This paper aims to develop a numerical model that can be used in sand control during production phase of an oil and gas well. The model is able to predict not only the onset of sand production using critical bottom hole pressure inferred from geomechanical modelling, but also the mass of sand produced versus time as well as the change of porosity versus space and time using hydromechanical modelling. A detailed workflow of the modelling was presented with each step of calculations. The empirical parameters were calibrated using laboratory data. Then the modelling was applied in a case study of an oilfield in Cuu Long basin. In addition, a sensitivity study of the effect of drawdown pressure was presented in this paper. Moreover, a comparison between results of different hydromechanical models was also addressed. The outcome of this paper demonstrated the possibility of modelling the sand production mass in real cases, opening a new approach in sand control in petroleum industry.
Sand production occurs in many oil fields across the world and it is especially common in the porous sediments. Sand production observed on the surface occurs as a series of three events that happen at downhole area: (1) formation failure, (2) sand erosion due to flow, and (3) sand transport.
During and after sand production, wells can sand-up and that has different effects on the productivity. At first, the productivity seems to increase due to the increase of permeability. However, after a while, the sand produced can obstruct the entrance of hydrocarbon into the wellbore, especially for cased perforated wells using sand screens or gravel pack. Disposal of produced sand is also a significant cost associated with sand production. Finally, sand can be transported to the surface which causes erosion of pipe lines, joints, chokes, and valves. So, if the prediction of sand production is identified, it will help operators to manage the situation properly and prepare suitable treatment methods for the well.
This study focuses on the first and second steps of sand production because they have the most impact and they are not well understood. The main reasons why in petroleum industry nowadays we still do not predict the mass of sand production are because of the complexity of numerical models (hence the lack of professional software in this domain) and the unavailable real data of sand production due to the difficulty in collecting this kind of data in the oil field. Therefore, most of the studies predicting the mass of produced sand still stay at the laboratory step. In real life, in petroleum companies’ reports, only Geomechanical models are being used to predict the onset of sand production. This papers aims to bring the application of the Hydromechanical models into a real case in petroleum industry and to combine the use of Geomechanical model, which predicts the onset of sand production, and the Hydromechanical model, which predicts the mass of produced sand.
Several studies released models predicting the onset of sand production and the amount of sand produced. Parameters affecting sand production have been discussed for decades. However, there is no clear consensus. In this section, a brief review of major conclusions of these past studies is presented.
Willson et al. [
The criterion for sanding is
where CBHP is critical bottomhole pressure;
The model predicts the rate of sand production by utilizing the nondimensionalized concepts of Loading Factor, LF (near-wellbore formation stress normalized by strength), Reynolds number (Re), and water production factor. An empirical relationship between Loading Factor, Reynolds number, and the rate of sand production incorporating the effect of water production was proposed as follows:
In this formula given by Willson et al. [
Although the Willson et al. model [
In 1996, Vardoulakis et al. [
The Vardoulakis model is difficult to solve because of the complexity of the equations. Moreover, the model does not take into account the different phases of the fluid, so the fluid is considered as single phase, which does not reflect the reality of petroleum fluid. Furthermore, the coefficients were not calibrated due to lack of experimental data.
Papamichos et al. [
The dimension of the above equation is that
Variation of sand mass due to erosion is given as
The main advantage of Papamichos et al. model [
Chin and Ramos [
The main inconvenient of this model [
The analytical model developed by Fjær et al. [
Gravanis et al. [
In 2010, Isehunwa and Olanrewaju [
The volume of sand produced can be expressed as
Among these models, the ones of Fjær et al. [
The models were solved using the workflow developed in Section
According to Gravanis et al. [ Fluid flow can be described by Darcy’s law. We define the mathematical time The function The whole region is divided into a plastic region and an elastic region. In elastic region, we apply Hooke’s law and in plastic region we consider the Mohr-Coulomb failure criterion. Under the condition that plasticity of the material is damaged and subject to decohesion, it can be eroded under weak hydrodynamic forces. It happens when drawdown pressure (DP) exceeds a critical drawdown pressure (CDP).
