Introduction

Coal seam water injection is a multifunctional mine disaster prevention and control technology that integrates dust reduction and removal, gas control, and coal seam softening1,2,3,4,5,6,7, and it has been widely applied in most coal mines in China and abroad8. The water migration range of coal rock after water injection is the key index to measure the effect of water injection. How to predict the wetting radius of coal seam effectively is a problem that has been studied by scholars both in China and abroad.

Seepage is the flow of fluid through porous media or fractured media. Coal is a typical porous medium, which is why experts and scholars have conducted extensive research on the seepage characteristics of water-injected coal around the classical linear Darcy seepage and nonlinear seepage Forchheimer equation. At sufficiently low flow rates, fluid flow in porous media is usually described by the famous Darcy’s law. Inertial resistance and viscous resistance at low flow rates are ignored; thus, Darcy’s law is a linear law. With the increase in flow velocity, the fluid in porous medium flows nonlinearly. The famous nonlinear seepage Forchheimer equation was established by Forchheimer9 based on experimental and observational experience in 1901. The Forchheimer formula can only predict the macroscopic transport of porous media with simple regular microstructure and cannot predict seepage flow in porous media with an irregular microstructure.

To describe the key role of interconnected pores and fissures in the seepage law, Ergun and Orning10 established the Ergun equation based on the combination of the parallel capillary bundle model and the hydraulic radius method to describe the nonlinear pressure drop of porous medium. With the influence of fluid viscosity on porous media seepage taken into consideration, the fluid seepage equations under different Reynolds numbers were derived theoretically by Irmay11 based on the dynamic Navier–Stokes equations and combined with Darcy’s law and the Forchheimer formula. To address the non-Darcy effect of high-speed fluid flow in porous media, Li et al.12 established a semi-theoretical equation for estimating the non-Darcy coefficient by comparing the derivation of equations by Ergun and Irmay. However, this equation contains an unknown constant. A serial capillary bundle model and a new equation to describe non-Darcy flow were established by Scheidegger and Adrian E13 to eliminate the shortcomings of parallel capillary models. With the heterogeneity and anisotropy of porous medium taken into consideration, a new expression of the non-Darcy coefficient was established by Macdonald14,15 by modifying the Ergun equation.

In addition, theoretical research on the spontaneous imbibition process under the action of capillary force in porous media has been conducted by scholars, and many related models have been proposed. The classic Lucas-Washburn (LW) infiltration model was proposed by Lucas and Washburn16,17 in the early twentieth century, but the equation had some initial errors. Subsequently, formulas for fluid flow in capillaries were proposed by scholars in the medical field considering factors such as inertial force, dynamic contact angle, and phase transition18,19,20,21,22,23. With the use of high-speed imaging technology and molecular dynamics simulation methods, a series of equations was obtained based on the capillary flow patterns in nanotubes to solve the problem of errors in the LW equation at the initial moment. Aside from that, based on different assumptions, many classical spontaneous imbibition models were established, such as the Terzaghi model, the Handy spontaneous imbibition model, the network model, and the perfect cross-flow model. Experts and scholars have conducted research on related models and proposed new models. A numerical model of dual porosity and dual permeability during water injection in tight sandstone formations was established by Liu Zhigang et al.24, who found that capillary pressure is not the only driving force that controls infiltration in tight sandstone. A new dynamic pore network model was established by Qin et al.25, which uses a variety of idealized pore units to represent complex pore space and has the potential to quantitatively predict the spontaneous absorption of real porous medium. An absorption model of spatiotemporal fraction was proposed by Yang et al.26 to capture abnormal absorption, which can characterize the heterogeneity of porous medium and the nonlocality of non-Newtonian fluid. They found the abnormal index is inversely proportional to the fractal dimension of capillary curvature. With the assumption of the fractal distribution of fractures and with the use of the bending plate fracture model, a semi-analytical model of spontaneous absorption and infiltration of the wetting phase into gas-bearing fractured porous medium was deduced by Wang et al.27, and the expression of immersed wet phase in gravity bearing and gravity-free fractured porous medium was obtained. Considering that the fractal characteristics of curved streamline have little influence on the cumulative imbibition mass/volume, a complete analytical model for spontaneous imbibition in porous medium was established by Wu et al.28 on the basis of the fractal characteristics of porous media, and gravity is included in the whole imbibition process. For porous medium with a fractal matrix and a pore matrix interface, Cai et al.29 proposed the modified Hagen-Poiseuille and Laplace–Young equations considering the tortuosity of laminar flow in media with non-circular capillary channels, which would be applicable to laminar flow in pre-fractal Menger sponges.

Previous studies achieved fruitful results. In the study of water-injected coal, the water migration mode is divided into seepage and spontaneous imbibition, and the transitional stage between seepage stage and spontaneous imbibition stage is neglected. In the research process, the seepage model and spontaneous imbibition model are often studied separately to simplify the model, and research on the transition stage between the two is not very in-depth. The connection points among the transition, seepage, and spontaneous imbibition stages cannot be accurately calibrated, resulting in misjudgment of these three stages. The study of spontaneous imbibition model usually adopts the unpowered spontaneous imbibition model. The initial speed of the spontaneous imbibition stage is often assumed to be 0. In the actual situation, before the fluid velocity drops to 0, the capillary force plays a role in attracting the spontaneous imbibition of the liquid. Assuming that the initial speed of spontaneous imbibition stage is 0 for those studies alone is unreasonable. Moreover, thinking that the water injection process is only two models of the seepage and spontaneous imbibition stages is also unreasonable. The existing water injection model cannot be well matched with the field experiment. The lack of a transition section model will reduce the prediction accuracy of the water injection effect.

In response to the above issues, the three water migration stage models (seepage, transition, and spontaneous imbibition) of water-injected coal is analyzed. The Darcy seepage stage model that contains many curved capillary coal samples is represented by area fractal dimension and tortuosity fractal dimension. The equation for solving the water migration distance in the Darcy seepage stage is provided. The transition stage of water injection from seepage to spontaneous infiltration is considered as non-Darcy flow, followed by spontaneous infiltration. On the basis of the classical Forchheimer model, the capillary force term is considered and added to construct the equation of the transition state between seepage and spontaneous imbibition. The upper critical velocity v1 from seepage to transition stage is calculated, and the lower critical velocity v2 from non-Darcy flow to spontaneous imbibition process is determined, which solves the problem that determining the critical point to calibrate the transition stage is difficult.

Theoretical model of seepage-transition-spontaneous imbibition

Water seepage stage model of water-injected coal

In the study of seepage theory, classical Darcy’s law and Hagen-Poiseuille’s seepage law are used. The pore structure in a porous medium greatly impacts seepage flow; thus, the capillary model of Hagen-Poiseuille’s seepage law is adopted, and the interior of the seepage area is regarded as the integral structure of countless capillaries. A single capillary is analyzed during the study, as shown in Fig. 1. First, a mechanical analysis is conducted on the Darcy flow stage of the fluid seepage process.

