Gel, which can be used to control profile and displace oil because its aqueous solution can block high-permeability channels and thus enlarge swept volume, plays an important role in enhanced oil recovery from heterogeneous reservoirs with low permeability. There were some experimental studies and field application in China and abroad^[1,2]. Gel damage may occur in a solution with high salinity, especially when divalent calcium and magnesium ions, or ions with higher valence, exist^[2,3]; it often manifests as weakened strength, impaired plugging performance, and low efficiency of oil-water displacement. Core flooding is often used to examine the impact of brine on gel, but it can’t reflect the distribution of displacing fluids and crude oil inside the core and the migration path and plugging area of gel. Thus, it is hard to understand damage of brine in formation pores to gel.

Microscopic visualization model, CT scan and magnetic resonance imaging (MRI) have been commonly applied to visualized core analysis^[4,5,6]. Di et al.^[6] investigated the characteristics of oil displacement by gel by using MRI and found that gel would initially flow into high-permeability and low- permeability channels, and have gravitational differentiation in high-permeability channels. In the imaging test, oil, water, and gel images can be discriminated using manganese chloride, which is cheap and easily acquirable^[7]. MRI is capable of detecting the decline of induced voltage caused by athwartship magnetization in the process of H-nucleus precession. The athwartship magnetization vector is in direct proportion to the number of H-nuclei; hence, the strength of nuclear magnetic signals represents the number of H-nuclei and signal attenuation rate represents the rate of H-nucleus relaxation. Mn²⁺ with paramagnetism can combine with OH^- in water to form coordination bond, accelerating the relaxation attenuation of water. But Mn²⁺ will not combine with H-nuclei in oil. When relaxation time is less than echo time, the imaging signals of water or gel cannot be detected; thus, it is possible to differentiate water or gel images from oil images. We found in a gel-water displacement test that Mn²⁺, as an indicator for signal differentiation, lessened the relaxation time of water phase, and meanwhile gel viscosity decreased with the rise of Mn²⁺ concentration. As SEM images represent, original dense net structure of gel became sparse and plugging performance of the gel declined significantly^[8]. This means we need to understand Mn²⁺ diffusion and migration in the core-gel system to minimize the negative impact of Mn²⁺. Rao et al.^[9] investigated OH^- diffusion in hydrogel by using a diffusion model with chemical reaction terms. Davison et al.^[10] obtained Mn²⁺ concentration distribution in gel by using atomic absorption spectrometry, which perfectly matched the result calculated using the Fick diffusion equation. But in a conventional core experiment, it is impossible to directly observe Mn²⁺ diffusion inside the core or measure Mn²⁺ concentration and its impact on gel. Here we present a visualization approach to investigate gel or weak gel damage in the process of profile control. We use MRI-based images of Mn²⁺ diffusion in a porous media-gel system to examine its impact on gel. We formulate a mathematical model, which involves the chemical reaction between gel and Mn²⁺, to simulate Mn²⁺ diffusion and concentration distribution in a porous media-gel system as well as the critical Mn²⁺ concentration for gel signal suppression.

1.1.1. Methodology

These experiments aim at obtaining the images of Mn²⁺ diffusion and distribution in a porous media-gel system. The reaction between Mn²⁺ in the porous media-gel system and OH^- in water may reduce the relaxation time of water and speed up signal attenuation. Relaxation attenuation increases with Mn²⁺ concentration. When Mn²⁺ concentration in gel is high enough, the relaxation time will be less than the echo time, and the signals of gel may not be detected, then it looks like the gel image is being "swallowed".

We used the MINI-MRI system for magnetic resonance imaging and displacement^[6], and magnetic induction was set at 0.5 T. The the experiment content is as follows: (1) The glass test tube of 20 mm in diameter and 80 mm long was filled with glass beads of 0.25 mm (60-mesh), which was taken as porous media with a porosity of 38%. The left part and right part of the tube, equally 40 mm long, was separated by a thin reticular membrane (which was permeable to ions). The schematic model is shown in Fig. 1. (2) Manganese chloride solution was injected into the left part of the tube; anionic polyacrylamide gel prepared with deionized water was injected into the right part. Manganese chloride solutions with the concentrations of 10 g/L and 20 g/L were tested by two sets of identical experimental models. (3) The model was put into the testing area of the MRI equipment. The MRI software system was started. Nuclear magnetic signals from the right part (aqueous gel) were acquired every 2-5 h to obtain the images of manganese chloride solution and gel. (4) We used a professional software system for image processing to find out the features of Mn²⁺ diffusion and Mn²⁺-gel reaction shown in the images.

