the radial eigenwaves of the core sample  26 ; which in one embodiment is at a frequency that is within about +/−5% of the first resonant frequency. Further in this example, the receiver  38  records the pressure of the radial acoustic wave R. Where in one example, the pressure of the radial acoustic wave is the increase in pressure due to the compressional waves. 
         [0024]    Referring back to  FIG. 2 , in one non-limiting example of operation, the coring tool  10  is rotated in the borehole  12  so that the bit  22  excavates an annular cut into the formation  14  thereby forming the core sample  26  that extends into the bore  24 . The transmitter  36  is actuated for a period of time to generate radial waves R ( FIG. 3 ) shown in the fluid F propagating towards the core sample  26 . A portion of the radial waves R reflect from the core sample  26  and are measured by the receiver  38 ; as indicated above the receiver  38  records the pressure of the radial acoustic wave R reflected from the core sample  26 . 
         [0025]    Referring now to  FIG. 4 , shown is a plot  44  that represents pressure in the fluid F measured by the receiver  38 , where the recorded pressure is a result of the acoustic signal in the fluid F generated by the transmitter  36 . Shown in  FIG. 4  is that the peaks  46  of the measured pressure decay over time. The decay of the pressure peaks  46  are modeled by plot  48 , which is represented by the expression Ae (−βt) . In one embodiment, the measured pressure decay over time takes place after the transmitter  36  no longer operates at the resonant frequency. It has been found that the attenuation factor of the pressure waves (the envelope of the maximal values of pressure) is also determined by permeability k and porosity φ of the formation, where: 
         [0000]      β=φ( k ,φ)  Equation (1).
 
         [0026]    Shown in  FIG. 5  is a plot  50  that represents a spectrum of sample eigenwaves. Analyzing the behavior of spectral component of pressure at the harmonic source at the minimal resonant frequency can yield useful information. For example, it has been found that the value of pressure at the first maximum (pressure at the frequency of the first resonant peak) is dependent upon formation permeability k and porosity φ, where: 
         [0000]        p   max =ψ( k ,φ)Equation  (2).
 
         [0027]    Equations (1) and (2) at given acoustic velocities in fluid and the formation, at corresponding porosities, constitute a set of equations for finding permeability and porosity. Measuring p max  and β and solving sets (1), (2), values can be obtained for permeability k and porosity φ. Provided below are how the functional dependences φ(k, φ) and p max =ψ(k, φ) were theoretically obtained. 
         [0028]    Functions φ(k, φ), ψ(k, φ) are found from the theory describing the radial waves in the borehole with saturated porous core sample. The acoustic equations of the two-velocity medium are a linearized version of the following equations, where Equations (3)-(7) describe the propagation of waves in the saturated porous core sample: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             ∂ 
                             ρ 
                           
                           
                             ∂ 
                             t 
                           
                         
                         + 
                         divj 
                       
                       = 
                       0 
                     
                     , 
                   
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             ∂ 
                             
                               g 
                               ik 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                         + 
                         
                           
                             g 
                             jk 
                           
                            
                           
                             
                               ∂ 
                               i 
                             
                              
                             
                               u 
                               j 
                             
                           
                         
                         + 
                         
                           
                             g 
                             ij 
                           
                            
                           
                             
                               ∂ 
                               k 
                             
                              
                             
                               u 
                               j 
                             
                           
                         
                         + 
                         
                           
                             u 
                             j 
                           
                            
                           
                             
                               ∂ 
                               j 
                             
                              
                             
                               g 
                               ik 
                             
                           
                         
                       
                       = 
                       0 
                     
                     , 
                   
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       4 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             ∂ 
                             
                               j 
                               i 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                         + 
                         
                           
                             ∂ 
                             k 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ρ 
                                   s 
                                 
                                  
                                 
                                   u 
                                   i 
                                 
                                  
                                 
                                   u 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   ρ 
                                   l 
                                 
                                  
                                 
                                   v 
                                   i 
                                 
                                  
                                 
