SYSTEMS AND METHODS FOR PREDICTING WELLBORE STIMULATION PERFORMANCE OF ACID JETTING THROUGH PRE-PERFORATED LINERS

Description

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to and the benefit of U.S. Provisional Patent Application Ser. No. 63/564,230, entitled “Systems and Methods for Predicting Wellbore Stimulation Performance of Acid Jetting Through Pre-Perforated Liners,” filed Mar. 12, 2024, which is hereby incorporated by reference in its entirety for all purposes.

BACKGROUND

The present disclosure generally relates to systems and methods for predicting wellbore stimulation performance of acid jetting through pre-perforated liners.

This section is intended to introduce the reader to various aspects of art that may be related to various aspects of the present techniques, which are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present disclosure. Accordingly, it should be understood that these statements are to be read in this light, and not as admission of prior art.

Acid stimulation treatments are widely used methods to stimulate productivity of hydrocarbon-producing wells in carbonate and sandstone reservoirs. Matrix acidizing and acid fracturing are two examples of these methods. Matrix acidizing is pumped below fracturing closure pressures and the mechanism of hydrocarbon delivery from reservoir to wellbore is the wormhole created by acid systems. This treatment is applicable in sandstones and carbonates. Acid fracturing is pumped above fracturing closure pressure to create fractures in the reservoir. The main function of acid is to increase conductivity of the created fractures by etching their walls.

Limited entry liners (LELs) have been deployed in the field to optimize acid coverage along long horizontal wells. Such liners are designed to choke back acid flow into higher injectivity zones to divert it into lower injectivity zones. The perforations created along LELs are relatively sparse and small to create sufficient pressure drops for choking and also to promote a desired jetting effect. The acid jet occurs in the annular space between the liner and the rock face, behind the liner's perforations. Typical jet velocities create an initial focus point for the acid on the rock face, which manifest itself as a small cavity or notch created by the dissolution of the carbonate rock with the acid.

Matrix acidizing of relatively long horizontal wells typically involves smaller volumes of acid per unit length of well along the pay zone, when compared to vertical wells. The main reason is cost. In addition, the injection rate per unit length of well along the pay zone is smaller. In this context, there is a risk that wormholes created by the matrix dissolution with acid may not extend deep enough into the reservoir to reach a satisfactory level of stimulation. Indeed, at lower rates, wormholes may form wide conical cavities that propagate slowly or that develop in the so-called face-dissolution regime where only the face of the rock is slowly dissolved.

BRIEF DESCRIPTION

A summary of certain embodiments disclosed herein is set forth below. It should be understood that these aspects are presented merely to provide the reader with a brief summary of these certain embodiments and that these aspects are not intended to limit the scope of this disclosure. Indeed, this disclosure may encompass a variety of aspects that may not be set forth below.

The embodiments described herein include systems and methods for predicting wellbore stimulation performance of acid jetting through pre-perforated liners (e.g., limited entry liners). In particular, the methods described include collecting data relating to a wellbore stimulation operation performed subsurface in a wellbore, and utilizing a physics-based model to predict an effect of jetting on efficiency of a reactive fluid during the wellbore stimulation operation based at least in part on the collected data. In addition, experimental and field treatment data, real-time telemetry, production logs, flow quantification logs or distributed sensing (e.g., temperature, acoustic, strain, and so forth) results may be used to calibrate tuning parameters of the physics-based model, which may be adjusted based on data analytics and machine learning methods. For example, the initial impact of acid jetting, calibration factors from the field measurements, reservoir and rock properties could also be tabulated in structured data to learn from the measurements and generate a robust ML model with strong physics-based fundamentals. Also, the methods described herein may include using jetting acids during the wellbore stimulation operation to create deep tunnels in geothermal wells to improve steam/heat recovery from geothermal reservoirs.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the present invention will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein:

FIG. 1 illustrates a well system that may employ certain analytic approaches, in accordance with aspects of the present disclosure;

FIG. 2 illustrates a 4×16 inches core inlet, with the right before injection and the left after acid jetting through a 0.03 inch diameter nozzle located at 0.2 inches from inlet face and with a jet velocity of approximately 150 feet/second;