1.1.2. Results and analysis

1.1.2.1. T₂ spectra of manganese chloride solution and aqueous gel in porous media

T₂ (transverse relaxation time) spectra were separated for all the components by algorithm inversion. We obtained a large amount of data, but here we only take part of them to show the change pattern of T₂ spectra during the experiment. Fig. 2 shows T₂ spectra at different time points obtained using manganese chloride solution of 10 g/L and 20 g/L, respectively.

As shown in Fig. 2, the relaxation time at the left peak is about 1 ms, which indicates the signals of manganese chloride solution. The relaxation time at the right peak is about 530 ms, this corresponds to aqueous gel. The peak intensity of gel signals decreases with relaxation time, whereas that of manganese chloride solution signals increases with relaxation time, this implies Mn²⁺ diffusing into gel with time. On the one hand, decreasing Mn²⁺ of manganese chloride solution in the left affected water relaxation and caused T₂ signals of water to get stronger; due to increasing amount of manganese chloride solution with diffusion, the peak intensity increased with time.

On the other hand, gel signals weakened and their peak intensity decreased due to the rise of Mn²⁺ concentration in gel. Comparison of the peak intensity of gel signals shows that Fig. 2a exhibits slower decline of gel peak than Fig. 2b. After 120 h, Fig. 2a exhibits much higher peak intensity than Fig. 2b. This means Mn²⁺ diffusion was faster at higher concentration and consequently there was more Mn²⁺ diffused into gel and affected gel signals greatly.

1.1.2.2. MRI images of manganese chloride solution and aqueous gel in porous media

Fig. 3a and 3b shows MRI images of manganese chloride and aqueous gel at different time obtained using two set of experimental models. Here we obtained the pseudocolor images (T₂ weighted images) by processed signals which are easily distinguishable. More and more gel signals were masked due to Mn²⁺ diffusion from the left part with high concentration to the right part.

As shown in Fig. 3, (1) the area of gel image decreases with time, indicating greater influence of Mn²⁺ diffusion on gel from 0 h to 120 h; (2) the high-concentration solution has larger impact on gel image than low-concentration solution in the same time comparing the Fig. 3a and 3b, this means Mn²⁺ moving faster at high concentration; (3) there is a transition zone between the blue zone and red zone, which indicates attenuated gel signals are not entirely sheltered because Mn²⁺ concentration is not high enough. Mn²⁺ concentration at the front edge of the transition zone is defined as the threshold C_th, which corresponds to the lowest concentration for gel signal suppression. The rate of front edge movement toward the right part is dependent on the rate of Mn²⁺ diffusion rate.

1.1.2.3. Mn²⁺ diffusion rate in porous media-gel system

Fig. 4 shows the variation of diffusion front edge with time obtained from MRI images. The diffusion of solution with the concentration of 20 g/L is faster than that with the concentration of 10 g/L. The relationship between diffusion distance and time was:

The distance of diffusion is proportional to the square root of time; this means Mn²⁺ diffusion in the porous media-gel system conforms to the Fick diffusion law^[11]. Here a_c is the scaling factor, it is equal to 1.12 mm/h^0.5 for the solution with concentration of 10 g/L and 1.87 mm/h^0.5 for the solution with concentration of 20 g/L.

1.2.1. Methodology

These experiments aimed at understanding Mn²⁺ impact on gel imaging and plugging performance in the process of manganese chloride solution displacing gel.

The displacing fluids were manganese chloride solutions with the concentration of 5 g/L and 10 g/L respectively. Artificial cores with similar petrophysical properties were used, the parameters are shown in Table 1. Refer to Reference [6] for detailed description of testing devices and workflow.

