PROPPANT FLOWBACK MITIGATION

Abstract

Proppant flowback during post-stimulation well clean up and production is a common occurrence in most hydraulically fractured wells. The production of the proppant from a propped fracture is related to the forces acting on the proppant pack during the well that is actively producing fluids. Flow rate and pressure data collected during the post-treatment flowback activity is used in simulating bottomhole (BH) production rates using a Gaussian solution scheme. The BH rate is distributed amongst the various perforation clusters while incorporating the effects of key hydraulic fracture characteristics in the presence of simulated effective bottomhole flowing pressures across different fluid entry points into the wellbore. The solution is updated at each time step during the simulation. The production allocation is then used in calculating effective flow velocities that are then compared with critical velocities to predict proppant flowback. Steps to mitigate or reduce the flowback are implemented.

Description

FIELD

Aspects of the present disclosure relate generally to systems and methods for using proppant in a fractured well and more particularly to optimizing a proppant flowback mitigation model.

BACKGROUND

Hydraulically fractured wells, especially those stimulated with large quantities of proppant are prone to producing proppant as routinely observed in the field. Proppant production during the initial flowback occurs frequently with the issue thought to be mostly arising from uneven distribution of proppant in the fracture, which leads to non-uniform loading on the proppant grains. Hence, the fluid movement through the pack during production can cause the proppant grains to dislodge and eventually flow into the well.

The treatment on a particular well stage is typically followed by a flush to clean up the wellbore, and preparations are made to isolate the treated zone by some mechanical means such as bridge plug, flow-through plug, packers, or other such devices, so that the next zone can readied up to receive the stimulation. While these actions are taking place, the hydraulic fractures placed in the stage that just terminated will eventually close but perhaps not fast enough to prevent settling of proppant. This can result in additional proppant, apart from that left behind in the wellbore during the stimulation process, to drop out into the wellbore. Thus, unless the fractures close instantaneously and lock the proppant in place or the proppant remains suspended, aided by other material such as network of fiber, or even neutrally buoyant proppant, the effect of gravity will dominate, and little can be done to prevent this fall back of proppant.

Once the wellbore, be it vertical or horizontal, undergoes the cleanup process after hydraulic fracturing treatment, a majority of the proppant and other debris is removed from the wellbore and the well is finally put on production. While it is desirable to obtain proppant-free production, the proppant now mainly trapped in the fracture, may flow into the wellbore along with produced fluids if conditions favoring this phenomenon exist downhole. Depending on the severity of the problem, this may either lead to frequent/periodic clean up runs or in extreme cases, a total loss of production if the proppant flows back uncontrollably.

It is with these observations in mind, among others, that aspects of the present disclosure were conceived and developed.

SUMMARY

Implementations described and claimed herein address the foregoing by mitigating proppant flowback. For example, a method of fracturing a hydrocarbon well can include calculating formation properties and production rates associated with a wellbore; calculating flow pressures at different production sleeves of the wellbore; using, by a simulator model, a material balance equation which considers the formation properties to determine pressure depletion for different zones of the wellbore; generating, by the simulator model, a simulated flow velocity value based at least partly on the production rates, flow pressures, and pressure depletion; determining, by the simulator model, that the simulated flow velocity value is within a predetermined threshold range of an actual flow velocity value; responsive to the simulated flow velocity value being within the predetermined threshold range of the actual flow velocity value, comparing the actual flow velocity value with a critical velocity value to form a normalized production rate; and/or optimizing mitigation of proppant production from the wellbore by adjusting or maintaining the normalized production rate.

In some implementations, the method further includes generating a z-factor vs. depth lookup table for a plurality of fluid segments and time intervals and optimizing mitigation of proppant production can be based at least partly on the z-factor vs. depth lookup table. The method can also include calculating, by the simulator model, a formation volume factor (FVF) and a specific fluid for the plurality of fluid segments. Furthermore, the method can include calculating, as a first output of the simulator model, a static column pressure of the wellbore. The method can also include calculating, as a second output of the simulator model, a tubular friction gradient of the wellbore. Additionally, the method can include calculating, by the simulator model one or more downhole production rates corresponding to the different production sleeves.

In some implementations, the method can further include calculating, by the simulator model, the flow pressures at different perforation clusters of the wellbore. Additionally, the method can include calculating, by the simulator model, a drawdown and fracture permeability of the wellbore. Calculating the drawdown and fracture permeability can be based on determining a fracture width and a fracture conductivity. Also, calculating the drawdown and fracture permeability can include updating the fracture width and the fracture conductivity for a closed boundary case. Furthermore, optimizing mitigation of the proppant production from the wellbore can include providing a recommended blowback guideline based on comparing the actual flow velocity value with the critical velocity value. Moreover, the method can further include normalizing a plurality of flow velocity values across production sleeves; and/or repeating a calculation to determine whether the simulated flow velocity value is within the predetermined threshold range of the actual flow velocity value.

In some implementations, a method of fracturing a hydrocarbon well includes determining one or more production rates associated with a wellbore; determining one or more flow pressures at one or more production sleeves of the wellbore; generating, by a simulator model, a simulated flow velocity value based at least partly on the one or more production rates and the one or more flow pressures; determining, by the simulator model, that the simulated flow velocity value is within a predetermined threshold range of an actual flow velocity value; comparing, responsive to the simulated flow velocity value being within the predetermined threshold range of the actual flow velocity value, the actual flow velocity value with a critical velocity value to form a normalized production rate; and/or optimizing mitigation of proppant production from the wellbore by adjusting or maintaining the normalized production rate.

In some implementations, the method can further include calculating formation properties associated with the wellbore, the simulated flow velocity value is at least partly based on the formation properties. The method can also include using, by the simulator model, a material balance equation which considers the formation properties to determine pressure depletion for different zones of the wellbore. Moreover, the method can include calculating, by the simulator model one or more downhole production rates corresponding to the one or more production sleeves. Also, the method can include calculating, by the simulator model, the one or more flow pressures at one or more perforation clusters of the wellbore. Additionally, the method can include calculating, by the simulator model, a drawdown and fracture permeability of the wellbore. Calculating the drawdown and fracture permeability can be based on determining a fracture width and a fracture conductivity.

In some implementations, one or more tangible non-transitory computer-readable storage media can store computer-executable instructions for performing a computer process on a computing system. The computer process can include determining one or more production rates associated with a wellbore; determining one or more flow pressures at one or more production sleeves of the wellbore; generating, by a simulator model, a simulated flow velocity value based at least partly on the one or more production rates and the one or more flow pressures; determining, by the simulator model, that the simulated flow velocity value is within a predetermined threshold range of an actual flow velocity value; comparing, responsive to the simulated flow velocity value being within the predetermined threshold range of the actual flow velocity value, the actual flow velocity value with a critical velocity value to form a normalized production rate; and/or optimizing mitigation of proppant production from the wellbore by adjusting or maintaining the normalized production rate.

In some implementations, proppant flow back is an association of the proppant production with critical flow velocities that are prevalent for given flowing conditions. This parameter can be used by the systems disclosed herein to predict proppant production during the well's producing life. Flow rate and pressure data collected during the post-treatment flowback activity can be used in simulating bottomhole (BH) production rates. Using Gaussian solution scheme, the simulated BH rate can be distributed amongst the various perforation clusters while incorporating the effects of hydraulic fracture characteristics in the presence of simulated effective bottomhole flowing pressures across each fluid entry point into the wellbore. The solution can be updated by the system at one or more steps, such as a plurality of different steps of the simulation. The production allocation can then be used in calculating effective flow velocities that are then compared with critical velocities (e.g., predetermined threshold values) to determine if proppant production can occur. Steps to mitigate or reduce the flowback, if present, can then be generated and provided as an output (e.g., instruction, recommendation, alert, etc.) to various types of computing devices and/or well drilling machines based on the analysis.

In some implementations, quantifying the flow parameters that may help in mitigating the proppant flowback includes determining physical conditions that can trigger the production of proppant. Calculations based on early time production conditions can be used in determining the flowback conditions leading to proppant production. These calculation methodologies and workflows to mitigate proppant flowback in producing wells are discussed in greater detail below.

In some implementations, the systems disclosed herein can include a method of fracturing a hydrocarbon well by perforating the wellbore to create fractures; calculating formation properties and production rates; calculating a flow pressure at each production sleeve; calculating drawdown to determine fracture width and fracture conductivity; using material balance to determine pressure depletion for each zone; determining actual flow rate at each production sleeve; and comparing actual velocity with critical velocity to normalize production rate, where the normalized production rate is optimized to mitigate proppant production.

In some implementations, a method of producing hydrocarbons from a hydrocarbon reservoir is disclosed where perforating a wellbore creates fractures; calculating formation properties and production rates; calculating a flow pressure at each production sleeve; calculating drawdown to determine fracture width and fracture conductivity; using material balance to determine pressure depletion for each zone; determining actual flow rate at each production sleeve; and comparing actual velocity with critical velocity to normalize production rate, where the normalized production rate is optimized to mitigate proppant production.

In some implementations, a method of normalizing production flow is provided where a fractured wellbore is used to calculate formation properties and production rates; flow pressure at each production sleeve; drawdown to determine fracture width and fracture conductivity; and using material balance to determine pressure depletion for each zone; determining actual flow rate at each production sleeve; and comparing actual velocity with critical velocity to normalize production rate, where the normalized production rate is optimized to mitigate proppant production.

