Online NMR Flowing Fluid Measurements

Abstract

Online nuclear magnetic resonance (NMR) is an efficient fluid detection method for fluid flowing measurement. In our previous work, the velocity effects on the motion NMR measurement were studied and the velocity correction method was proposed. However, the flow state effects on NMR measurement is more complicated than motion velocity effects. Thus, the influence of flow state on NMR measurement is investigated. A multi-functional low-field and online NMR fluid analysis system is developed to realize the rapid measurement of one-dimensional and two-dimensional NMR relaxation times of fluids under slow laminar flow state.

Appendix: The Flow Velocity Distributions in Circular Tube Under Different Flow Patterns

• (1) Bingham model [8].

Bingham mode velocity distribution function v(r)_b:

$$v(r)_{\text{b}} = \left\{ \begin{aligned} \frac{{\Delta P_{\text{b}} }}{{4\eta_{\text{p}} L_{\text{A}} }}(R - r)^{2} ,r \le r_{m} \hfill \\ \frac{{\Delta P_{\text{b}} }}{{4\eta_{\text{p}} L_{\text{A}} }}[(R - r_{m} )^{2} - (r - r_{m} )^{2} ],r > r_{m} \hfill \\ \end{aligned} \right\},$$

(8)

where △P _b is the Bingham mode fluid pressure difference at both sides of the pipe in the detection range:

$$\Delta P_{\text{b}} = \frac{{8\eta_{\text{p}} L_{\text{A}} u}}{{R^{2} }} + \frac{{8\tau_{0} L_{\text{A}} }}{3R},$$

(9)

L _A is the length of antenna, R is the diameter of the sample tube, η _p is the plastic viscosity of Bingham mode fluid:

$$\eta_{\text{p}} = 300\frac{{\theta_{2} - \theta_{1} }}{{N_{2} - N_{1} }},$$

(10)

θ ₁, θ ₂ represent the rheological coefficients is in rotary viscometer’s rotor with rotation rate N ₁, N ₂, respectively, τ ₀ is the yield value:

$$\tau_{0} = \frac{{K_{\tau } (\theta_{1} N_{2} - \theta_{2} N_{1} )}}{{B_{\text{b}} (N_{2} - N_{1} )}},$$

(11)

the constant K _τ is determined by rotary viscometer inner cylinder size and the characteristics of the spring, B _b is the Bingham fluid correction coefficient:

$$B_{\text{b}} = \frac{{2R_{0}^{2} }}{{R_{0}^{2} - R_{i}^{2} }}\ln \frac{{R_{0} }}{{R_{i} }},$$

(12)

R ₀, R _i represent the rotary viscometer’s outer and inner tube diameter, the flow core radius:

$$r_{m} = \frac{{2L_{\text{A}} \tau_{0} }}{\Delta P}.$$

(13)

• (2) Ostwald-de Waele power law model.

Power law mode velocity distribution function v(r)_p:

$$v(r)_{\text{p}} = \frac{n}{n + 1}\left( {\frac{{\Delta p_{\text{p}} }}{{2KL_{\text{A}} }}} \right)^{{\frac{1}{n}}} \left( {R^{{\frac{n + 1}{n}}} - r^{{\frac{n + 1}{n}}} } \right),$$

(14)

where △P _p is the Power law mode fluid pressure difference at both sides of the pipe in the detection range:

$$\Delta P_{\text{p}} = \frac{2KA}{R}\left( {\frac{3n + 1}{4n}} \right)^{n} \left( {\frac{4u}{R}} \right)^{n} ,$$

(15)

n is the liquidity index:

$$n = \frac{{\lg (\theta_{2} /\theta_{1} )}}{{{ \lg }(N_{2} /N_{1} )}},$$

(16)

K is the viscosity coefficient:

$$K = \frac{{0.511\theta_{600} }}{{(1022B_{\text{p}} )^{n} }},$$

(17)

B _p is Power law mode fluid correction coefficient:

$$B_{\text{p}} = \frac{{(R_{i} /R_{0} )^{2} - 1}}{{n[(R_{i} /R_{0} )^{2/n} - 1]}}.$$

(18)

• (3) Casson model [9].

Casson mode velocity distribution function v(r)_c:

$$v(r)_{c} = \frac{1}{{\eta_{k} }}\left[ {\frac{{\Delta P_{\text{c}} }}{{4L_{\text{A}} }}(R^{2} - r^{2} ) - \frac{4}{3}\tau_{\text{c}}^{0.5} \left( {\frac{{\Delta P_{\text{c}} }}{{2L_{\text{A}} }}} \right)^{0.5} (R^{1.5} - r^{1.5} ) + \tau_{\text{c}} (R - r)} \right],$$

(19)

where △P _c is the Casson mode fluid pressure difference at both sides of the pipe in the detection range:

$$\Delta P_{\text{c}} = \left[ {\frac{2A}{R}\left( {\frac{{4\eta_{k} u}}{R} - \frac{4}{147}\tau_{\text{c}} } \right)^{2} + \frac{8}{7}\tau_{\text{c}}^{0.5} } \right]^{2} ,$$

(20)

η _k is the plastic viscosity of Casson mode fluid:

$$\eta_{k} = \left[ {\frac{{\sqrt {K_{\tau } } (\sqrt {\theta_{2} } - \sqrt {\theta_{1} } )}}{{\sqrt {K_{\gamma } } (\sqrt {N_{2} } - \sqrt {N_{1} } )}}} \right]^{2} ,$$

(21)

K _γ according to the rotary viscometer’s outer and inner tube diameter, B _c is Casson mode fluid correction coefficient:

$$B_{\text{c}} = 2\frac{{R_{0} }}{{R_{0} + R_{i} }},$$

(22)