The steps are as follows. (1) Core 1 was vacuum-saturated with manganese chloride solution of 5 g/L and put in the core holder, and manganese chloride solution of 5 g/L was injected for displacing, then pressure measurement, T₂ spectrum extraction and imaging were done. (2) Aqueous gel of 0.4% was injected at the rate of 0.5 mL/min till the volume of injection reached 0.4 PV (pore volume). The confining pressure was 5 MPa. T₂ spectrum was measured and imaged and displacement pressure was recorded every 0.1 PV of displacement. (3) T₂ spectrum was measured and imaged after 2 h waiting on gelling. (4) Manganese chloride solution of 5 g/L was injected into the core sample at a constant rate of 0.5 mL/min till the volume of injection reached 0.8 PV. MRI was performed and displacement pressure was recorded every 0.1 PV of displacement. (5) Steps (1) to (4) were repeated on Core 2 with manganese chloride solution of 10 g/L.

1.2.2. Experiment result and analysis

1.2.2.1. MRI image variation with different concentrations of manganese chloride solution displacing gel

Fig. 5 shows the pseudocolor images converted from the grey-scale maps of core sagittal plane acquired in the process of displacement without color reversal. Fig. 5a and 5b shows the images of core flooding by manganese chloride solutions of 5 g/L and 10 g/L, respectively. As shown in Fig. 5a, gel moved quickly in the lower part when 0.4 PV was injected. The area of gel image accounts for 2/5 of the sagittal plane, which agrees with the volume of injection. The image basically remained unchanged after 2 h waiting on gelling. After 0.2-0.4 PV injection of manganese chloride solution of 5 g/L, the gel moved forward as a whole, but it did not reach the outlet. The area of gel image reduced gradually. A notch occurred at the side being displaced and the displacing fluid showed finger advance, this is more obvious especially at 0.6 PV displacement. The area of manganese chloride solution image exceeded what it should be occupied by calculation according to the volume of injection, this means Mn²⁺ already moved into gel and changed gel signals.

As shown in Fig. 5b, the downside of the front end of gel image became dim after 2 h waiting on gelling. In the process of manganese chloride solution displacement from 0.2 to 0.4 PV, gel advanced, but its image area decreased gradually while that of manganese chloride solution enlarged. Water breakthrough occurred at the upside in the incoming direction, the image of manganese chloride solution exhibited an "upper triangle", and a "lower triangular" in the outgoing direction. The decrease in gel image at both sides indicates large-scale Mn²⁺ diffusion into gel, which inhibited gel imaging.

The differences in MRI images obtained at two concentrations of manganese chloride solutions show that the area of gel image decreases quickly, indicating that Mn²⁺ diffusing faster at higher concentration of manganese chloride solution.

1.2.2.2. Impact of manganese chloride concentration on pressure difference of gel plugging

Fig. 6 shows the displacement pressure variation with time in the experiments with manganese chloride solutions of 5 g/L and 10 g/L respectively. (1) In the periods of gel injection and waiting on gelling, the injection pressures were almost the same. The maximum injection pressure differences in the experiments were 2.04 MPa and 2.12 MPa respectively, indicating similar petrophysical properties of cores and gel strength in two displacement experiments. (2) In the period of displacement by manganese chloride solution, the injection pressures declined quickly, especially in the experiment with manganese chloride solution of 10 g/L, the injection pressure dropped faster and more widely. This means high Mn²⁺ concentration led to small residual pressure difference and stronger gel damage.

In summary, Mn²⁺ can diffuse in the porous media-gel system and followed the Fick diffusion law. The occurrence of Mn²⁺ in aqueous gel would reduce the relaxation time and attenuate signals of aqueous gel, consequently the visible area of gel would be reduced. Mn²⁺ can damage the gel structure and lowering the plugging performance, and the higher the Mn²⁺ concentration, the lower the residual pressure difference and the poorer the plugging performance will be.

With fluid components distribution in the porous media revealed by MRI images, we formulated a mathematical model to describe Mn²⁺ diffusion and movement in the porous media. Fig. 7 shows the computational model for Mn²⁺ diffusion experiments above. The total length of the model was 2L. The origin of coordinates was at the separation plane. The model is of slender tube shape, hence, Mn²⁺ diffusion may be regarded as a 1D movement.

Mn²⁺ movement in the porous media-gel system is expressed using the 1D convective-diffusive equation with chemical reaction terms.

This model could be simplified. As the diffusion process is static, the velocity of flow is zero, i.e. u=0. The porosity remains unchanged and thus can be removed from equation (2). Then we get equation (3).