FIG. 3 illustrates examples of dissolution wormholes extending into the core's matrix from the dissolution notch tip during jetting (from the left side) experiments in linear cores, in accordance with aspects of the present disclosure;

FIG. 4 illustrates an example limited entry liner (LEL) completion having four annular sections and a plurality of packers, in accordance with aspects of the present disclosure;

FIG. 5 is a schematic of a notch extending in front of an LEL's perforation due to acid jetting, in accordance with aspects of the present disclosure;

FIG. 6 is a plot of Equation 10 described herein with χ₀=6, in accordance with aspects of the present disclosure;

FIGS. 7 and 8 illustrate the lateral extents over which Equations 16 and 18 described herein apply, in accordance with aspects of the present disclosure;

FIG. 9 illustrates Pore Volume to Break Through (PVB curves with and without jetting, in accordance with aspects of the present disclosure;

FIG. 10 illustrates the difference of wormhole radial penetration depths near and away from a perforation, in accordance with aspects of the present disclosure;

FIG. 11 illustrates the lateral extent δMD_wh,it around perforation #I over which Equation 26 described herein applies, in accordance with aspects of the present disclosure; and

FIG. 12 illustrates a flow diagram of a workflow that enables engineers to assess the effect of acid jetting across liner perforation of wellbore simulation performance, in accordance with aspects of the present disclosure.

DETAILED DESCRIPTION

One or more specific embodiments will be described below. In an effort to provide a concise description of these embodiments, not all features of an actual implementation are described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers' specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure.

When introducing elements of various embodiments of the present disclosure, the articles “a,” “an,” and “the” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. Additionally, it should be understood that references to “one embodiment” or “an embodiment” of the present disclosure are not intended to be interpreted as excluding the existence of additional embodiments that also incorporate the recited features.

In addition, as used herein, the terms “real time”, “real-time”, or “substantially real time” may be used interchangeably and are intended to describe operations (e.g., computing operations) that are performed without any human-perceivable interruption between operations. For example, as used herein, data relating to the systems described herein may be collected, transmitted, and/or used in control computations in “substantially real time” such that data readings, data transfers, and/or data processing steps occur once every second, once every 0.1 second, once every 0.01 second, or even more frequent, during operations of the systems (e.g., while the systems are operating). In addition, as used herein, the terms “continuous”, “continuously”, or “continually” are intended to describe operations that are performed without any significant interruption. For example, as used herein, control commands may be transmitted to certain equipment every five minutes, every minute, every 30 seconds, every 15 seconds, every 10 seconds, every 5 seconds, or even more often, such that operating parameters of the equipment may be adjusted without any significant interruption to the closed-loop control of the equipment. In addition, as used herein, the terms “automatic”, “automated”, “autonomous”, and so forth, are intended to describe operations that are performed are caused to be performed, for example, by a computing system (i.e., solely by the computing system, without human intervention). Indeed, it will be appreciated that the data processing system described herein may be configured to perform any and all of the data processing functions described herein automatically.

In addition, as used herein, the terms “near” and “away from” may be used to describe proximities of two physical elements that are closer to each other by a predetermined distance or less and farther away from each other by the predetermined distance or more, respectively. For example, two physical elements may be considered “near” each other when they are close to each other by no more than the predetermined distance, whereas the two physical elements may be considered “away from” each other when they are farther away from each other by at least the predetermined distance. As but one non-limiting example, physical elements may be considered “near” each other when they are directly adjacent (e.g., abutting) or within 50 millimeters, 25 millimeters, 15 millimeters, 10 millimeters, 5 millimeters, 1 millimeter, or even less of each other. Conversely, physical elements may be considered “away from” each other when they are more than 1 millimeter, 5 millimeters, 10 millimeters, 15 millimeters, 25 millimeters, 50 millimeters, or even further, apart from each other.

