Abstract:
The specification discloses a downhole tool that determines hydraulic permeability of a downhole formation taking into account one or both of the dip angle of the borehole relative to the formation, and damage to the borehole wall and invasion of borehole fluid—collectively referred to as skin.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit of Provisional Application Ser. No. 60/325,903, which is incorporated herein by reference as if reproduced in full below. 
   STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
   Not applicable. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The preferred embodiments of the present invention generally relate to determining hydraulic permeability of earth formations traversed by a borehole. More particularly, the preferred embodiments are directed to determining permeability anisotropy of the earth formations. More particularly still, the preferred embodiments are directed to determining the hydraulic permeability, permeability anisotropy and skin using an analytic model that considers the storage effect downhole, as well as the dip angle of the formation relative to the borehole. 
   2. Description of the Related Art 
   It is well known that some earth formations exhibit anisotropic properties. That is, certain downhole parameters may have more distinctive qualities, or may be more pronounced, in one physical direction than another. While there may be many properties that exhibit this characteristic, this specification is directed to determining the hydraulic permeability anisotropy of the earth formations. 
   Permeability is a measure of how easily fluids flow through a particular environment. Earth formations having a very high permeability may flow greater volumes of liquids than formations having a low permeability for the same pressure differentials. Because of the way earth formations are formed, typically horizontal layer upon horizontal layer, the permeability of earth formations is generally higher in a direction substantially parallel to the layers of earth formation. Likewise, the permeability is generally lower in directions perpendicular to the layers of the earth formation. While it is generally true that the horizontal permeability is greater than the vertical permeability, this need not necessarily be the case. 
   Permeability of a formation generally may be determined by inducing a fluid flow from the formation into a test apparatus, and measuring the pressure differential created by the induced flow. If the testing is performed with only a single probe, then the permeability determined is spherical permeability and indistinguishably contains both the horizontal and vertical permeability components. That is, using a single probe, and inducing a flow, the flow of formation fluids comes not only from locations within the formation on the same plane as the probe, but also from above and below. Thus, using only a single probe, while giving the ability to determine the permeability generally, is not sufficient to ascertain the horizontal and vertical components of the permeability. Related art devices compensate for this inability to determine both the horizontal and vertical permeability by having two probes vertically spaced apart. 
   In particular, related art devices may include a source probe at a first elevation, and a second probe vertically displaced from the first probe. Further, some devices may also include a third probe at the same elevation as the first probe, but azimuthly rotated therefrom, for example, U.S. Pat. No. 5,247,830. By inducing either positive or negative pressure within the formation at the source probe (by forcing fluid flow into or out of the formation respectively), and detecting pressure response at the other probes, it is possible to determine both the horizontal and vertical components of the overall permeability of the formation. However, there are aspects of a borehole traversing an earth formation that are not determined or compensated for in devices such as those described in the U.S. Pat. No. 5,247,830. 
   Drilling of earth formations typically involves a drill bit at the end of a drill string cutting or chipping away pieces of the formation. Drilling fluid, also known as “mud,” flows through an inside diameter of the drill string, through jets on the drill bit, and then back up through the annular region between the drill string and the borehole wall. The mud serves several purposes. First, the mud moving past the drill bit acts to cool and lubricate the bit. Secondly, the circulation of drilling mud through the annular region carries cuttings away from the drill bit. Finally, the drilling mud has a specific density such that the pressure within the borehole as it traverses earth formations is greater than the pressure of fluid or gas within the formations, thereby forcing the downhole hydrocarbons to remain within the formation rather than entering the borehole. If the pressure of the drilling fluid is not carefully maintained, it is possible for the downhole hydrocarbons to enter the borehole and/or expand under the reduced pressure pushing the drilling mud back toward the surface, known as “kick.” 
   The rather violent process of drilling through an earth formation, in combination with the drilling mud present during the process, affects the downhole formation&#39;s ability to produce hydrocarbons. In particular, the act of drilling tends to damage, even if slightly, the formation immediately adjacent to the borehole wall. This damage may affect the permeability of the formation in this location. Further, the presence of the drilling mud at pressures greater than the formation results in invasion of the mud into the formation. This too tends to affect the permeability of the formation at locations adjacent to the borehole. While related art devices have advanced in their ability to determine both the horizontal and vertical components of the permeability in downhole formation, they are not capable of accounting for the affects of the formation damage and invasion of the drilling fluid near the borehole wall—which combination of factors is collectively known in the industry as “skin.” The effect of the skin on the measured permeability may be as high as an order of magnitude, thus contributing substantially to error in related art permeability determinations, as they do not take skin into account. Other factors too introduce error into related art determinations of permeability anisotropy, like compressibility of formation fluids and dip angle of the formation. 
   With the advent of directional drilling, it is now possible, indeed probable, that any particular borehole may not be substantially vertical. That is, as the direction and inclination of the drilling process changes, the borehole may cross otherwise substantially horizontal earth formations at an angle. Thus, the axis of the borehole at any particular location may have an angle of inclination, also known as the “dip angle,” with respect to the direction of horizontal permeability. This difference in the angle between the axis of the borehole and the direction of horizontal permeability may also manifest itself where an otherwise vertical borehole crosses an earth formation itself having an inclination. Regardless of why the dip angle is present, the difference between an assumed horizontal permeability normal to the borehole axis and the actual horizontal permeability affects the determination of the actual horizontal and vertical permeability. Related art permeability testing devices do not compensate for the dip angle. 
   Thus, what is needed in the art is a structure and related method for determining horizontal and vertical permeability, and thus the anisotropy of the earth formation, that takes into the account skin and dip angles of the well bore, as well as any other downhole parameter which may affect a measurement, such as the storage effect caused by compressibility of the formation fluids. 
   SUMMARY OF SOME OF THE PREFERRRED EMBODIMENTS 
   The problems noted above are solved in large part by a formation tester comprising two probes. Collecting data regarding formation pressure starts by fluidly coupling the probes to the formation walls. At least one of the probes creates a pressure gradient which is sensed by the related probe. The pressure data obtained is then applied to a series of analytic models which take into account the skin of the formation, the dip angle encountered and the storage effects downhole. 
   Using numerical regression analysis techniques, the preferred embodiments manipulate the parameters of the analytic model until the pressure response predicted by the model matches the actual pressure response. Once this is complete, the formation parameters such as permeability and skin are available in the solved model. In embodiments where one probe is used, the analytic model calculates spherical permeability taking into account one or both of the skin and dip angle. Where two probes are used, the analytic model calculates both horizontal and vertical permeability (and therefore anisotropy) taking into account one or both of skin and dip angle. 
   The disclosed device and methods comprise a combination of features and advantages which enable them to overcome the deficiencies of the prior art devices. The various characteristics described above, as well as other features, will be readily apparent to those skilled in the art upon reading the following detailed description, and by referring to the accompanying drawings. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a more detailed description of the preferred embodiments, reference will now be made to the accompanying drawings, wherein: 
       FIG. 1  shows a wireline formation testing tool of the preferred embodiments; 
       FIG. 2A  shows an internal schematic of the relevant portions of the wireline formation tool for small draw-down tests; 
       FIG. 2B  shows an internal schematic of the relevant portions of the wireline formation tool for large draw-down tests; and 
       FIG. 3  shows an exemplary set of data for a small draw-down permeability test. 
   