Mn²⁺ will be adsorbed onto the gel in the process of diffusion. The complexation reaction between Mn²⁺ and carboxylate radicals on gel surface^[12,13] will decrease free Mn²⁺ in the solution. This complex adsorption reaction process is described using the source term, w.

The capacity of gel adsorption to Mn²⁺ is dependent on C_R, which is related to gel concentration and degree of hydrolysis. The source term is equal to zero in the left part of the model without gel. The rate of adsorption reaction is proportional to the concentration of reactant, and we assume gel will not move in this process. Thus,

Assuming that one Mn²⁺ is combined with n carboxylate radicals and k₂=nk₁, where n is set to be 2. The initial conditions are

The model is sealed on both sides and the diffusion flux is equal to zero; thus, the boundary condition is

We can use this model to numerically calculate Mn²⁺ concentration anywhere at any time.

The initial concentration of carboxylate radical, C_R0 (7.7 mmol/L), was estimated by polymer concentration and its degree of hydrolysis^[14,15]. In view of Reference [16], the two reaction rate constants, k₁ and k₂, were set at 1.2 and 2.4 L/(mmol·h), respectively. The other parameters, including diffusion coefficient of Mn²⁺ in porous media-gel, were fitted using experimental data. Mn²⁺ has similar diffusion coefficient in gel and water, so equal diffusion coefficients were set for both parts of the model^[10]. To make the calculation easier, some parameters were non-dimensionalized, dimensionless concentration was C_Mn/C_Mn0 and dimensionless distance was x/L.

2.3.1. Comparison of distances of manganese ion diffusion from simulation and experiment

Gel signals would be masked by Mn²⁺ when Mn²⁺ concentration reaches the threshold; this manifests as the transition from red to blue on the pseudocolor images. But it is hard to determine the concentration threshold directly using nuclear magnetic signals. Experiments showed that gel significantly changed in properties when it was immersed in manganese chloride solution with concentration of above 2 g/L, over crosslinking and even polycondensation dehydration may occur^[8]. Gel signals were entirely masked when the concentration of manganese chloride solution exceeded 5 g/L. Thus, C_th was set at 2-5 g/L.

Fig. 8 shows the location variation of threshold concentration with time. The discrete points derived from MRI images in it denote the variation of diffusion distance of different concentrations of manganese chloride solutions with time, or the variation of points at C_th with time. These points basically agree with the solid lines calculated using the mathematical model. C_th calculated from simulation is about 2.5 g/L, which is between 2 and 5 g/L, in accord with with the MRI result. In conclusion, the mathematical model can describe Mn²⁺ diffusion in the media effectively.

The simulated concentration threshold is above 2 g/L, indicating that the transition front on the pseudocolor images corresponds to the significant change in gel properties. The diffusion coefficient of Mn²⁺ in the porous media-gel system calculated is 1.6 mm²/h, which is smaller than that (of 3.6 mm²/h) in water^[10]. This is attributed to larger diffusional resistance in the porous media and chemical reaction between Mn²⁺ and gel.

For a specific concentration threshold, the higher the initial concentration of Mn²⁺, the faster the front movement, the larger the area of gel damage, and the larger the difference between gel distribution on MRI images and actual distribution will be. This means Mn²⁺ concentration should be kept as low as possible provided that signals of different components can be differentiated.

2.3.2. Impacts of chemisorption and reaction rate on manganese ion diffusion

From the experiment results, Mn²⁺ diffusion conforms to the Fick diffusion law. The adsorption reaction, coexisting with diffusion, may bias the concentration distribution curve. Fig. 9 shows C_Mn distribution of manganese chloride solution of 20 g/L at 30 h with and without adsorption reaction.

Compared with the scenario without adsorption reaction, Mn²⁺ concentration is lower when there is adsorption reaction. This means chemisorption leads to decrease of Mn²⁺ concentration at the front edge of diffusion, which is equivalent to reduction of Mn²⁺ diffusion rate in the right part.

The adsorption capacity of gel to Mn²⁺ is related to its degree of hydrolysis. Carboxylate radical concentration in gel increases with the rise of hydrolysis degree, thus, more Mn²⁺ may be adsorbed. The experiments show that the impact of metallic ions on gel is a slow-moving process, as proved by gel aging^[13]. This means chemisorption may occur at a small reaction rate constant, which may be too small to be measured directly. For this reason, an estimated value was used in accordance with the work by Li et al.^[16].