In some implementations, the systems disclosed herein can use various input values/parameters to calculate different output values/parameters. The nomenclature and abbreviations for some of these values/parameters are defined below in Table 1:

TABLE 1
Nomenclature & Abbreviations
a                unknown vector in Eq. (21), various
b                matrix output in Eq. (21)
A-D, Y           constants of Eq. (5) to Eq. (7), dimensionless
A10, A11         constants in Eq. (4), dimensionless
Bg               gas volume factor, ft3/SCF or bbl/SCF
Bo               oil formation volume factor, rb/STB
Bob              oil formation volume factor at bubble point, rb/STB
Bt               composite two phase formation volume factor, rb/STB
Bti              initial oil volume factor, rb/STB
Bw               water formation volume factor, rb/STB
cf               formation compressibility, M−1L1t2, psi−1 (Pa−1)
co               oil compressibility, M−1L1t2, psi-1 (Pa-1)
c1 to c4         constants in Eq. (2), unitless
Cd(i)            discharge coefficient at ith sleeve, dimensionless
Ct               total compressibility, M−1L1t2, psi-1 (Pa-1)
D, Dp            proppant diameter, M0L1t0, in. (mm)
DPperf(i, t)     pressure drop across the ith perforation at time t, M1L−1t−2, psi (Pa)
Ds(i)            sleeve diameter of ith sleeve, M0L1t0, in. (mm)
DVwt             constant of Eq. (14)
DVwp             constant of Eq. (14)
fn, fn′          function and its derivative for Newton Raphson solution, shown in relation to Eq. (4)
f(Y)             function fof Y, Eq. (6)
F(z)             function F of z, Eq. (4)
F                coefficient of correlation - function of Rso, Eq. (12), ° F.
FBHP(i, t)       flowing bottomhole pressure at the ith sleeve at time t, M1L−1t−2, psi (Pa)
FBHPg(, t)       flowing bottomhole pressure gauge at time t, M1L−1t−2, psi (Pa)
FcDi             dimensionless fracture conductivity at ith sleeve, dimensionless
G                original gas in place, M0L−3t−0, SCF (m3)
Grad(fric, t)    tubular fluid friction gradient at given time t, M1L−2t−2, psi/ft (Pa/m)
hf               fracture height, M0L1t0, ft (m)
i                number of sleeves, number
k                permeability, M2L0t0, md (m2)
n                number of segments, dimensionless
N                original oil in place, M0L−3t−0, STB (m3)
Np               cumulative produced oil, M0L−3t−0, STB (m3)
p                pressure, M1L−1t−2, psi (Pa)
pb               bubble point pressure, M1L-1t−2, psi (Pa)
ppc              pseudocritical pressure, M1L−1t−2, psi (Pa)
ppr              pseudoreduced pressure, dimensionless
pwD(t,i)         dimensionless pressure at a given dimensionless time (tDXf) for the ith sleeve,
                 dimensionless
PDDN(i, t)       drawdown pressure at ith sleeve at time t, M1L−1t−2, psi (Pa)
Pri              reservoir pressure at the ith segment of reservoir associated with ith sleeve, M1L−1t−2,
                 psi (Pa)
Pstat(t)         the final static pressure at a time t, M1L−1t2, psi (Pa)
Ptf(i, t)        total tubular friction till the ith sleeve at time t, M1L−1t2, psi (Pa)
q(tot, t)        total downhole flow rate at time t, , M0L3t−1, bbl/d (m3/s)
Qo               oil production rate, M0L3t−1, bbl/d, (m3/s)
Qw               water production rate, M0L3t−1, bbl/d, (m3/s)
RFCD(i, t)       ratio of initial and local dimensionless fracture conductivity
SGl              effective liquid specific gravity, dimensionless
t                elapsed time, M0L0t1, minutes (s)
tp               production time, M0L0t1, minutes (s)
tDXf             dimensionless time associated with fracture half-length Xf, dimensionless
srtLi            tubular section length for a specific sleeve, M1L010, ft (m)
Sleevefric(t, i) pressure drop across ith sleeve at time t, M1L−1t2, psi (Pa)
T                temperature, °R or ° F.
Tpc              pseudocritical temperature, °R
Tpr              pseudoreduced temperature, dimensionless
Volseg           segment volume, M0L310, bbl (m3)
w, wp            propped width of the fracture, M0L1t0, in. (mm)
wf               hydraulic width of the fracture, M0L1t0, in. (mm)
We               cumulative encroached water, M0L−3t−0, STB (m3)
Wp               cumulative produced water, M0L−3t−0, STB (m3)
x                solution root, various
xn, n+1          root of function shown in Eq. (4). dimensionless
Xf               fracture half-length, M0L1t0, ft (m)
Yg               correlation exponent in Eq. (9), various
z                gas deviation factor, dimensionless
Δp               change in average reservoir pressure, , M1L-1t2, psi (Pa)
∂                mathematical operator, derivative
f                porosity, decimal, dimensionless
gg               gas specific gravity, dimensionless
go               oil specific gravity, dimensionless
gw               water specific gravity, dimensionless
mo(t, i)         viscosity of fluid at ith sleeve for a given time t, M1L−1T−1, cP (Pa · s)
rr               reduced density, dimensionless
GLR              gas liquid ratio, unitless
WOR              water oil ratio, unitless

Other implementations are also described and recited herein. Further, while multiple implementations are disclosed, still other implementations of the presently disclosed technology will become apparent to those skilled in the art from the following detailed description, which shows and describes illustrative implementations of the presently disclosed technology. As will be realized, the presently disclosed technology is capable of modifications in various aspects, all without departing from the spirit and scope of the presently disclosed technology. Accordingly, the drawings and detailed description are to be regarded as illustrative in nature and not limiting.

BRIEF DESCRIPTION OF DRAWINGS

The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawings will be provided by the Office upon request and payment of the necessary fee. A more complete understanding of the presently disclosed technology and benefits thereof may be acquired by referring to the follow description taken in conjunction with the accompanying drawings.

FIG. 1 depicts an example proppant flowback mitigation system including forces acting on particles at the surface of the perforation tunnel.

FIG. 2 depicts an example proppant flowback mitigation system including measured critical flowback velocities for various fracture widths along with extrapolated values in dashed curves.

FIG. 3 depicts an example proppant flowback mitigation system including calculated gas deviation factors (z) from example well production data.

FIG. 4 depicts an example proppant flowback mitigation system including calculated flowing pressure in the wellbore at various time intervals.

FIG. 5 depicts an example proppant flowback mitigation system including calculated fluid densities for various produced volumes filling up the wellbore.

FIG. 6 depicts an example proppant flowback mitigation system including cumulative static pressure exerted by columns of varying lengths totaling to the downhole gauge setting depth.

FIG. 7 depicts an example proppant flowback mitigation system including a proppant performance chart for ISP depicting changes in proppant conductivity and permeability with stresses.

FIG. 8 depicts an example proppant flowback mitigation system including a loss of FcD with production and/or increase of drawdown pressures (red curve).

FIG. 9 depicts an example proppant flowback mitigation system including a transient production for horizontal well completions with longitudinal fractures.

FIG. 10 depicts an example proppant flowback mitigation system including a transient production for horizontal well completions with transverse fractures.

FIG. 11 depicts an example proppant flowback mitigation system including post-treatment production data for first case history.

FIG. 12 depicts an example proppant flowback mitigation system including static and tubular friction pressures calculated for the first case history using the simulator.

FIG. 13 depicts an example proppant flowback mitigation system including a simulated flowback and fluid distribution for different sleeves after 57.5 hours of flowback, showing the prevalent fluid velocities were low.

FIG. 14 depicts an example proppant flowback mitigation system including a simulated flowback and fluid distribution for different sleeve after 118 hours of flowback which shows closing the gap between produced and critical velocities used to initiate the proppant flowback.

FIG. 15 depicts an example proppant flowback mitigation system including simulated flowing bottomhole pressures at different producing sleeves and showing gauge tubing and Bottomhole Flowing Pressure (BHFP).

FIG. 16 depicts an example proppant flowback mitigation system including a plot showing calculated hydrostatic and tubular friction from production data of the second case history.

FIG. 17 depicts an example proppant flowback mitigation system including a simulated flowback and fluid distribution for different sleeve after 210 hours of flowback.

FIG. 18 depicts an example proppant flowback mitigation system including a simulated flowback and fluid distribution for each sleeve after 580 hours of flowback.

FIGS. 19A and 19B depict an example method performed by any of the proppant flowback mitigation system(s) depicted in FIGS. 1-18, including a workflow for a flowback optimizer.

FIG. 20 depicts an example computer system for implementing a proppant flowback mitigation system, which can form at least a portion of any of the system(s) depicted in FIGS. 1-19B.

DETAILED DESCRIPTION OF THE INVENTION

A proppant flowback mitigation system can address the issues discussed above by collecting data from multiple data points to predict proppant flowback. Some of the data used by the systems disclosed herein to address these issues can include input data values such as a sand packing value (e.g., lbm/ft²), a fluid velocity value (e.g., cm/s), a proppant flowback pressure value (e.g., psi) for closure stresses, a critical velocity value (cm/s), a change in critical velocity value cm/s, an increase in proppant size value, and/or critical flowback velocities for a range of psi effective stresses on proppant pack. Furthermore, an average production rate (e.g., bbl/d/perforation), which can be based on whether proppants were treated with any specialty chemicals to mitigate flowback, can be generated and compared to critical flow rates.

The system can determine that proppant flowback initiates at higher cleanup rates when proppant size is increased. In some scenarios, if a low closure pressure is maintained on the pack, the system can flowback and clean the fractures at higher rates. This calculation may be used in the initial phase of flowback where higher closure stresses can cause the proppant to be “pinched out” of the fracture into the wellbore. The critical flowback rates can have increases corresponding to the reduction of fracture width to proppant diameter (e.g., w/D ratio discussed in next paragraph) with the highest rates observed when the w/D ratio reaches 3.2. In other words, the proppant flowback can initiate at higher values if proppant size is increased.

In some instances, the bridging criteria can generally be calculated as the ratio of propped fracture width (w_p) and proppant diameter (D_p) and can play a role in preventing proppant from being produced. A w_p/D_p value of around 2.5 can be prevalent for some stable packs and can have higher critical velocities before which the proppant production would initiate. Incidentally, proppant bridging can also take place during the fracturing treatment and can cause proppant to assimilate into clusters if the fracture widths are not wide enough to allow smooth transport. A default value of bridging factor in facture simulators can be around 3.5 to provide non-hindered proppant movement in the fracture. During flowback conditions, the effective width can be propped width (w_p) whereas during the fracturing conditions the effective width can be taken as hydraulic width, wf. Other relevant factors taken into consideration by the systems that influence proppant production include proppant pack permeability and/or fluid viscosity. In other words, proppant production can be linked to prevalent or expected flow velocities. Control of proppant flowback by means of choke control can be used and the system can forecast probable proppant production based on the choke settings. Such solutions can be specific to given fields and completion types and beneficial in mitigation of the problem since a direct measurement of recovered solids can be possible during field operations. Recommendations of choke settings can be optimized to avoid issues with taking the problem farther away from where the action of solids production is taking place.