   NOTATION AND NOMENCLATURE 
   Certain terms are used throughout the following description and claims to refer to particular system components. This document does not intend to distinguish between components that differ in name but not function. In the following discussion and in the claims, the terms “including” and “comprising” are used in an open-ended fashion, and thus should be interpreted to mean “including, but not limited to . . . ”. 
   The following table of exemplary terms is provided to aid in understanding the various parameters of the equations given below: 
   
     
       
             
             
           
         
             
                 
                 
             
           
           
             
                 
               c = total compressibility (l/psi) 
             
             
                 
               c d  = dimensionless compressibility 
             
             
                 
               k s  = skin permeability (md) 
             
             
                 
               k f  = formation spherical permeability (md) 
             
             
                 
               k z  = vertical permeability (md) 
             
             
                 
               k r  = horizontal permeability (md) 
             
             
                 
               p = pressure (psi) 
             
             
                 
               p d  = dimensionless pressure 
             
             
                 
               P(t) = measured pressure (psi) 
             
             
                 
               ΔP p  = source probe pressure differential (psi) 
             
             
                 
               ΔP z  = vertical probe pressure differential (psi) 
             
             
                 
               Q(t) = volume flow rate (cm 3 /s) 
             
             
                 
               Q o  = draw-down flow rate (cm 3 /s) 
             
             
                 
               r = spherical coordinate (cm) 
             
             
                 
               r d  = dimensionless radius 
             
             
                 
               r dv  = dimensionless probe spacing 
             
             
                 
               r p  = physical source radius 
             
             
                 
               r s  = equivalent spherical source radius (cm) 
             
             
                 
               r v  = vertical probe spacing length (cm) 
             
             
                 
               S = skin factor 
             
             
                 
               S d  = dimensionless skin factor 
             
             
                 
               t = time (sec) 
             
             
                 
               t d  = dimensionless time 
             
             
                 
               x, y, z = Cartesian coordinates (cm) 
             
             
                 
               μ = formation fluid viscosity (cp) 
             
             
                 
               φ = formation porosity 
             
             
                 
               λ = anisotropy (k z /k r ) 
             
             
                 
               τ = constant related to borehole radius 
             
             
                 
               τ s  = source probe geometric shape factor 
             
             
                 
               τ v  = vertical probe geometric shape factor 
             
             
                 
                 
             
           
        
       
     