As discussed above, acid fracturing is widely used method to stimulate productivity of hydrocarbon-producing wells in carbonate and sandstone reservoirs. The main function of acid is to increase conductivity of the created fractures by etching their walls. The embodiments described herein provide methods and algorithms that provide engineers with tools that capture the main effects of jetting acids on treatment fluid placement and wellbore post-treatment productivity. The tools may be calibrated using field and laboratory data. The jetting model described herein can help model and calibrate the effect of limited entry liner (LEL) completions and jetting bottom hole assemblies. The effect captures the differential wormhole growth in front of the jetting element (such as nozzle, perforation holes, and so forth) and shows the impact of these technologies. These completions and downhole tools are becoming more popular in the stimulation domain.

FIG. 1 illustrates a well system 10 that may employ the systems and methods of this disclosure. As illustrated, an LEL 12 may be conveyed through a wellbore 14 extending through a geological formation 16 via a conveyance mechanism 18 (e.g., wireline, e-line, slickline, coiled tubing, and so forth) that is deployed into the wellbore 14, for example, using a winch system 20. Although the winch system 20 is schematically shown in FIG. 1 as a mobile winch system carried by a truck, in other embodiments, the winch system 20 may be substantially fixed (e.g., a long-term installation that is substantially permanent or modular). The conveyance mechanism 18 may be spooled and unspooled on a drum 22 and an auxiliary power source 24 may provide energy to the winch system 20 and/or the LEL 12.

As illustrated, in certain embodiments, a data processing system 26 may be configured to perform the data processing techniques described herein, and to send and receive signals 28 from the winch system 20 and/or the LEL 12 to enable control of the well system 10 described herein. In certain embodiments, the data processing system 26 may include a processor 30, which may execute instructions stored in memory 32 and/or storage 34. As such, the memory 32 and/or the storage 34 of the data processing system 26 may be any suitable article of manufacture that can store the instructions. For example, the memory 32 and/or the storage 34 may be ROM memory, random-access memory (RAM), flash memory, an optical storage medium, or a hard disk drive, to name a few examples. In certain embodiments, a display 36, which may be any suitable electronic display, may provide a visualization, a well log, or other indication of properties in the geological formation 16 or the wellbore 14.

As discussed above, LELs 12 have been deployed in the field to optimize acid coverage along long horizontal wells. In general, LELs 12 are liners with a plurality of holes 38 distributed along their lengths, which may be used to introduce acid into the rock face of the formation 16. Such LELs 12 are designed to choke back acid flow into higher injectivity zones to divert it into lower injectivity zones. The perforations created along LELs 12 are relatively sparse and small (e.g., 3-4 millimeters in diameter) to create sufficient pressure drops for choking and also to promote a desired jetting effect. The acid jet occurs in the annular space between the LEL 12 and the rock face, behind the liner's perforations. Typical jet velocities range between 5 and 70 meters/second. Such jets create an initial focus point for the acid on the rock face which manifest itself as a small cavity or notch 40 as illustrated in FIG. 2, a few inches in length, created by the dissolution of the carbonate rock with the acid.

FIG. 2 illustrates a 4×16 inches core inlet, with the left before injection and the right after acid jetting through a 0.03 inch diameter nozzle located at 0.2 inches from inlet face and with a jet velocity of approximately 150 feet/second. In particular, FIG. 2 illustrates that jetting acid at high velocity on the face of a carbonate core can produce a notch caused by the dissolution of the carbonate rock by the jetted acid. The left picture shows the situation after the experiment is complete and what the notch 40 looks like. The right picture shows what the same face of the core looked like before acid was jetted.

Matrix acidizing of relatively long horizontal wells typically involves smaller volumes of acid per unit length of well along the pay zone, when compared to vertical wells. The main reason is cost. In addition, the injection rate per unit length of well along the pay zone is smaller. In this context, there is a risk that wormholes created by the matrix dissolution with acid may not extend deep enough into the reservoir to reach a satisfactory level of stimulation. Indeed, at lower rates, wormholes may form wide conical cavities that propagate slowly or that develop in the so-called face-dissolution regime where only the face of the rock is slowly dissolved. It is believed that the jetting effect achieved with LELs 12 may help avoid this problem:

• • The jet creates a notch in the rock at the jet impingement point, which may by-pass by itself some of the formation damage.
  • Pressure inside the notch is larger, thereby promoting additional acid flux at the tip of the notch, resulting in faster wormhole propagation into the matrix from that point. FIG. 3 illustrates examples of dissolution wormholes extending into the core's matrix from the dissolution notch tip during jetting (from the left side) experiments in linear cores (4×16 inches).