In some scenarios, the system can use smaller w_p/D_p ratios in combination with higher effective stresses which can stabilize the pack. If only proppant pack porosity and density are used for calculating the propped width (taking solids-packing approach), the loss of porosity with increasing effective stresses on proppant pack can lead to smaller propped widths which for the same proppant size can reduce the w_p/D_p ratio in a favorable way. If during the production phase, the effective bottomhole flowing pressures decrease, the system can expect a higher stress on the pack. These conditions can be detrimental initially, especially for wider propped fractures, because the proppant may extrude out into the wellbore if inadequately bonded due to lack of frictional forces and/or inter-proppant grain cohesion. Eventually though, the action of increasing stresses on the proppant pack can lock the proppant in place.

It should be understood that the features and concepts disclosed herein may be implemented in other arrangements and that the scope of the presently disclosed technology is not limited to the embodiments described or illustrated.

Turning to FIG. 1, a proppant flowback mitigation system 100 is depicted, including one or more stable packs 102 in which the three dominant forces 104 existing in the pack 102, such as a hydrodynamic or drag force, can cause the proppant to extrude out of the fracture. Frictional forces that originate from relative motion of the particles in contact with each other and fracture-walls can help lock the proppant in place. Also, the gravitational forces that constantly act on the pack causing it to pack even further via settling, can be seen in balance in FIG. 1. Contrary to the general observation that the increased closure stresses can cause the packs to destabilize and thus produce proppant, this can occur in some scenarios only if the frictional resistance and cohesion between the proppant material is small or weak. An increase or decrease in critical flow velocities can depend on the friction coefficient. With increasing closure stresses and subsequently the normal forces on the particles, friction can increase and result in improved stability of the pack. The destabilizing hydrodynamic or drag forces are successfully countered if ample resistive forces are available, though the latter will depend on the proppant grain make up, shape, innate particle flowability and inter-particle friction.

In the process of well fracturing and completion, the production of the proppant from a propped fracture can be related to the forces 104 acting on the proppant pack 102 during in a well that is actively producing fluids. Some of these, such as increase of effective stresses on the proppant pack 102, can be stabilizing in nature, whereas others can be related to inertial flow from fluid velocity, viscosity and others can destabilize the pack. Synthetic fibers or resin coated proppant can trap at least some of the proppant to attempt to mitigate the proppant's flowing back into the wellbore. Proppant flowback during the post-stimulation well clean up and/or during is a common occurrence in most hydraulically fractured wells. Whilst some proppant is expected to be produced during the well cleanup operations that typically follow the treatment execution, the long-term production of proppant can be problematic from an economic viewpoint. The proppant flowback mitigation system 100 disclosed herein can aid in minimizing proppant flowback and associated issues.

FIG. 2 shows an example proppant flowback mitigation system 200 including results from experiments 202 for 40/70 proppant, where the dashed lines 204 represent extrapolated values as the stresses on the pack increase. The proppant flowback mitigation system 200 can determine that for a given closure stress a narrow fracture can hold the proppant more effectively than a wide fracture because of the closer packing in narrow fractures, which can cause the system to use higher critical flowback velocities to dislodge the proppant.

In some scenarios, the proppant flowback mitigation system 200 can generate one or more proppant flowback prediction calculations.

For example, there can be various conditions that the system 200 determines favor production of proppant into the wellbore. From the perspective of defining a unique variable that can be used to warn of possible proppant production, the flowback fluid velocity can be selected as a useful parameter. However, since this is a calculated value which can depend on the production rates, available flow area and the prevalent pressure and temperature conditions, it can in some situations serve as only a qualitative indicator. In comparison to this, the measured bottomhole flowing pressure (BHFP) or the effective drawdown pressures (e.g., which can be obtained from BHFP without much loss of accuracy), can be used instead for controlling the flowback, because the changes in their values can affect producing fluid velocities.

In some examples, the proppant flowback mitigation system 200 can include deployment of downhole measurements devices in the wellbore such as pressure and temperature gauges that can also relay data in a live mode via cable. In horizontal wells, these gauges can be installed close to the heel of the lateral and/or in the section above the kick-off point or a location. The gauge data can also be used in generating various treatment parameters (e.g., critical treatment parameters) such as formation face pressure and/or net pressures and may provide pressures during the well production phase in live mode. Because of the significant distance between the downhole pressure gauge where the measurements are made and the production sleeve or perforations where the reservoir fluid is entering, the pressure may not provide adequate data to fully represent the flowing pressures at perforation or production sleeve locations. Furthermore, since the production into the wellbore can take place at various sections and points, the system may have pressure measurement sensors in the wellbore at multiple discrete point rather than a single location (or measured depth) of down hole gauge. Thus, with the final objective to calculate flowing fluid velocity at multiple, different, producing perforation clusters or sleeves, a simulator 206 can be developed to calculate a sequence of values and operations based on those values, as discussed in greater detail below.

In some scenarios, the proppant flowback mitigation system 200 can perform, with the simulator 206 a wellbore details input operation. This step can involve obtaining and/or inputting wellbore and/or completion related data. The input data can include wellbore information such as various tubular depths and internal diameters (ID), length and IDs of any wellbore hardware that may be in the path of production, such as sleeves, constrictions, downhole safety valves, etc., wellbore deviation, gauge depth, gas lift valve depth, perforation or production sleeve details, e.g., diameter, number of perforations, estimated discharge coefficients, tunnel length, measured depths, combinations thereof and other such data.

Furthermore, the proppant flowback mitigation system 200 can perform, with the simulator 206, a wellbore calculations operation. In this step, a wellbore tubular capacity and/or volume can be calculated for an entire wellbore at unit length intervals (e.g., 1 ft or 1 m intervals). By performing this wellbore calculations operation, the proppant flowback mitigation system 200 can track the production fluid movement as it is produced up the wellbore, such that the production fluid movement can be accounted for when calculating hydrostatic pressures of various time-interval-segmented fluid columns. The results from these calculations can be saved in virtual strings, arrays, or databases such that the results can be calculated once and/or are not recalculated unless in response to any of the wellbore related inputs being altered.

In this wellbore calculations operation, fluid PVT properties and/or other data related to well completion can be input. The inputs associated with hydraulic fracturing completion can be obtained from a pressure history match of various treatment stages. These can include fracture parameters for different zones or perforation sets, such as fracture orientation (e.g., transverse or longitudinal), fracture half length, height, fracture permeability, propped width, proppant distribution, and/or combinations thereof. Reservoir pressures, permeability, and/or porosity associated with different perforation sets (or zones) can also be used as input to facilitate transient calculations by the simulator 206. Inputs can also include the actual production data where the oil, water and gas rates are obtained from physical flowmeters during well test phase, and other items such as surface and bottom hole pressures and temperature, which can be obtained as continuous data points. In some scenarios, to provide the computation times at reasonable values, the production data can be input at periodic time interval, such as every ten minutes, every 15 minutes, every one-half hour, every hour, or another frequency. For instance, oil, gas, and water produced in one such time interval (e.g., ½ hour) can be denoted as a “fluid segment” in the calculations.

In some scenarios, the proppant flowback mitigation system 200 can perform one or more pressure and/or volume calculations.

For example, the objective of computing flowing bottomhole pressures at the perforation depths can be performed by determining a reasonably good estimate of hydrostatic and composite fluid. Since most wells may flow with all 3 constituents such as oil, gas and water, such calculations can use continuous updating of fluid properties with the changing conditions in the wellbore at each time interval. The workflow to carry out the pressure and/or volume calculations is described below.

The proppant flowback mitigation system 200 can perform a calculation of a z factor. To initiate the calculations, the wellbore can be divided into a finite number of sections that are coarser than a wellbore fluid movement interval (e.g., 1 ft), which is the interval used to track the wellbore fluid movement. The purpose of generating additional grid segments can be to speed up the calculations, especially because for the segments and 30 min production time interval, the gas deviation factor (z) can be determined using iterative technique such as Newton Raphson (NR), which may be a time-consuming action and can increase dramatically with finer segment lengths. For example, 4,000 segments of well (typical value) vs. 24,000 segments for a well with measured depth of 24,000 ft (7315.2 m) could result in nearly 5 to 6 folds in computation time. It may be noted that these calculations, aimed at creating a wellbore pressure map can be refined later in the workflow.

The one or more pressure and/or volume can be initiated by calculating the pressure distribution inside the wellbore of a flowing well with the help of surface and bottomhole pressures data. This solution scheme may be initialized by assuming that the pressure at the depth of a first node (e.g., or segment) in the wellbore starting from the surface in the downward direction, is nearly the same as that of the recorded surface pressure if the segment is small enough, for example, 5 ft. A similar assumption of linear temperature distribution between the surface and downhole gauge depth can be made. With the help of these, pseudoreduced pressure (p_pr) and pseudoreduced temperature (T_pr) at the first node can be calculated using the following relationship:

$\begin{array}{cc}{p}_{\mathrm{pr}}=\frac{p_n}{p_{\mathrm{pc}}};& \mathrm{Eq}.1\end{array}$ $T_{\mathrm{pr}}=\frac{t_n}{T_{\mathrm{pc}}}$

where, p_n and t_n are pressures and temperatures for the first node n, and p_pc and T_pc are pseudocritical pressure and temperature values in psia and ° R respectively, that may be obtained with the help of the correlations such as those dependent on gas specific gravity (g_g).