The impact of chemical reaction rate may be numerically examined. Fig. 10 shows Mn²⁺ concentration distribution at 100 h in two scenarios with 10-time difference in reaction rate.

It can be seen from the figure that chemical reaction rate mainly affects the front edge of diffusion. A higher rate may lead to higher probability of collision and higher diffusion resistance, thus, the concentration may decline.

2.3.3. Impact of diffusion time on Mn²⁺ diffusion

Fig. 11 shows Mn²⁺ concentration distribution at 4 moments of manganese chloride solution with 20 g/L, indicating the Mn²⁺ diffusion process.

Due to Mn²⁺ diffusion from left to right with time, Mn²⁺ concentration decreased in the left and increased in the right until an equilibrium was reached. The distance of diffusion was short at 1 h and exceeded one half of L at 30 h, but the concentration was low. In Fig. 3, the transition zone between the blue zone and red zone expands with time, which is consistent with the result in numerical simulation that the concentration distribute tends to become gentle with time.

The results show the concentration decreased in the left and increased in the right with time. The concentration distribution curve presents power exponential relation.

2.3.4. Impacts of initial concentration on Mn²⁺ diffusion

Initial Mn²⁺ concentration may affect MRI signal differentiation and gel damage degree. Fig. 12 shows Mn²⁺ concentration distribution after 120 h diffusion with different Mn²⁺ initial concentrations.

As shown in Fig. 12, the concentration at the diffusion front edge of the manganese chloride solution of 20 g/L is much higher than that of 5 g/L; this means the lower the initial concentration, the lower the concentration at the front edge will be. In spite of equal difference of 5 g/L between the 4 initial concentrations, the difference in concentration at the front edge is unequal. The lower the initial concentration, the larger the concentration difference at the diffusion front edge, the lower the relative concentration will be. Thus, lowering initial concentration will slow down the diffusion and reduce the concentration at the front edge of diffusion, this may decrease gel damage.

Mn²⁺ diffusion in the porous media-gel system conforms to the Fick diffusion law. The occurrence of Mn²⁺ in aqueous gel reduces the relaxation time and suppresses gel signals. We established a chemical reaction model to describe Mn²⁺ diffusion in the porous media-gel system. The simulated diffusion rate and concentration distribution with the model agree with those from MRI experiments. The diffusion coefficient calculated was 1.6 mm²/h, which is smaller than that in water. The concentration threshold of manganese chloride solution which can bring gel damage was calculated as 2.5 g/L.

Chemisorption between gel and Mn²⁺ can slow down Mn²⁺ diffusion. A method of multi-stage profile control is recommended. Salt-tolerant gel with high degree of hydrolysis can be used at the front and rear ends to increase gel adsorptive capacity and reaction rate at the front and rear interfaces, so that Mn²⁺ diffusion in gel may be mitigated to avoid inner gel damage. The initial concentration of manganese chloride solution should also be reduced as much as possible to decrease the concentration at the front edge of diffusion and preserve gel signals. In other words, there should be a compromise between signal differentiation and gel strength preservation.

We developed a visualization approach to investigate gel or weak gel damage in the process of profile control. This approach can be used to study the mechanism of gel profile control as well as the diffusion rate and features of surfactant, nano-fluid, compound injectants, air, and CO₂ in core.

a_c—scaling factor, mm/h^0.5;

C_Mn—molar concentration of manganese chloride, mmol/L;

C_Mn0—initial concentration of manganese chloride solution before diffusion, mmol/L;

C_R—molar concentration of carboxylate radicals in gel, mmol/L;

C_R0—initial concentration of carboxylate radicals in gel before diffusion, mmol/L;

C_th—concentration threshold, mmol/L;

D—diffusion coefficient of Mn² in core, mm²/h;

k₁—reaction rate constant corresponding to C_Mn, L/(mmol·h);

k₂—reaction rate constant corresponding to C_R, L/(mmol·h);

L—half model length, 80 mm;

t—diffusion time, h;

u—average flow velocity in core, mm/h;

w—reaction term, mmol/(L·h);

x—diffusion distance, mm;

ϕ—porosity, %.