As discussed above, the embodiments described herein provide methods and algorithms that provide engineers with tools that capture the main effects of jetting acids on treatment fluid placement and wellbore post-treatment productivity. The tools may be calibrated using field and laboratory data. The jetting model described herein can help model and calibrate the effect of LEL completions and jetting bottom hole assemblies. The effect captures the differential wormhole growth in front of the jetting element (such as nozzle, perforation holes, and so forth) and shows the impact of these technologies. These completions and downhole tools are becoming more popular in the stimulation domain.

FIG. 4 illustrates an example LEL completion 42 having four annular sections 44 and a plurality of packers 46. The jetting effect is caused by acid flow across the liner's perforations 48, and it is associated with significant pressure drops. While flow along conventional wellbores 14 are generally considered a one-dimensional problem, the use of LELs 12 introduces a two-dimensional flow problem. Indeed, the injected wellbore fluids 50 may flow along the inner part of the LEL 12, along the annular gap 52 between the LEL 12 and the formation 16, and across the LEL 12 through the perforations 48. In general, the annular packers 46 are typically deployed along the pay zone to isolate the various annular sections 44 and to support the LEL 12.

To simulate flow in and around the LEL 12, the (physics-based) flow model described herein must determine:

• • Fluid flow velocities along the inner part of the LEL 12
  • Fluid flow velocities along the annulus 52 between the LEL 12 and the reservoir rock of the formation 16. The flow velocities across each liner perforation 48. The flow rates into the reservoir rock of the formation 16

For that purpose, the flow model must account for:

• • Friction and hydrostatic pressure gradients along the LEL 12
  • Friction and hydrostatic pressure gradients along the annulus 52 between the LEL 12 and the reservoir rock of the formation 16
  • Pressure discharge across each liner perforation 48
  • Flow discontinuity at the packers 46
  • The impingement pressure on the rock face in front of each liner perforation 48 due to the fluid jetting through the perforation 48 under consideration
  • The pressure profiles in the formation rock along the production interval

The pressure p_l along the inner part of the LEL 12 may be determined using the frictional and gravity contributions as follows:

∂ p l ∂ MD ⁢ ( t , MD ) = ( 2 ⁢ ρ l ⁢ f l d l ⁢ v l 2 + ρ l ⁢ g ⁢ cos ⁢ θ ) ⁢ ( t , MD ) ( 1 ) ∂ p a ∂ MD ⁢ ( t , MD ) = ( 2 ⁢ ρ a ⁢ f a d a ⁢ v a 2 + ρ a ⁢ g ⁢ cos ⁢ θ ) ⁢ ( t , MD ) ( 2 )

where:

• • t: time(s)
  • MD: measured depth along wellbore axis (m)
  • p_l: pressure in the inner part of the LEL 12 (Pa)
  • p_a: pressure in the outer part of the LEL 12 (the annulus 52) (Pa)
  • ρ_l: fluid density in the inner part of the LEL 12 (kg/m³)
  • ρ_a: fluid density in the outer part of the LEL 12 (the annulus 52) (kg/m³)
  • f_l: friction factor in the inner part of the LEL 12 (dimensionless).
  • f_a: friction factor in the outer part of the LEL 12 (the annulus 52) (dimensionless)
  • d_l: inner diameter (m) of the LEL 12
  • d_a: annulus external diameter (m) of the LEL 12
  • g: gravity acceleration (m/s²)
  • θ: wellbore deviation angle from vertical axis (radians)

Many correlations exist to determine f_l and f_a. For instance, a friction model for non-Newtonian power-law fluids (Equations 3-6) may be used.