Fig. 1
figure 1

Model of the capillary.

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\(\Delta \user2{F}\) is the force exerted by the pressure difference between the two ends of the fluid on the equivalent unit, and its calculation formula is

$$\Delta F = \Delta P \times \pi r^{2}$$
(1)

where \(\Delta \user2{F}\) is the pressure difference between the two ends of the capillary, Pa, is the cross-sectional radius of the fluid, m, and f is the viscous resistance formed during the seepage process, N, whose calculation formula is

$$f = 2\mu \pi rl\frac{{dv}}{{dr}}$$
(2)

where μ is the liquid dynamic viscosity coefficient, Pa·s, v is the infiltration velocity, m/s, and l is the apparent length of the capillary tube, m.

The following can be obtained from Formulas (1) and (2):

$$\Delta P \times \pi {r^2}= - 2\mu \pi rl\frac{{dv}}{{dr}}$$
(3)

Formula (3) is integrated to obtain an expression for velocity

$$\nu = - \frac{{\Delta P}}{{4\mu l}}r^{2} + n$$
(4)

The default seepage velocity of the capillary wall is 0, and the value of the constant n will be

$$n = \frac{{\Delta P}}{{4\mu l}}\lambda ^{2}$$
(5)

where \(\lambda\) is the cross-section radius of capillary, m.

Formula (5) is substituted into Formula (4) to obtain the formula

$$v = \frac{{\Delta P}}{{4\mu l}}\left( {\lambda ^{2} - r^{2} } \right)$$
(6)

The analysis unit is integrated along the capillary axis, and the integration formula is

$$dQ = \frac{{\pi r\Delta P}}{{2\mu l}}\left( {\lambda ^{2} - r^{2} } \right)dr$$
(7)

Where Q is the flow rate of capillary tube, m3/s. The ideal flow rate of a single capillary tube after integration is

$$Q = \frac{{\pi \lambda ^{2} \Delta P}}{{8\mu l}}$$
(8)

The total flow rate of capillary flow is represented by the diameter φ of the capillary

$$Q=\frac{{\pi {\varphi ^4}\Delta P}}{{128\mu l}}$$
(9)

For Eq. (9), the influence of the bending resistance of the capillary on the seepage distance is not taken into account in the analysis of the internal seepage force inside the capillary. A new method needs to be introduced to represent the complexity of the distribution of capillaries within the coal because of the complex distribution of porosity within the coal body and the different interface sizes of capillaries in the model.

The distribution of capillaries inside the analysis volume can be well reflected by the average diameter of capillary pores involved in the fractal theory of porous medium. The complexity of the curvature of the capillary tube inside the coal body can be reflected by the concept of tortuosity introduced in analysis theory. Therefore, the expression of the fractal dimension of the tortuosity of the porous medium is established according to the structural characteristics of coal

$${D_\tau }=1+\frac{{\ln \overline {\tau } }}{{\ln \frac{l}{{\overline {\varphi } }}}}$$
(10)

Where \({D_\tau }\) is the fractal dimension of tortuosity of curved capillary, \(\bar {\tau }\) is the average tortuosity rate, and \(\bar {\varphi }\) is the average cross-sectional diameter of capillaries, m. The expression for the average tortuosity rate is30

$$\overline {\tau } =[\frac{1}{2}+\frac{1}{4}\sqrt {1 - M} +\frac{1}{2}\sqrt {1 - M} \frac{{\sqrt {{{(\frac{1}{{\sqrt {1 - M} }} - 1)}^2}+\frac{1}{4}} }}{{1 - \sqrt {1 - M} }}]$$
(11)

Where M is the porosity of the coal sample, %.

Porous medium cannot be accurately described by Euclidean geometry because of the relatively complex microstructure parameters such as roughness and specific surface area of the coal. As a self-similarity theory based on statistics, the fractal method has been widely used in fluid transport in porous medium by researchers. U et al. proposed a criterion to judge whether porous medium can be characterized by fractal theory. The literature indicates that the advantages of using fractal theory to solve infiltration problems are mainly reflected in the following aspects: (1) The tortuosity of pores and the distribution of different pore sizes can be considered, and the calculated results are in good agreement with the experimental results; (2) the calculated results are mainly controlled by fractal dimension, pore curvature, and pore size without calculating the effective permeability, capillary pressure, or fluid flow rate; and (3) the reason the time index in the imbibition scale model is less than 0.5 is identified. The expression for the average pore size \(\bar {\varphi }\) of the internal capillaries in coal samples is as follows30:

$$\overline {\varphi } =\frac{{{D_f}{\varphi _{\hbox{max} }}}}{{{D_f} - 1}}\left[ {\frac{{{\varphi _{\hbox{min} }}}}{{{\varphi _{\hbox{max} }}}} - {{(\frac{{{\varphi _{\hbox{min} }}}}{{{\varphi _{\hbox{max} }}}})}^{{D_f}}}} \right]=\frac{{{D_f}{\varphi _{\hbox{min} }}}}{{{D_f} - 1}}\left[ {1 - {{(\frac{{{\varphi _{\hbox{min} }}}}{{{\varphi _{\hbox{max} }}}})}^{{D_f}}}^{{ - 1}}} \right]$$
(12)

where \({\varphi _{\hbox{max} }}\) is the maximum cross-sectional diameter of the capillary tube (m), \({\varphi _{\hbox{min} }}\) is the minimum cross-sectional diameter of the capillary tube, m, and \({D_f}\) is the area fractal dimension. The calculation principle originates from the Menger sponge model, and the values can be obtained from mercury intrusion experiments. In the data fitting image of \(\log \frac{{ - {d_v}}}{{{d_\varphi }}} - \log {d_\varphi }\), the coefficient of the first term is the value of \({D_f}\), where V is the amount of mercury intrusion and \(\varphi\) is the pore diameter.

