An Improved NMR Permeability Model for Macromolecules Flowing in Porous Medium

Abstract

The extraction of macromolecules from nano-self-assembled material can be used as a laboratory model for enhancing oil recovery in reservoirs. By combining Darcy’s law and Poiseuille equation, an improved nuclear magnetic resonance (NMR) permeability model, suitable for macromolecular flow in mesopores is obtained. The calibration coefficients in the Coates equation are expressed in terms of the physical parameters of pore throat ratio r_b/r_t, tortuosity, and thickness of bond film in the improved model. The results show that the proportion of irreducible fluid to total fluid obtained through NMR characterization can reflect the variation tendency of irreducible macromolecule and water. By simplifying the pores of the extracted samples, the thickness model of irreducible macromolecule and water is established with the total thicknesses calculated as 1.482 nm, 1.585 nm, 1.674 nm, and 1.834 nm. The corresponding permeability results obtained from the NMR characterization (K_NMR) are 7.39 mD, 6.02 mD, 5.27 mD, and 6.25 mD. The permeability results obtained from mercury intrusion experiment (K_HG) are 5.10 mD, 4.73 mD, 5.82 mD, and 5.56 mD, and those from the Darcy flow experiment (K_D) are 4.1 mD and 5.19 mD. The absolute deviation between K_NMR and K_HG varies from 0.69 to 2.29 mD, while that between K_NMR and K_D is 1.58 mD. This method can be applied to the enhanced recovery of shale oil.

Appendix

When incompressible viscous fluid flows along a horizontal pipe, it conforms to Poiseuille’s law, where q (m³/s) is fluid flow; ΔP (Pa) is the pressure drop at both ends of the tube; L (m) is the length of a porous medium model; η (Pa s) is fluid viscosity; r (m) is tube radius; and τ is tortuosity, which is the ratio of pore channel length l and porous medium L

$$q = \frac{\pi }{8\eta }r^{4} \frac{1}{\tau }\frac{\Delta p}{L}$$

(17)

The Darcy law is the experimental law reflecting the flow of fluid in porous media. It describes the linear relationship between flow velocity and hydraulic gradient

$$q = \frac{k}{\eta }A\frac{\Delta p}{L}$$

(18)

The permeability formula of a single capillary can be derived from combining the above two formulas

$$k = \frac{\pi }{8A}r^{4} \frac{1}{\tau }$$

(19)

The expression of the porosity of a cylindrical pore in porous media is given by

$$\emptyset = \frac{{V_{\text{pore}} }}{{V_{\text{total}} }} = \frac{{\pi r^{2} l}}{AL} = \frac{{\pi r^{2} \tau }}{A}.$$

(20)

Combining formulas (19) and (20),

$$k = \frac{1}{8}\emptyset \frac{{r^{2} }}{{\tau^{2} }}.$$

(21)

The specific surface to volume ratio of cylindrical channels is,

$$\frac{S}{V} = \frac{2\pi rl}{{\pi r^{2} l}} = \frac{2}{r}$$

(22)

Combining formulas (21) and (22),

$$k = \frac{1}{2}\emptyset \frac{1}{{\tau^{2} }}\left( {\frac{V}{S}} \right)^{2}$$

(23)

The cylindrical pore is an ideal pore shape for porous media. In fact, the aperture changes constantly. We take this variable factor into account and assume that the variable aperture conforms to a linear function

$$r\left( x \right) = r_{\text{t}} + ax,$$

(24)

where a is the pore shape factor,

$$a = \frac{{r_{\text{b}} - r_{\text{t}} }}{l}$$

(25)

The tortuosity can be expressed as $\tau = l/L = 1/\cos \left( \gamma \right)$. Equation (19) is integrated along the X-axis of the pore channel

$$k = \frac{\pi }{8A}\frac{1}{\tau }\frac{1}{l}\mathop \int \limits_{0}^{l} r^{4} \left( x \right){\text{d}}x$$

(26)

The integral yields the following equation,

$$k = \frac{{\pi r_{\text{b}}^{4} }}{40A\tau }\left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{3} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{4} } \right]$$

