As mentioned in the previous section, one of the main reasons why visualization of traditional gravel displacement experiment cannot be realized is that there is no visible color difference between porous medium and particles. To meet the experimental requirements of visible and measurable, the grain size of porous medium and particles in the prototype are enlarged in this experiment. Glass beads are packed in the sand packing tube instead of gravel to form a transparent granular media as well. A particular criterion should be established in the process of hydrodynamic experiment design to ensure a similar flow and particle motion state between the model and the prototype. Therefore, a fixed proportional relationship of physical quantities such as geometrical, kinematical and dynamical quantities is required. For solid-liquid two-phase flow, flow and particle Reynolds criterion, Stokes criterion and Freud criterion are the four dominant similarity criteria in designing experiments.

(1) lpρpupμfp=lmρmumμfm(Re)p=(Re)m

(2) dspρspurpμfp=dsmρsmurmμfm(Res)p=(Res)m

(3) dsp2ρspurpμfplp=dsm2ρsmurmμfmlm(Stk)p=(Stk)m

(4) ρplp2up2ρpglp3=ρmlm2um2ρmglm3(Fr)p=(Fr)m

where, subscript p represents prototype and subscript m represents model; Re and Re_s are the flow and particle Reynolds number, dimensionless; Stk is the Stokes number, dimensionless; Fr is the particle Freud number, dimensionless; l is the geometric characteristic length, m; d_s is the particle diameter, m; ρ and ρ_s are the fluid and particle density, kg/m³; u and u_r are the fluid velocity and relative velocity between solid and liquid phases, m/s; μ is the fluid viscosity, N⋅s/m².

Because it is impossible to satisfy all the criteria simultaneously during the design of hydromechanics experiment, secondary factors that have less influence on similarity should be neglected [23]. The similarity issue of Reynolds number (Re) can be ignored because the Reynolds number of prototype and model is in the same self-modelling region (Re < 2000). The Froude number (Fr) indicates the ratio of inertia force to the gravity of particles. However, due to the height limitation of the sand-packing tube, the vertical distance that gravity acts on is short, and its similarity can be ignored as well. Therefore, the two criteria that the prototype and model should obey are the particle Reynolds and Stokes criteria.(5) λv=λl−1λt=λlλv=λl2 where, λ_l is the length similarity coefficient which equals to l_p/l_m; λ_v is the velocity similarity coefficient which equals to u_p/u_m; λ_t is the time similarity coefficient which equals to t_p/t_m.

Fig. 1 shows the set-up of the visual experiment. A peristaltic pump with a flow range of 1–1300 ml/min was used for displacement. The measuring container placing at the outlet collect both the outflow liquid and the produced particles. A clear cylindrical PMMA pipe packed with scale-up glass beads and quartz sand were selected to achieve the visualization of particle migration and blockage in the gravel pack. To ensure the similarity of particle motion between the experiment and real situation during the flooding, the injection rate should determine by the hydrodynamic similarity criterion (seeing Chapter 2.1). The particle penetration depth within the simulated gravel pack was photographed and recorded every 10 min once the flooding began. Once the particle penetration depth is stabilized, open the sand-pack tube from the outlet side and extract the packed gravel and sand particles in sequence. By weighing the mass of glass beads and quartz sand in each section, the distribution of the retained sand mass in the simulated gravel pack can be obtained.

According to the selected similarity criterion, the geometric magnification of the experiment was determined to be ten times. Take sand with 0.1 mm median grain size as an example, the quartz sand with ten times enlarged median grain size was selected to simulate the intrusive sand in a real reservoir, which is 1 mm. The glass beads with specific particle size were selected to simulate gravel packs to ensure the visibility and measurability of intrusive particles. The particle size distribution of the experimental sand sample is demonstrated in Fig. 2. The flooding fluid was polyacrylamide solution, and the viscosity adjustment was achieved by configuring different solutions with various polyacrylamide concentrations.

The experimental scheme was designed as shown in Table 1.

The gravel-to-sand grain size design adopts in this paper mainly refers to the study carried out by Saucier [24]. Taking both conductivity and sand retention rate into account, he proposed that the average grain size of packed gravel used for sand control operation should be 5–6 times of that of the intrusive sand. The design of the injection rate and fluid viscosity refers to the experimental data of Li et al. [5] in the conventional gravel packing displacement experiments.

