On the Pore Geometry and Structure Rock Typing

Rock typing is a vital step in oil and gas reservoir development to achieve predictions of hydrocarbon reserves, recovery, and underground storage capacity for CO2 or hydrogen. To address inaccurate initial hydrocarbon-in-place prediction and improper rock property distribution in a reservoir model, a recent rock typing method, pore geometry and structure (PGS), has revealed a more accurate prediction on connate water saturation and better grouping of capillary pressure. However, the current state still needs physical interpretations of the PGS rock typing. We have compiled thousands of experimentally measured hydraulic properties, such as permeability k within 12 orders of magnitude, porosity ϕ up to 0.9, specific surface area SS within 4 orders of magnitude, and pore size R ranges around 3 orders of magnitude. We conduct the first-ever holistic physical interpretations of the PGS rock typing using gathered data combined with analytical theory and the Kozeny–Carman equation. Surprisingly, our physics-inspired data-driven study reveals advanced findings on the PGS rock typing. These include (i) why PGS method prevails over the hydraulic flow unit rock typing, (ii) explanations to distinguish between causality and indirect relationships among hydraulic properties, rock type number, and electrical resistivity, (iii) a proposed novel method: permeability prediction from the resistivity and rock type number relationship, and (iv) a suggestion and criticism on how to avoid a recursive prediction on permeability.


APPENDICES 1. Appendix A: Permeability
Newton's Viscosity Law expresses that viscosity  is a measure of its resistance to deforming at a certain shear rate / and the fluid deformation itself is due to a shear stress   constituted by the drag force   working on a tangential surface   along the pipe with a length of ℓ: The drag force   is quantifiable through a given pressure drop ∆ working on a cross-sectional  Assign  = 0 at the pipe edge ( = ) and  =   at the center o thef pipe ( = 0).
Therefore, we have velocity  at an arbitrary  derived as follow Compute the volumetric rate  passing through into a pipe with a crossectional area   : We call Eq.A-3d as Hagen-Poiseuille equation.Tortuosity  is the ratio between fluid path ℓ and the straight bulk length , which has a dimensionless unit and typical values of There are  numbers of bundled pipes that contribute to the total rate  passing through the rock: . Recall the definition of porosity  to define the bulk crossectional area : Arrange Eq.A-5d using A-6b, such that Recall the definition of Darcy's equation: Then, equate the formulations in Eq.A-7 and A-8; therefore, the absolute permeability  equals to be Note: The  is the pore diameter.The dimension of absolute permeability  is [L 2 ] with the unit of m 2 , Darcy, or milli-Darcy mD.
We will derive the conversion of permeability in petroleum engineering  into geotechnical or civil engineering's permeability, known as hydraulic conductivity  ℎ , with the dimension of [L/T] or unit of cm/s or m/s.The total fluid velocity   in porous media is driven by the total head gradient  ≡ ℎ/ through a geomaterial with a hydraulic conductivity  ℎ : The studied liquid in geotechnical engineering is mainly water.So, it is not important to explicitly state the viscosity .Recall  in Eq.A-8 to be Recall the definition of static pressure  = ℎ and the total head gradient  = ℎ/.Then, equate the petroleum engineering version on the left-hand-side with the geotechnical engineering version on the right-hand-side: Note that viscosity  has a unit of mPa.s and dimension of [ML -1 T -1 ].Thus, we can convert the hydraulic conductivity  ℎ [LT -1 ] into permeability  [L 2 ] using Eq.A-12c (unit in m 2 ).Then, we need the value of 0.9869233×10 −12 m 2 /D to convert into Darcy or 9.869233×10 −16 m 2 /mD (milli-Darcy).

Appendix B: Specific surface area
Volumetric specific surface area   [1/m] or [1/cm] is defined as The substitution of  2  in Eq.A-6b into B-1e results in Gravimetric specific surface area   [m 2 /g], also just called Specific surface with a known mineral density   is defined as follows: We evaluate a volumetric composition of mineral   that is obtained from total   and substracted by pore volume   .Substitute Eq.B-2b into B-3e in the process of derivation.Some works come up with the terminology of volumetric grain specific surface    , which is an analogue of   (Eq.B-3f):  Suppose  >  ≫  which results in 1/ ≫ 1/ > 1/, thereby the   is Assume the structure is formed by a few platy minerals, i.e., parallel sheets.Therefore, the porosity and pore space   established in the parallel sheet emerge to be Input the mineral thickness  from Eq. B-5c into Eq.B-6.Accordingly, the pore space in the parallel sheet   is then defined as We recall B-3f in diameter variable  = 2 (cylindrical pore) to be See the comparison between Eqs.B-7 and B-8 designate parallel sheet vs cylindrical pores.We can generalise the factors of 2 (parallel sheet) or 4 (cylindrical) to be the geometrical factor  as in the pore diameter