The relationship between the length L and the equivalent length l of the bent capillary tube can be expressed by the fractal dimension of tortuosity31,32

$$L={\phi ^{1 - {D_\tau }}}{l^{{D_\tau }}}$$
(13)

The area fractal dimension and tortuosity fractal dimension are used to express the Darcy percolation part of the coal sample rich in curved capillaries

$$Q = \frac{{\pi \Delta PD_{f} \varphi _{{\max }}^{{D_{f} }} (\varphi _{{\max }}^{{3 + D_{\tau } - D_{f} }} - \varphi _{{\min }}^{{3 + D_{\tau } - D_{f} }} )}}{{128\mu l^{{D_{\tau } }} \left( {3 + D_{\tau } - D_{f} } \right)}} = \frac{{\pi \Delta PD_{f} \varphi _{{\max }}^{{3 + D_{\tau } }} }}{{128\mu l^{{D_{\tau } }} \left( {3 + D_{\tau } - D_{f} } \right)}}\left[ {1 - (\frac{{\varphi _{{\min }}^{{}} }}{{\varphi _{{\max }}^{{}} }})^{{3 + D_{\tau } - D_{f} }} } \right]$$
(14)

With l<\({D_\tau }\)< 3 and 1<\({D_f}\)< 2, and because fractal theory of porous medium stipulates that \(\frac{{{\varphi _{\hbox{max} }}}}{{{\varphi _{\hbox{min} }}}}\)~ 10− 2, Eq. (14) can be simplified as

$$Q = \frac{{\pi \Delta PD_{f} \varphi _{{\max }}^{{3 + D_{\tau } }} }}{{128\mu l^{{D_{\tau } }} \left( {3 + D_{\tau } - D_{f} } \right)}}$$
(15)

The formula for calculating the seepage distance of Darcy’s law is derived as follows:

$${L_D}=\sqrt[{{D_\tau }}]{{\frac{{\pi \Delta P{D_f}\varphi _{{\hbox{max} }}^{{3+{D_\tau }}}}}{{128\mu Q(3+{D_\tau } - {D_f})}}}}$$
(16)

Water spontaneous imbibition stage model of water-injected coal

Spontaneous imbibition of porous media is widely studied. At present, research mainly includes the LW spontaneous imbibition and expansion model, Terzaghi’s spontaneous imbibition model based on Darcy’s law, Handy’s body model based on piston hypothesis, and the dimensionless scale equation model.

However, in the study of the spontaneous imbibition process of porous medium, the LW spontaneous imbibition and expansion model starts from the mechanical point of view and attributes the wetting power to capillary force. Therefore, the whole process is regarded as continuous to realize the connection with fluid migration at the end of the transition phase, and the existing model is optimized. The basic assumptions for the LW model are as follows:

(1) The spontaneous imbibition in porous media is fully developed.

(2) The capillary force is the main driving force.

(3) The fluid is an incompressible Newtonian fluid.

(4) The fluid is regarded as an incompressible Newtonian fluid.

$${H^2}=\frac{{c(\frac{{\bar {\varphi }}}{2})\delta \cos \theta t}}{{2\mu }}$$
(17)

where H is the spontaneous imbibition distance, m, \(\frac{{\bar {\varphi }}}{2}\) s the average capillary radius, m, \(\delta\) is the surface tension, N, \(\theta\) is the solid-liquid contact angle, °, and c is the capillary shape coefficient, t is the spontaneous imbibition time, s.

The vertical upward spontaneous imbibition distance changed with time inside the capillary can be expressed as33

$$\frac{{dh}}{{dt}} = \frac{{\varphi ^{2} }}{{32\mu h}}\left( {P_{0} - P_{e} + \frac{1}{2}\rho v_{s}^{2} } \right)$$
(18)

The expressions for \({P_0}\) and \({P_e}\) are as follows:

$$P_{0} = \frac{{4\delta \cos \theta }}{\varphi }$$
(19)
$$P_{e} = \rho gh_{e}$$
(20)

where \({P_0}\) is the capillary force, Pa, \({P_e}\) is the hydrostatic pressure, Pa, \(\rho\) is the density of the inhalation liquid phase, kg/m3, g is the gravitational acceleration, m/s2, \({h_e}\) is the vertical height of the position of the meniscus in the pipe, m, and \(\delta\) is the interfacial tension between the gas and the liquid, N/m.

In considering the bending effect of the capillary tube, the bending degree is expressed by fractal dimension, and the spontaneous imbibition speed is expressed by the fractal principle. The velocity in a curved capillary is34

$$v_{s} = D_{\tau } l^{{D_{\tau } - 1}} \varphi ^{{1 - D_{\tau } }} v_{2}$$
(21)

Formulas (13), (19), (20), and (21) are substituted into Formula (18), obtaining the following formula:

$$\frac{{dl}}{{dt}}=\frac{{{\varphi ^{2{D_\tau } - 1}}\delta \cos \theta }}{{8\mu {D_\tau }{l^{^{{2{D_\tau } - 1}}}}}} - \frac{{{\varphi ^{2{D_\tau }}}\rho g}}{{32\mu {D_\tau }{l^{^{{2{D_\tau } - 2}}}}}}+\frac{{\rho {D_\tau }{\varphi ^2}{v_2}^{2}}}{{64\mu l}}$$
(22)

When the spontaneous imbibition distance within the capillary tube changes to 0 over time (\(\frac{{dl}}{{dt}}=0\)), the spontaneous imbibition distance is considered to have reached an equilibrium state. Then, Formula (22) can be converted to

$$\frac{{{\varphi ^{2{D_\tau }}}\rho g}}{{32\mu {D_\tau }{l^{^{{2{D_\tau } - 2}}}}}}=\frac{{{\varphi ^{2{D_\tau } - 1}}\delta \cos \theta }}{{8\mu {D_\tau }{l^{^{{2{D_\tau } - 1}}}}}}+\frac{{\rho {D_\tau }{\varphi ^2}{v_2}^{2}}}{{64\mu l}}$$
(23)

Formula (23) can be simplified to

$$\rho l\left( {2{\varphi ^{^{{2{D_\tau } - 2}}}}g - {l^{2{D_\tau } - 3}}{D_\tau }^{2}{v^2}} \right)=8{\varphi ^{2{D_\tau } - 3}}\delta \cos \theta$$
(24)

The influence of gravity is ignored in the horizontal direction, and the belief is that as long as time is infinite, the spontaneous imbibition distance will increase infinitely. Therefore, the LW equation is used for analysis, and the analysis process follows the speed \({v_2}\) at the end of the transition phase. The momentum change of fluid motion in a capillary is as follows:

$$\frac{{d(m\bar {v})}}{{dt}}={F_{ch}}$$
(25)

Where \(\bar {v}\) is the average velocity during the entire spontaneous imbibition stage. The initial velocity is \({v_2}\). Thus, the formula is changed to

$$\frac{{d(m\bar {v} - m{v_2})}}{{dt}}={F_{{\text{c}}h}}$$
(26)

where

$$\bar {v}=\frac{{dl}}{{dt}}$$
(27)

The above equation can be expressed as

$$\frac{{d(\pi {\lambda ^2}\rho l\frac{{dl}}{{dt}} - \pi {\lambda ^2}\rho l{v_2})}}{{dt}}=2\pi \lambda \delta \cos \theta - 8\pi \mu l\frac{{dl}}{{dt}} - \pi {\lambda ^2}l\rho g$$
(28)

When gravity and inertial forces are ignored, the original equation can be simplified as

$$2\pi \lambda \delta \cos \theta =8\pi \mu l\frac{{dl}}{{dt}}$$
(29)

