Modeling Tracer Flow Characteristics in Different Types of Pores: Visualization and Mathematical Modeling

Structure of porous media and fluid distribution in rocks can significantly affect the transport characteristics during the process of microscale tracer flow. To clarify the effect of micro heterogeneity on aqueous tracer transport, this paper demonstrates microscopic experiments at pore level and proposes an improved mathematical model for tracer transport. The visualization results show a faster tracer movement into movable water than it into bound water, and quicker occupancy in flowing pores than in storage pores caused by the difference of tracer velocity. Moreover, the proposed mathematical model includes the effects of bound water and flowing porosity by applying interstitial flow velocity expression. The new model also distinguishes flowing and storage pores, accounting for different tracer transport mechanisms (dispersion, diffusion and adsorption) in different types of pores. The resulting analytical solution better matches with tracer production data than the standard model. The residual sum of squares (RSS) from the new model is 0.0005, which is 100 times smaller than the RSS from the standard model. The sensitivity analysis indicates that the dispersion coefficient and flowing porosity shows a negative correlation with the tracer breakthrough time and the increasing slope, whereas the superficial velocity and bound water saturation show a positive correlation.

Mennella et al. [Mennella, Bryant and Lockhart (1999) ;Li, Liu, He et al. (2012); Krogstad, Lie, Nilsen et al. (2017)] applied the Continuous-Time-Random-Walk (CTRW) method, proposed by Scher et al. Lax (1973a, 1973b); Stalgorova and Babadagli (2012)], to describe the interactions between tracer and polymer and model tracer transport in polymer flooding reservoirs. Li et al. [Li, Jiang, Li et al. (2011)] showed in the experiment that the surface of rocks and retained polymers can adsorb tracers simultaneously. Yang et al. [Yang and Liu (2007)] ignored adsorption of tracers during tracer modeling, because the used aqueous tracer is close to ideal state and hardly adsorbed by rock surface. For microelement tracers, however, it is unwise to ignore the adsorption effect because of the trivial amount of injected tracers. Yang [Yang (2012)] reported that the adsorbed concentration can significantly affect the produced tracer concentration. Cox et al. [Cox, Alsenani, Miller et al. (2017)] found that besides using novel polymeric nanoparticles to mitigate memory effects and deploy multiple tracers simultaneously in complex oil reservoir, retention or adsorption of it has even greater influence on the produced tracer concentration. Nevertheless, Dindoruk et al. [Dindoruk and Dindoruk (2008); Widiatmojo, Sasaki, Yousefi-Sahzabi et al. (2015)] did not consider the adsorption term in analytical modelling of tracer transport.
Additionally, Semra et al. [Semra, Sardin and Simonnot (2008)] ignored the differences of tracer velocity in different types and sizes of pores, which were considered by Boving et al. [Boving and Grathwohl (2001)] and resulted in discrepancies compared to experimental data for low-porosity cores. Bellin et al. [Bellin and Rinaldo (1995)] found porous media structures and fluid distribution can cause physical and chemical microscale heterogeneity, respectively. Semra et al. [Semra, Sardin and Simonnot (2008)] showed experimentally that microscale heterogeneity is responsible for solute distribution and has a visible effect on breakthroughs. Liu et al. ] demonstrated that dispersion can affect tracer distribution on flow and latitudinal direction when cores have strong microscale heterogeneity in the microscopic experiments. Brigham et al. [Brigham, Reed and Dew (1961)] summarized the above features macroscopically into dispersion, which is a function of tracer velocity. It demonstrates the necessity to consider the differences of tracer velocity between different types of pores to obtain accurate results of tracer concentration.
Li et al. ] discovered the "Chromatographic effect" in microscopic experiments. It indicates the injected aqueous tracer can be adsorbed by rock surface and enter the bound water, resulting in the tracer flows being slower than movable water. Furthermore, Liu et al. ] showed this phenomenon becomes conspicuous in highly heterogeneous porous media. It means that water and oil distribution can also affect the tracer distribution during diffusion and dispersion.
The objective of this paper is to improve tracer transport modeling by considering the effects of micro-heterogeneity, i.e., different types of pores and fluid saturation. In Section 2, microscopic experiments are conducted to analyze tracer flow characteristics in flowing and storage pores, especially in movable and bound water. In Section 3, based on the findings in microscopic experiments, the mechanisms of tracer in different types of pores are summarized and characterized mathematically. In Section 4, applying the equations of physical properties, we propose the mathematical model for tracer transport and obtain the analytical solution of tracer concentration, which better matches the tracer concentration curve than old model. In Section 5, sensitivity analysis is conducted to find the trend of tracer concentration with different tracer and fluid properties.
2 Microscopic tracer flow characteristics 2.1 Experimental setup and procedure In this section, microscopic experiments are conducted to study tracer flow characteristics at pore level. Fig. 1 illustrates the procedure of the microscopic experiments, which is mainly Figure 1: Experimental setup scheme of the tracer transport microscopic experiment 1 Screen, 2 Signal-collecting system, 3 Support unit, 4 Container for tracer, 5 6 Container for water, 7 Glass sheet, 8 Microscopic model, 9 Microscope, 10 Automatic coupling, 11 Container for product liquid, 12 Light source, 13 Power source constructed by Screen (1), Signal-collecting system (2), Microscopic model (8) and Microscope (9).
The signal-collecting system (2) has imaging speed of 25 frames per second and image resolution of 576×768. The microscopic model (8) is laser etched according an electronic microscope image of real cores. The model is square with length 5 cm, permeability 10-50 D, and pores radius 30-200 μm, which pore volumes (PV) are 0.149 mL measured by weighing method. The microscope (9) has objective multiple as 0.32, 1, and 2, and stepless zoom range from 5 to 40 times.
According to the industry standard SY/T5925-94, methylene-blue solution is recommended to be used in microscopic experiments, which are easy to obtain and convenient for observation in microscopic experiments.
The major experimental procedures are as follows.
(I) Evacuate the microscopic model (8) and then saturate it by water.
(III) Inject 5 PV tracer solutions with same rate and temperature in the Step (II). In the displacement Process (II) and (III), Signal-collecting system takes images of the microscopic model with 25 frames per second. Some of the images are shown in Section 2.2.