(27)

The porosity of the model (28) can be obtained by (Eq. 20)

$$\varphi = \frac{\pi \tau }{Al}\mathop \int \limits_{0}^{l} r^{2} \left( x \right){\text{d}}x$$

(28)

The integral along the X-axis including pore factor (Eq. 25) yields

$$\varphi = \frac{\pi \tau }{A} \frac{{\left( {r_{\text{b}}^{3} - r_{\text{t}}^{3} } \right)}}{{3\left( {r_{\text{b}} - r_{\text{t}} } \right)}} = \frac{{\pi \tau r_{\text{b}}^{2} \left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} } \right]}}{3A}$$

(29)

The surface area and volume of the model can be written as follows,

$$S = 2\pi \tau \mathop \int \limits_{0}^{l} \left( {r_{\text{t}} + ax} \right){\text{d}}x = \pi \tau l\frac{{\left( {r_{\text{b}}^{2} - r_{\text{t}}^{2} } \right)}}{{\left( {r_{\text{b}} - r_{\text{t}} } \right)}}$$

(30)

$$V = \pi \tau \mathop \int \limits_{0}^{l} \left( {r_{\text{t}} + ax} \right)^{2} {\text{d}}x = \frac{\pi \tau l}{3}\frac{{\left( {r_{\text{b}}^{3} - r_{\text{t}}^{3} } \right)}}{{\left( {r_{\text{b}} - r_{\text{t}} } \right)}}$$

(31)

The expression of the specific surface area can be derived as

$$\frac{S}{V} = \frac{{3\left( {r_{\text{b}}^{2} - r_{\text{t}}^{2} } \right)}}{{\left( {r_{\text{b}}^{3} - r_{\text{t}}^{3} } \right)}} = \frac{{3\left( {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)}}{{r_{\text{b}} \left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} } \right]}}$$

(32)

The formula (32) can be obtained in the following formula,

$$r_{\text{b}} = \frac{{3\left( {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)}}{{\frac{S}{V}\left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} } \right]}}$$

(33)

Combining Eqs. (27) and (29) to obtain Eq. (34)

$$k = \frac{{\varphi r_{\text{b}}^{2} }}{{8\tau^{2} }}\left[ {\frac{{3\left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{3} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{4} } \right]}}{{5\left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} } \right]}}} \right].$$

(34)

Equation (33) is put into Eq. (34) to obtain Eq. (35).

$$k = \frac{1}{2}\varphi \frac{1}{{\tau^{2} }}\frac{1}{{\left( {\frac{S}{V}} \right)^{2} }}\left\{ {\frac{27}{20} \cdot \frac{{\left( {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} \left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{3} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{4} } \right]}}{{\left[ {1 + \frac{{r_{\text{t}} }}{{r_{\text{b}} }} + \left( {\frac{{r_{\text{t}} }}{{r_{\text{b}} }}} \right)^{2} } \right]^{3} }}} \right\}$$

(35)

The relationship between the surface area to volume ratio and film thickness, BVI, and BVM is as follows.

$$\frac{S}{V} \approx \frac{1}{d}\frac{\hbox{BVI}}{\text{BVM}}$$

(36)

Equation (36) is put into Eq. (35) to obtain Eq. (37).

$$k = \frac{{\varphi d^{2} }}{{2\tau ^{2} }}\left( {\frac{{{\text{BVM}}}}{{{\text{BVI}}}}} \right)^{2} \left\{ {\frac{{27}}{{20}} \cdot \frac{{\left( {1 + \frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }}} \right)^{2} \left[ {1 + \frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }} + \left( {\frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }}} \right)^{2} + \left( {\frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }}} \right)^{3} + \left( {\frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }}} \right)^{4} } \right]}}{{\left[ {1 + \frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }} + \left( {\frac{{r_{{\text{t}}} }}{{r_{{\text{b}}} }}} \right)^{2} } \right]^{3} }}} \right\}$$

(37)