A gas deviation factor (z) can be calculated next using the equation of state below which can provide reasonably good results if T_pr≠1.0 and p_pr>1.0:

$\begin{array}{cc}z=1+c_1\left(T_{\mathrm{pr}}\right){\rho }_r+c_2\left(T_{\mathrm{pr}}\right){\rho }_r^2-c_3\left(T_{\mathrm{pr}}\right){\rho }_r^5+c_4\left({\rho }_r,T_{\mathrm{pr}}\right)& \mathrm{Eq}.2\end{array}$ $\begin{array}{cc}{\rho }_r=0.27p_{\mathrm{pr}}/\left({\mathrm{zT}}_{\mathrm{pr}}\right)& \mathrm{Eq}.3\end{array}$

where, the various coefficients c₁ to c₄ for given pseudoreduced temperature can be obtained. It may be noted that the solution of Eq. (2) may be not straightforward and explicit since z occurs on both sides of the equation as shown in the expression for reduced density r, and hence iterative schemes such as NR may be adopted. Eq. (2) can be rearranged, and its derivative was obtained as follows:

$\begin{array}{cc}\left(\partial F\left(z\right)/\partial z\right)T_{\mathrm{pr}}=1+c_1\left(T_{\mathrm{pr}}\right){\rho }_r/z+2c_2\left(T_{\mathrm{pr}}\right){\rho }_r^2/z-5c_3\left(T_{\mathrm{pr}}\right){\rho }_r^5/z+\frac{2A_{10}{\rho }_r^2}{T_{{\mathrm{pr}}^z}^3}\left(1+A_{11}{\rho }_r^2-{\left(A_{11}{\rho }_r^2\right)}^2\right)\exp \left(-A_{11}{\rho }_r^2\right)& \mathrm{Eq}.4\end{array}$

where, F(z)=0 is the rearranged form with z on the right-hand side (RHS) of Eq. (2) and A₁₁=0.7210. The new NR root (xn+1) can now be determined as x_n+1=x_n−f_n/f_n′ where, f_n and f_n′ are the values of function and its derivative respectively, at initial guess. The convergence can occur in just a few iterations (e.g., two iterations, three iterations, four iterations, five iterations, etc.) with a difference between the subsequent roots of less than 1×10⁻¹⁰.

In an alternative approach z factor can also be determined. Here the constitutive equation to determine z is given below:

$\begin{array}{cc}z=\frac{{\mathrm{Ap}}_{\mathrm{pr}}}{Y}& \mathrm{Eq}.5\end{array}$

where, Y is the reduced density function given as:

$\begin{array}{cc}f\left(Y\right)=\frac{Y+Y^2+Y^3-Y^4}{{\left(1-Y\right)}^3}-A{p}_{pr}-B{Y}^2+C{Y}^D=0& \mathrm{Eq}.6\end{array}$

To employ the NR technique discussed above, the derivative of above function, shown in Eq. (7) below can be used:

$\begin{array}{cc}\frac{\partial f\left(Y\right)}{\partial Y}=\frac{1+4Y+4Y^2-4Y^3+Y^4}{{\left(1-Y\right)}^4}-2BY+CD{Y}^{D-1}& \mathrm{Eq}.7\end{array}$

As the solution progresses, a gas deviation factor, obtained at the very first step can be applied to solve for average fluid density of the segment and subsequently the static pressure gradient, from which the pressure at the next node in the downward direction can be obtained. At the end of this step thus, a multi-dimensional table of gas deviation factors for various depths in the wellbore can be generated for each or multiple different time steps, along with a preliminary estimate of pressure distribution in the wellbore at various time intervals in production history.

FIGS. 3 and 4 depict example proppant flowback mitigation systems 300 and 400 showing calculated values of z-factors 302 at surface and downhole conditions, respectively. Measured values of temperature and pressure are also shown in these plots for comparison purposes. The values of z-factors 302 obtained from calculation procedure discussed above can be verified with verification data including a chart of compressibility factors for natural gases. An average deviation of 0.02% can be observed between calculated and chart values, which can be consistent with a predetermined error range.

In a next step, the pressure exerted by the static column of fluid (assuming the flowing fluid column is momentarily stationery) can be calculated. This calculation can use an initial calculation of the fluid specific gravity of oil and water mixture at the downhole gauge depth. Here, fluid implies only oil and water. The calculations for pressure exerted by a gas column (e.g., in solution or free) can be done separately. For a system comprising oil and water, this may be calculated using the following equation:

$\begin{array}{cc}S{G}_l=\left[\frac{{\gamma }_o}{B_o}\frac{1}{1+WOR}+\frac{{\gamma }_w}{B_w}\frac{1}{1+WOR}\right]& \mathrm{Eq}.8\end{array}$

where, SG_l is the effective liquid specific gravity, g_o and g_w are oil and water specific gravities respectively, and B_o and B_w are oil and water formation volume factors in rb/STB, respectively. The volume factors may be calculated with the help of correlations (see Eq. (12) and (13) below) for given temperature and pressure conditions, at above or below bubble point pressure. WOR is water-oil ratio that can be obtained from production data. Calculation of B_o can use the knowledge of solution gas oil ratio (R_so) which can be estimated from correlation for pressure equal to or less than bubble point pressure:

$\begin{array}{cc}{R}_{so}={\gamma }_g\left(\frac{p}{18{\left(10\right)}^{Y_g}}\right)& \mathrm{Eq}.9\end{array}$

where Y_g can be determined using an available correlation that is a function of temperature and oil density, and p is applicable pressure; at or above bubble point conditions, the pressure p can be substituted with bubble point pressure. The gas volume factor (B_g), which may be used later to determine gas volume at bottomhole conditions in barrels, can be calculated using the following equation:

$\begin{array}{cc}{B}_g=0.00504\frac{zT}{p}\mathrm{bbl}/\mathrm{SCF}& \mathrm{Eq}.10\end{array}$

Where the gas deviation factor (z), temperature at the point of interest (T) in ° R and pressure (p) are known, the gas volume factor can be readily calculated.

Once the effective liquid specific gravity and volume factors are obtained for a given time step, an interim loop can be introduced in the calculation where the production data “fluid segments” defined above, which can fill up the production tubular from the surface to the downhole gauge location. Since each of these sections may have different values of oil, gas and water production, and also varying flowing bottomhole and surface temperatures and pressures, the fluid can occupy column heights that will change with time. To initiate these calculations, first only the liquid volumes can be accounted for to obtain an initial value of column segment (Vol_seg) for a first row of production data, if the production data for at least a recovered liquid volume that exceeds the tubular displacement volume the downhole gauge is available.

$\begin{array}{cc}Vo{l}_{seg\left(n\right)}=\left(t_{p\left(n+1\right)}-t_{p\left(n\right)}\right)\times \left\{\left(B_o{Q}_o\right)+\left(B_w{Q}_w\right)\right\}& \mathrm{Eq}.11\end{array}$

where, t_p is the production time, and n is the data row denoting various production history. Q_o and Q_w are oil and water production rates in appropriate units. If the time is in hours, then the rate in STB/day can be divided by 24 to obtain the segment volume in STB since the time interval is 0.5 hours. For instance, for one-half hour production interval, if the oil and water rates were observed to be 1,200 and 1,750 STB/D having B_o and B_w values of 1.21 and 1.0 rb/STB respectively, then the volume segment can be nearly 66.71 STB. If the well is flowing up 3.98 in (101.1 mm) internal diameter (ID) casing, this can occupy nearly 4,224.84 ft (1,321.26 m). With the depth now known, the gas deviation factor (z) can be determined from the z vs. depth table generated earlier, for the given production interval t_p(n).

With the z determined, the segment volume can be recalculated by the proppant flowback mitigation systems 400, in this case by accounting for gas volume that may be added to the total tubular volume and potentially alter the segment length. To account for this refinement in calculation, the gas volume factor shown in Eq. (10) can be incorporated in the Eq. (11) and multiplied with the measured gas production rate. The gas production rate can also include the injected gas; and exclusion of the latter can increase the static pressure nominally depending on the gas injection rate. The proppant flowback mitigation systems 400 can then determine the volume of the next segment, by analyzing the production data in this segment and the process continues, until for the given time step interval, all the data rows that occupy the tubular volume to the downhole gauge are accounted for. In the case of partial volumes, adjustments can be made accordingly. This adjustment step can result in a library of start and end row numbers in the production data table and any incremental volume that is accounted for when totaling up to displacement volume. Also, the individual length of segments and their corresponding formation volume factors can be obtained and stored in multi-dimensional arrays for the different time intervals and depths.

In some instances, the total mass produced, inclusive of produced gas, at any given time interval can be calculated from the measured volume collected at the surface or from the production rates measured for that time interval. Even though the mass remains constant, the corresponding volume can change as this “batch” of fluid was traversing up to be produced at the surface.

FIGS. 5 and 6 depict proppant flowback mitigation systems 500 and 600 with the densities of various fluid columns for given producing time intervals in hours at the x-axis. Since the change in volume can be continuously updated in the calculations above, it is now possible to calculate the effective density of the different fluid segments at a given time as shown in FIG. 5. As such, the proppant flowback mitigation systems 500 can perform simultaneous calculation of the pressure exerted at the base of each fluid column. Summing up the individual pressures exerted by the different columns can result in a final “static” pressure at the downhole gauge depth shown by the red dashed line in FIG. 6.

The calculated cumulative pressure values in FIG. 6 show that if the flow were to momentarily stop, then the flowing pressures being measured by the downhole gauge can recede to the value calculated before any segregation or re-distribution of fluid occurs. Although this pressure can be denoted as “hydrostatic,” the term is not used here deliberately since the column may comprise of fluids other than water alone. Now that surface, bottomhole and static fluid pressures are known, effective friction pressure gradient of the fluid mixture in the wellbore at any given producing interval time can be easily calculated.

In some instances, the solution provided by the proppant flowback mitigation systems 600 can vary to some extent if slip is considered in the calculations. Slip can occur due to expansion of gas and consequent speeding up relative to oil movement; if this occurs the effective density can be higher. The phenomenon is also influenced by fluid properties and flow conditions. In some embodiments, because of the choked flow conditions, which is the case for many of the initial flowbacks, slip may not apply as much and can be ignored.

In the next step, the production rates measured at the surface can be converted to corresponding rates at the downhole gauge depth with an objective to calculating the same at the perforation depths. There is no significant use of these rates, but they are generated mostly for the purpose of comparison to ensure the validity of assumptions and calculation results. To perform the calculation, fluid properties such as solution gas oil ratio (R_so), and gas formation volume factors (B_g) described above can be obtained for downhole conditions of pressure and temperature at the gauge depth location. As an example, oil formation volume factor (B_o) can be obtained for pressures less than bubble point pressure, as follows:

$\begin{array}{cc}{B}_o=0.972+0.000144F^{1.175}& \mathrm{Eq}.12\end{array}$

where, F is a function of R_so, and oil and gas gravities as described in the reference and T is temperature in ° F. For pressures above bubble point pressure, B_o can be determined using the following:

$\begin{array}{cc}{B}_o=B_{ob}\exp \left[c_o\left(p_b-p\right)\right]& \mathrm{Eq}.13\end{array}$

where, B_ob is an oil formation volume factor at bubble point pressure in rb/STB and c_o is oil compressibility in 1/psi. The current calculator of the proppant flowback mitigation systems 600 can also be equipped to obtain these values from tables if so desired. The volume factor for water (B_w) can be determined using the following correlation:

$\begin{array}{cc}{B}_w=\left(1+D{V}_{wt}\right)\left(1+D{V}_{wp}\right)& \mathrm{Eq}.14\end{array}$

where, values for DV_wt and DV_wp can be obtained from the detailed expressions presented in the reference.