Microscopic images
Figs. 2 and 3 illustrate the microscopic model images at different tracer injection volumes. The red and yellow boxes in Figs. 2 and 3 mark the flowing and storage pores, respectively. In the flowing pores, which is connected to more than one adjacent pores, the injected tracer flows along with a very tortuous path and is distributed into movable water as early as 0.04 PV ( Fig. 2(a)). This is due to advection of tracer. At 0.89 PV ( Fig. 2(b)), the adjacent pores become blue and are occupied by the tracer, while the bound water in relative tight pores is still white. It indicates there are different tracer transport characteristics between the movable and bound water, even in the same flowing pores. The difference between tracer velocities in movable and bound water results in different mechanisms of tracer transport. Lan et al. [Lan (2011);Li (2014)] considered that mechanical dispersion rather than  (Fig. 2(c)), the tracers occupy all pores, which are blue completely.
In the storage pores (yellow box in Fig. 3), which has only one adjacent pore, it is still white after 3.02 PV. Since water cannot flow into the storage pores, there is no advection or dispersion, but only slow diffusion.

Results analysis
From Fig. 2, we observe that the injected tracer first occupies movable water and then bound water in the flowing pores. It indicates that there are different forces or mechanisms controlling the tracer concentration distribution in movable and bound water. Based on Fig. 3, the injected tracer cannot occupy the storage pores completely until 5 PV tracer injections. It implies that the mechanism in storage pores are very different from that in flowing pores, because the water flow velocity is zero in storage pores. Therefore, we should consider these characteristics and treat different types of pores separately in tracer transport mathematical modelling.
3 Mechanisms of tracer transport at pore level Tracers have different transport characteristics not only in movable and bound water, but also in flowing and storage pores. It should be considered in mathematical modelling of tracer transport. Fig. 4 and Tab. 1 illustrate main conceivable action of tracer, which are diffusion and adsorption in storage pores, and advection, dispersion, diffusion and adsorption in flowing pores, respectively. In storage pores, tracer diffuses into the bound water by concentration differences and can be adsorbed onto rock surface. In flowing pores, tracer can adsorb onto rock surface and disperse into movable water by advection, as well as diffuses into bound and movable water.
3.1 Interstitial flow velocity in flowing pores Interstitial flow velocity in flowing pores can affect the dispersion and play a key role in tracer concentration distribution. The inaccurate real velocity will induce mistakes on evaluation of tracer concentration distribution. As Fig. 3 shows, the aqueous tracer flows only in movable water in flowing pores, rather than in the bound water, the oil, and storage pores. Therefore, the tracer interstitial flow velocity in flowing pores should be characterized as follows where u and u′ are interstitial and superficial velocities of tracer, respectively, cm/s; u´is the quotient of injection rates and sectional area of cores; Φ f is flowing porosity, i.e., porosity of flowing pores; S o and S wc are oil saturation and bound water saturation in flowing pores, respectively. Figure 4: Schematic diagram of a flowing pore and a storage pore 1, 2, 3 are bound water, movable water and oil in flowing pore, respectively; 4, 5 are rock surface, bound water, respectively; 6 is oil in storage pore  [Brigham, Reed and Dew (1961)] considered that both the dispersion and diffusion can affect the distribution of tracer concentration in flowing pores. The mixing coefficient resulting from dispersion and diffusion is expressed as where D is mixing coefficient, cm 2 /s; D m is molecular diffusion coefficient, cm 2 /s; l is rock tortuosity factor; Φ is total porosity; α is dispersion coefficient, cm.
Li et al. [Li, Jiang and Liu (2010)] found that the dispersion is in line with tracer flow direction while the diffusion is perpendicular to the direction. Li et al. [Li (2014); Huseby, Sagen, Viig et al. (2013)] considered that molecular diffusion is dominant when fluid velocities are close to zero. However, tracers usually transport fast because of high permeability channels or fractures between wells. Therefore, it is reasonable to neglect diffusion and only consider dispersion in flowing pores. Hence, Eq.
(2) can be simplified as follows

