DPG is prepared with polymer and cross-linker, which can swell up to 30 times its beginning size in brine. DPG is a particle system formed by the mechanical shearing of the dried gel on the ground. Swollen gel strength and particle size are controllable, overcoming some distinct drawbacks of in situ gel systems. And as a new profile control agent, DPG is environmentally friendly, and not sensitive to reservoir minerals and formation brine (Dai et al., 2014b). With mechanical shearing for 6 min, the DPG particles are prepared on the shearing equipment. The microstructure of prepared DPG particles is observed by scanning electron microscope (SEM), which is shown in Figure 1.

It can be seen from Figure 1 that the DPG solution is composed of regular pseudo spheres particles. When the DPG solution is prepared with high salinity produced water and aged at 90°C for 5 days, 10 days, and 15 days respectively, SEM is used to study the microscopic morphology of DPG particles. Figure 1 reveals the microscopic morphology of DPG particles before and after aging at high temperature. Scanning electron micrographs show that the DPG solution is uniformly distributed as a single particle before aging, and single DPG particle size is mainly dispersed between 1.8 and 2.2 μm. After 5 days aging under high salinity and temperature conditions, the size of the individual particle begins to be bigger and the shape tends to be irregular, and the expansion of DPG particles is limited, which enables the particles to maintain a high strength; when the DPG is further aging, the particles aggregates with each other, forming larger DPG particle aggregates. After aging for 30 days, the aggregates size can be up to 20 μm. When the DPG begins to age at high temperature, the presence of electrolytes and high temperature accelerates particles aggregation. Because the surface of DPG particles is negatively charged, and high salinity water contains a high concentration of ions, there will be more ions in the adsorption layer so as to reduce or even completely neutralize the charge on the DPG particles surface, which decreases mutual repulsive force between particles. This is the main reason for the increase of particle size.

The macroscopic pictures of DPG solution before and after aging are showed in Figure 2. As we can see, no degradation or dehydration occurs in the DPG solution after ageing in Figure 2, which shows DPG good temperature and salt resistance. These properties of DPG are beneficial to the effective profile control of high permeable zones. When pore throat radius is small, a single particle can swell to implement water shutoff; when pore throat radius is large, multiple particles will form a larger particle aggregation and achieve conformance control of high water-cut reservoir.

Figure 3A points out that the viscosity of the gel system increases as the DPG concentration increases. This is because the DPG particles will clump together when the number of DPG particles increases, and free movement of particles becomes difficult with the increase of interaction force between particles, resulting in a big increase in the viscosity of the gel system. But the increase in polymer viscosity is mainly due to the random entanglement of groups of the polymer. Bond strength between polymer groups is very weak. Therefore, the viscosity of polymer solution at the same concentration is lower than that of DPG.

Conventional polymer used for enhancing oil recovery is affected by shear of ground injection equipment and underground percolation. As shown in Figure 3B, the viscosity of polymer can be reduced by more than 50% after 20 min of ground equipment shearing. So the shear thinning effect should be considered in the numerical simulation model of polymer flooding. In other words, the shear resistance of polymer is poor, and most injected water easily reaches production wells along high permeable regions when implementing polymer flooding in heterogeneous reservoirs. In view of shear thinning, the shear resistance of DPG is investigated by comparing with polymer. At room temperature, shearing equipment is respectively used to DPG particles and conventional polymer shear at 1,000 rpm for 5 min, 10 min, 15 min, and 20 min. The corresponding viscosity is measured at 30°C, and both of them have a mass concentration of 0.2%. Figure 3B illustrates that the viscosity of polymer decreases sharply with the increase of shear time, however the viscosity of DPG decreases slightly with increasing shear time. DPG solution is a stable system formed by the high viscoelasticity body gels with high-speed mechanical shear, which can still maintain effective viscosity after a period of shearing. So it is necessary to take shear resistance effect into account and it can be approximated that the viscosity of DPG is constant, when the mathematical model of DPG profile control is proposed.

For simplicity DPG in conformance control, the main assumptions are as follows: 1) The process of DPG profile control is isothermal and energy exchange is negligible in the reservoirs; 2) the extended Darcy’s law applies when fluid flows through porous medium; 3) the fluid underground only contains two phases: The water phase and the oil phase, and the water phase contains three components: pure water, DPG particles and salt; 4) the process of DPG particles adsorption is irreversible; 5) no chemical reaction takes place; 6) diffusion and dispersion for DPG particles are negligible.