   
   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   This application is directly related to the Society of Petroleum Engineers Conference Paper SPE 64650 titled “Advanced Dual Probe Formation Tester with Transient, Harmonic, and Pulsed Time-Delay Testing Methods Determines Permeability, Skin and Anisotropy,” which is incorporated herein by reference as if reproduced in full below. This application is also related to the Society of Petroleum Engineers Paper SPE 62919 titled “Advanced Permeability and Anisotropy Measurements While Testing and Sampling in Real-Time Using a Dual Probe Formation Tester,” which is also incorporated herein by reference as if reproduced in full below. 
     FIG. 1  shows a wireline formation tester  10  constructed in accordance with the preferred embodiments. In particular, the wireline formation tester is preferably disposed within a borehole  12  traversing earth formations. The wireline formation tester  10  is preferably suspended by an armored multi-conductor cable  14 , which not only supports the formation tester  10  within the borehole, but also provides electrical communication pathways between the borehole  10  and a surface computer  11 . In the preferred embodiments, the surface computer  11  controls activities of the various downhole devices, and records parameters monitored during testing. The surface computer  11  also performs the modeling, described below, to determine the formation parameters in a post processing system (after the probe response are recorded). The volume between the wireline formation tester  10  and the borehole  12  may be filled with drilling fluid  16  whose specific gravity creates pressure at each depth which is greater than the formation pressure. 
   In order to make the permeability determinations, the wireline formation tester  10  preferably comprises a lower or source probe  18  which may be extended from the tool  10  body to be in fluid communication with the formation, as shown in  FIG. 1 . Likewise, the tool  10  comprises an upper or vertical probe  20 , which is also extendable away from the tool body to be in fluid communication with the formation  12 . In order that sufficient pressure may be placed on the probes  18 ,  20  to seal them against the formation wall, the wireline formation tester  10  also preferably comprises one or more stabilizers  22  placed on the opposite side of the tool body. While only one such stabilizer is shown, any number of stabilizers may be used, and their placement is not critical so long as sufficient force may be applied to the probes  18 ,  20  to seal them against the formation wall. 
   Referring now to  FIG. 2A , there is shown a schematic diagram of the relevant portions of the wireline formation tester  10  for performing small draw-down tests. In particular, the tool  10  comprises the source probe  18  and vertical probe  20 , as discussed with respect to  FIG. 1 . On the ends of each of the probes  18 ,  20  are seals  24 , of which one of ordinary skill in the art is fully aware, that aid in sealing of the probes  18 ,  20  against the formation wall  12 . Each of the probes  18 ,  20  are preferably fluidly coupled to a cylinder or chamber  26 . The chamber  26  preferably has disposed therein a piston  28 . By displacing the piston  28  within the chamber  26 , formation fluids are pulled through one or more of the probes  18 ,  20  into the chamber  26 . Alternatively, each probe could have its own piston and chamber. Except for acceleration and deceleration times of the piston  28 , it is preferred that the movement of the piston, and thus the increasing volume within its respective chamber, provides a constant fluid flow rate from the formation. Referring still to  FIG. 2 , preferably each of the probes  18 ,  20  has an associated pressure transducer, being pressure transducer  30  for the source probe  18  and transducer  32  for the vertical probe  20 . 
   A sequence for determining the permeability of the formation will now be described referring simultaneously to  FIGS. 2A and 3 . Preferably, the wireline formation tool  10  is placed within the borehole at a particular depth level of interest. The probes  18 ,  20  are extended to be in fluid communication with the formation, and likewise stabilizer  22  is extended to aid in providing sufficient force for sealing the probes  18 ,  20  against the formation. Initially, valve  34  is open and thus both the source probe  18  and the vertical probe  20  are in fluid communication with the chamber  26  and piston  28 . As shown in  FIG. 3 , the pressure transducers  30 ,  32  initially sense the hydrostatic pressure within the borehole, which in the exemplary drawing is shown by line  50  and is between 6,050 and 6,150 pounds per square inch (psi). With valve  34  open, the piston  28  is displaced within its chamber  26  causing a fluid flow from the formation through the probes  18 ,  20 . Because of the permeability or resistance to flow of the formation, the pressure transducers  30 ,  32  sense a pressure drop, as indicated by the terms “First Drawdown with Both Probes” in  FIG. 3 . After approximately five seconds, the initial flow or draw-down ceases and each of the pressure transducers  30 ,  32  begin to sense formation pressure, which is lower than the hydrostatic pressure of the drilling mud within the borehole. In the