That is,

$$l = \sqrt {\frac{{\delta \varphi t\cos \theta }}{{4\mu }}}$$
(30)

Water migration model considering the transitional stage

The derivation principle of the seepage model is based on mechanical equilibrium. Throughout the entire process, viscous resistance, inertial resistance, and capillary resistance play a resistance role. However, in the spontaneous imbibition stage after the seepage stage, capillary resistance plays a dynamic role, and during the unpowered spontaneous imbibition experiment, the initial water migration speed is relatively high. These two processes are continuous in the actual water injection process, but significant gaps exist in the theoretical analysis process. The main reason is that the capillary force plays a resistance role in the seepage stage, but when entering the spontaneous imbibition stage, the capillary force does not gradually increase from 0, but remains at a certain fixed value for spontaneous imbibition, which is unreasonable. Therefore, in the later stage of seepage analysis, when the seepage pressure decreases to a small value and the seepage velocity decreases to a low range, the fluid properties change from Darcy flow to non-Darcy flow, and the capillary resistance gradually decreases. Finally, when the injection pressure is 0, the transition process completely transforms into capillary force. In this process, capillary force has both resistance and partial power processes. The Hagen-Poiseuille equation, which takes capillary resistance as resistance, is not applicable to the transition stage. Therefore, determining the critical point from the seepage to the transition stage is necessary to determine the seepage distance of the front section.

The seepage process of the porous medium is complicated. Many scholars are proposing nonlinear seepage equations to describe the seepage process, and the most widely used among them is the Forchheimer Eqs35,36,37. Xiong Feng et al. applied the Forchheimer equation to describe the coupling characteristics of pore fracture seepage. In this section, the Forchheimer flow of pore fractures is considered to modify the Forchheimer equation, and a continuous equation of the transition stage is established for the water migration mode of water-injected coal.

The main formula of the Forchheimer equation is as follows:

$$J=Av+B{v^2}$$
(31)

Where J is the hydraulic slope, v is the infiltration velocity, m/s, and , A,B are empirical constants. Extensive experiments proved that A is directly proportional to viscous resistance and inversely proportional to porosity and particle size. B is proportional to porosity and independent of viscous resistance. For consistency with the first-order term of Darcy’s law speed, the Forchheimer formula is often expressed as

$$- \frac{{dP}}{{dX}}=\frac{{\mu v}}{k}+\beta \rho {v^2}$$
(32)

where k is the permeability; \(\rho\) is the fluid density (kg/m3); \(\beta\) is the influence coefficient of non-Darcy flow, which is mainly related to the structure and fluid properties of porous medium; P is the pressure, N; and X is the direction of seepage. \(\frac{{dP}}{{dX}}\) is the pressure change of the fluid in the direction of seepage

$$\rho gJ=\frac{{\mu v}}{k}+\beta \rho {v^2}$$
(33)
$$J=\frac{{\mu v}}{{k\rho g}}+\frac{{\beta {v^2}}}{g}$$
(34)

\(\beta\) is a parameter that represents the deviation degree of non-Darcy flow from linear flow under the action of inertial force, which is mainly related to the structure and fluid properties of porous medium. When the seepage velocity is small enough to be ignored without timing, the velocity v on the right side of Eq. (34) tends to 0, and Eq. (34) is converted into the Darcy seepage equation.

Determination of the lower critical point

The Forchheimer model considers only the viscous resistance and inertial resistance. In the process of liquid movement from seepage to spontaneous imbibition, the Reynolds number changes from 1 to less than 1. In this process, the capillary resistance can be ignored from the seepage stage to the dominant force of spontaneous imbibition in the later stage. Only by reflecting this change process in the formula can seepage stage be connected to the spontaneous imbibition stage as a continuous process. Therefore, the Forchheimer model was modified by adding a capillary force term and considering its variation to be related to inertial force and capillary radius \(\lambda\).

$$J+\frac{{\alpha {P_0}}}{{\beta \rho {v^2}\lambda }}=\frac{{\mu v}}{{k\rho g}}+\frac{{\beta {v^2}}}{g}$$
(35)

where \({P_0}\) is the capillary force.

The assumption is that \(\alpha\) is a very small constant whose value is close to the minimum value of inertial resistance at this stage. Therefore, at the lower critical point, where J is 0 and the inertial resistance term is small enough, the final velocity \({v_2}\) can be expressed as follows:

$${v_2}=\frac{{k\rho g}}{{\mu \lambda }}{P_0}$$
(36)

Determination of upper critical point

The force analysis shows that the pressure formed by the water pressure gradient J is equal to the sum of viscous and inertial resistance. When the two are no longer equal, the capillary force begins to change, and this is the beginning of the transition phase. At this point, capillary force begins to generate and gradually increases. When the seepage stage transitions to the transitional stage, the following conditions are met:

$$- \frac{{dP}}{{dX}}<\frac{{\mu v}}{k}+\beta (\rho {v^2}+C)$$
(37)

C represents a very small equivalent capillary force because the fluid has just transitioned from the percolation stage to the transition stage. J can be expressed by Darcy’s law when calculating the critical position. The above formula can be transformed as follows:

$$\frac{v}{K}<\frac{{\mu v}}{{k\rho g}}+\frac{{\beta {v^2}}}{g}+\frac{{\beta C}}{{\rho g}}$$
(38)

where K is the permeability coefficient, m/s.

The coefficient of the quadratic term of the above unary quadratic inequality is greater than 0. Thus, the opening of the quadratic function is upward. Two situations need to be analyzed to find the critical velocity condition for the equation to transition from equal to unequal. First, the equation has an intersection with the x-axis, and except for this point, all conditions for the equation are greater than 0, which does not meet the requirements. Second, the equation has two intersection points with the x-axis. As the velocity v decreases, the entire equation changes from less than 0 to equal to 0, and after passing this point, the root that becomes greater than 0 becomes the critical point.

$$v_{1} = \frac{{\frac{g}{K} - \frac{\mu }{{k\rho }} - \sqrt {(\frac{\mu }{{k\rho }} - \frac{g}{K})^{2} - \frac{{4C\beta ^{2} }}{\rho }} }}{{2\beta }}$$
(39)

The critical velocity \({v_1}\) is determined by defining the transition identification value of the equivalent capillary force C.