                                   v 
                                   k 
                                 
                               
                               + 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 
                                   δ 
                                   ik 
                                 
                               
                               + 
                               
                                 
                                   g 
                                   jk 
                                 
                                  
                                 
                                   h 
                                   ij 
                                 
                               
                               + 
                               
                                 π 
                                 ik 
                               
                             
                             ) 
                           
                         
                       
                       = 
                       0 
                     
                     , 
                   
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       5 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           
                             v 
                             i 
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             v 
                             , 
                             ∇ 
                           
                           ) 
                         
                          
                         
                           v 
                           i 
                         
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             1 
                             ρ 
                           
                         
                          
                         
                           
                             ∂ 
                             i 
                           
                            
                           p 
                         
                       
                       + 
                       
                         
                           
                             ρ 
                             s 
                           
                           
                             2 
                              
                             ρ 
                           
                         
                          
                         
                           
                             ∂ 
                             i 
                           
                            
                           
                             
                               ( 
                               
                                 u 
                                 - 
                                 v 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       - 
                       
                         
                           1 
                           
                             2 
                              
                             ρ 
                           
                         
                          
                         
                           h 
                           jk 
                         
                          
                         
                           
                             ∂ 
                             i 
                           
                            
                           
                             g 
                             jk 
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             ρ 
                             l 
                           
                         
                          
                         
                           f 
                           l 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   and 
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       6 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           ∂ 
                           S 
                         
                         
                           ∂ 
                           t 
                         
                       
                       + 
                       
                         div 
                          
                         
                           ( 
                           
                             
                               S 
                                
                               
                                 j 
                                 ρ 
                               
                             
                             + 
                             
                               q 
                               T 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         R 
                         T 
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       7 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0029]    The condition of the positive dissipative function R is as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       - 
                       R 
                     
                     = 
                     
                       
                         f 
                          
                         
                           1 
                           
                             ρ 
                             l 
                           
                         
                          
                         
                           ( 
                           
                             j 
                             - 
                             
                               ρ 
                                
                               
                                   
                               
                                
                               u 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         q 
                          
                         
                           
                             ∇ 
                             T 
                           
                           T 
                         
                       
                       + 
                       
                         
                           A 
                           ik 
                         
                          
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 i 
                               
                                
                               
                                 v 
                                 k 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 k 
                               
                                
                               
                                 v 
                                 i 
                               
                             
                             - 
                             
                               
                                 2 
                                 3 
                               
                                
                               
                                 δ 
                                 ik 
                               
                                
                               divv 
                             
                           
                           ) 
                         
                       
                       + 
                       adivv 
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
         [0030]    and it determines the form of the dissipative flows: 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     = 
                     
                       
                         
                           α 
                           12 
                         
                          
                         
                           ( 
                           
                             
                               ρ 
                                
                               
                                   
                               
                                
                               u 
                             
                             - 
                             j 
                           
                           ) 
                         
                       
                       + 
                       
                         κ 
                          
                         
                           1 
                           T 
                         
                          
                         
                           ∇ 
                           T 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       9 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       f 
                       + 
                       
                         
                           ∂ 
                           k 
                         
                          
                         
                           π 
                           ik 
                         
                       
                     
                     = 
                     
                       
                         b 
                          
                         
                           ( 
                           
                             
                               ρ 
                                
                               
                                   
                               
                                
                               u 
                             
                             - 
                             j 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ρ 
                           l 
                         
                          
                         
                           α 
                           12 
                         
                          
                         
                           1 
                           T 
                         
                          
                         
                           ∇ 
                           T 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       10 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       π 
                       ik 
                     
                     = 
                     
                       
                         A 
                         ik 
                       
                       + 
                       
                         a 
                          
                         
                             
                         
                          
                         
                           δ 
                           ik 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       A 
                       ik 
                     
                     = 
                     
                       - 
                       
                         η 
                          
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 i 
                               
                                
                               
                                 v 
                                 k 
                               
                             
                             + 
                             
                               
                                 ∂ 
                                 k 
                               
                                
                               
