Due to its long lifespan and high sand-removal efficiency, gravel packing is one of the most applied sand control methods during the recovery of reservoirs with sanding problems. The blockage and retention of injected sand in a gravel pack is a complex process affected by multiple mechanisms. The majority of existing studies based on the phenomenological deep bed filtration (DBF) theory focused on the gravel pack’s overall permeability damage and failed to obtain the inner-pore particle distribution pattern. In this work, experiments and simulations were carried out to reveal the particle distribution in a gravel pack during flooding. In particular, through real-time monitoring of particle migration, the penetration depth and distribution pattern of invaded particles with different gravel-sand particle ratios, fluid viscosities and injection rates could be determined. By simplifying each unit bed element (UBE) into a pore-throat structure with four tunnels (two horizontals for discharge and two verticals for sedimentation), a new network simulation method, which combines deep bed filtration with a particle trajectory model, was implemented. Cross comparison of experimental and numerical results demonstrates the validity and accuracy of the model.
During the development of reservoirs with high sand production rate and silt content such as heavy oil, hydrate and unconsolidated sandstone reservoirs, sanding is a severe issue that restrict efficient development. Gravel packing completion technique has been widely applied to solve this problem because of its high sand control efficiency and long validity period [
It is inadequate and inaccurate to describe the physical process of particle transportation in granular media by only applying theoretical and numerical methods considering the non-negligible heterogeneity of pore throat distribution and particle grain size. Therefore, experimental research is indispensable in this study. Previous researchers have studied the plugging and deposition pattern of particles in the granular medium through a laboratory test. Based on these experimental results, the prediction model of the retained particle concentration was established, and a series of empirical coefficients for characterizing the particles plugging in the pore space has been obtained [
Nevertheless, the traditional approach is the lack of intuitiveness. Most of the sand-filling pipe or sandbox used in traditional flooding experiments is invisible, and the simulation materials of particles and the porous medium have no color difference. The particle penetration front in the gravel pack cannot be monitored during the experiment. It is hard to separate retained particles from gravels after the experiment as well. Thus, the current study using traditional approach failed to obtain particle distribution in the gravel pack.
The predecessors mainly used the simulation method base on the phenomenological model and trajectory analysis model to study particle transportation characteristics in gravel pack numerically [
In this study, visual experiments were carried out to study the dynamic distribution of intrusive particles in the gravel pack. Based on the improved particle trajectory model and stochastic algorithm, self-compiled numerical simulation software was developed to simulate the dynamic particle blockage of the gravel bed. In view of the experiments and numerical simulation results, a more intuitive and accurate method for studying dynamic blockage law of gravel-bed was proposed.
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.
where, subscript
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 [
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
The experimental scheme was designed as shown in
Group number | Gravel-to-sand grain size ratio | Flooding rate/(ml/min) | Fluid viscosity/mPa⋅s | Packing length (sand)/cm | Packing length (gravel)/cm |
---|---|---|---|---|---|
1# | 4 | 400 | 3 | 15 | 15 |
2# | 5 | 400 | 3 | 15 | 15 |
3# | 6 | 400 | 3 | 15 | 15 |
4# | 7 | 400 | 3 | 15 | 15 |
5# | 6 | 200 | 3 | 15 | 15 |
6# | 6 | 600 | 3 | 15 | 15 |
7# | 6 | 800 | 3 | 15 | 15 |
8# | 6 | 400 | 1 | 15 | 15 |
9# | 6 | 400 | 5 | 15 | 15 |
10# | 6 | 400 | 7 | 15 | 15 |
The gravel-to-sand grain size design adopts in this paper mainly refers to the study carried out by Saucier [
During the experiment, the transportation and distribution of particles in the granular medium were continuously monitored. Take the experimental groups 1 to 4# in
According to the experiment results, which were dynamic particle penetration depth and the mess of the retained particles in the granular medium, the sensitivity analysis of three factors that affect, i.e., gravel particle size ratio, displacement velocity and liquid viscosity, was carried out.
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
The distribution of retained particles inside the gravel pack shows in
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. [
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
The total mass of retained particles increased linearly with the flooding rate, as shown in
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
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
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 [
As one can observe from
A modified Hagen-Poiseuille equation can determine the viscous pressure gradient of a blocked element:
The kinetic pressure gradient
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 [
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:
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.
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
Based on the DLVO theory, the forces acting on particles of the porous media internal surface can be characterized by the following equation:
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:
where
By integrating
The trajectory of injected particles is characterized by
The phenomenological model of filter media describes the macroscopic distribution characteristics of particles in the porous media during the process of deep bed filtration [
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 (
where,
For a single two-dimensional UBE as shown in
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.
where
The particle transportation model in a two-dimensional scale consists of
Pore damage by particle intrusion can be divided into the following two modes, as shown in
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.
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:
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
The particle distribution characteristic obtained by numerical simulation is shown in
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.
In this work, the distribution pattern of injected particles in the gravel pack during flooding was carried out by visual experimental and numerical approaches. The inner-pore particle distribution, particle penetration depth and mass of the retained particles were obtained experimentally. By analysis of the experimental results under the influence of G-S size ratio, injection rate and fluid viscosity, the following conclusion can be drawn: When it comes to the influence of G-S ratio on particle migration, there is a particular threshold value that exists. The sand arch that formed at the sand-gravel interface can alleviate sand intrusion effectively when the G-S ratio is less than five, while massive particle intrusion occurs with the increase of the G-S ratio. When the superficial fluid velocity does not reach the critical velocity of particle initiation, both the penetration depth and mass of the intrusive sand were relatively low. But as the fluid velocity further increased above the critical sand production velocity, the gravel pack blockage could be more severe. For the experimental group with low viscosity, the drag force acting on particles does not reach the critical sand migration force, resulting in a decrease of overall sand production rate and alleviate the gravel pack blockage.
By analyzing the characteristics of the sand blockage process in the gravel pack, the limitation of using tradition DBF model in solving this engineering problem were proposed. A new network simulation method was proposed by the combination of DBF theory and particle trajectory model. The experimental validation showed that the new simulation method is of high reliability in predicting both particle penetration depth and distribution pattern. The experimental and simulation method proposed by this work can provide practical guidance for gravel packing design in-field operation.
This work was financially supported by National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2016ZX05011004–003).