Transition range calculation

With the Forchheimer optimization equation combined with the Hagen-Poiseuille equation, Eq. (40) is obtained

$$\frac{{\mu v}}{{k\rho g}} + \frac{{\beta v^{2} }}{g} - \frac{{2\alpha P_{0} }}{{\beta \rho v^{2} \varphi }} = \frac{{\varphi ^{2} \Delta P}}{{32K\mu l}}$$
(40)

Considering the influence of capillary curvature on the length of the transition zone, Eq. (15) can be transformed into

$$\begin{gathered} Q = \frac{{\pi \Delta PD_{f} \varphi _{{\max }}^{{3 + D_{\tau } }} }}{{128\mu l^{{D_{\tau } }} \left( {3 + D_{\tau } - D_{f} } \right)}}\begin{array}{*{20}c} = & {\frac{{K\pi D_{f} \varphi _{{\max }}^{{3 + D_{\tau } }} }}{{4\varphi ^{2} l^{{D_{\tau } - 1}} \left( {3 + D_{\tau } - D_{f} } \right)}}} \\ \end{array} \left( {\frac{{\mu v}}{{k\rho g}} + \frac{{\beta v^{2} }}{g} + \frac{{8\alpha \delta \cos \theta }}{{\beta \varphi ^{2} v^{2} }}} \right) \hfill \\ \hfill \\ \end{gathered}$$
(41)

According to the above formula, the calculation formula for the length of the transition zone is derived as follows:

$$L = \sqrt[{D_{\tau } - 1}]{{\begin{array}{*{20}c} {\frac{{K\pi D_{f} \varphi _{{\max }}^{{3 + D_{\tau } }} }}{{4\varphi ^{2} Q\left( {3 + D_{\tau } - D_{f} } \right)}}} & {\left( {\frac{{\mu v}}{{k\rho g}} + \frac{{\beta v^{2} }}{g} + \frac{{8\alpha \delta \cos \theta }}{{\beta \varphi ^{2} v^{2} }}} \right)} \\ \end{array} }}$$
(42)

It can be inferred from the formula that the length of the transition region in the transition model is primarily affected by the coal body and fluid. Related parameters were selected for the sensitivity analysis, as illustrated in Fig. 2. Steps 1 to 10 represent the ten selected data groups, each corresponding to different step lengths, reflecting the changes of each parameter at various values. The length of the transition region increases with the maximum pore size of the coal body, while it decreases with increases in the capillary tortuosity, coal body permeability, capillary diameter, and capillary flow. For the fluid within the coal body, the transition region length increases as the dynamic viscosity of the fluid rises, but it decreases as the fluid movement speed increases. Moreover, the differences in the relative rate of change under different step lengths in Fig. 2 indicate that capillary tortuosity significantly influences the transition region length, followed by the maximum pore size of the coal body and the capillary diameter. Capillary flow and fluid movement speed have a lesser impact on the transition region length. The permeability of the coal body and the dynamic viscosity of the fluid exert minimal effects on the length of the transition region.

Fig. 2
figure 2

Sensitivity analysis of relevant parameters

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With velocity \({v_1}\) substituted into the formula, the range of the transition region can be obtained. Figure 3 shows a schematic of the seepage-transition-spontaneous imbibition of coal with water injection.

Fig. 3
figure 3

Schematic of seepage-transition-spontaneous imbibition of coal with water injection.

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Seepage-transition-spontaneous imbibition model verification experiment of water-injected coal

Spontaneous imbibition experiments were conducted to verify the applicability of the transition section and the accuracy of the transition section model in the process of coal and rock water injection. These spontaneous imbibition experiments were conducted using coal powder and coal sample. The spontaneous imbibition experiment on coal powder regards the coal body as an isotropic homogeneous porous medium, and the fluid undergoes spontaneous imbibition from one end of the coal body. The spontaneous imbibition experiment on coal sample considers the anisotropy of the coal body. The fluid flows from the injection port and gradually permeates the interior of the coal sample. Notably, ultrasonic technology was used for monitoring.

A water injection process monitoring experiment was performed by applying low-pressure water injection to the coal sample, with nuclear magnetic resonance (NMR) technology used to monitor the imbibition process. The applicability of the transition section and the accuracy of the transition section model were verified through the analysis and calculation of the experimental data and the analysis and comparison with the spontaneous imbibition experimental results.

Analysis experiment of coal spontaneous imbibition stage in the water injection process

The study of water migration in water-injected coal rock is usually divided into two processes: the seepage stage and the spontaneous imbibition stage. After the seepage stage is completed, non-dynamic spontaneous imbibition is studied separately38,39,40,41. The initial velocity of spontaneous imbibition is usually assumed to be 0, which is not in line with the actual water migration law. With the changes in capillary force and the speed of water migration during the transition phase being ignored, the capillary force is assumed to be generated instantaneously, resulting in the inability to obtain the initial velocity of the spontaneous infiltration phase accurately. The coal spontaneous imbibition experiment was conducted in a hollow test tube to explore the difference between the seepage-transition-spontaneous imbibition process of water-injected coal and the spontaneous imbibition process without considering the transition section and to verify the necessity of considering the transition section. The geometric shape of the seepage channel formed by the accumulation of pulverized coal particles can be approximated as a capillary tube. The particle size and accumulation state of coal particles determine the pore size of the capillary tube. Referring to the data and conclusions in references38,39, one can conclude that the stable height achieved by spontaneous imbibition is closest to the theoretical calculation results when selecting large-particle-size coal powder. The coal powder is placed in a test tube and shaken to simulate the spontaneous infiltration channel of the coal capillary. The water body spontaneously infiltrates upwards, thus allowing the variation law of the spontaneous imbibition speed to be obtained without considering the transition section.

Three spontaneous imbibition experiments were performed under the same conditions by selecting 40–60 mesh pulverized coal and a glass tube with an inner diameter of 15 mm and a height of 200 mm. The abnormal experimental data were eliminated, and the average values of the three experiments were calculated as the experimental results. The process of spontaneous imbibition experiment is shown in Fig. 4. The specific experimental steps are as follows: First, the pulverized coal suitable for the spontaneous imbibition experiment is selected and then placed in a drying oven for drying. The temperature is adjusted to 105–110 °C, dried for 1.5 h, and cooled to room temperature for later use. The test tube is washed with distilled water, and the bottom of the test tube is sealed with filter paper. A certain amount of pulverized coal is added until 100 mm is reached, and the glass tube is shaken to form a pulverized coal column with uniform density. Finally, the bottom of the glass tube is immersed in a glass vessel filled with distilled water. When the glass tube coal pillar hits the water surface, the timing is started, and the experimental data are recorded. When the suction height no longer changes, the experiment ends.

Fig. 4
figure 4

Spontaneous imbibition experiment process.

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Ultrasonic technology is widely used in the monitoring of water migration in water injection coal rock42,43. The spontaneous imbibition experiment of pulverized coal in a test tube assumes the coal body as an isotropic homogeneous medium without considering the pore and fracture structure inside the coal body. Thus, the water migration experiment of coal sample based on ultrasonic method was conducted. The water migration during the spontaneous imbibition process for coal sample was monitored using an ultrasonic monitoring system. The ultrasonic emission frequency was set to 28 kHz, and the excitation time was 120 min. The samples were divided into groups A, B, C, D and E, and the corresponding ultrasonic power was 264, 374, 484, 594, and 704 W, respectively. The experimental design table is shown in Table 1.