                                 v 
                                 i 
                               
                             
                             - 
                             
                               
                                 2 
                                 3 
                               
                                
                               
                                 δ 
                                 ik 
                               
                                
                               divv 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     a 
                     = 
                     
                       
                         - 
                         Ϛ 
                       
                        
                       
                           
                       
                        
                       
                         divv 
                         . 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       11 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0031]    The equation of the state of the saturated porous medium is selected in the following form: 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                     = 
                     
                       
                         p 
                         0 
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           Kg 
                           ll 
                         
                       
                       + 
                       
                         
                           1 
                           
                             ρ 
                             0 
                           
                         
                          
                         
                           ( 
                           
                             K 
                             + 
                             γ 
                           
                           ) 
                         
                          
                         ρ 
                       
                       + 
                       
                         
                           α 
                           p 
                         
                          
                         S 
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         
                           
                             
                               ρ 
                               s 
                             
                              
                             
                               ( 
                               
                                 u 
                                 - 
                                 v 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       12 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       T 
                       = 
                       
                         
                           T 
                           0 
                         
                         + 
                         
                           
                             α 
                             s 
                           
                            
                           S 
                         
                         + 
                         
                           
                             α 
                             p 
                           
                            
                           
                             1 
                             
                               ρ 
                               0 
                             
                           
                            
                           ρ 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                         
                     
                      
                     and 
                   
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       13 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     h 
                     ik 
                   
                   = 
                   
                     
                       h 
                       
                         0 
                         , 
                         ik 
                       
                     
                     + 
                     
                       μ 
                        
                       
                           
                       
                        
                       
                         g 
                         ik 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       λ 
                        
                       
                           
                       
                        
                       
                         g 
                         ll 
                       
                        
                       
                         δ 
                         ik 
                       
                     
                     - 
                     
                       
                         ρ 
                         
                           ρ 
                           0 
                         
                       
                        
                       
                         Kg 
                         ll 
                       
                        
                       
                         δ 
                         ik 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                        
                       
                         g 
                         ik 
                       
                        
                       
                         
                           
                             
                               ρ 
                               s 
                             
                              
                             
                               ( 
                               
                                 u 
                                 - 
                                 v 
                               
                               ) 
                             
                           
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       14 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0032]    In the formulae above, ρ l , ρ s  are partial densities of the saturating fluid and porous matrix, respectively, ρ=ρ l +ρ s  is density of the saturated medium; u, v are velocities of the porous matrix and saturating fluid; j=ρ s u+p l v is the momentum; h ik  is the stress tensor, g ik  is the metric tensor; p is pressure, T is temperature, S is entropy; λ, μ and γ are elastic moduli of the saturated porous medium; α s , α p  are thermodynamic parameters of the medium. The dissipative coefficient of the interphase friction b may be found from the ratio b=η/(kρ), where η is dynamic viscosity of saturating fluid, k is permeability of the porous medium. 
         [0033]    The linearized isothermal version of Equations (3)-(6) enables one to find the motion equations of the porous matrix u and saturating fluid v in the acoustic approximation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           u 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                       - 
                       
                         
                           c 
                           t 
                           2 
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         u 
                       
                       - 
                       
                         
                           a 
                           1 
                         
                          
                         
                           ∇ 
                           divu 
                         
                       
                       + 
                       
                         
                           a 
                           2 
                         
                          
                         
                           ∇ 
                           divv 
                         
                       
                       + 
                       
                         
                           
                             ρ 
                             l 
                           
                           
                             ρ 
                             s 
                           
                         
                          
                         
                           b 
                            
                           
                             ( 
                             
                               
                                 
                                   ∂ 
                                   u 
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                               - 
                               
                                 
                                   ∂ 
                                   v 
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     0 
                   
                   , 
                   
                     
 
                   
                    
                   
                       
                   
                    
                   and 
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       15 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           v 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                       + 
                       
                         
                           a 
                           3 
                         
                          
                         
                           ∇ 
                           divu 
                         
                       
                       - 
                       
                         
                           a 
                           4 
                         
                          
                         