If the well is under gas injection and the gas lift valve is located above the gauge depth, then the injection gas rate can be ignored when calculating the total effective production rate at gauge depth, shown by Eq. (13) below:

$\begin{array}{cc}{q}_{\left(tot,t\right)}=Q_o{B}_o+\left(GLR-R_{so}\right)Q_o{B}_g+Q_w{B}_w& \mathrm{Eq}.15\end{array}$

where, q_(tot,t) is total downhole flow rate in (bbl/d) at given time interval t; Q_o is the oil flow rate at standard conditions in STB/d; B_o is in rb/STB; B_g is in bbl/scf; GLR is gas liquid ratio; Q_w is water flow rate in STB/d; and B_w is formation volume factor for water in rb/STB—all at the specified time interval. For uniformity in units, appropriate constants may be used.

Next, the bottomhole flowing pressures adjacent to various perforations can be calculated for the different time steps (e.g., or production intervals) as shown in Eq. (16) to provide the calculation of effective drawdown pressures and eventually the theoretical contribution from the different sleeves. The first allocation of total fluid produced can be carried out under bottomhole conditions:

$\begin{array}{cc}FBH{P}_{\left(i,t\right)}=FBH{P}_{g\left(t\right)}+P_{stat\left(t\right)}+R_{FCD\left(i,t\right)}{P}_{\mathrm{tf}\left(i,t\right)}& \mathrm{Eq}.16\end{array}$

where, FBHP_(i,t) is the flowing bottomhole pressure at the i^th sleeve at time t; FBHP_g(t) is gauge pressure at time t; P_t(i,t) is the total tubular friction till the i^th sleeve; P_stat(t) is the final static pressure at time t; and R_FCD(t) is a ratio of initial and local dimensionless fracture conductivity (F_cDi) of the fracture corresponding to the i^th sleeve, to the total sum of F_CDs to differentiate the contribution of fluid flow only on the basis of stimulation treatment.

The formation face pressure at each sleeve can then be calculated by accounting for the pressure drop across the perforations (DP_perf) for an F_CD apportioned flow rate at the gauge. Here, solutions for open perforations or gravel pack conditions can be obtained. For strictly liquid phase flows such as oil and/or water in absence of gas an equation for frictional pressure drop in circular orifice may be used. Finally, the effective drawdown pressure with respect to time (_PDN(l,t)) across each perforation set or sleeve can be calculated as:

$\begin{array}{cc}{P}_{DDN\left(i,t\right)}=P_{ri}-FBH{P}_{\left(i,t\right)}-D{P}_{perf\left(i,t\right)}& \mathrm{Eq}.17\end{array}$

where P_ri is the reservoir pressure at the i^th segment of reservoir associated with i^th sleeve. The effective stress on the proppant at a given time interval t can then be calculated for the i^th fracture if the closure pressures are known. For closed boundaries, the effective reservoir pressures can be updated with the help of material balance calculations at every time interval.

For closed boundaries with no water influx, the effective reservoir pressures can be updated with the help of material balance calculations at every time interval. These calculations can depend on original oil (N) and gas (G) in place for each of the reservoir segments that the horizontal well (in the case studies discussed) intersects. The material balance equation can account for changes in oil, free gas, water, and void space volumes with the help of appropriate formation volume factors, material compressibility and saturation. One such equation, obtained after rearranging the general volumetric equation for undersaturated oil wells, to highlight the change in average reservoir pressure (Δp) with production, is presented below. Note that this relation can be for cases where there is an absence of gas cap.

$\begin{array}{cc}\Delta \overline{p}=\left[N_p\left\{B_t+\left(R_p-R_{soi}\right)B_g\right\}+B_w{W}_p-N\left(B_t-B_{ti}\right)-W_e\right]÷{\mathrm{NB}}_{ti}\left(\frac{c_w{s}_{wi}+c_f}{1-S_{wi}}\right)& \mathrm{Eq}.18\end{array}$

In the Eq. (18) above, N_p and W_p are the produced oil and water volumes (STB) respectively, B_g (bbl/SCF) and B_w (rb/STB) are the formation volume factors for gas and water respectively, R_p is the ratio of cumulative produced gas to oil, R_soi is the initial solution gas oil ratio (SCF/STB), B_ti is the initial oil volume factor and B_t is the composite two phase formation volume factor, both expressed in rb/STB. For a reservoir with no water influx/encroachment, the term (W_e) may be ignored in calculations. Also, c_w is water compressibility (1/psi), Cr is formation compressibility (1/psi) and S_wi is the initial water saturation. The change in average reservoir pressure can be calculated in psia at the different time intervals using Eq. (18) and for each of the production “compartments” in the wellbore defined by the sleeves and/or perforation spacing. This is possible because the simulator 206 can be set up to accept petrophysical properties for individual zones that are treated independently during the fracture stimulation treatment. Depending on the inputs to the simulator 206, there may be minor variations in how the individual zones deplete in case there is lack of pressure support; while the overall stresses can recede with depletion, the effective stresses on the proppant pack increase. These calculations thus can help in accounting for that effect, especially in estimation of critical flow velocities. The simulated change in reservoir pressure can be shown in the plot for one of the case histories, where the effective reservoir pressures are averaged for all zones for a given time interval for the sake of simplicity in plotting.

FIG. 7 depicts an example proppant flowback mitigation systems 700 including a proppant performance charts to determine a fracture permeability if the effective stresses on the proppant are known (e.g., for an intermediate strength proppant). When using such plots, the effective conductivity can be calculated for a given proppant distribution in lbm/ft²; in absence of such data the value can be normalized by taking a ratio of input lbm/ft² and the distribution for which the chart was developed, e.g., 2.0 lbm/ft². The latter may not be a complete accuracy method since the changes in conductivity may not be linear, but still can be a more reasonable approach than assuming default values that do not represent the actual distribution. Note that the lbm/ft2 can also be updated if the history values of embedment losses with effective stresses are available.

FIG. 8 depicts an example proppant flowback mitigation system 800 including a changing dimensionless fracture conductivity (F_cD) 802. In some scenarios, the effective FcDi for the different perforation sets or sleeves can finally be obtained for any given time interval as the flowback progresses. Calculations performed by the proppant flowback mitigation systems 700 can show that F_cD evolves with time, as shown in FIG. 8, primarily depending on how the propped fracture width and/or proppant conductivity change with variations in drawdown or effective proppant stresses. The various curves shown in FIG. 8 represent separate fractures, also highlighting the fact that treatment placements in a single well may not result in fractures that have identical characteristics. Thus, the treatments in individual stages can be evaluated independently to obtain fracture geometries and flow capacities.

In some instances, similar calculations as shown from Eq. (12) to (14) are carried out prior to calculating the production rates at individual sleeve or perforation depths, since the PVT properties can be revised for conditions existing at the sleeve production depths. The formation volume factors can be calculated at the different sleeves; in addition to this perforation/sleeve relation friction pressure drops can be re-calculated with the help of dedicated equations for pressure drop across perforations for oil and/or gas phase. As may be appreciated, these pressures drop calculations can be rate dependent. If the rate calculations are in progress, an explicit solution may be unavailable, and a Newton-Raphson (NR) method can be used to determine the production rate entering the wellbore from a single perforation cluster to account for this. The process can be repeated for all clusters until theoretical rates from the different clusters are obtained. Transient calculations to determine rates can be carried out after Bo and Bg are re-calculated using Eq. (10), (12) and (13), and an effective mixture viscosity can be obtained for a given time interval. The solution for the function G(x₀) where x0 or q(t, i) in Eq. (19) can be the first initial guess of the production rate at ith sleeve, and is expressed in Eq. (19) with sequence (i,t) now switched to (t,i) for data tabulation purposes only

$\begin{array}{cc}G\left(x_0\right)=q_{\left(t,i\right)}-\frac{\left(k_i{h}_i\left(P_{\mathrm{ri}}-{\mathrm{FBHP}}_{\left(t,i\right)}-{\mathrm{DP}}_{\mathrm{stat}\left(t,i\right)}-{\mathrm{Sleeve}}_{\mathrm{fric}\left(t,i\right)}-\left\{{\mathrm{Grad}}_{\left(\mathrm{fric},t\right)}str{L}_{\left(i-\left(i-1\right)\right)}\right\}\right)\right)}{\left(1412B_{o\left(ti\right)}{\mu }_{o\left(ti\right)}{p}_{wD\left(ti\right)}\right)}& \mathrm{Eq}.19\end{array}$ $\begin{array}{cc}{\mathrm{Sleeve}}_{fric\left(t,i\right)}=\frac{0.2369{\left\{\left(q_{\left(t,i\right)}-q_{\left(t,i-1\right)}\right)/1440\right\}}^2{\rho }_{\left(t,i\right)}}{\left(C_{d\left(i\right)}^2{D}_{s\left(i\right)}^4\right)}& \mathrm{Eq}.20\end{array}$

To carry out the calculations, a solution can also be generated by the proppant flowback mitigation system 800 for G(x₁) to obtain the numerical derivative (f_n′) and the next root (x_n+1) can be calculated as x_n+1=x_n−f_n/f_n′. Grad_(fric,t) can be a tubular fluid friction gradient at given time t and srtL is the tubular sectional length from previous production cluster (or sleeve) to the current one. Sleeve_fric(t,i) is the pressure drop across any sleeve in the tubular. C_d(i) and Ds(i) denote a discharge coefficient and a sleeve diameter of i^th sleeve, respectively. p_wD(t,i) is the dimensionless pressure at a given dimensionless time (t_DXf) for the ith sleeve.

FIG. 9 depicts an example proppant flowback mitigation system 900 including data represented by one or more type curves 902. p_wD(t,i) can be obtained for a calculated value of t_DXf by using the type curves, such as those shown in FIG. 9 for single longitudinal fracture in a horizontal well completion, for a given F_cDi.