Table 1 Experimental design table.
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The clamping force between the probe and the sample during the measurement has a great influence on the measurement results because of the long migration time of the experimental water. A shaped coal sample holder was designed to ensure achieve accurate experimental results and consistency of the experiment, as shown in Fig. 5a, b.

Fig. 5
figure 5

Ultrasonic instruments and operating principles.

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The gripper can adjust the tightness of the sensor and the specimen and realize the angle rotation of the specimen. The ultrasonic NM-4B wave velocity detector (measurement accuracy 0.1 us, measurement range 0.1–210000.0 us) was used to detect the wave velocity amplitude of the sample under different power excitations at the same acoustic emission frequency. The working principle of the ultrasonic instrument is shown in Fig. 5c, d. Different water migration conditions have different effects on ultrasonic propagation. The monitored ultrasonic wave can characterize the internal characteristics of coal samples. The propagation of ultrasonic wave in water-bearing porous media is based on the propagation process of water migration. In the process of ultrasonic monitoring, the monitoring wave velocity is the average wave velocity between the two transducers. During the experiment, all the equipment was installed and debugged, and then the sensitivity experiment of water migration was conducted. The initial wave velocity was measured before water migration, and then the data were collected and recorded in turn at an interval of 12 h according to experience. The influence range of water migration range on acoustic wave transmission was determined, and the specimen was compared and analyzed to determine the sensitivity standard under different conditions.

Monitoring experiment of water injection process based on low-field NMR

A monitoring water migration experiment based on low-field NMR was conducted to verify the accuracy of the transition section model. Low-field NMR technology uses NMR phenomena for analysis. It has high sensitivity and does not require sample processing42,43,44. Low-field NMR analysis technology mainly detects the hydrogen and other nuclei in the sample, and it uses the radio frequency excitation source to apply a specific frequency radio frequency signal to the sample. When the hydrogen nucleus in the sample is excited by the radio frequency signal, the NMR phenomenon will be generated, and the energy will be absorbed. In this process, the nuclei in the sample will relax according to a certain frequency and amplitude, and energy is released during the relaxation process. By measuring the width of the relaxation time of each nucleus in the sample, the concentration of each nucleus and the content of other chemical components in the sample can be calculated. Given that the water contains hydrogen nuclei, the moisture information in the sample can be directly measured by low-field NMR analysis technology, and accurate results can be obtained without any loss of the sample. In this study, the low-field NMR experimental equipment, known as the MacroMR12-110 H-I, was used for the water injection experiment on coal samples, as shown in Fig. 6.

Fig. 6
figure 6

Low-field NMR experimental equipment.

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Methods: Coal samples of 25 mm × 100 mm are prepared and dried at a constant temperature of 80 °C for at least 24 h. The coal sample is heated in a heat shrinkable tube until the tube completely encapsulates the coal sample. The coal sample is placed behind the ceramic holder and installed in the nuclear magnetic equipment, as shown in Fig. 7. The resonance frequency is 12.00 MHz, the magnet temperature is controlled at 32.00 ± 0.01 °C, the probe coil diameter is 28 mm, the confining pressure is loaded at a rate of 2 MPa/min until 5 MPa, and then 1 MPa water injection pressure is applied. The water injection time is 24 h, and sampling is performed every 30 min. With the help of spatial frequency coding technology, the real-time test of coal sample imbibition process is conducted to obtain the change in water content in different layers of the sample with time. Then, the migration speed of water body in coal is analyzed. This approach is a more advanced accurate method for evaluating water body migration in coal.

Fig. 7
figure 7

Low-field NMR experimental process.

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Result and discussion

Variation law of water migration velocity in non-dynamic spontaneous imbibition

The experimental model for spontaneous imbibition of pulverized coal in a test tube and the curve of the experimental results are shown in Fig. 8. The experimental model for spontaneous imbibition of pulverized coal in a test tube is specifically shown in Fig. 8a. Pulverized coal particles accumulate to form capillary channels, and the fluid spontaneously imbibes in the capillary channels. The fluid uniformly drawn up from the lower end of the pulverized coal, and finally, the liquid reaches the equilibrium position and forms a stable state at a certain height. The relationship between water migration time and distance is shown in Fig. 8b. The initial position of water migration is 0. The water body undergoes spontaneous imbibition under the action of capillary force. The pulverized coal particles accumulate, the pore structure is abundant, the adsorption capacity of liquid molecules is strong, and the affinity with water is strong. The surface wettability is good, and the regularity of coal particle imbibition rate is good. Within 100 h, the spontaneous imbibition height of the water body is 80 mm. When the surface tension and contact angle of the liquid remain unchanged, the smaller channel diameter corresponds to a greater driving force of the liquid entering the capillary45,46. The diameter of the channel at the contact area between the pulverized coal particles and the tube wall is smaller than that of the internal channel. Therefore, the spontaneous imbibition velocity of the fluid at the tube wall is greater than that inside the pulverized coal during the spontaneous imbibition process. The pulverized coal in the test tube has a certain height. The extrusion of the pulverized coal at the bottom of the test tube is greater than that at the upper part of the test tube. The diameter of the channel between the particles is smaller than that at the upper part of the test tube. Accordingly, the spontaneous imbibition rate increases rapidly from zero to a certain value at the initial stage of spontaneous imbibition. with the increase in water migration distance, the spontaneous imbibition speed gradually decreases. When the spontaneous imbibition height is about 20 mm, the downward trend of the suction speed tends to be gentle. About 25 h, the spontaneous imbibition height of the water body is 40 mm. After 70 h, the rate of increase in water migration distance declines, and the seepage speed tends to 0.

Fig. 8
figure 8

Relationship between water migration time and water migration distance.

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The experimental data of ultrasonic monitoring stage of coal spontaneous imbibition were sorted out and calculated, and the relationship curve between water spontaneous imbibition speed and distance in coal spontaneous imbibition experiment was drawn. The relationship curve between water migration speed and distance in coal imbibition experiment was drawn, as shown in Fig. 9. The diagram shows that the spontaneous imbibition process of coal water injection is studied separately. In the initial stage of spontaneous imbibition, the water migration speed is 0. Under the action of capillary force, the spontaneous imbibition speed gradually increases, and the speed increase rate is about 0.0069 m/h. After the spontaneous imbibition speed reaches the maximum value, it begins to decrease gradually. When the spontaneous imbibition height is about 1 cm to 1.8 cm, the speed reduction rate is 0.0045 m/h. When the spontaneous imbibition height is about 1.8 cm to 3.5 cm, the speed reduction rate is 0.0018 m/h. When the suction height exceeds 3.5 cm, the downward trend of the suction speed tends to be gentle until the suction height does not change, and the spontaneous imbibition process ends. The above unpowered spontaneous imbibition model is usually used, ignoring the study of the transition section before spontaneous imbibition. The default capillary force is instantaneously generated, the initial velocity of spontaneous imbibition is 0, and the velocity has a process of acceleration from 0 to deceleration.The variation law of fluid migration displacement and velocity in the spontaneous imbibition experiment on coal samples is consistent with the experimental results for pulverized coal.