Schematic of the hollow cylinder [
Calculate function where Calculate pressure The depth of plastic region
In plastic region, we consider the Mohr-Coulomb failure criterion and combine with the boundary condition
where
At this stage, there are two continuity conditions and two unknowns, the integration constant
The porosity field is calculated from erosion model equation ( The formula involves the integral Calculation of porosity and radius of sand production cavity: Radius of sand production cavity
Calculate function
The erosion strength is
where
Calculate sand production mass rate:
Note that this equation must be used in the next time step along with
The porosity field is calculated from ( This involves the values of the depth
The analytical model is based on these assumptions: The driving mechanism for continuous sand production is erosion from plastified material in the vicinity of the production cavity. The sand production rate depends on (1) how much the well pressure is reduced below the critical sand production pressure, (2) the fluid flow rate and the fluid viscosity, and (3) the cementation of the rock. The sand in place is fully degraded from the beginning and the production is due only to the hydrodynamic forces. These forces are proportional to the fluid pressure drop over the volume element, and the pressure drop is proportional to the fluid flow rate as specified by Darcy’s law; moreover the permeability is given by the Kozeny-Carman equation; we have where Firstly, it is required that DP > CDP, which expresses the fact that stress induced damage of the rock is a necessary condition for sand production. Secondly, it is required that
Following Fjær et al. [
Calculate porosity from ( Calculate volumetric sand production where Sand production mass:
Illustration of the plastified zone and the sand producing zone around a cavity [
Once the entire part of the rock that has been producing sand collapses, the remaining solid material in that part is produced in one burst.
The collapse of the sand producing zone implies that the radius of the cavity increases, from Cumulative sand production: sand mass is given as the initial amount of solid material in the sand producing zone; that is, Now the stresses around the cavity are redistributed, and the situation is the same as it was at
We use experimental data profile of Papamichos et al. [
Experimental data of Papamichos et al. [
Variable | Value |
---|---|
Cylinder internal radius, | 0.01 |
Cylinder external radius, | 0.1 |
Cylinder height, | 0.2 |
Internal pore pressure, | 0 |
External pore pressure, | 0.15 |
Inner radial stress, | 0 |
Outer radial stress, | 11 |
Flow rate, | 0.5 |
Ratio | 0.03 |
Young’s modulus, | 6750 |
Poisson ratio, | 0.19 |
Biot’s ratio, | 1 |
Cohesion, | 3.7 |
Friction angle, | 37.4 |
Initial porosity, | 0.3 |
Permeability, | 500 |
Kozeny-Carman parameter, | 8.96 |
Solids density, | 2640 |
Dynamic viscosity, | 5 |
We have empirical parameters:
Sand mass produced over time while maximum erosion strength
From the results presented from Figures
Sand mass produced over time while changing exponent coefficient
Sand mass produced over time while changing ratio of initial to maximum erosion strength
Sand mass produced over time while changing ratio of threshold depth and initial plastic region
The model of Fjær et al. has three empirical parameters:
Sand mass produced over time while changing critical porosity
Sand mass produced over time while changing critical fluid flux
Sand mass produced over time coefficient sand production
Figures
In fact, the result of the model seems very different than the experimental result because Fjaer’s model considers a step of “collapse” (Step
These results also allow us to choose suitable values of empirical parameters which are critical porosity (
An application for the oilfield X in Cuu Long basin using Geomechanical model of Willson et al. [
After collecting and processing data by analyzing log and other parameters, input data are shown in Table
Data of Well X1.
Input data | Value |
---|---|
TVD (ft) | 9479.5 |
| 52 |
| 244 |
| 9300 |
| 8670 |
| 9448 |
| 0.3 |
UCS (psia) | 2450 |
| 1 |
| 193 |
| 90 |
TWC (psia) | 4936 |
| 60 |
| 0.3 |
| 3 |
| 0.0328 |
| 1.6405 |
| 979004 |
The critical bottomhole pressure CBHP is important information during production phase because it indicates the lowest bottomhole pressure for sand production to not occur. For a specific well in oil field, the only production data that we control is the bottomhole pressure, which is adjusted using surface chokes. The pressure is controlled; hence the flowrate is controlled. For this reason, before predicting the sand production using Hydromechanical models of Gravanis et al. and Fjær et al., we firstly calculate the CBHP using Geomechanical model of Willson et al.