Using the above assumptions, mass balance equations are given for each component. Considering the physicochemical phenomena including weak shear thinning effect, permeability reduction, DPG particles adsorption and dead pore volume, the basic equations of the DPG profile control model are as follows:

Continuity equation for oil component: ∇ · K K r o μ o B o ∇ p o − γ o ∇ D + q o = ∂ ∂ t ϕ S o B o (1)

Continuity equation for water component: ∇ · K K r w μ w B w ∇ p w − γ w ∇ D + q w = ∂ ∂ t ϕ S w B w (2)

Continuity equation for DPG particles component: ∇ · K K r w R k μ w B w ∇ p w − γ w ∇ D C m + q m = ∂ ∂ t ϕ S w C m + C m , a d s B w (3)

Well flow equations for multiphase flow: q s c , l = 2 π K K r l μ l B l ln r e / r w + s p e − p w f (4) where p is the pressure, Pa; K is the absolute permeability, m²; K _r is the relative permeability, dimensionless; C _m is the concentration of DPG particles in the water phase, kg/m³; C _m,ads is the concentration of DPG adsorbing on pore throat surface, kg/m³; γ is the gravity factor, N/m³; μ is the viscosity, Pa·s; D is the depth, m; q is the flow rate, m³/s; t is the time, s; ϕ is the porosity, dimensionless; B is the formation volume factor, dimensionless; r _w is the wellbore radius, m; r _e is the effective radius, m; s is the skin factor, dimensionless; subscript l represents the liquid phase, l = o, w, o represents oil, w represents water; subscript m is the DPG component.

Constrained condition of saturation: s o + s w = 1 (5)

Constrained condition of capillary force: p c o w = p o − p w = p c S w , σ w o (6)

Based on the capillary force function (J function), the capillary force (Qin and Li, 2003): p c o w = J s w σ w o ⁡ cos ⁡ θ φ k (7) where p _cow is the capillary pressure between water and oil, Pa; J (s _w ) is the Leverett J-function; σ _wo is interface tension between water and oil, N/m; θ is the wetting angle.

DPG particles flowing through the reservoir can interact with the rock surface, which causes DPG particles to bind to the surface of the rock, reducing DPG concentration in the aqueous phase. In the process of numerical simulation, empirical formula calculation and data table interpolation are given to determine the DPG adsorption concentration.

(1) Formula method:

The amount of DPG particles adsorption on rock surface can be calculated by Langmuir adsorption isotherm. C m , a d s = min C m , a d s max , a p C m 1 + b p C m (8) where C _m,ads is the absorption concentration of DPG particles, kg/m³; Cmax m, ads is the maximum saturation adsorption concentration of DPG particles, kg/m³; a _p and b _p are the coefficients changing with ion concentration, obtained by fitting the experimental data, dimensionless.

(2) Data table interpolation method:

According to the relationship between DPG concentration and its adsorption concentration at a certain salt concentration in related experiments, DPG adsorption concentration is attained by interpolating DPG concentration.

For both of these methods, this paper assumes that DPG particles adsorption is an irreversible process. The following steps are adopted in the actual calculation: 1) Determine the maximum adsorption concentration of DPG, which is usually given by experiments; (2) the adsorption concentration of DPG in the current time step was calculated by Langmiur isothermal adsorption equation; (3) update DPG particles adsorption concentration if it is greater than the adsorption concentration of the previous time step, the adsorption concentration remains unchanged if the adsorption concentration is less than that of the previous time step.

PEBI grid has the following advantages: Flexible grid division, any grid points can be arranged in place, without considering other grid points; Easy to change the shape and size of the grid, suitable for describing complex geological boundaries, tectonic faults, pinch-outs, etc. Compared with regular grid, the grid orientation effect can be reduced. The near-well area can be locally encrypted and the coarse and fine mesh is smooth, which is suitable for describing the radial flow in the near-well area. So PEBI mesh is used for division instead of corner point grid in this paper. The governing equations are discretized by finite volume method, linearized by Taylor expansion and time term conservation form expansion, and the coefficient matrix of the equations is obtained. Finally, GMRES solver is used to solve the coefficient matrix. Since GMRES algorithm is a mature solution technique for solving large sparse matrices, this section focuses on the discretization and full implicit linearization of equations (Zha et al., 2018).

Here, we take Eq. 3 as an example and its discrete scheme is given: ∑ j T i j , w C m ∆ p w − γ w ∆ Z n + 1 = C m m δ p + C m w δ S w + C m c δ C m + C m a δ C m , a d s + q w s c n + 1 (9)

In Eq. 9, the coefficients of the terms are shown below: T i j , w = K K r w μ w B w R k i j · w i j d i j , C m m = V i ∆ t 1 B w n ∂ ϕ m ∂ p C m n S w n + C m , a d s n + ϕ m n + 1 ∂ 1 / B w ∂ p C m S w + C m , a d s , C m w = V i ∆ t ϕ m C m B w n + 1 , C m c = S w n V i ∆ t ϕ m B w n + 1 , C m a = V i ∆ t ϕ m B w n + 1 . where w _ij is the area of interface between adjacent grids, m²; d _ij is the distance between the center points of adjacent grids, m; V _i is the volume of grid i, m³; n is the time step.

Fully implicit linearization of Eq. 9 is as follows: ∑ j T i j . w ν δ p j − δ p i + p j ν − p i ν − γ w Δ Z + ∂ T i j . w C m ∂ p + ν δ p + p j ν − p i ν − γ w Δ Z + ∂ T i j . w C m ∂ C m + ν δ C m + p j ν − p i ν − γ w Δ Z = C m m ν + 1 δ p + p ν − p n + C m w ν + 1 δ S w + S w − S w n + C m c δ C m + C m ν − C m n + C m a δ C m , a d s + C m , a d s ν − C m , a d s n + q w s c n + 1 (10) where ν is the iterative step, the subscript + is the upstream grid.