Gradation design of the packing gravel is one of the most critical parts for the gravel packing operation. The gravel pack should have adequate sand control ability to avoid extensive sand production. Meanwhile, the gravel pack’s conductivity has a significant influence on productivity and should be considered, as well. Four groups of experiments were designed to study the influence of the gravel-to-sand grain size ratio (G-S ratio) on particle penetration depth and retained sand mass, see groups 1–4#.

The trend of particle penetration depth for the four groups with different G-S ratios shown in Fig. 4a. The ratio of particle penetration depth (L_pd) and gravel packing length (L_pack, which equals 15 cm) was utilized to non-dimensionalize the experimental data. The group 1# with a G-S ratio of four exhibits excellent sand control performance. The penetration depth of particles was only 1.7 cm, and it reached a stable state within 150 min. Group 2# with a G-S ratio of five had a faster growth rate in particle penetration depth in the early stage of flooding, but also reach stability quickly with a depth of 2.6 cm. For groups with a G-S ratio of six and seven, the velocity and depth of particle penetration front increase. This phenomenon explains that with the enlargement of pore space, the formation of the sand bridge at the gravel-sand interface became slower. More particles intruded into the gravel pack without the restrain by a stable sand bridge, resulting in severe pore blockage.

The distribution of retained particles inside the gravel pack shows in Fig. 4b. Same as the particle penetration depth, the ratio of the retained sand mass in gravel pack (m_retained) and the mass of the total packing sand (m_pack, which equals 268 g) was proposed to non-dimensionalize the intrusive particle mass. For experimental groups with a G-S ratio of four and five, the total amount of intrusion particles is much smaller than the other two groups. The retained particles were concentrated at the gravel-sand interface as well. The invasion depth and particle concentration increased significantly as the enlargement of packing grain size. The cumulative particle mass of groups 3#, 4# with a G-S ratio of six and seven is 14.5 g, and 24.6 g, respectively, which is much higher than that in groups 1# and 2#. The intrusive sand of these two groups covered a much deeper gravel bed in longitudinal depth as well.

It can be concluded that a particular threshold value exists when it comes to the influence of G-S ratio on particle transportation and blockage. The formation of sand bridges in the pore space kept particles from further intrusion when the G-S ratio is less than five. The G-S ratio should not be higher than six to avoid massive particle intrusion. The experimental results are in good agreement with those proposed by Saucier et al. [24].

The flooding rate in the experiment simulated the flow rate of the near region of the wellbore during the production. The effect of flooding rate on particle migration can be concluded into the following three items: sand production rate, the formation of the sand-bridge at the interface and the particle re-migration rate. In this experiment, the migration and blockage of particles in porous media at the flooding rate of 200–800 ml/min were studied.

The particle penetration chart of various flooding rate was shown in Fig. 5a. The penetration depth is shorter than the other groups when the flooding rate is 200 ml/min. That is because certain drive speed is necessary for the formation of loose particles, and fewer intrusive particles generated under the case of low flooding rate. The intrusion depth development of groups with flooding rate of 400 and 600 ml/min show similar law. But when the flooding rate reaches 800 ml/min, re-migration of the retained particles occurred in the pore space, resulting in a noticeable increase of particle penetration depth.

The total mass of retained particles increased linearly with the flooding rate, as shown in Fig. 5b. The blocking degree near the gravel-sand interface was low due to a less mass of intrusive sand in the group of 200 ml/min. For the other three groups of experiments with the flooding rate of 400–800 ml/min, the mass of retained particles near the gravel-sand interface was similar, but the mass of particles that remained in the inner region of the gravel pack increase significantly with the rise of flooding rate.

It can be concluded from the above experimental results that the penetration depth and mass of intrusive particles are relatively low if the flooding velocity does not reach the critical velocity of the particle migration. While when the flooding velocity is above the critical velocity, further velocity increase could lead to a deeper penetration depth of particles in the gravel pack.

The fluid viscosity mainly affects the drag force and settling velocity acting on the intrusive sand. A higher fluid viscosity means a greater particle start-up force and a longer horizontal migration distance. As the fluid viscosity increase, the particles have a more considerable invasion distance in the gravel pack and settled or blocked particles in the pore have a higher chance of re-migration as well. Three experimental groups were carried out to study the sensitivity of fluid viscosities to particle migration and pore blockage.