Fig. 9
figure 9

Relationship between water migration speed and distance in the coal spontaneous imbibition experiment.

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The spontaneous imbibition experiments on coal powder and coal sample are both investigated by separating the spontaneous imbibition stage from the seepage stage. The initial velocity of spontaneous imbibition is 0. The results of spontaneous imbibition experiments on isotropic coal powder and anisotropic coal samples show that the spontaneous imbibition velocity increases with time and displacement during the spontaneous imbibition process. The speed shows a trend of increasing first and then decreasing, and the spontaneous imbibition acceleration is larger in the initial stage.

Before the actual water migration speed of the water-injected coal reduced to 0, the capillary force has played a role in attracting liquid spontaneous imbibition. In the seepage stage, the capillary force acts as a resistance. Although the capillary force could be ignored unlike the larger water pressure gradient in the initial stage, it could no longer be ignored in the later stage of the seepage stage where the water pressure per unit area significantly decreases. At this point, it enters a non-Darcy seepage process, which is a transition from the seepage stage to the spontaneous imbibition stage. This stage is signaled by the process in which the role of capillary force gradually changes from resistance to power. Until the spontaneous imbibition stage is entered, the capillary force and residual kinetic energy are the main driving forces for water migration. In the study of water migration of water-injected coal, the experiment that ignores the transition section and studying the spontaneous imbibition process alone cannot achieve the connection with the seepage stage. The study of coal and rock water injection is divided into two stages: seepage and self-absorption. However, this model is unsuitable for the actual process of coal and rock water injection. Therefore, the seepage–transition–spontaneous imbibition model is more suitable for the study of actual coal and rock water injection, as shown in Fig. 10.

Fig. 10
figure 10

Process of percolation-transition-spontaneous imbibition.

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Change law of water migration velocity in water-injected coal considering the transition stage

The T2 inversion diagram of coal sample water injection can be obtained through a water injection experiment based on low-field NMR, as shown in Fig. 11a. Low-field NMR technology tracks H element and can detect H element in free water and capillary water. The peak area of T2 distribution reflects the amount of water content. Figure 10 shows that the moisture content of coal samples changes with time, and the increment in water at different times is different, which indicates that the water migration speed varies at different times.

Fig. 11
figure 11

T2 relaxation spectrum of coal NMR during water injection process.

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The relationship between water injection time and water content signal in low-field NMR experiments is shown in Fig. 11b. The figure intuitively shows the change in moisture content of coal samples with time. The slope of the curve represents the rate of change in water content in the coal sample, which can be regarded as the migration speed of water within the coal sample. From 0 h to 1 h, water is injected into the coal body, and the transport speed of the water increases to a certain value. Thereafter, the slope of the curve gradually decreases, which implies that the transport speed of the water gradually decreases. After 4 h, the rate of decrease in water transport declines, which suggests that the water transport rate tends to be gentle. At 16 h, the rate of decrease in water transport speed increases, and the water transport speed reduces to 20 h. Next, the transport speed approaches 0 and no longer changes. The analysis reveals a significant change in the water transport rate at 4 and 16 h. Therefore, the transition period from 4 h to 16 h is the upper and lower critical points of the transition period.

Water injection experiments based on low-field NMR were conducted to capture the signal quantities of water at different positions of coal bodies at different times for studying the entire water transport process and the water transport speed at different positions. Then, the corresponding water transport distance at that time was determined. Next, the water transport speed at different positions was ascertained. The conventional T2 relaxation spectrum is the excitation of the entire sample and cannot reflect the signal of a certain level. NMR imaging can be used to select layers. Spatial frequency coding technology was used to obtain the changes in water content of different layers of the sample with time, and the dynamic study of the spatial direction of water migration was performed. Based on the GR-HSE pulse sequence data obtained by the experiment, the relationship curve between the water migration position and the signal amplitude at different times was drawn, as shown in Fig. 12.

Fig. 12
figure 12

Relationship curve between water transport position and signal amplitude at different times.

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As shown in Fig. 12, the horizontal axis, which ranges from 0 mm to 100 mm, corresponds to the starting and ending of axial water injection into the coal sample. The amplitude of the ordinate signal represents the level of water content. At the initial stage of water injection, the water content of coal sample increases, and nearly no water inflow is found in the middle and rear sections. After water enters the coal sample from the injection end, two migration directions are observed: longitudinal migration along the length direction and transverse migration along the cross-section direction. After water enters the coal sample, a small part of water advances along the connected path with small capillary resistance and reaches the outlet first, which forms a dominant channel. Therefore, a small amount of water content signal is detected in the middle and rear sections of the coal sample. Most of the water diffuses along the transverse direction to the same horizontal plane. When the pores in this plane are close to saturation, they move forward along the longitudinal direction. After 2 h of water injection, the water content signal at a distance of 8 mm from the injection end significantly increases, which implies that the water has infiltrated there. After 4 h of water injection, the water content signal at a distance of 12 mm from the injection end significantly increases, and the water seepage rate slows down. From then on to 16 h, the water transport of the coal sample slows down and tends to flatten out. At this time, the water transports to a distance of about 70 mm from the injection port. At 20 h, the water migrates to a distance of about 78 mm. After about 24 h of further migration, the signal change becomes insignificant, which suggests that infiltration and absorption have ended.

Establishing the relationship between water transport distance and velocity is necessary to verify the accuracy of the transition model. By analyzing the matrix data of NMR imaging, a two-dimensional color filling contour map was established. Based on the boundary criteria of the contour map, the original matrix data were filtered, and the proportional relationship between pixel sizes was used to determine the migration distance of water bodies in coal samples at different times. The corresponding time velocity was obtained by time differentiation. The relationship curve between water migration distance and migration velocity in NMR experiment was drawn, and the fitting curve of transition section model was drawn. The fitting curve of transition model was compared with the experimental data of NMR, as shown in Fig. 13a. The figure shows that the fitting curve of the transition model is consistent with the experimental results of the transition section of the NMR experiment.

Fig. 13
figure 13

Spontaneous imbibition experimental data, water injection monitoring experimental results, and transition model fitting curve.