exemplary drawings of  FIG. 3 , the formation pressure is approximately 6,000 psi. After giving the formation time to stabilize from the initial draw-down, a second flow or draw-down is performed. Prior to the second draw-down however, the valve  34 , which was open in the initial draw-down, is closed. Once the vertical probe  20  is isolated from the source probe  18  by way of valve  34 , the piston  28  is again displaced within its chamber  26 , thereby drawing fluids from the formation through the source probe  18 . The piston  28  moving downward within its chamber  26  draws fluid from the formation by creating a differential pressure between the formation and the piston. The differential pressure created is that which is necessary to move the additional volume created by the piston moving within its cylinder. Because the volume rate of flow in the second draw-down test is preferably greater than the volume rate of flow in the initial draw-down test, in one embodiment the second draw-down preferably taking approximately four cubic centimeters per second, and also because the vertical probe  20  is isolated from the piston during the second draw-down test by way of closed valve  34 , the drop in pressure sensed by pressure transducer  30  associated with the source probe  18  is typically greater than that pressure drop sensed during the initial draw-down test. The pressure sensed as a function of time by the transducer  30  is shown in the exemplary drawing of  FIG. 3  in dashed lines. 
   Although the vertical probe  20  is isolated during the second draw-down test, because of its relatively close spacing to the source probe, in the preferred embodiments from six to twelve inches, and because the vertical probe  20  is in fluid communication with the formation, it too senses a pressure drop which has propagated through the formation from the source probe  18  to the vertical probe  20 . The exemplary drawing of  FIG. 3  shows a pressure drop sensed by the pressure transducer  32 , on the order of 9 psi, compared to a source probe sensed pressure drop of approximately 318 psi in the exemplary system. 
   In a second sequence for determining the permeability of the formation, pumping of fluid from the source probe  18  during the second draw-down takes place for an extended period of time. That is, after each probe is coupled to the formation and reads formation pressure (first draw-down), the two probes are isolated from each other using valve  34 , and fluid is pumped through the source probe  18  for an extended period of time.  FIG. 2B  shows tool  10  for performing large or extended draw-down tests having the chamber  26  and piston  28  replaced by a pump  27 . In this arrangement, pump  27  is preferably a positive displacement pump, which could comprise a chamber and piston arrangement similar to that of  FIG. 2A . In the extended pump down procedure, preferably the fluid is pumped through the source probe  18  (discharged into the borehole) at a rate of approximately one-half gallon a minute for approximately one-half hour. Thereafter, the pumping is stopped and the pressure responses sensed by the source probe  18  and the vertical prove  20  (pressure transducer  30  and  32  respectively) are monitored and recorded. For the extended pump-down test, recording times may range, depending on the formation, from ten to thirty minutes, but preferably fifteen minutes. 
   Determining the formation properties of interest of the preferred embodiment involves applying the waveforms representing the pressure sensed by the pressure transducers  30 ,  32  during the second draw-down test (whether the short draw-down taking only a few cubic centimeters or the extended draw-down test taking several gallons), such as those exemplified in  FIG. 3 , to an analytic model. In particular, the analytic model takes into account permeability of the skin, the storage effect downhole, as well as the dip angle of the formation relative to the borehole. The equation of the analytic model for the pressure response of the source probe during a draw-down test is: 
                   Δ   ⁢           ⁢       P   p     ⁡     (       r   p     ,   t     )         =       (           (     1   +     S   d       )     ⁢   μ       k   f       ⁢       Q   0       4   ⁢   π       ⁢       τ   s       r   p         )     ⁢     {         p   ds     ⁡     (       c   d     ,     S   d     ,     t   d       )         (     1   +     S   d       )       }               (   1   )               
where ΔP p  is the pressure response detected at the source probe  18  as a function of source probe radius r p  (which is related to equivalent source probe radius r s ) and time t, and where S d  is a dimensionless skin constant, μ is the formation fluid viscosity, k f  is the spherical permeability, Q 0  is the draw-down flow rate, τ s  is a geometric shape factor to make corrections to the source probe radius, and p ds  is the source pressure transient as a function of the dimensionless compressibility factor c d , the dimensionless skin S d  and dimensionless t d . Before going on to more thoroughly define the various terms of equation (1) above, it is noted that the portion of the equation within parenthesis represents the static response of the analytic model, and the portion within the brackets represents the transient response of the analytic model.
 