The viscosity and concentration relationship of DPG is the key factor affecting water plugging in high permeability areas. The application of DPG can be limited by the relevant parameters of high water-containing heterogeneous layers such as water saturation and permeability ratio. In order to investigate the influences of DPG properties and reservoir conditions on water-cut reduction, a PEBI grid system (500 × 500 × 5.9 m) is established. In the basic simulation case, there is an injection well and four production wells in the reservoir, as shown in Figure 5. The injection rates respectively are 72 m³/d for 6 days, 48 m³/d for 3 days, 0 m³/day for 7 days and 60 m³/day for 720 days; the production rates separately are 18 m³/d for 6 days, 12 m³/day for 3 days, 0 m³/day for 7 days and 15 m³/day for 720 days, every producing well produces same quantitative liquid. The DPG concentration is 1.6 kg/m³, the absolute permeability of high permeable layer is .4145 μm², and the permeability ratio between high and low permeable regions is 8.4. The main numerical simulation parameters for sensitivity analysis are listed in Table 2.

In addition to the above DPG solution viscosity, DPG concentration, water saturation of high permeable regions and permeability ratio between high and low permeable regions, there are other factors affecting the water shutoff effect of DPG profile control. In order to further understand the water plugging principle of DPG, the following numerical simulation examples are given.

In this section, a 500 × 500 × 5.9 m single-layer heterogeneous reservoir is studied. The green and brown areas are high permeable areas and the other white areas are low permeable areas. There is a five-point well pattern with one injection well and four producing wells, and the brown high permeable zone is closer to the injection well, as shown in the Figure 11 below. The injection well is firstly injected with the DPG for 6 days and then the water for 3 days at a constant rate of 72 m³/d, and the concentration of DPG is 1.6 kg/m³, and the four production wells last 9 days with a constant rate of 18 m³/d at surface condition. The main numerical simulation parameters are the same as the sensitivity analysis example.

In order to reveal effective factors of water shutoff, the mobility ratio between the high and low permeability zones is adjusted in the process of numerical simulation. The following Figure 12 show the distribution of DPG concentration after profile control for 9 days when the mobility ratios are 24 and 84 respectively. As can be seen from the figures below, the DPG concentration closer to the injection well in the high permeability zones is lower than that in other high permeability zones, so the injected water is easier break through the DPG sealing zones closer to the injection well. When the mobility ratio is relatively high, it is more unfavorable to water plugging, as shown in Figure 12B.

After a long time water flooding in mature oil fields, the reservoir has high permeable channels or fractures, and the injected water can easily enter into the producing well along high permeable regions, if we continue to apply water flooding treatment. In order to displace out remaining oil in low permeable regions, polymer flooding has been widely used in high water-cut oil fields. Polymer flooding does not significantly increase oil production when the permeability ratio of high permeability zones to low permeability zones increases. But the effect of DPG profile control becomes better, as the permeability ratio between high and low permeable regions increases. In order to clarify the difference between DPG profile control and polymer flooding in heterogeneous reservoirs, a series of geological models with the permeability ratios of 5, 10, 15, and 20 between high and low permeable regions are investigated. A 432 cubic DPG solution is injected with the concentration of 1.6 kg/m³, and then switch to water flooding for 3 days, afterwards shut down the wells for 7 days. After 15 days of DPG profile control, the reservoir begins to be injected with water for 240 days. The following operation of polymer flooding is the same as that of DPG profile control except displacement fluid, the comparison results are seen in Figure 13. From Figure 13A, the water-cut reduction effect of polymer flooding becomes worse, when the permeability ratio between high and low permeable regions increases. This is because that polymer is easier to channel into the production well along the high permeable zones, leading to less oil displaced out in low permeable areas, as the permeability ratio increases. In comparison with polymer flooding, there is more obvious water-cut reduction funnel for DPG profile control with the increase of permeability ratio. This is because polymer flooding cannot effectively plug water in high water cut areas when the permeability ratio increases. Cumulative oil production for polymer flooding will decrease by degrees as permeability ratio between high and low permeable areas increases. However, DPG profile control comes to the opposite conclusion in Figure 13B. So the DPG profile control is more suitable for heterogeneous reservoirs than polymer flooding.

The stability of the numerical simulation largely independent on the grids is very important. Therefore, it is necessary to investigate the impacts of the number of grids on numerical simulation results. After a long period of water flooding, a 432 cubic DPG solution is injected with water saturation of .8 in high permeable regions, and then switch to water flooding for 3 days, afterwards shut down the wells for 7 days, and the main numerical simulation parameters are the same as the sensitivity analysis example except for the number of grids. After 15 days of DPG profile control, the reservoir begins to be injected with water for 720 days, the results are shown in Figure 14. It can be seen from Figure 14 that the difference of calculated water cut in production well near high permeability areas under different grid numbers is in the range of permitted errors and does not affect the actual numerical simulation results.