Adsorption on rock surface
Hanaor et al. [Hanaor, Ghadiri, Chrzanowski et al. (2014)] showed that tracers can be adsorbed onto rock surface when they flow through porous media. The adsorption could be thought to obey the Langmuir isothermal adsorption law where C r is adsorbed tracer concentration, mg/g; a, b are the adsorption coefficients; C is tracers concentration in aqueous, mg/L. (2013)] showed that the tracer concentration in porous media is relatively small compared with the huge water swept volume by most tracer tests. Meanwhile, aiming to reduce cost, engineers look forward to injection tracers as little as possible to just meet detection demand. In some field cases, very small tracer amounts are recovered. When the tracer concentration is low, therefore, the adsorption law could be simplified as follows

Diffusion in bound water and storage pores
The tracers are contacted to bound water with a large area and long time, especially in the case of reservoir scale. Therefore, it is assumed that the diffusion makes the distribution in equilibrium ultimately, i.e., concentrations in bound and water phase are equal where C swc is tracer concentration in bound water, mg/L.
Considering large sweep pore volume in interwell channels, it is also assumed that concentrations in storage pores and aqueous are equal where C nonp is tracer concentration in storage pores, mg/L.

Mathematical models 4.1 Governing equation
Considering one-dimensional flow through the pores shown in Fig. 4, the dispersion occurs in flowing pores only and can be expressed as follows where u i is tracer flux velocity caused by dispersion, cm/s; x is distance, cm.

The advection occurs in flowing pores only and can be expressed as follows
Àu The adsorption occurs on rock surface and can be expressed as follows where ρ r is rock density, g/cm 3 .
Assuming the storage pores have the same S wc with flowing pores, then the diffusion term in bound water can be expressed as follows @C swc @t fS wc dxdt The accumulative mass change in time dt consists of the mass changes in flowing pores and storage pores, respectively According to the principle of mass conservation, the governing equation of tracer in flowing and storage pores can be expressed as follows Substituting Eqs. (5) to (7) into the governing Eq. (14), yields The tracer mixture velocity can be expressed as The governing equation of tracer flow at pore level is finally obtained as

Initial and boundary conditions
In the tracer tests, t=0 corresponds to the time of tracer injection. In one-dimensional case, the initial and boundary conditions can be expressed as follows where C o is tracer concentration in injected water, mg/L.

Analytical solution
Eqs. (17) to (20) complete the mathematical model of tracer flow at pore level. To obtain the analytical solution, the modified time t′ is defined as By introducing the modified time t′, the mathematical model of tracer flow at pore level is simplified to a typical convective-dispersive solute transport equation consequently [Nihoul and Jacques (1983); Yang, Shao, Zhu et al. (2019)]. Applying the Laplace transform, which applied by Kocabas [Kocabas (2011)], the analytical solution in real space is obtained as follows where C/C o is the tracer concentration ratio.
It is worth noting that replacing t′ by t in Eq. (22) leads to the standard model for tracer concentration. The difference between the new model (22) and previous standard model lies in the variable transformation in Eq. (21), which accounting for the effects of flowing porosity and bound water.