   The k f  term within the denominator of the static response portion of equation (1) above is the spherical permeability. The spherical permeability is made up of both horizontal and vertical permeability components. Generally, k f  is defined as: 
                   k   f     =         k   r   2     ⁢     k   z       3             (   2   )               
where k r  is the horizontal permeability, and k z  is the vertical permeability. The permeability k f , including its horizontal and vertical components k r  and k z  respectively, are some of the unknowns to be determined using equation (1) above, as well as equation (14) with regard to the vertical probe, discussed below. The dimensionless skin factor, S d  is given by the equation:
 
                   S   d     =     S   ζ             (   3   )               
where S is the skin factor and ζ is calculated by:
 
                 ζ   =       1     λ   3       ⁢     (       1   2     +         λ     ⁢   arc   ⁢           ⁢   sin   ⁢           ⁢     (       1   -     1   /   λ         )         2   ⁢       1   -     1   /   λ               )               (   4   )               
where λ is the anisotropy defined as the ratio of the vertical to horizontal permeability:
 
                 λ   =       k   z       k   r               (   5   )               
and where S is defined as:
 
                 S   =       δ     r   s       ⁢         k   f       k   s       .               (   6   )               
where δ is the skin thickness. The Q o  term from equation (1) is the flow rate used in the draw-down test. In the exemplary plot shown in  FIG. 3 , the flow rate Q o  for the second draw-down used is 4 cm 3 /s. The term τ s  in the static portion of equation (1) is a geometric shape factor used to make corrections to the source probe radius r p  that include geometric considerations in actual wireline formation testing applications. In particular, the analytic model exemplified, in part, by equation (1) above, was tested in the context of a finite element model and analysis. In that finite element model, the physical source used was a circular area projected against the side of a cylindrical borehole. The analytical solution, however, was developed assuming the physical source as a spheroid. The geometric shape factor τ s  thus compensates the analytical model for the difference in source geometries. Finally the geometric shape factor can be defined as
 τ s =2τζ  (7) 
where τ is determined by the finite element model and ζ is as given above. The value of τ changes based, at least in part, on the diameter of the particular borehole. In general, τ ranges from approximately 1 for large diameter boreholes, to approximately 1.5 for small diameter boreholes, e.g. six inch diameter. Finally, with respect to the static portion of equation (1) above, the r p  is related to the equivalent source radius r s  by the equation:
 
                   r   s     =       r   p       τ   s               (   8   )               
where τ s  is as defined above.
 