Appendix C: Kozeny-Carman equation
The substitution of  = (  ) from Eq. B-2b,  = (  ) from Eq. B-3f, and  = (  , ) from Eq. B-9 into the definition of permeability  (Eq.A-9a) yields in the Kozeny-Carman equation in these forms: We will compare the Kozeny-Carman equation in petroleum and geotechnical engineering.The equation in geotechnical engineering is presented as follows 1 : Convert the hydraulic conductivity in geotechnical engineering  ℎ into petroleum engineering  (Eq.A-12c); alter the void ratio  into porosity  with the relationship of  = /(1 − ), and expand the kinematic fluid viscosity   to a dynamic fluid viscosity  =   ; those lead to Later, compare Eqs.C-2 and C-4c, which results in The value of pore topology constant   of 0.2 1,2 suggests that the tortuosity of soil samples  is around 1.58.
If we move a porosity variable  into the left-hand side of Eq.C-2 and then perform square root operation, we get the concept of hydraulic flow unit HFU: The HFU concept consists of Reservoir Quality Index , normalised porosity   , and flow zone indicator .This concept is just the way to plot where the  is not explicitly calculated from ,   , and   but rather /  .Note: the unit of  in Eq.C-6a is m 2 .

Appendix D: Electrical conductivity
Suppose the value of one is expandable to a ratio of volume to volume: where the subscript : total or bulk, : fluid, : mineral, and : surface.Note that the volume is approached with a tube model with a cross-sectional area  and length .Recall the definition of the resistance-resistivity - which is equivalent to conductance-conductivity -: Electronic conductions in solid minerals, ionic conductions in pore fluid, and counterion transports due to mineral-fluid interactions can be modelled as a parallel circuit if each phenomenon is in the parallel orientation within the applied electrical field [3][4][5] .We use the basic equation (Eq.D-2) to model the parallel circuit among the fluid, mineral and surface components: Suppose the tube length over the bulk length is defined as the tortuosity .The expression of surface-related cross-sectional area   is constructed from the diffuse double layer with the thickness of  and covers along the pore perimeter 2 (Figure S 3).Together with the definition of   , substitute Eq.D-1 into Eq.D-3d to obtain this set of equations:  The internal surface  can be described with  =       =     (1 − )  .Then, expand the   = ; therefore, The thickness of diffuse double layer  could be approached by the Debye-Huckel characteristic length 6 : with the associated variables are bulk fluid relative permittivity  ′ , vacuum permittivity  0 , Boltzmann constant , temperature , Avogadro number   , elementary charge , ionic strength , the concentration of each ion   , and valence number   .
We simplify the form of Eq.D-5c with assumed tortuosity   =   = 1, generally called as , and define a surface conduction factor Γ =     /  2 : In the clayey sample, the contribution of   is huge compared to   and the mineral conductivity is minute,   ≪   .
The first term of Eq.D-5c prevails in a clean sample, which we name Archie's equation where   > 0. Note: electrical conductivity  [S/m] is the reciprocal of resistivity  [Ω.m].

Appendix E: PGS Permeability fitting
The pore geometry and structure (PGS) concept is defined as the plot between √   and   3 that is connected by positive constants of  and : We will show the derivation of permeability prediction formulation using PGS constants  and .The fitting method consists of two other constants  and  which correlate the water saturation   and measured permeability .The predicted permeability  ̂ indeed uses a recursive method and non-explicitly forward model: The following derivations detail formulations derived from routine core analysis RCA and special core analysis SCA data.The data derived from the SCA technique provides a negative slope − between the (irreducible) water saturation   and measured permeability  9,10 : We set constants  and  as positive values.Substitution of Eq.E-3 into the left-hand side of Eq. E-1 results in The -th root operation on both sides yields in Rearrange Eq.E-5 to state the predicted permeability  ̂ explicitly: Note: the constants {, } are from RCA data, while {, } are from SCA tests.

6.
Appendix F: Effects of specific surface area and tortuosity onto the PGS Rock Type area   .The tangential shear stress   would resist the fluid flow and hinder velocity  across the radius of pipe .The negative sign denotes the reduction of velocity with the growth of radius from the centre of the pipe.Note: vectors of  and  are perpendicular (Figure S 1).

Figure S 1 .
Figure S 1. Illustration of fluid flow in the pore.The drag force   on cylindrical pore with a length of ℓ.The distribution of velocity  decreases as the radius  reaches the edge of pipe at .
Next, let us derive another form of a specific surface   according to the physical structures of the mineral, for example, a long platy mineral as depicted in Figure S 2.

Figure S 2 .
Figure S 2. Depiction of platy minerals to illustrate parallel clay.

Figure S 3 .
Figure S 3. Double layer  emergence between the bulk pore fluid and solid surface.The thickness of  ≪  hence we can assume that  is the distance of solid surface to the centre of pore.

Figure S 4 .Figure S 5 .%
Figure S 4. Conventional permeability-porosity cross-plot with known PGS rock type RT lines.PGS stands for the Pore Geometry and Structure.Permeability is computed from the Kozeny-Carman equations with porosity from 0.0001 to 0.99.Circles are to show the intersection points.Symbol of Ss for specific surface area.
we use arbitrary positive constants   and   (Eq.D-8b) to fit the data and accommodate the surface conduction (2 nd term of Eq.D-7).We call the squared-tortuosity  2 as the tortuosity factor   > 0. Saturated sample   = 1 alters Archie's equation to be and