As can be seen from Figs. 6a and 6b, the particles penetration depth was linearly correlated with the viscosity of the injecting fluid, but not for the mass of retained particles. The experimental groups with a fluid viscosity of 1 and 3 mPa⋅s had the similar retained particle mass, but a different penetration depth. When the viscosity raised to 5 and 7 mPa⋅s, the total mass of blocked particles and the plugging degree of porous media increase significantly. That is because, for the experimental group with lower fluid viscosity, the drag forces acting on the particles have not met the start-up force for particle migration. Therefore, these groups had a lower particle production rate and a smaller gravel-to-sand mixing region. However, when the forces are big enough to separate particles off, a large mass of particles could intrude into the gravel pack and cause a severe blockage.

Compared with other deep filtration applications, the particle blockage problem in the gravel pack has the following characteristics, which leads to the invalidation or inaccuracy of the traditional model. Firstly, the density of the sand particles is much higher than that of the reservoir fluid, so the sedimentation of particles plays a much more important role than that of other cases. Therefore, the network model is chosen to simulate particle distribution in the gravel pack instead of the traditional one-dimensional deep bed filtration model.

Secondly, the gravel-to-sand size ratio is much larger than that in many other industrial fields. Therefore, the influence of particle sedimentation on pore throat characteristics changes and subsequent particle migration cannot be neglected. In the classical deep filtration model, due to the small-suspended particles, the deposition coefficient of particles in porous media is generally constant, that is, the effect of deposited particles on subsequent particle migration was not taken into consideration.

Lastly, gravel packing is an engineering issue that needs large-scale simulation. The existing approaches cannot satisfy the requirements for obtaining inner-pore particle distribution and solving large-scale model efficiently at the same time.

In this work, a two-dimensional network was applied to simulate the process of particle transportation and distribution in the gravel pack. The unit bed element (UBE) has been simplified to a pore-throat structure. The elements either have one pore and four throat channels (two vertical and two horizontal) in the middle parts, or one pore and three throat channels (one vertical and two horizontal) at the boundary.

The diagram of the network model used in the simulation is shown in Fig. 7, from which one can obtain the simplified UBE structure. The pressure drop of a single UBE can be demonstrated as follows:(6) ΔPu=ΔPth+ΔPpo+pf where ΔP_u stands for the pressure drop of the whole UBE; ΔP_th and ΔP_po is the pressure gradient caused by viscous energy loss in throat and pore channel; p_f is the pressure gradient caused by internal fluid kinetics.

Assuming that the flow diffusion rate is much lower than the injection flow rate for a saturated porous media. The relationship between superficial flow velocity in the pore and throat channel is [25]:(7) utup=(dhp−2δavepdht−2δavet)2 where u is the flow velocity, m/s; d_h is the equivalent diameter of throat and pore channels, m; δ_ave(i,j) is the average thickness of deposited layer of retained particle along the channel, m; the subscript t and p denote the throat and pore.

As one can observe from Fig. 7, d_hp equals d_ht + d_p. Therefore, the non-homogeneity of the pore structure could be achieved by replacing fixed d_ht and d_p values with a randomly distributed array.

A modified Hagen-Poiseuille equation can determine the viscous pressure gradient of a blocked element:(8) ΔP=ΔPpo+ΔPth=128μ(Lpup(dhp−2δavep)2+Ltut(dht−2δavet)2) where d_h is the equivalent diameter of throat and pore channels; δ_ave(i,j) is the average thickness of deposited layer of retained particle along the channel.

The kinetic pressure gradient p_f is caused by the sudden expansion and contraction of the inner-pore geometry. According to Wu et al. [26] p_f can be represented as functions of pore-throat diameter ratio β:(9) pf=ρghf=(ξe+ξc)up2ρ2=(32+1β4−52β2)up2ρ2 where β is the pore-throat diameter ratio, β=dhp−2δavepdht−2δavet.

The forces acting on a single solid particle in the fluid phase can be divided into the following three categories: forces independent of the relative motion of fluid-particle, including gravity and the pressure gradient force; forces in the same direction as the relative motion between fluid and particle, including resistance, additional mass force and Basset force; forces perpendicular to the relative motion direction between fluid and particle, such as Magnus force and Saffman force [27,28].