FIG. 10 depicts an example proppant flowback mitigation system 1000 including similar type curves for transverse fractures, which can be generated by a calculator of the simulator 206. Depending on the application type curves, various geometric settings can be generated and input in the simulator for lookup purposes. At the end of this step, theoretical production rates at the different sleeves or perforation clusters can be obtained. These production rates can be generated with an objective of establishing a fracture F_cD based distribution of production across the horizontal wellbore, highlighting the contributions from the various zones.

In some scenarios, the flow rate solutions for individual sleeves can then be transformed into a matrix with n number of linear equations as the total number of sleeves (summation from i=1 to n) or perforation clusters. The sum of production rates from the different sleeves can total to the known (or calculated) bottomhole production at the shallowest sleeve, and can be inclusive of a gas rate in bbl/d. The equations can take into consideration various flowing-well related pressure drops occurring in the tubular and through any other flow constrictions at the different time steps and can be solved simultaneously instead of solving for individual sections to minimize inaccuracies and mismatch from the known values. This calculation can be carried out by summing up the results from Eq. (18) for the different sleeve/perforation cluster. Thus, the solution matrix (M) can take the form:

$\begin{array}{cc}\begin{array}{ccc}M& a& =b\end{array}& \mathrm{Eq}.21\end{array}$

where, a is the unknown vector relation representing Eq. (18), that can be solved simultaneously for the different sleeves using techniques such as a Gauss elimination method. In the final calculation step, the flow velocities corresponding to the different sleeves or perforation clusters can be obtained by dividing the rates by associated flow areas.

In some examples, the simulator 206 can be used to carry out the analysis of some of the flowback conducted after the well stimulations. In these cases, the flowmeter based measured production data can be updated in 12 hour increments, whereas the pressure and temperature data, for both surface and bottomhole measurements was available at normal per second frequency. Prior to inputting in the simulator, the data can be synchronized and adjusted for every 30 min frequency. The analysis for some of the treatment case histories is presented in the next section.

The following examples of certain embodiments are given. Each example is provided by way of explanation of the disclosed technology, and the following examples should not be read to limit, or define, the scope of the invention. Case histories presented here pertain to fracture stimulations carried out in multistage horizontal wells in low to medium permeability predominantly oil-bearing sandstones. In both these cases, only marginal amounts of proppant were produced though the reason for that is uniquely different. The details are presented below.

In one example involving a multistage horizontal oil well completion, a well can be drilled and completed in medium permeability sandstones of 3.25 md with an average reservoir thickness of nearly 150 ft (45.72 m). The well can be completed in 11 fracture stimulation stages with 20/40 U.S. Mesh ISP (intermediate strength proppant) proppant having a S.G. of 2.71. Fracture geometry and related details obtained after pressure history match are presented in Table 2 below.

TABLE 2
Fracture and formation properties for oil well
completion in heterogenous sand reservoir.
No.	Xf (ft)	(lbm/ft2)	wp (in)	hf (ft)	hp (ft)	f (decimal)
1	192.1	1.99	0.217	147.7	147.7	0.120
2	168.0	2.25	0.246	144.6	144.6	0.170
3	215.4	2.10	0.229	145.4	145.4	0.150
4	188.3	2.52	0.275	148.0	148.0	0.110
5	168.1	2.63	0.287	160.3	160.3	0.090
6	219.7	2.19	0.239	166.9	166.9	0.120
7	264.1	1.77	0.193	150.7	150.7	0.110
8	192.0	1.76	0.192	148.4	148.4	0.110
9	173.0	2.33	0.254	149.8	149.8	0.060
10	198.6	2.56	0.279	151.0	151.0	0.070
11	261.9	1.94	0.212	155.0	155.0	0.091

In the table above X_f denotes fracture half length, lbm/ft² denotes proppant distribution in the fracture, w_p is propped fracture width, h_f is fracture height, h_p is pay zone thickness, f is zone porosity in decimal. Average reservoir pressure of 2,445 psi (16.86 MPa) can be determined from falloff analysis. The simulator 206 can accept reservoir permeability, compressibility, and other pertinent inputs discretely for each zone. Oil API gravity can be around 27° API with average viscosity of 2.0 cP and bubble point pressure reported at 2,135 psi (14.72 MPa).

FIG. 11 depicts an example proppant flowback mitigation system 1100 including a “Gas” curve 1102. As per the simulation of the simulator 206, the flowing pressure at the first sleeve can reach this value nearly 40 hours into the production, which can also be marked by increase in gas production rates, as shown in the “Gas” curve 1102. At bubble point the oil viscosity can increase to 2.34 cP. Oil compressibility of 1.163×10⁻⁵ psi⁻¹ (0.081 Pa⁻¹) can be reported whereas the total system compressibility (C_t) can be taken as 3.3×10⁻⁵ psi⁻¹ (0.227 Pa⁻¹). The specific gravity (S.G.) of the produced gas can be 0.7.

As shown in FIG. 11, bottomhole pressures can be obtained from the downhole gauges permanently installed in the well. Temperature data can also be measured. As the flow testing progresses, a larger drawdown can be applied which can be evident from the bottomhole flowing pressure trends (the brown curve). The changes in the choke settings can be implied from the measured flowing tubing pressure at the surface (the red curve) since this measurement is closer to the downstream choke. The sudden increase in gas rate seen after 70 hours of flowback (beige curve) can show increased gas injection rates for lifting purposes. Both oil and water can continue to increase at nearly the same ratio.

FIG. 12 depicts an example proppant flowback mitigation system 1200 including data represented by a plot 1202 showing the calculated curves for static column pressures (discussed above) and calculated tubular friction. For the flow coming out of the wellbore, the frictional pressures can be minimum at the surface and maximum at the deepest point from where the production is obtained. The friction pressure curve shown in the plot 1202 can be at the depth location. As may be noted, the calculated static pressure (P_Hyd) that the column of fluid can exert at gauge depth can tend to be stable around 1,500 psi (10.34 MPa) with stabilized gas and liquid rates at a constant choke setting and surface pressure of 615 psi (4.24 MPa); with increase in gas, mostly due to the injection gas, the column can be subsequently lightened and P_Hyd can recede. The presence of gas and liquid mixture can also increase the tubular friction pressure.

The increased water production (e.g., as depicted in FIG. 11) after ˜70 hours of flow also can call for this action, however with loading of the well with heavier fluid (water), the static pressure can continue to increase, and the surface pressures can eventually decline in a flowing well. If the static pressures exceed the flowing bottomhole pressure (brown curve), the production can stop unless additional gas is injected into the system. No appreciable amount of proppant flowback can be reported in some circumstances during a flow test period, though trace amounts of reservoir sediments can be continuously collected at the surface.

FIGS. 13 and 14 depict example proppant flowback mitigation systems 1300 and 1400 including results 1302 of the simulator 206. In some scenarios, the critical velocity calculations can depend on the propped-fracture width which can be dynamic in nature as it changes with effective stresses on the proppant pack. The dynamic width can also account for the loss of width due to embedment (e.g., which can be assumed at 8% and can gradually increasing with drawdown) and nearly 35 to 40% damage to the pack, which can be determined based on conductivity testing. The results 1302 from the simulator 206 presented in FIGS. 13 and 14 can represent data indicating one or more of (a) normalized production contribution (blue vertical bars) from various fractures being uneven as expected; (b) critical velocities (red curves) for the zones being not identical for all the zones; (c) fluid distribution continuing to evolve disproportionately as production continues; and/or (d) one zone having a higher tendency to produce proppant than the other zones.

FIG. 14 depicts an example proppant flowback mitigation system 1400 including a late time analysis 1402. The late time analysis 1402 illustrated in FIG. 14 shows that zones associated with particular sleeves (e.g., sleeves 2, 4 and 9) may be prone to producing proppant with their high fluid flow velocities exceeding critical values. Referring to Table 2, these can be the zones with higher proppant distribution of greater than 2.0 lbm/ft² and higher corresponding propped widths, which can be consistent with most experimental findings that suggest that higher widths can promote proppant flowback. The role of embedment of proppant resulting in loss of propped width and thus stopping the proppant from flowing into the wellbore can be investigated. In contrast, fractures corresponding to other sleeves (e.g., sleeves 7 and 8), due to their low proppant distribution, may not witness proppant production at all—the dashed red curve can indicate their critical flow velocities as amongst the highest in the group.

In some scenarios, the effective drawdown in early time production can be nearly 350 psi (2.41 MPa) and can assist in maintaining a stable pack. With a further increase in drawdown up to 450 psi (3.10 MPa) as show in FIG. 14, and an ensuing increase in production rate, the conditions can be more favorable for proppant production. However, since this condition does may not apply to all the sleeves, the well may not produce an appreciable amount of proppant. Thus, constantly monitoring the dynamic wellbore conditions including individual production rates and pressures can provide date which can be used to generate a range of key parameters that may assist in preventing excessive production of proppant. In some cases, flowing the well back while not exceeding 450 psi drawdown can hold off the solids production and yet the well can still clean up and produce in flow test period.

FIG. 15 depicts an example proppant flowback mitigation system 1500 including simulated bottomhole pressures 1502 generated by the simulator 206 for multistage horizontal oil well completion. In some examples, the well can be drilled and completed in medium permeability sandstones of 2.25 md and with an average reservoir thickness of nearly 125 ft (38.1 m). The well can be completed in 7 fracture stimulation stages with 20/40 U.S. Mesh ISP proppant having a S.G. of 2.71. Fracture geometry and related details obtained after pressure history match are presented in Table 3 below.

TABLE 3
Fracture and formation properties for oil well
completion in heterogenous sand reservoir.
No.	Xf (ft)	(lbm/ft2)	wp (in)	hf (ft)	hp (ft)	f (decimal)
1	324.0	1.13	0.123	147.7	125.0	0.220
2	192.0	1.32	0.144	144.6	125.0	0.210
3	206.0	0.96	0.105	145.4	125.0	0.230
4	230.7	0.80	0.087	148.0	125.0	0.090
5	204.0	0.53	0.058	160.3	125.0	0.185
6	241.7	0.94	0.103	166.9	125.0	0.200
7	240.0	1.42	0.155	150.7	125.0	0.210

Simulated bottomhole pressures 1502 for the different sleeves are presented in FIG. 15 along with measured surface and bottomhole pressures.