   With respect to the transient portion of equation (1) above, the model equation for the pressure transient is: 
                     p   ds     ⁡     (       c   d     ,     S   d     ,     t   d       )       =         1   +     s   d           S   d     ⁢     c   d         ⁢       ∑     n   =   1     3     ⁢     {       1     y   n       ⁢     (     1   -       (     1   +         x   n     ⁢     S   d         1   +     S   d           )     ⁢     e       x   n   2     ⁢     t   d         ⁢     erfc   ⁡     (       -     x   n       ⁢       t   d         )           )       }                 (   9   )               
where y n  is one of:
   y   1   =x   1 ( x   1   −x   2 )( x   1   −x   3 )  (10)   y   2   =x   2 ( x   2   −x   1 )( x   2   −x   3 )  (11)   y   3   =x   3 ( x   3   −x   1 )( x   3   −x   2 )  (12) 
and where x n  are the three roots of the equation:
 
                     x   3     +       (       1   +     S   d         S   d       )     ⁢     x   2       +     x       S   d     ⁢     c   d         +     1       S   d     ⁢     c   d           =   0           (   13   )               
Derivation of equation (9) is provided in the appendix of Society of Petroleum Engineers Paper No. SPE 64650, which is incorporated by reference herein as if reproduced in full below, and in Society of Petroleum Engineers Paper No. SPE 62919, which is also incorporated herein by reference as if reproduced in full below.
 
   In similar fashion to the source probe and its related equations described above, the analytic model of the preferred embodiment is also capable of modeling the time series pressure sensed at the vertical probe  20 . In the particular case where there is a dip angle between the formation and the well bore, and considering skin, the analytic model of the vertical probe pressure during a draw-down test is: 
                   Δ   ⁢           ⁢       P     z   ⁢               ⁡     (     z   ,   t     )         =       (         Q   0       4   ⁢           ⁢   π       ⁢       τ   v       r   v       ⁢     μ     k   r       ⁢     1         λ   ⁢           ⁢       sin   ⁡     (   θ   )       2       +       cos   ⁡     (   θ   )       2             )     ⁢     {       r   dv     ⁢       p   dv     ⁡     (       r   dv     ,     c   d     ,     S   d     ,     t   d       )         }               (   14   )               
where Q 0  is the flow rate defined above, μ is the formation fluid viscosity, k r  is the horizontal permeability, λ is the ratio of the vertical to horizontal permeability as defined above, θ is the dip angle, and p d  is the dimensionless transient pressure response expected for the vertical probe as a function of the dimensionless vertical probe radius r dv , the dimensionless compressibility constant c d , the dimensionless skin factor S d  and dimensionless time t d , where the dimensionless t d  is given by the equation:
 
                   t   d     =       t     ϕ   ⁢           ⁢     cr   s   2         ⁢       k   f     μ               (   15   )               
where φ is the formation porosity, and c is the total compressibility, and where the dimensionless vertical probe radius r dv  is given by:
 
                   r   dv     =         r   v       r   s       ⁢       (       k   r       k   z       )       1   3                 (   16   )               
where r v  is the probe spacing.
 
   Similar to the model equation of the source probe equation (14) above is logically divided into a static portion, contained in parenthesis, and a transient portion, contained in brackets. The transient portion of the analytic model representing the vertical probe response is given by: 
                     p   dv     ⁡     (       r   dv     ,     c   d     ,     S   d     ,     t   d       )       =       1       S   d     ⁢     r   dv     ⁢     c   d         ⁢       ∑     n   =   1     3     ⁢     {       1     y   n       ⁢     (             erfc   ⁡     (       1   -     r   dv         2   ⁢       t   d           )       -                 e     (         -     (     1   -     r   dv       )       ⁢     x   n       +       x   n   2     ⁢     t   d         )       ⁢     erfc   ⁡     (         1   -     r   dv         2   ⁢       t   d           -       x   n     ⁢       t   d           )               )       }                 (   17   )               
where r dv  is dimensionless probe spacing, and where the remaining parameters are as defined above.
 
   By combining equations (1) and (14), the anisotropy for the particular formation is thus as follows: 
                 λ   =         k   z       k   r       =         (           (     1   +     S   d       )     ⁢   Δ   ⁢           ⁢     P   z         Δ   ⁢           ⁢     P   p         ⁢           ⁢         τ   s     ⁢     r   v           τ   v     ⁢     r   s         ⁢         λ   ⁢           ⁢       sin   ⁡     (   θ   )       2       +       cos   ⁡     (   θ   )       2           )     3     ⁢       {         p   dv     ⁡     (       c   d     ,     S   d     ,     t   d       )           (     1   +     S   d       )     ⁢     r   dv     ⁢       p   dv     ⁡     (       r   dv     ,     c   d     ,     S   d     ,     t   d       )           }     3                 (   18   )               
and if the dip angle is zero, this reduces to
 