The testing showed that a relationship between the effective in situ strength of the formation U and the TWC (Thick wall cylinder) strength relative to a specimen with an OD/ID ratio (Outer/Inner diameters) would be equivalent to
Figure
Prediction of sand production with variation of UCS (Unconfined Compressive Strength).
The results showed that if UCS gets smaller, the sand production window is greater. This can be explained by the reason that the formation rocks around the wellbore are weakened when UCS decreases. Thus, risk of sand production increases.
We consider a case study with UCS = 2450 psi,
We use Hydromechanical Erosion models of Fjær et al. [
Results of two models in the case study of Well X1.
Figure
Finally, it is important to recall that, due to the lack of validation data in reality, it is impossible to choose which model to use in practical use. At this state, these results can only be used for illustration in study, and to demonstrate the possibility of using Hydromechanical models in real cases to predict eroded sand mass over time. In the future if real data is available, these models can be revised and recalibrated. Unfortunately, until now, measuring sand mass rate is technically impossible in the oilfields. An idea was proposed to solve this problem; the determination of real sand rate may be based on the erosion rate of the choke. However, this study has never been realized and may constitute a new objective in our future study.
This study combines Geomechanical model and the two Hydromechanical Erosion models of Fjær et al. [
The experimental data (sand production’s mass in function of time) is rarely done not only in Vietnam but also in the world (only some laboratories and authors mentioned in the paper have done this kind of experience because of their research in this field). Hence, it is currently impossible to obtain empirical data for Cuu Long basin (sand mass in function of time), not only data in laboratory but also production data due to difficulties in collecting and measuring sand production rate. Therefore, application of the calibrated model for Cuu Long basin in this study is destined to demonstrate the possibility of using Hydromechanical models in real cases and the result can be used for illustration in research. In the next studies when we have available data from Cuu Long basin, we can do a revision/recalibration of these models. We already thought about the determination of real sand rate data based on the erosion rate of the choke. This idea will be the objective of our future study.
Critical bottomhole pressure (psia)
Drawdown pressure (psia)
Loading Factor (near-wellbore formation stress normalized by strength)
Sand Production Rate (g/s)
True vertical depth (ft)
Thick wall cylinder strength (psia)
Unconfined compressive strength (psia)
Notation representing the norm of a vector
Biot’s constant
Exponent parameter
Plastic shear strain
Usual Euler Gamma function
Experimentally evaluated sand production coefficient (1/m)
Proportionality constant (s/m3)
Initial erosion strength (m−1)
Maximum erosion strength (m−1)
Depth of the plastic region (m)
Threshold depth of the
Viscosity (MPa·s)
Porosity
Initial porosity
Critical porosity
Perforation orientation from the top of the wellbore in deviated wells (degree)
Solid density (g/m3)
Maximum principal stress (psia)
Minimum principal stress (psia)
Vertical stress (psia)
Minimum principal horizontal stress (psia)
Maximum principal horizontal stress (psia)
Minimum horizontal stress direction (degree)
Poisson ratio
Integration constant
Young’s modulus
Well inclination (degree)
Permeability (mD)
Principal stress ratio
Perforation length (ft)
Mass of produced sand
Rate of eroded solid mass per unit volume (g/s·m3)
Rate of eroded solid mass (g/s)
Initial amount of solid material in the sand producing zone (g)
Pore pressure (psia)
Internal pore pressure (Mpa)
External pore pressure (Mpa)
Bottomhole pressure
Reservoir Pressure
Flow rate (m3/s)
Specific flow rate in the
Critical flow rate (m3/s)
Perforation radius (ft)
Cylinder internal radius (m)
Cylinder external radius (m)
Location of the plastic zone boundary (m)
Radius of the cavity (m)
Radius of the sand production zone (m)
Radius of the plastic zone (m)
Reynolds number
Material cohesion (MPa)
Real time
Mathematical time
Initial time
Solid velocity (m/s)
Volumetric sand production (m3)
Well azimuth (degree).
The authors declare that there are no conflicts of interest regarding the publication of this paper.