FIG. 16 depicts an example proppant flowback mitigation system 1600 including pressure data 1602. The increase in the bottom-hole pressures at around 250 hours of production may be attributed to an increase in frictional pressure, as shown FIG. 16, with additional gas in the system (between 250 and 300 hrs) which can remain steady despite a drop in liquid production. The average gas-liquid-ratio (GLR) can be steady around 1,320 scf/bbl which can imply that loss of static pressure may be gradual, and the tubular friction pressure of the gas laden fluid can increase.

Eventually, after a short intervention procedure to clean up the wellbore around 300 hrs, higher injection gas rates can be employed, which can result in improved production rates. With the uniform increase of both liquid production and gas rate, the flowing conditions can result in a uniform density column in the wellbore which manifests as stable static column pressure. With hydrostatics now stable, the drop in bottomhole flowing pressures implies a gradual decrease of tubular friction. In some scenarios, with more than 700 hours of flow, the downhole conditions can be determined to be relatively steady.

FIG. 17 depicts an example proppant flowback mitigation system 1700 including data represented by a production allocation plot 1702. The production allocation plot 1702 shows the production at 210 hours, which can be prior to the intervention event. Based on the analysis performed by the proppant flowback mitigation system 1700, proppant production from one or more particular sleeves (e.g., sleeve numbers 2 and 7) can be possible for the given drawdown pressures of 380 psi (2.62 MPa) and 420 psi (2.90 MPa) respectively, which can lead to an increase in bottomhole pressure due to the presence of produced solids. As shown in Table 2, these fractures can have higher proppant distribution and hence can be wider than others. Based on the simulation it is apparent that fractures at some sleeves (e.g., sleeves 3 and 6) can outperform others, though it may be noted that wellbore hydraulics can also play a role. As the well continues to produce, the overall production rate can improve.

FIG. 18 depicts an example proppant flowback mitigation system 1800 including data represented by a plot 1802. The plot 1802 of FIG. 18 shows that the drawdown can increase to 500 psi (3.45 MPa) but can also lead to the conditions that can cause proppant production from fractures located at some of the sleeves (e.g., sleeves no. 2 and 7) since the producing fluid velocity can exceed the critical velocities in both these cases. Thus, if bottom hole pressures are being monitored, the drawdown could be limited if the proppant is detected at the surface.

In some scenarios, the task of controlling proppant flowback in producing wells can be tedious. However the data analysis techniques disclosed herein, which can be based on constant monitoring of flowback related pressures and/or producing fluids can be used to perform operations which mitigate the issue. The data can also be analyzed physically closer to the measuring point, e.g., at the downhole gauge level. Attempting to correlate and predict choke settings at the surface can sometimes introduce errors because the action is based on setting points that are farther away from the location where the key parameter like flowing pressures would have the maximum impact. To avoid this, the controlling parameters can be obtained by moving closer to the bottomhole gauge. The disclosed techniques of employing the simulator 206 can help in minimizing the ill effects of proppant production by optimizing the flowback to its fullest potential and still prevent proppant production.

In some examples, the flowback simulator 206 can calculate downhole conditions that can lead to proppant flowback during stimulated well flowback period. The simulator 206 can further be applied in the field to provide guidelines and/or generate recommendations (e.g., textual or audio recommendations to change a setting) to help mitigate the issue. The simulator 206 can account for general observations made by researchers that the fractures with higher widths are more prone to producing proppant initially, even with increasing closure stresses. The critical velocity versus closure stresses can be provided to the simulator 206 as input as look-up tables for various widths. These inputs can be updated in the simulator 206 based on live production data as the production data becomes available.

In some instances, the simulator can use the actual production data, bottomhole and surface pressure and temperature data, wellbore details including hardware and wellbore deviation, basic PVT properties, and/or combinations thereof to calculate the changing bottomhole production rates for multiple (e.g., every) time step of production data, as discussed herein.

Furthermore, bottomhole rates calculated by the system(s) 100-1800 and method 1900 disclosed herein can be distributed amongst various perforation clusters or sleeves to determine the theoretical contribution from one or more of the plurality of zones (e.g., some or all of the zones). Hydraulic fracture characteristics such as X_f, h_f, w_p, F_cD, and reservoir characteristics such as P_r, f, c_o, m and others can be used as inputs for these calculations.

In an example case where proppant distributions of 2.0 lbm/ft² or higher were determined from history match, controlling bottomhole flowing pressures during the flowback period can help in limiting the production of proppants.

In some examples, where the proppant distribution is low and nearly 1.0 lbm/ft² on average, little proppant production can be observed even at higher drawdown pressures. However, compromising the proppant distribution may be avoided to prevent the proppant flowback problem because (a) this action could lead to underperforming wells, and/or (b) depending on tendencies to embed (or lack thereof), even low proppant distributions in the fracture can result in proppant production.

In some scenarios, the system(s) 100-1800 and 2000 and method 1900 disclosed herein can generate flowback guidelines. For instance, the use of the bottomhole pressure gauge data (if available) may be used to calculate flowing pressures at individual perforation clusters. This approach can keep the control closer to the production depths and can, in some instances, eliminate potential correlation-based uncertainties associated with choke setting guidelines. Even if the flow may be manipulated by choke settings, the controlling input parameter can still be bottomhole conditions. Additionally, inter-grain friction between the proppant grains can be considered and incorporated into the simulator 206 as a means of predicting proppant production.

FIGS. 19A and 19B illustrate an example method 1900 for mitigating proppant flowback, which can be performed by any of the systems 100-1800 or 2000 disclosed herein. As shown in FIGS. 19A and 19B, inputs for performing the method 1900 can include at least wellbore details data, production data, formation properties data, and/or completion details data.

In some examples, at operation 1902, the method 1900 can generate a z-factor vs. depth lookup table. At operation 1904, the method 1900 can calculate a formation volume factor (FVF) and specific gravity (SG) for fluid segments. At operation 1906, the method 1900 can obtain an appropriate fluid segment z value. At operation 1908, the method 1900 can calculate/obtain a first output value being a static column pressure value. At operation 1910, the method 1900 can calculate/obtain a second output value being a tubular friction gradient value. At operation 1912, the method 1900 can calculate one or more downhole production rates. At operation 1914, the method 1900 can calculate flow pressures for the different sleeves and/or perforations. At operation 1916, the method 1900 can calculate a drawdown and fracture permeability value. At operation 1918, the method 1900 can calculate a material balance value by calculating a pressure depletion for the different zones and updating a fracture width and a dimensionless fracture conductivity value for a closed boundary case. At operation 1920 the method 1900 can calculate flow rate values for the different sleeves or perforation clusters. At operation 1922, a simulated flow rate/velocity value can be compared to an actual or measured flow rate/velocity value. If these two values do not match within a predefined error range, the method 1900 can proceed to operation 1924, in which the method 1900 can normalize the flow rates across the different sleeves to recalculate the simulated flow rate values (e.g., by repeating operation 1920). If operation 1922 results in a match between the simulated flow rate/velocity value and the actual or measured flow rate/velocity value, the method 1900 can proceed to operation 1926 by calculating a velocity value for one or more of the perforations, comparing the velocity values to critical velocity values (e.g., one or more predetermined threshold values), and/or making a recommendation of flowback guideline based on this comparison.

FIG. 20 illustrates an example proppant flowback mitigation system 2000 including a computing system 2002 having one or more computing units that may implement the various systems 100-1800 and methods 1900 discussed herein.

In some examples, the computing system 2002 may be included in at least one of the one or more control devices, the string assembly, wireless communication units, and/or combinations thereof forming at least a portion of the systems 100-1800. The computing system 2002 may be a computing system capable of executing a computer program product to execute a computer process. Data inputs and/or program files discussed herein may be input to the computing system 2002, which reads the files and executes the programs therein. Some of the elements of the computing system 2002 are shown in FIG. 20, including one or more hardware processors 2004, one or more data storage devices 2006, one or more memory devices 2008, and/or one or more ports 2010-2012. Various elements of the computing system 2002 may communicate with one another by way of one or more communication buses, point-to-point communication paths, or other communication means, as discussed in greater detail below.

The processor 2004 may include, for example, a central processing unit (CPU), a microprocessor, a microcontroller, a digital signal processor (DSP), a graphics processing unit (GPU) and/or one or more internal levels of cache. There may be one or more processors 2004, such that the processor 2004 comprises a single central-processing unit, or a plurality of processing units capable of executing instructions and performing operations in parallel with each other, referred to as a parallel processing environment.

The computing system 2002 may be a computer, a distributed computer, or another type of computer, such as one or more external computers made available via a cloud computing architecture. The presently described technology including the simulator 206 can optionally be implemented in software stored on the data stored device(s) 2006, stored on the memory device(s) 2008, and/or communicated via one or more of the ports 2010-2012, thereby transforming the computing system 2002 in FIG. 20 to a special purpose machine for implementing the operations of the proppant flowback mitigation systems 100-1800 described herein. Examples of the computing system 2002 can include personal computers, terminals, workstations, mobile phones, tablets, laptops, personal computers, multimedia consoles, gaming consoles, wearable devices, internet of thing devices, vehicle devices, set top boxes, and the like. As such, the computing system 2002 can implement the presently disclosed technology into various practical applications.

The one or more data storage devices 2006 may include any non-volatile data storage device capable of storing data generated or employed within the computing system 2002, such as computer executable instructions for performing a computer process, which may include instructions of both application programs and an operating system (OS) that manages the various components of the computing system 2002. The data storage devices 2006 may include, without limitation, magnetic disk drives, optical disk drives, solid state drives (SSDs), flash drives, and the like. The data storage devices 2006 may include removable data storage media, non-removable data storage media, and/or external storage devices made available via a wired or wireless network architecture with such computer program products, including one or more database management products, web server products, application server products, and/or other additional software components. Examples of removable data storage media include Compact Disc Read-Only Memory (CD-ROM), Digital Versatile Disc Read-Only Memory (DVD-ROM), magneto-optical disks, flash drives, and the like. Examples of non-removable data storage media include internal magnetic hard disks, SSDs, and the like. The one or more memory devices 2008 may include volatile memory (e.g., dynamic random-access memory (DRAM), static random-access memory (SRAM), etc.) and/or non-volatile memory (e.g., read-only memory (ROM), flash memory, etc.).

