Modeling Proppant Transport in Casing and Perforations Based on Proppant Transport Surface Tests

Abstract

The results of a series of proppant transport surface tests (PTSTs) were used in conjunction with Eulerian multiphase-computational fluid dynamics (EMP-CFD) modeling to develop an engineering model of proppant distribution. The PTSTs were carried out to evaluate proppant placement through perforated casing. In these tests, sand slurry was pumped at realistically high flow rates through perforated casing and the distribution of sand and slurry from each perforation cluster was observed. The tests show that gravitational settling in horizontal casing, proppant slip past perforations and the visco-elastic properties of slickwater fluids strongly affect the distribution of proppant from the heel to the toe of the completion. The EMP-CFD modeling was used to estimate the gravitational settling of sand in fully-developed turbulent slurry flow in horizontal casing as a function of casing velocity. A survey of 36 calculations was carried out to generate tables of sand concentration in a cross section through the casing as a function of flow rate and particle size. A single-phase CFD analysis showed how sand exiting each perforation is taken from a limited ingestion area which is proportional to the ratio of flow through the perforation to total flow in the casing. A detailed EMP-CFD analysis of flow through single perforations showed how sand slips past the perforation. The results of 28 EMP-CFD calculations provided slip factors as a function of particle size, casing flow velocity, and perforation flow velocity in straight and angled perforations. The EMP-CFD settling tables and a parameterization of the slip factors were integrated into an engineering model. The model predicts the distribution of slurry and sand through each perforation based on the proppant size, perforation phase angle, and pump rate. The engineering model was used to predict the sand distributions observed in the PTSTs. The PTSTs were conducted with a range of sand sizes and with low-viscosity friction reducing polymer (FR) additives, while the EMP-CFD analysis assumed water. A weight factor is introduced in the settling model to account for the increased dispersion of sand in water with low viscosity FR and to match the observed sand distributions in the PTSTs. The observed slip of 100 Mesh and 40/70 Mesh sand is consistent with the EMP-CFD calculations in water. The model reflects the PTST observations that fine sand is distributed relatively uniformly throughout the length of a perforated completion while coarser sand tends to slip past the heel perforations and concentrate on the bottom towards the toe of the completion.

A model for proppant dynamics in a perforated wellbore

“…In order to better understand qualitative behavior of the solution, the viscosity factor η s is further approximated as η s ( φs ) ≈ (1− φ) −2 , which introduces maximum error of up to 40% for concentrations φ < 0.9. Since η 1/2 s occurs in the equation (19), then the error is practically under 20%. In this case, the differential equation can be solved as…”

“…While the system of equations (18) gives the complete or full model, it requires solving a two-dimensional equation, which needs significantly more computational effort than classical wellbore simulation approaches. Therefore, the purpose of this section is to outline a reduced model, that is based on the steady-state solution (19), but yet still captures the dominant effect of time dependence.…”

“…To proceed with the reduced model, let's assume that φ( t, z) = φs (G a ( t), z), i.e. the solution can always be represented as a steady-state solution (19) with an apparent G a . The primary governing equation for the reduced model is the vertically averaged particle balance equation ( 10)…”

“…The reason for that lies in the slip velocity between the fluid and proppant. The displacement of the latter slip is denoted by s. Note that the fact that proppant located only near the perforation leaves the wellbore was observed in [19] via CFD simulations and the area outlined by l p was called the "ingestion area". Similarly, authors in [18] plotted the fluid and proppant streamlines, which confirmed that the "ingestion" area is nearly circular and that there is a difference between those areas for fluid and proppant, as outlined by l f and l p .…”

“…This is a very important observation and it will be used for constructing the model in this paper. It is also important to mention field scale measurements discussed in [19,20]. Authors considered field scale stage length, wellbore diameter, and particle diameters.…”