When the solid phase composition in the particle flow is dominant, inter-particle collision is the primary mechanism affecting particle motion, and the influence of the interaction force between particles should be clarified. In the issue of gravel packing sand control, the particle concentration is low, and the inter-particle collision can be ignored, the particle can be regarded as approximate random free motion. For larger pore space, short-range forces such as electrostatic force and double-layer repulsion force have less effect on particles than gravity force, thus can be neglected. The particles are assumed to move by sliding due to the force controlling the rolling of the particles (Magnus force) is relatively small compared to Stokes force. To take both calculation efficiency and accuracy into consideration, the effects of gravity, buoyancy, viscous force and wall-particle interaction on particle motion are the main considerations in this study.

The joint force of gravity and buoyancy on the vertical direction of particles in pore-throat of the granular porous media is:(10) Fg=4π3g(ρp−ρl)rp3 where r_p is the particle size, m; ρ_p is the particle density, kg/m³.

For solid-liquid two-phase flow with a low Reynolds number, the resistance of particles in a Newtonian fluid is characterized by the Stokes equation.(11) FD=Dcur where D_c is the viscosity coefficient, D_c = 6πμr_p; u_r denotes the difference between the particle velocity (v) and the flow velocity (u), i.e., u_r = v – u.

For a saturated porous media, it can be assumed that the inner-pore fluid flows only in the horizontal direction, while the flow rate in the vertical direction can be assumed to be 0. When particles move under the domination of Stokes force, the expression of u_r is as follows:

(12) urx=u−vx

(13) ury=vy

Based on the DLVO theory, the forces acting on particles of the porous media internal surface can be characterized by the following equation:

(14) FA=1.13×10−14D⋅rp⋅rs[1D2−(rp+rs)2−1D2−(rp−rs)2] where r_s is the particle size of porous media skeleton, m; D = r_s + r_p + d, d is the minimum distance between particle and the internal surface.

When the solid phase composition in the particle flow is dominant, interparticle collision is the primary mechanism affecting particle motion, and the influence of the interaction force between particles should be clarified. In the issue of gravel packing sand control, the particle concentration is low, and the interparticle collision can be ignored, the particle can be regarded as approximate random free motion. For larger pore space, short-range forces such as electrostatic force and double-layer repulsion force have less effect on particles than gravity force, thus can be neglected.

Force analysis of particles in fluids in longitudinal and transverse directions:

(15) Fg+FA1−Fdy−FA2=ρpVpdvydt

(16) Fdx=ρpVpdvxdt

where V_p is the particle volume, m³; v_x and v_y are the particle velocity components in X and Y direction, respectively, m/s; F_dx and F_dy are the viscous force components in X and Y directions, respectively, N; F_A1 and F_A2 are DLVO forces from top and bottom surfaces of the pore internal wall, respectively.

By integrating Eqs. (15) and (16), the expressions of particle velocities in X and Y direction can be obtained

(17) vy=(FA1+Fg−FA2)Dc[1+exp(−DcρpVpt)]

(18) vx=v−v⋅exp(−(Dc⋅t)/ρpVp)

The trajectory of injected particles is characterized by Eqs. (17) and (18), and the trajectory model based on the real force of particles can be established.

The phenomenological model of filter media describes the macroscopic distribution characteristics of particles in the porous media during the process of deep bed filtration [6,29–31]. Assuming that the flow inside a UBE is one-dimensional and linear, and the dispersion effect of the porous media is negligible. The particle mass conservation equation of one-dimensional deep filtration model is as follows:(19) −us∂c(x,t)∂x+D∂2c(x,t)∂2x=∂∂t(ϕ(x,t)c(x,t))+∂σ(x,t)∂t where, c is the particle concentration per unit bed volume; σ is the retained particle volume per unit bed volume; D is the flow diffusivity rate.

To obtain the distribution characteristic of particles in the porous media, transportation model of particles in two-dimensional scales is required. A two-dimensional UBE has two entrances and exits in the horizontal and vertical directions. For a UBE with coordinates (x, y), the volume of particles passing through the entrance and exit in the horizontal direction at t time is C_inn1(x,y,t) and C_{inn1(x+1,y,t)}, respectively. Similarly, these variables are C_inn2(x,y,t) and C_{inn2(x+1,y,t)} for vertical entrance and exit. Consider each row of the UBEs as a column cell, the c and σ in Eq. (20) can be written as the sum of intrusion and retained particles all over the row, as shown in Eqs. (20) and (21).