Computer program products containing mechanisms to effectuate the systems and methods in accordance with the presently described technology may reside in the data storage devices 2006 and/or the memory devices 2008, which may be referred to as machine-readable media. It will be appreciated that machine-readable media may include any tangible non-transitory medium that is capable of storing or encoding instructions to perform any one or more of the operations of the present disclosure for execution by a machine or that is capable of storing or encoding data structures and/or modules utilized by or associated with such instructions. Machine-readable media may include a single medium or multiple media (e.g., a centralized or distributed database, and/or associated caches and servers) that store the one or more executable instructions or data structures.

In some implementations, the computing system 2002 includes one or more ports, such as an input/output (I/O) port 2010 and a communication port 2012, for communicating with other computing systems, network, or devices. It will be appreciated that the ports 2010-2012 may be combined or separate and that more or fewer ports may be included in the computing system 2002.

The I/O port 2010 may be connected to an I/O device, or other device, by which information is input to or output from the computing system 2002. Such I/O devices may include, without limitation, one or more input devices, output devices, and/or environment transducer devices.

In one implementation, the input devices convert a human-generated signal, such as, human voice, physical movement, physical touch or pressure, and/or the like, into electrical signals as input data into the computing system 2002 via the I/O port 2010. Similarly, the output devices may convert electrical signals received from computing system 2002 via the I/O port 2010 into signals that may be sensed as output by a human, such as sound, light, and/or touch. The input device may be an alphanumeric input device, including alphanumeric and other keys for communicating information and/or command selections to the processor 2004 via the I/O port 2010. The input device may be another type of user input device including, but not limited to: direction and selection control devices, such as a mouse, a trackball, cursor direction keys, a joystick, and/or a wheel; one or more sensors, such as a camera, a microphone, a positional sensor, an orientation sensor, a gravitational sensor, an inertial sensor, and/or an accelerometer; and/or a touch-sensitive display screen (“touchscreen”). The output devices may include, without limitation, a display, a touchscreen, a speaker, a tactile and/or haptic output device, and/or the like. In some implementations, the input device and the output device may be the same device, for example, in the case of a touchscreen.

The environment transducer devices can convert one form of energy or signal into another for input into or output from the computing system 2002 via the I/O port 2010. For example, an electrical signal generated within the computing system 2002 and/or by the components of the string assembly and/or the wellhead assembly may be converted to another type of signal, and/or vice-versa. In one implementation, the environment transducer devices sense characteristics or aspects of an environment local to or remote from the computing system 2002, such as, light, sound, temperature, pressure, magnetic field, electric field, chemical properties, physical movement, orientation, acceleration, gravity, operational machine characteristics, and/or the like. Further, the environment transducer devices may generate signals to impose some effect on the environment either local to or remote from the example computing system 2002, such as, physical movement of some object (e.g., a mechanical actuator), heating or cooling of a substance, adding a chemical substance, and/or the like.

In one implementation, a communication port 2012 is connected to a network by way of which the computing system 2002 may receive network data useful in executing the methods 1900 and systems 1000-1800 set out herein as well as transmitting information and network configuration changes determined thereby. Stated differently, the communication port 2012 can connect the computing system 2002 to one or more communication interface devices configured to transmit and/or receive information between the computing system 2002 and other devices by way of one or more wired or wireless communication networks or connections. Examples of such networks or connections include, without limitation, Universal Serial Bus (USB), Ethernet, Wi-Fi, Bluetooth®, Near Field Communication (NFC), Long-Term Evolution (LTE), and so on. One or more such communication interface devices may be utilized via the communication port 2012 to communicate one or more other machines, either directly over a point-to-point communication path, over a wide area network (WAN) (e.g., the Internet), over a local area network (LAN), over a cellular (e.g., third generation (3G), fourth generation (4G), or fifth generation (5G)) network, or over another communication means. Further, the communication port 2012 may communicate with an antenna or other link for electromagnetic signal transmission and/or reception.

In an example implementation, the proppant flowback mitigation systems 100-1900, the simulator 206, input data discussed herein, output data discussed herein, other data files, other software, and/or other modules or services may be embodied by data files and/or instructions stored on the data storage devices 2006 and/or the memory devices 2008 and retrieved and/or executed by the processor 2004.

The system set forth in FIG. 20 is but one possible example of a computer system that may employ or be configured in accordance with aspects of the present disclosure. It will be appreciated that other non-transitory tangible computer-readable storage media storing computer-executable instructions for implementing the presently disclosed technology on a computing system may be utilized.

In the present disclosure, the methods disclosed may be implemented as sets of instructions or software readable by a device such as the computing system 2002. Further, it is understood that the specific order or hierarchy of steps in the methods disclosed are instances of example approaches. It is understood that the specific order or hierarchy of steps in the method can be rearranged while remaining within the disclosed subject matter. The disclosed methods present elements of the various steps in a sample order and are not necessarily meant to be limited to the specific order or hierarchy presented.

It is to be understood that the specific order or hierarchy of steps in the method(s) depicted throughout this disclosure are instances of example approaches and can be rearranged while remaining within the disclosed subject matter. For instance, any of the operations depicted throughout this disclosure may be omitted, repeated, performed in parallel, performed in a different order, and/or combined with any other of the operations depicted throughout this disclosure.

Although the systems and processes described herein have been described in detail, it should be understood that various changes, substitutions, and alterations can be made without departing from the spirit and scope of the invention as defined by the following claims. Those skilled in the art may be able to study the preferred embodiments and identify other ways to practice the disclosed technology that are not exactly as described herein. The description, abstract and drawings are not to be used to limit the scope of the disclosed technology.

While the present disclosure has been described with reference to various implementations, it will be understood that these implementations are illustrative and that the scope of the present disclosure is not limited to them. Many variations, modifications, additions, and improvements are possible. More generally, embodiments in accordance with the present disclosure have been described in the context of particular implementations. Functionality from any of the FIGS. 1-21 may be separated or combined differently with any other functionalities of FIGS. 1-21 in various embodiments of the disclosure or described with different terminology. These and other variations, modifications, additions, and improvements may fall within the scope of the disclosure as defined in the claims that follow.

Claims

1. A method of fracturing a hydrocarbon well comprising: calculating one or more formation properties and one or more production rates associated with a wellbore;calculating one or more flow pressures at different production sleeves of the wellbore;determining pressure depletion for different zones of the wellbore using a simulator model, the simulator model determining the pressure depletion for the different zones based on a material balance equation considering the one or more formation properties;generating, by the simulator model, a simulated flow velocity value based at least partly on the one or more production rates, the one or more flow pressures, and the pressure depletion;determining, by the simulator model, that the simulated flow velocity value is within a predetermined threshold range of an actual flow velocity value;forming a normalized production rate responsive to the simulated flow velocity value being within the predetermined threshold range of the actual flow velocity value, the normalized production rate formed by comparing the actual flow velocity value with a critical velocity value; andoptimizing mitigation of proppant production from the wellbore by controlling the normalized production rate.

2. The method of claim 1, further comprising: generating a z-factor vs. depth lookup table for a plurality of fluid segments and time intervals, optimize mitigation of proppant production is based at least partly on the z-factor vs. depth lookup table.

3. The method of claim 2, further comprising: calculating, by the simulator model, a formation volume factor (FVF) and a specific fluid for the plurality of fluid segments.

4. The method of claim 1, further comprising: calculating, as a first output of the simulator model, a static column pressure of the wellbore.

5. The method of claim 4, further comprising: calculating, as a second output of the simulator model, a tubular friction gradient of the wellbore.

6. The method of claim 1, further comprising: calculating, by the simulator model one or more downhole production rates corresponding to the different production sleeves.

7. The method of claim 1, further comprising: calculating, by the simulator model, the one or more flow pressures at different perforation clusters of the wellbore.

8. The method of claim 1, further comprising: calculating, by the simulator model, a drawdown and fracture permeability of the wellbore.

9. The method of claim 8, wherein calculating the drawdown and fracture permeability is based on determining a fracture width and a fracture conductivity.

10. The method of claim 9, wherein calculating the drawdown and fracture permeability includes updating the fracture width and the fracture conductivity for a closed boundary case.

11. The method of claim 1, wherein optimizing mitigation of the proppant production from the wellbore includes providing a recommended blowback guideline based on comparing the actual flow velocity value with the critical velocity value.

12. The method of claim 1, further comprising: normalizing a plurality of flow velocity values across the different production sleeves; andrepeating a calculation to determine whether the simulated flow velocity value is within the predetermined threshold range of the actual flow velocity value.

13. A method of fracturing a hydrocarbon well comprising: determining one or more production rates associated with a wellbore;determining one or more flow pressures at one or more production sleeves of the wellbore;generating, by a simulator model, a simulated flow velocity value based at least partly on the one or more production rates and the one or more flow pressures;determining, by the simulator model, that the simulated flow velocity value is within a predetermined threshold range of an actual flow velocity value;forming a normalized production rate, responsive to the simulated flow velocity value being within the predetermined threshold range of the actual flow velocity value, by comparing the actual flow velocity value with a critical velocity value; andoptimizing mitigation of proppant production from the wellbore by controlling the normalized production rate.

14. The method of claim 13, further comprising: calculating s, the simulated flow velocity value is at least partly based on the formation properties.

15. The method of claim 14, further comprising: using, by the simulator model, a material balance equation which considers the formation properties to determine pressure depletion for different zones of the wellbore.

16. The method of claim 13, further comprising: calculating, by the simulator model one or more downhole production rates corresponding to the one or more production sleeves.

17. The method of claim 13, further comprising: calculating, by the simulator model, the one or more flow pressures at one or more perforation clusters of the wellbore.

18. The method of claim 13, further comprising: calculating, by the simulator model, a drawdown and fracture permeability of the wellbore.

19. The method of any of claim 18, wherein calculating the drawdown and fracture permeability is based on determining a fracture width and a fracture conductivity.

20. One or more tangible non-transitory computer-readable storage media storing computer-executable instructions for performing a computer process on a computing system, the computer process including: determining one or more production rates associated with a wellbore;determining one or more flow pressures at one or more production sleeves of the wellbore;generating, by a simulator model, a simulated flow velocity value based at least partly on the one or more production rates and the one or more flow pressures;determining, by the simulator model, that the simulated flow velocity value is within a predetermined threshold range of an actual flow velocity value;forming a normalized production rate, responsive to the simulated flow velocity value being within the predetermined threshold range of the actual flow velocity value, by comparing the actual flow velocity value with a critical velocity value; andoptimizing mitigation of proppant production from the wellbore by controlling the normalized production rate.