Comparison with standard model
In this section, we conduct a tracer displacement experiment in a sand pack model. Tab. 2 shows the main parameters in the experiment, in which the core porosity is 0.372 while the flowing porosity is 0.23.
Based on the proposed tracer transport concentration solution (Eq. (22)), that accounting for flowing porosity and bound water effects, the calculated tracer concentration (blue line in Fig. 5) fits the measured tracer concentration very well, and the RSS is 0.0005. When neglecting the effect of flowing porosity and bound water as the standard model does, the calculated tracer concentration (green line in Fig. 5) cannot fit the measured tracer concentration and corresponding RSS is 0.0515, about one hundred times larger of the proposed model.
As Fig. 5 shows, the tracer concentration ratio shows deviation like higher dispersion effect when ignoring effect of flowing porosity and bound water. This phenomenon will result in error of interwell-heterogeneity estimation in tracer field application. To validate the new model in this paper, the new model in this paper is used to the tracer test interpretation with necessary streamline method and optimization algorithm. Fig. 6 illustrates the field matching curves from H shale oil reservoir in China, which reaching application demand. The interpretation results show the average permeability and swept volume of interwell channeling as 105.6 mD and 457 m 3 , respectively. These parameters are consistent with the knowledge from the wells production performance. To figure out effect of physical actions and fluid distribution on tracer concentration, Section 5 shows sensitivity analysis by proposed model.
5 Sensitivity analyses 5.1 Dispersion As Fig. 7 shows, when mixing coefficient D increases from 0.01 to 0.06, the corresponding tracer concentration ratios break through earlier but increase more slowly. It results from that the dispersion tends to make tracer concentration distribution more uniform in water phase in flowing pores. Meanwhile, dispersion does not induce any mass loss, as shown in Fig. 7(b).

Adsorption
As Fig. 8(a) shows, when the adsorption coefficients increase from 0.01 to 0.08, the corresponding tracer concentration ratios break through later and increase more slowly. When the surface of flowing and storage pores adsorbs more tracers, it results in less cumulative tracer concentration, as shown in Fig. 8(b).

Superficial velocity
As Fig. 9(a) shows, when tracer superficial velocity u′ increases from 0.003 to 0.007 cm/s, the corresponding tracer concentration ratios break through later and increase more quickly, just contrast to the change laws of mixing coefficient D. On high superficial velocity, since the dispersion has less time to control tracer concentration distribution, the tracer breakthrough point becomes later versus PVI. Superficial velocity does not affect cumulative tracer concentration, as shown in Fig. 9(b).

Bound water saturation
As Fig. 10(a) shows, when bound water saturation S wc increases from 0 to 0.4, the corresponding tracer concentration ratios break through later and increase more quickly, just same with the change law of tracer superficial velocity u′. Same with the laws of tracer superficial velocity, S wc has a positive correlation with the interstitial flow velocity and results that it is less time to control tracer concentration distribution for the dispersion as well. In continuous injection situation, bound water saturation has little effect on cumulative tracer concentration, as shown in Fig. 10(b).

Flowing porosity
As Fig. 11(a) shows, when flowing porosity Φ f increases from 0.1 to 0.3, the corresponding tracer concentration ratios break through earlier and increase more slowly. A larger Φ f contributes more space for dispersion and results in same change law with higher dispersion coefficient. Fig. 11(b) illustrates that change of flowing porosity does not induce any mass loss. (1) The microscopic experiments illustrate that injected tracer enters movable water first and then bound water in the flowing pores, indicating that it is necessary to consider the effect of bound water on tracer concentration distribution in mathematical modelling. Moreover, the injected tracer cannot occupy the storage pore completely until 5 PV injected, which is longer than that in the flowing pores. It indicates that flowing porosity has different effect on tracer concentration distribution if compared with total porosity. As microscopic experiments show, flowing and storage pores have different tracer behaviors, i.e., diffusion and adsorption in storage pores and advection, dispersion, diffusion and adsorption in flowing pores.
(2) The proposed mathematical model improves tracer transport description by considering bound water and flow porosity separately. The analytical solution of tracer concentration fits the experimental data better than that ignoring the effects of bound water and flowing porosity, with corresponding RSS=0.0005 and 0.0515, respectively.
(3) The dispersion coefficient and flowing porosity have a negative correlation with the tracer breakthrough time and the concentration growth rate, whereas the superficial velocity and bound water saturation have a positive correlation with them. The adsorption leads to a later breakthrough of tracer and a slower increase of concentration.