                           ∇ 
                           divv 
                         
                       
                       - 
                       
                         b 
                          
                         
                           ( 
                           
                             
                               
                                 ∂ 
                                 u 
                               
                               
                                 ∂ 
                                 t 
                               
                             
                             - 
                             
                               
                                 ∂ 
                                 v 
                               
                               
                                 ∂ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       16 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0034]    Coefficients a i  (i=1, . . . , 4) are determined by the elastic moduli of the saturated porous medium λ, μ and γ: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       a 
                       1 
                     
                     = 
                     
                       
                         1 
                         
                           ρ 
                           s 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 ρ 
                                 s 
                                 2 
                               
                               
                                 ρ 
                                 2 
                               
                             
                              
                             γ 
                           
                           + 
                           
                             
                               
                                 ρ 
                                 l 
                                 2 
                               
                               
                                 ρ 
                                 2 
                               
                             
                              
                             K 
                           
                           + 
                           
                             
                               1 
                               3 
                             
                              
                             μ 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       a 
                       2 
                     
                     = 
                     
                       
                         
                           ρ 
                           l 
                         
                         
                           ρ 
                           s 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 ρ 
                                 l 
                               
                               
                                 ρ 
                                 2 
                               
                             
                              
                             K 
                           
                           - 
                           
                             
                               
                                 ρ 
                                 s 
                               
                               
                                 ρ 
                                 2 
                               
                             
                              
                             γ 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       17 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       a 
                       3 
                     
                     = 
                     
                       
                         
                           
                             ρ 
                             l 
                           
                           
                             ρ 
                             2 
                           
                         
                          
                         K 
                       
                       - 
                       
                         
                           
                             ρ 
                             s 
                           
                           
                             ρ 
                             2 
                           
                         
                          
                         γ 
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       a 
                       4 
                     
                     = 
                     
                       
                         
                           
                             ρ 
                             l 
                           
                           
                             ρ 
                             s 
                           
                         
                          
                         K 
                       
                       + 
                       
                         
                           
                             ρ 
                             l 
                           
                           
                             ρ 
                             2 
                           
                         
                          
                         γ 
                       
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       18 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0035]    The three moduli K, μ and γ, where K=λ+2μ/3, may be found from three acoustic velocities c p1 , c p2 , c i  of the ideal hydrodynamic approximation, in accordance with the formulae given below: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       μ 
                       = 
                       
                         
                           ρ 
                           s 
                         
                          
                         
                           c 
                           t 
                           2 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       λ 
                       = 
                       
                         
                           1 
                           2 
                         
                          
                         
                           
                             ρ 
                             s 
                           
                           
                             ρ 
                             l 
                           
                         
                          
                         
                           ( 
                           
                             
                               ρ 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 2 
                               
                             
                             + 
                             
                               ρ 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 
                                   p 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 2 
                               
                             
                             - 
                             
                               4 
                                
                               
                                 ρ 
                                 l 
                               
                                
                               
                                 c 
                                 t 
                                 2 
                               
                             
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ρ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           p 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                         2 
                                       
                                     
                                     + 
                                     
                                       ρ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         c 
                                         
                                           p 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   
                                     64 
                                     9 
                                   
                                    
                                   
                                     ρ 
                                     s 
                                   
                                    
                                   
                                     ρ 
                                     l 
                                   
                                    
                                   
                                     c 
                                     t 
                                     4 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                         
                     
                      
                     and 
                   
                 
               
               
                 
                   
                     Equation 
                      
                     
                         
                     
                      
                     
                       ( 
                       19 
                       ) 
                     
                   
                   , 
                 
               
             
             
               
                 
                   γ 
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       
                         ( 
                         
                           
                             ρ 
                              
                             
                                 
                             
                              
                             
                               c 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               2 
                             
                           
                           + 
                           
                             ρ 
                              
                             
                                 
                             
                              
                             
                               c 
                               
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               2 
                             
                           
                           - 
                           
                             
                               8 
                               3 
                             
                              
                             