(20) c(x,t)=∑i=0ymaxCinn1(x,i,t)V0⋅∑i=0ymaxε(x,i,t)

(21) σ(x,t)=∑i=0ymaxσ1(x,i,t)+Csettle(x,t)V0⋅∑i=0ymaxε(x,i,t)

where, V₀ is the average initial pore volume of the UBE, m³; C_settle(x,t) is the total volume of retained particles in a column cell, m³; ε_(x,y,t) is the UBE’s porosity.

For a single two-dimensional UBE as shown in Fig. 7, the mass conservation equation is as follows:(22) −∂Cinn1(x,y,t)∂x=∂Cinn2(x,y,t)∂y+∂σ1(x,y,t)∂t

The volume of inlet and outlet particles of each cell can be expressed as the function of the particle retained and settled rate of particles, which can be determined from the particle trajectory model. The retained particle volume in the UBE at time t can be defined by the accumulation of captured particles by multiple mechanisms.

(23) Cinn1(x,y,t)=C0−∑i=0x−1(∂σ1(i,y,t)∂t+λsettle(i,y,t)Cinn1(i,y,t)−λre(i,y,t)Cinn2(i,y,t))

(24) Cinn2(x,y,t)=∑i=y−1ymax(λsettle(x,i,t)Cinn1(x,i,t)−λre(x,i,t)Cinn2(x,i,t))

(25) σ1(x,y,t)=σ1(x,y,t−1)+∑i=1nλcapture(i)Cinn1(x,y,t)

where λ_settle is the settlement coefficient of injected particles; λ_re is the coefficient of re-migration of settled particles; λ_capture is the particle capture coefficient; n is the number of particle capture mechanisms considered in the model, such as adsorption, blockage and so on. The particle trajectory model in Chapter 4.3 can determine the above coefficients.

The particle transportation model in a two-dimensional scale consists of Eqs. (22) and (25). The macroscopic distribution characteristic of particles in the porous media can be obtained by solving the model using the finite difference method.

Pore damage by particle intrusion can be divided into the following two modes, as shown in Fig. 8. Some intrusive particles were trapped on the pore surface due to the inertia force and adsorption. The pore damage caused by these particles can be characterized utilizing the equivalent hydrodynamic radius, that is, particles accumulation on the surface area equivalent to the enlargement of pack grain size of the porous media [14,16].(26) dg=(1−ε1−ε0)1/3dg0 where d_g0 is the initial packing grain size, m; ε₀ is the initial porosity.

Also, some intrusive particles settled at the bottom of the pore space under gravity. The settled particles in the pore affect the flow channel width and the subsequent migration of the inflow particles of the UBE. Therefore, the pore damage caused by particle settling can be characterized by the shrinkage of flow channels in the UBE.(27) D=(1−Vp(1−εp)ε0V0)D0 where Vp is the volume of settling particles, m³; V₀ is the initial volume of the pore space, m³; ε_p is the porosity of granular matter formed by settling particles.

An UBE should be regarded as the source term of the particles if it was completely blocked by the intrusive particles. The particle production rate is as follows:(28) Cinn1(x0,y0)=Cinn1(x0+1,y0)=λrmu0Vp

Based on the theoretical model proposed in the above sections, the transportation and distribution of particles in the gravel pack under the same parameter setting as the experiments were simulated by the in-house code.

A comparison of numerical simulation with the result of the flooding experiments was conducted to verify the accuracy of the simulation method. The accuracy of the model is validated by determining whether the particle distribution morphology and the particle penetration depth in the numerical simulation are similar to those in the physical simulation. According to the parameters listed in Table 1, the numerical simulation is operated with the experimental groups 1–4# with the G-S ratio of 4–7 as an example.

The particle distribution characteristic obtained by numerical simulation is shown in Fig. 9. The simulated particle distribution characteristic in the gravel pack is consistent with that of the monitored figures during the experiment. The depth and developing velocity of particle intrusion frontier increase with the G-S ratio so does the frontier inclination. Fig. 10 shows the comparison of steady particle penetration depth between the numerical and physical simulation, which shows good agreement as well.

From the validation, it can be concluded that the numerical model has high reliability, and it is applicable in the simulation of particle transportation and distribution pattern in the gravel pack.