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Abstract

Archie empirical law is the basis of quantitative Petrophysics. The physical significance of this law is not well understood. This issue involves substantial uncertainty in oil in place.

Similarly, Carman-Kozeny (C-K) relationship is source of several permeability models. C-K is derived from Poiseuille equation, applicable in laminar viscous flow in straight non-communicating uniform tubes. Neither Poiseuille nor C-K formulae consider convective/inertial accelerations caused by changes in either cross section or flow direction. Implications include sizeable limitations in permeability modeling.

In this paper, appropriate hydrodynamic and electrical models are created using fluid mechanics analytical methods. The models delineate the velocity and electrical potentials, streamlines and controls, represented by C-K and Archie equations. This approach theoretically verifies both relations and demonstrates that:

• Fluid circulation and permeability are optimally modeled invoking superposition of viscous and inertial regimes in nozzles, throats, diffusers, pipe networks, and arrays of solid particles. These hydraulic components, once assembled, emulate porous media well.

• Changes in fluid and electric flow direction are characterized by the flow around a corner solution of Laplace differential equation.

• The solution precisely defines rock frame, conductive phase, and hydraulic tortuosities, enabling direct ties to pore geometry. This facilitates permeability calculations utilizing the “perfect permeability transform” procedure.

Several experiments/examples are discussed to show validity. Some conclusions and technical contributions/applications are:

• Electric measurements can predict hydraulic and petrophysical rock properties.

• Cementation exponent can be continuously computed employing acoustic tools and rock physics principles.

• Saturation exponent can be geometrically related to wettability and saturation using pore-scale modeling.

• A quantitative rock catalog can be implemented applying practical rock type definitions.

• Supported by these concepts, synthetic production logs can be generated to anticipate well performance, and to effectively assist in history matching.

• Archie law ceases being merely empirical. We gain a thorough physical understanding of its power law (fractal?) behavior. Rock responses can now be clearly explained and evaluated.

Introduction

Archie and C-K are very popular/famous equations. Their usage is widespread despite a common lack of full comprehension of these relationships.[1]  Different versions of Archie and C-K equations exist. Their parameters adopt varied values based on simplifying assumptions, which affect the accuracy of water saturation and permeability estimates. 

Both oversimplification and ambiguity, created by the diversity of Archie and C-K equation forms, will be addressed first. Then, fluid mechanics and electric field considerations will be employed, to quantify the influence of both the viscosity of the fluids and the geometry of porous media. This quantification is the main difficulty in permeability modeling, and in understanding fluid and electric flow.

A general tactic to solve the problem dictates a blend of theory and experiment. A wealth of experimental results is already available in the literature. Although this paper is more centered in the theory, the purpose is to join these notions with existing experimentation, relieving doubts and gaining exactitude in the calculations.

Archie. A tortuosity coefficient (a) always equal to 1 is an unreasonable assumption according to Adisoemarta et al.[1] They sustain that a ≥ 1 should be used, based on core measurements and the definition of tortuosity. Other authors claim that a should be 1 to comply with physical bounds at 100% porosity.[2,3] Archie himself assumed a = 1.4 Some publications promote values of a ≤ 1.5 Mavko et al. mention that a > 1 is usually needed to fit the data of non-Archie rocks.[6] In general, a = 1 is employed for simplification purposes. There is not an universal agreement about the correct value of a.

Numerous hypotheses about the variation of cementation exponent (m) values are available, especially for non-Archie rocks. Formation factor (F) exhibits noticeable curvature close to the origin on the Co-Cw (rock vs. brine conductivities) plot for shaly sands, suggesting that m should vary if formation water salinity changes.[7]