                               ρ 
                               s 
                             
                              
                             
                               c 
                               t 
                               2 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   
                                     ρ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       
                                         p 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       c 
                                       
                                         p 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               - 
                               
                                 
                                   64 
                                   9 
                                 
                                  
                                 
                                   ρ 
                                   s 
                                 
                                  
                                 
                                   ρ 
                                   l 
                                 
                                  
                                 
                                   c 
                                   t 
                                   4 
                                 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       20 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0036]    The acoustic field in borehole fluid is described by the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         v 
                       
                       
                         ∂ 
                         
                           t 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           c 
                           
                             p 
                              
                             
                                 
                             
                              
                             0 
                           
                           2 
                         
                          
                         
                           ∇ 
                           divv 
                         
                       
                       = 
                       0 
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       21 
                       ) 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0037]    Where c p0  is the velocity of sound in borehole fluid. 
         [0038]    The following boundary conditions are shown in  FIG. 3 : (1) On the surface of the acoustic transmitter, the source with the amplitude p (0) =p 0 (t) is given; (2) At the boundary between the borehole and saturated porous sample, the continuity condition is applied to the mass flow, stress tensor Σ rr , and partial pressures: (1−φ)u r   (1) +φv r   (1) =v r   (0) , Σ rr   (1) =Σ rr   (0) , p (1) /ρ (1) =p (0) /ρ (0)  where φ is porosity. The latter condition follows from a more general continuity condition of the partial stress tensor in fluid Σ rr   1(1) =φΣ rr   (0) , where Σ rr   (1) =Σ rr   s(1) +Σ rr   l(1) . 
         [0039]    Time derivatives of stress tensor components and pressure can be found via the displacement velocities: 
         [0000]                      -       ∂     p     (   1   )           ∂   t         =         d   1          divu     (   1   )         +       d   2          divv     (   1   )             ,             Equation                   (   22   )       ,                     ∂     ∑   rr     (   1   )           ∂   t       =         d   3          divu     (   1   )         +       d   4          divv     (   1   )         +     2        d   5          u   r   ′           ,     
        and             Equation                   (   23   )       ,                     ∂     ∑   rr     (   0   )           ∂   t       =       -       ∂     p     (   0   )           ∂   t         =       d   0          divv     (   0   )             ,           Equation                     (   24   )     .                 Where:  d   1 =(ρ s   (1) γ−ρ l   (1)   K )/ρ (1)   ,d   2 =(ρ l   (1) γ+ρ l   (1)   K )/ρ (1) ,  Equation (25)
 
         [0000]        d   3 =ρ s   (1) γ/ρ (1) −2μ/3, d   4   =ρl   (1) γ/ρ (1) ,  Equation (26), and
 
         [0000]      π 5 =μ,π 0 =ρ (0)   c   p0   2 .  Equation (27).
 
         [0040]    The set of equations presented with corresponding boundary conditions was solved numerically. The result is the functional dependences φ(k, φ), ψ(k, φ). 
         [0041]    Referring back to  FIG. 1 , a controller  52  is shown in communication with the coring tool  10 , wherein the controller  52  can be housed within the coring tool  10  or on surface. In an example, the controller  52  receives the values of pressure measured in the coring tool  10 . Yet further optionally, the controller  52  is equipped with a processor for calculating the values of porosity and/or permeability as described above. The controller  52  may include an information handling system (IHS), where the IHS optionally includes one or more of a processor, memory accessible by the processor, nonvolatile storage area accessible by the processor, and logics for performing each of the steps above described. 
         [0042]    The present invention described herein, therefore, is well adapted to carry out the objects and attain the ends and advantages mentioned, as well as others inherent therein. While a presently preferred embodiment of the invention has been given for purposes of disclosure, numerous changes exist in the details of procedures for accomplishing the desired results. These and other similar modifications will readily suggest themselves to those skilled in the art, and are intended to be encompassed within the spirit of the present invention disclosed herein and the scope of the appended claims.