   
     
       
         
           
             
               
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                 19 
                 ) 
               
             
           
         
       
     
   
   Applying the actual sensed pressure time series to the analytic models exemplified in equations (1) and (14) will not yield the desired formation parameters information in only one calculation. That is, the variables within these equations are manipulated to make the analytic model predicted pressure response match the actual formation test data thus yielding the parameters of interest. Preferably, fluid viscosity μ is supplied by an external source, such as by MWD tools, or preferably by a viscosity analysis tool or meter coupled to the tool  10 . An initial estimate as to the various parameters for the analytical model is made and the results predicted by the analytical model are compared to the actual time series. The parameters are accordingly adjusted, and the model run again using regression analysis techniques known to those of ordinary skill in the art. The regression analysis could take many hundreds or thousands of iterations. The regression analysis stops, in the preferred embodiments, when the modeled pressure response substantially matches the actual pressure response, within approximately five percent. The final state of the manipulated parameters when the regression analysis completes are the formation parameters determined by the model. 
   In embodiments having two probes, a source probe  18  and a vertical probe  20 , the dip angle must be provided for there to be a solution to the equations. Obtaining dip angle prior to running the wireline formation testing of the preferred embodiment may be obtained by many known techniques, including measuring-while-drilling (MWD) devices capable of taking these measurements. Other parameters such as the viscosity of the formation fluid and compressibility of the formation fluid (which manifest itself as the storage effect) may be determined by other downhole tools, including the wireline formation tool itself prior to, during, or after the permeability test described herein. In yet another embodiment, only the source probe could be used, along with the analytical model of equation (1), and in so doing the spherical permeability, along with the skin, could be determined. 
   In the preferred embodiments, two probes and an externally supplied dip angle are used to determine the horizontal permeability and the permeability anisotropy. The analytic models may be manipulated, however, such that the determination as to permeability may consider only skin, or only dip angle. In the case of a permeability determination taking into consideration skin of the formation, but not dip angle, equation (1) above for the source probe would remain unchanged, but equation (14) modeling the vertical probe response reduces to: 
                   Δ   ⁢           ⁢       P   Z     ⁡     (   t   )         =       (         Q   0       4   ⁢   π       ⁢       τ   v       r   v       ⁢     μ     k   r         )     ⁢     {       r   dv     ⁢       p   d     ⁡     (       r   dv     ,     c   d     ,     S   d     ,     t   d       )         }               (   20   )               
with P d  as defined in equation (9) above.
 
   In the case of a permeability determination taking into dip angle but not skin, equation (1) becomes: 
                   Δ   ⁢           ⁢       P   p     ⁡     (       r   p     ,   t     )         =       (           Q   0     ⁢     τ   s         4   ⁢   π   ⁢           ⁢     r   p         ⁢     μ     k   f         )     ⁢       p   d     ⁡     (       c   d     ,     t   d       )                 (   21   )               
where p d  is given by:
 
                     p   d     ⁡     (       c   d     ,     t   d       )       =       1       r   d     ⁢       1   -     4   ⁢     c   d               ⁢       ∑     n   =   1     2     ⁢           (     -   1     )       n   +   1         β   n       ⁡     [     1   -       e     (       β   n   2     ⁢     t   d       )       ⁢     erfc   ⁡     (       β   n     ⁢       t   d         )           ]                   (   21   )               
where β n  is given substantially by:
 
                   β   1     =       1   -       1   -     4   ⁢     c   d               2   ⁢     c   d                 (   23   )                 β   2     =       1   +       1   -     4   ⁢     c   d               2   ⁢     c   d                 (   24   )               
In this second case, taking into account dip angle but not skin, the equation for the vertical probe is the same as equation (14) above.
 
   Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. While the preferred embodiment is a wireline formation tester, the system and methods described in this specification could be implemented in tool within a drill string. In an drill string embodiment, drilling ceases during tested, and solving the analytic models described above takes place on a microprocessor or microprocessors within the tool. The results may be stored for later retrieval, or the results or summaries of the results telemetered to the surface using known or after developed techniques. It is intended that the following claims be interpreted to embrace all such variations and modifications.