Advances in Materials Science and Engineering

Volume 2017, Article ID 2195404, 10 pages

https://doi.org/10.1155/2017/2195404

## Estimation of Sand Production Rate Using Geomechanical and Hydromechanical Models

Faculty of Geology & Petroleum Engineering, Department of Drilling & Production Engineering, Ho Chi Minh City University of Technology-Vietnam National University, Ho Chi Minh City, Vietnam

Correspondence should be addressed to Son Tung Pham; nv.ude.tumch@gnutnosmahp

Received 16 May 2017; Revised 8 August 2017; Accepted 24 August 2017; Published 17 October 2017

Academic Editor: Antonio Riveiro

Copyright © 2017 Son Tung Pham. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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.

*(1) Formation Failure*. In situ stresses and pore pressure act on formation sands and under certain conditions; the criteria for failure are met. The presence of the wellbore and perforations causes a concentration of stresses near these cavities and deformation and failure can occur under certain well-known conditions. This criterion is bottomhole pressure that makes the maximum effective tangential compressive stress equal or higher than the rock strength (failure criteria); significant sanding begins at some point (the onset).

*(2) Sand Erosion due to Flow*. Damaged regions that have failed (meeting the failure criteria) face additional stresses caused by pore pressure gradients. The process of sand erosion is essential for the sand to be removed from the failed region and then to be entrained with the fluid.

*(3) Sand Transport*. Sand erosion detaches sand grains into a perforating cavity or wellbore. Some of the grains are transported to the surface while others settle into the perforation tunnel or into the well hole.

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.

#### 2. Literature Review

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. [1] developed a model to predict the sand production rate from the onset of sanding model. The onset of sanding is predicted using a stress–based model of shear failure around a perforation or an open hole. Sand production is assumed to occur once the maximum value of the effective tangential stress around the perforation exceeds the apparent rock strength (rock strength is strength with is boost factor, for cased perforated wells = 2; TWC is the Thick Wall Cylinder which is a measure of rock’s strength and is used in sand production study instead of Unconfined Compressive Strength, because TWC reflects more closely the reality of in situ stress sustained by the borehole or the perforation channels than the UCS). No consideration was given to sand transport by drag forces. The model of Willson et al. [1] requires TWC test data for hole size the same as the perforation size. If the perforation is of different size, the application of the model must be careful. Hence the boost factor exists to compensate for the difference between the real data and the test data.

The criterion for sanding is

where CBHP is critical bottomhole pressure; is maximum principal stress; is minimum principal stress; is pore pressure. is poroelastic constant given by with : Poisson ratio and : Biot’s constant.

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:(see [1]).

In this formula given by Willson et al. [1], SPR is the sand production rate; water cut is the ratio of water produced compared to the volume of total liquids produced.

Although the Willson et al. model [1] takes into account the different phases of the fluid via water cut, the model did not give clear expression of the Sand Production Rate SPR in function of the Loading Factor LF, the Reynolds number, and the water cut.

In 1996, Vardoulakis et al. [2] proposed the following sand production model usingmixture theory, assuming that the sand in place is fully degraded from the beginning and the production is due only to the hydrodynamic forces. Equilibrium equation for solid phase is often ignored. The process is initiated with a very small solid concentration given as a boundary condition. The results are insensitive to this value as long as it is small:where is the porosity, is the rate of eroded solid mass per unit volume, is the solid density, *λ* is the experimentally evaluated sand production coefficient, is the transport concentration, is the specific discharge in the th direction, and is the notation representing the norm of a vector.

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. [3] developed the following model based on the assumption that failure is due to erosion and porosity increases until it reaches unity. Below is the relation between sand mass and the porosity:

The dimension of the above equation is that is the rate of eroded solid mass per unit volume (g·s^{−1}·m^{−3}), is the solid density (g·m^{−3}), is the porosity, and is time in second (s).

Variation of sand mass due to erosion is given aswhere is sand production coefficient and is plastic shear strain.

The main advantage of Papamichos et al. model [3] is that the authors provided experimental data in order to calibrate the experimental parameters. In this model, not all the plastic deformation areas will produce sand. The sand is produced only when the plastic deformation reaches a limit value (). However, it is practically difficult to determine this limit value in reality. Besides, the system of differential equations of this model is extremely complex with plenty of empirical parameters.

Chin and Ramos [4] developed a sand production model considering that erosionoccurs during sand production as follows, where is the solid velocity:

The main inconvenient of this model [4] is that it was developed only for weak formation. Besides, the model retains only primary physics of rock failure and coupled rock deformation and fluid flow. The model does not include the effects of well configuration and completion, wellbore storage, erosion, interaction of disaggregated solid and flowing fluid, and solid transport through the porous medium.

The analytical model developed by Fjær et al. [5] combined a theoretical model with an empirical relationship as shown below:where the porosity increases with time:where is the flow rate, is the critical flow rate, is the initial porosity, is the initial time, and is proportionality constant; has the dimension of s/m^{3}.

Gravanis et al. [6] developed a coupled stress-fluid flow erosion model: where , with dimension of inverse length, represents the strength of the erosive processes that lead to sand production. is an exponent parameter of the model. is an erosion function defined as follows:

is the usual Euler Gamma function and is the depth of the plastic region. The profile function approaches a step function as increases. The exponent is fixed at the beginning of the analysis and can be tuned further as part of the calibration procedure. In this work we will fix , which is the same value used in the work of Fjær et al. [5].

In 2010, Isehunwa and Olanrewaju [7] proposed a new model for sand production, considering the effects of flow rate, fluid viscosity and density, grain size, and cavity height. Sand is produced by drag and buoyancy forces which predominantly act on the sand particles. The radius of sand production cavity is

The volume of sand produced can be expressed aswhere is the fluid flow rate, is the grain radius, is the cavity height, and is the gravitational acceleration.

Among these models, the ones of Fjær et al. [5] and Gravanis et al. [6] were chosen for this study because of their clarity in the explanation and equations, which are necessary for us to be able to numerically solve the problems. In addition, these models have some advantages in comparison with others. They proposed the basic theory for hydrodynamic erosion of sandstone which is based on filtration theory. They adopted full strength hardening/softening behavior of reservoir stone. They also do take into account the grain size, the gradient elastoplasticity for thick wall cylinders and the cavity failure around boreholes.

The models were solved using the workflow developed in Section 3 of this paper; then the coding was made using MATLAB software.

#### 3. Workflow for Calculation and Calibration

##### 3.1. Hydromechanical Erosion Model of Gravanis et al. [6]

According to Gravanis et al. [6], the basic assumptions are as follows:(1)Fluid flow can be described by Darcy’s law.(2)We define the mathematical time by for simple calculations. is the real time while is the mathematical time which is introduced to facilitate the resolution of the problem. (Figure 1). The function is related to pressure drop as follows: is the permeability.(3)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.(4)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).