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Figure 13a shows that the water migration rate decreases at a faster rate before the water body migrates to 12 mm from the injection port. Corresponding to Fig. 11b and 11, the injection time is 4 h. From 12 mm to 70 mm, the decrease rate of water migration tends to be gentle. After about 16 h, the velocity continues to decrease at a faster rate, and the water body moves into the spontaneous imbibition stage. The experimental data related to water injection process monitoring cover three process, including water migration, seepage, transition, and spontaneous imbibition. In the seepage stage, the capillary force plays a resistance role. With the decrease in water conservancy gradient, the water migration speed slows down. Specifically, with the increase in water migration distance, the seepage speed gradually decreases to the minimum speed in the seepage stage, that is, the lower critical point. The water migration mode transits from the seepage state to the nonlinear seepage state. At this time, the capillary force is the main driving force of water migration. As the resistance works, the hydraulic gradient continues to decrease, which further reduces the water migration speed, and the reduction speed is slower than that in the seepage stage. As the migration distance of water body continues to increase, the water body migration mode changes from non-Darcy seepage to spontaneous imbibition stage when the viscous resistance is equal to the capillary force and inertial force, that is, the upper critical point is reached. Under the action of capillary force, the water body is spontaneously imbibed. However, after a certain distance, the spontaneous imbibition process ends. The fitting curve of the transition section is the water migration in the transition stage of water-injected coal. With the increase in water migration distance, the migration speed gradually decreases from a certain value (upper critical speed) to a non-zero value (lower critical speed). A comparison between the fitting data curve of the transition model and the results of the water injection experiment shows transition stages and the upper and lower critical points of the transition section in the actual coal water injection process. The transition section can be calibrated by the upper and lower critical speeds and the water migration distance of the transition section. The shapes of the transition model fitting curve and of the transition section curve of the water injection experiment are highly similar, indicating that the transition section model established in this paper is basically in line with the actual situation and has a certain accuracy.

The experimental results of coal spontaneous imbibition stage analysis were compared with the experimental data of NMR monitoring in water injection process and the fitting curve of transition model, as shown in Fig. 13b. A comparison of the results of coal spontaneous imbibition experiment with the results of the water injection monitoring experiment shows that the water migration speed changes continuously during the actual water injection process. In the spontaneous imbibition experiment, the velocity of the water migration increases from 0 and then decreases, which is inconsistent with the change in the actual water migration velocity. In the velocity reduction section, the trend of the curve is basically consistent with the water injection monitoring experiment results. The findings confirm that the spontaneous imbibition stage and the seepage stage are studied separately without considering the transition stage, which will lead to the separation of the spontaneous imbibition stage and the seepage stage in the water migration process. This condition cannot accurately reflect the actual movement of the fluid in the initial stage of spontaneous imbibition. Considering the existence of the transition section between the seepage and the spontaneous imbibition stage, the seepage-transition-spontaneous imbibition process of the water-injected coal is proposed. The transition section model, which can establish the connection between the seepage and the spontaneous imbibition, can more accurately describe the velocity change of the fluid during the water injection process.

The experimental curve of the water injection process monitoring is compared with the fitting curves of different models( the mathematical model considering the transition section and the mathematical model that does not consider the transition section), as shown in Fig. 14. The error distribution at each point on the curves of different models was calculated, allowing us to derive the following: the fitting precision of the mathematical model that considers the transition section reaches 0.995, while that of the model not considering the transition section is 0.875. Therefore, the mathematical model considering the transition section is more accurate than the model that does not consider the transition section under the same water injection condition. Comparisons indicate that the mathematical model accounting for the transition section aligns more closely with the actual results obtained from the experiments. The seepage-transition-spontaneous imbibition model considering the transition section takes into account that the capillary force plays different roles at different times, and the capillary force water pressure gradient is low. The transition section model reflects the conversion of capillary force from the hindrance of the high water pressure gradient in the seepage stage to the capillary force, which is consistent with the continuous change of the actual water migration speed. A comparison of the water migration model of water injection coal considering the transition section with the model that does not consider the transition section shows that the seepage-transition-spontaneous imbibition model can more accurately reflect the water migration situation.

Fig. 14
figure 14

Experimental curve of the water injection process monitoring compared with the fitting curves of different models.

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Conclusion

The analysis and research in this work found that a transition stage exists between water migration seepage and spontaneous imbibition of water-injected coal, and a transition section model considering capillary force is proposed. On the basis of the classical Forchheimer model, the capillary force term is considered and added to construct the equation of the transition stage between seepage and spontaneous imbibition. The critical point of the upper and lower velocity of the transition section is determined, and the spontaneous imbibition experiment and the water injection process monitoring experiment based on the ultrasonic method are completed. Through theoretical derivation and experimental verification, the following conclusions are drawn:

  • For water-injected coal, the capillary force changes continuously in the actual process of water migration from seepage to spontaneous imbibition, and the water migration velocity is continuous. The water migration process is divided into seepage and spontaneous imbibition processes, which cannot realize the docking of capillary force and water migration velocity. A theoretical model of water seepage-transition-spontaneous imbibition of water injection coal rock is proposed. The Forchheimer model of nonlinear seepage equation is modified, and the transition stage equation between seepage and spontaneous imbibition is constructed. The critical velocities of upper and lower points and the formula of water migration range in the transition stage are derived. The transition stage is calibrated by the critical point of velocity and the distance of water migration.

  • The spontaneous imbibition experiment verifies that the initial water migration velocity is 0 when the spontaneous imbibition process is studied separately. Under the action of capillary force as the main driving force, the water migration velocity increases first and then decreases, which is inconsistent with the actual water body migration. Without the transition stage taken into consideration, the spontaneous and seepage stages are studied separately, which will lead to the separation of the spontaneous and seepage stages in the water migration process. The water migration in the initial stage of spontaneous imbibition cannot be accurately reflected. Not applicable to the study of actual coal rock water injection process, experimental verification of water injection process monitoring based on NMR ultrasonic method shows that the fitting curve of the transition section model is basically consistent with the trend of experimental data. Transition sections and upper and lower critical points exist in the actual water migration process of water-injected coal. The upper and lower critical speeds of the transition section model and the water migration distance of the transition section can be used to calibrate the transition section. Considering the seepage-transition-spontaneous imbibition process of the transition section is more in line with the actual water migration of water-injected coal.

  • The accuracy of the water migration model considering the transition section is above 0.99, which is 12% higher than that of the model that does not consider the transition section, thus being more accurate. The seepage-transition-spontaneous imbibition model can fill the shortcomings of the existing model and describe the water migration process of water-injected coal.

  • The theoretical model of water seepage-transition-spontaneous imbibition can provide theoretical support for both underground and surface mining in coal seam gas extraction through water injection, thereby enhancing the extraction efficiency. Additionally, this model is also applicable to natural gas extraction in the petroleum industry, offering a valuable basis for adjusting water injection parameters to improve recovery efficiency.