τ _c is the Casson yield value:

$$\tau_{\text{c}} = \left[ {\frac{{\sqrt {K_{\tau } } (\sqrt {\theta_{1} N_{2} } - \sqrt {\theta_{2} N_{1} } )}}{{B_{\text{c}} (\sqrt {N_{2} } - \sqrt {N_{1} } )}}} \right]^{2} .$$

(23)

• (4) Herschel–Bulkley model [10]

H-B mode velocity distribution function v(r)_h:

$$v(r) = \frac{n}{n + 1}\left( {\frac{{\Delta P_{\text{h}} }}{2KA}} \right)^{{\frac{1}{n}}} [(R - r_{m} )^{{\frac{n + 1}{n}}} - (r - r_{m} )^{{\frac{n + 1}{n}}} ],$$

(24)

make the coefficients a = θ ₆₀₀ − θ ₃₀₀, b = θ ₆ − θ ₃, a′ = θ ₃₀₀ + θ ₆, b′ = θ ₆₀₀ + θ ₃, c = θ ₆₀₀ × θ ₃, d = θ ₃₀₀ × θ ₆, here θ ₃, θ ₃₀₀, θ ₆₀₀ represent the rheological coefficients is in rotary viscometer’s rotor with rotation rate N ₃ = 3 r/min, N ₃₀₀ = 300 r/min and N ₆₀₀ = 600 r/min, respectively, △P _h is the H-B mode fluid pressure difference at both sides of the pipe in the detection range:

$$\Delta P_{\text{h}} = \frac{{2KL_{\text{A}} }}{R}\left( {\frac{3n + 1}{4n}} \right)^{n} \left( {\frac{4u}{R}} \right)^{n} + \left( {\frac{3n + 1}{2n + 1}} \right)\frac{{2\tau_{0} L_{\text{A}} }}{R},$$

(25)

yield value τ ₀:

$$\tau_{0} = 0.511\frac{{\theta_{600} \theta_{3} - \theta_{x}^{2} }}{{C_{1} (\theta_{600} + \theta_{3} - 2\theta_{x}^{{}} )}},$$

(26)

θ _x  = τ _x /0.511, the coefficient C ₁:

$$C_{1} = \frac{1}{1 - n}\frac{{1 - (R_{i} /R_{0} )^{2/n - 2} }}{{1 - (R_{i} /R_{0} )^{2/n} }},$$

(27)

n is the liquidity index:

$$n = \frac{{\lg [(\theta_{600} - \theta_{x} )/(\theta_{x} - \theta_{3} )]}}{{\lg (N_{600} /N_{3} )}},$$

(28)

K is the viscosity coefficient:

$$K = \frac{1}{{C_{2} }}\frac{{(\theta_{600} - \theta_{3} )}}{{(N_{600}^{n} - N_{3}^{n} )}},$$

(29)

coefficients C ₂:

$$C_{2} = \left\{ {\frac{0.20944}{{n[1 - (R_{i} /R_{0} )^{2/n} ]}}} \right\}^{n} ,$$

(30)

the flow core radius r _m :

$$r_{m} = \frac{{2A\tau_{0} }}{{\Delta P_{\text{h}} }}.$$

(31)

• (5) Robertson–Stiff model [11]

R-S mode velocity distribution function v(r)_l:

$$v(r)_{\text{l}} = \frac{{B^{{\prime }} }}{{B^{{\prime }} + 1}}\left( {\frac{{\Delta P_{l} }}{{2A^{{\prime }} L_{\text{A}} }}} \right)^{{\frac{1}{{B^{{\prime }} }}}} (R^{{\frac{{B^{{\prime }} + 1}}{{B^{{\prime }} }}}} - r^{{\frac{{B^{{\prime }} + 1}}{{B^{{\prime }} }}}} ) - C^{{\prime }} (R - r).$$

(32)

where, △P _l is the R-S mode fluid pressure difference at both sides of the pipe in the detection range:

$$\Delta P_{\text{l}} = \frac{{2A^{{\prime }} A}}{R}\left[ {\frac{3B + 1}{{B^{{\prime }} R}}\left( {u + \frac{{C^{{\prime }} R}}{3}} \right)} \right]^{{B^{{\prime }} }} ,$$

(33)

the viscosity coefficients A′:

$$A^{{\prime }} = 0.511\left[ {\frac{{B^{{\prime }} (\theta_{600}^{{1/B^{{\prime }} }} - \theta_{x}^{{1/B^{{\prime }} }} )(1 - 0.93646^{{2/B^{{\prime }} }} )}}{{0.20944(600 - N_{x} )}}} \right]^{{B^{{\prime }} }} ,$$

(34)

the liquidity index B′:

$$B^{{\prime }} = \frac{{\lg \left( {\frac{{\theta_{600} }}{{\theta_{x} }}} \right)}}{{\lg \left( {\frac{{600 - N_{x} }}{{N_{x} - 3}}} \right)}},$$

(35)

the shear rate correction value C′:

$$C^{{\prime }} = \frac{{B^{{\prime }} \left( {\frac{{0.511\theta_{600} }}{A}} \right)^{{1/B^{{\prime }} }} (1 - 0.93646^{{2/B^{{\prime }} }} ) - 125.664}}{0.13128672},$$

(36)

N _x according to the θ _x , N ₁ and N ₂:

$$N_{x} = N_{1} + \frac{{\tau_{x} - \tau_{1} }}{{\tau_{2} - \tau_{1} }}(N_{2} - N_{1} ),$$

(37)

R-S yield value τ _x :

$$\tau_{x} = 0.511\sqrt {\theta_{600} \theta_{3} } .$$

(38)