1 f = 4 n 0 . 7 ⁢ 5 ⁢ log ⁡ ( Re ⁢ f 1 - n 2 ) - 0 . 4 n 1.2 ( 3 ) Re = ρ ⁢ V 2 - n ⁢ d n k ⁡ ( a + bn n ) n ⁢ 8 n - 1 ⁢ a = ( 1 - κ ) 2 4 ⁢ { 1 - 1 - κ 2 2 ⁢ ln ⁡ ( 1 / κ ) [ 1 - ln ⁡ ( 1 - κ 2 2 ⁢ ln ⁡ ( 1 / κ ) ) ] } ⁢ a + b = ( 1 - κ ) 2 1 + κ 2 - 1 - κ 2 ln ⁡ ( 1 / κ ) ( 4 ) For ⁢ f i : d = d l ⁢ κ = 0 ( 5 ) For ⁢ f a : d = d a ( 1 - κ ) ⁢ κ = d l / d a ( 6 )

• • n: the power-law exponent of the fluid
  • k: the consistency index of the fluid (Pa·sⁿ/m²)

As an example, the pressure-drop across a liner perforation 48 may be determined using the following model:

( p l - p a ) ⁢ ( t , MD p , i ) = q p , i 2 d p , i 4 ⁢ c p , i 2 ( d p . i ) × { ρ l ( t , MD p , i ) if q p , i > 0 ρ a ( t , MD p , i ) if q p , i < 0 ( 7 )

• • MD_p,i: Measured depth along the wellbore 14 at which perforation #i is located (m)
  • q_p,i: flow rate through perforation #i, positive if from the LEL 12 towards the annulus 52 (m3/s)
  • d_p,i: perforation #i diameter (m)
  • c_p,i: perforation #i discharge coefficient (dimensionless), a function of d_p

The maximum impingement pressure p_n,iof the acid jet 54 (e.g., passing through the steel 55 of the LEL 12) at the rock face of the formation 16 once the notch tip 56 has reached a distance x from the perforation 48 (see FIG. 5) is typically estimated using Bernoulli's equation.

p n , i ( t , x i ) = { p a ( t , MD p , i ) + 1 2 ⁢ ρ l ( t , MD p , i ) ⁢ v jry , t 2 ( t , x i ) if ⁢ q p , i > 0 0 if ⁢ q p , i < 0 ( 8 )

• • p_n,i: impingement pressure at the rock face of the formation 16 in front of perforation #i (Pa)
  • v_jet,i²: effective jet velocity from perforation #i (m/s)
  • x_i: distance between perforation #i outlet and its associated notch tip 56 (m)

v jet , i = 4 ⁢ q p , i π ⁢ d p , i 2 ( 9 )

Submerged water jets have been observed to maintain self-preservation with standoff ranging between 5 to 10 times the nozzle diameter. That is, the jet velocity remains relatively constant until it reaches 5 to 10 times the nozzle diameter. Then, the velocity decays due to the viscous drag exerted by the surrounding water. A general expression for the effective jet velocity v_jet(x) at a distance x from the perforation 48 could take the following form:

v jet , i ( t , x i ) = v jrt , i 0 ( t ) 1 + ( max ⁢ ( , x i ( t ) / d p , i χ 0 ) ) n d ( 10 )

• • χ₀: a coefficient with value in the range of 5-10 (dimensionless)
  • v_jet,i⁰: the fluid velocity in perforation #i (m/s)
  • n_d: the decay exponent (dimensionless)

A plot of Equation 10 is represented in FIG. 6. It has been observed experimentally that the rate of growth of the notch extension x tends towards zero as the notch grows. The rate of growth of the notch x_i could be modeled by the following equation:

dx i dt = f ⁡ ( x i ( t ) , v jet , i 0 ( t ) , Φ acid , Φ rock ) ( 11 )

Φ_acid, Φ_rock are sets of acid and rock properties including acid dissolving power, acid density, acid viscosity, rock mineralogy, rock porosity, rock permeability. To match the observation that x_i stagnates, it is imposed that:

df ⁡ ( x i ( t ) , v jet , i 0 ( t ) , Φ acid , Φ rock ) dx i < 0 ( 12 )

For instance, f could take the following form:

f ⁡ ( x i ( t ) , v jey , i 0 ( t ) , Φ acid , Φ rock ) = g ⁡ ( v jet , i 0 ( t ) , Φ acid , Φ rock ) x i n n ( t ) ( 13 )

with n_n>0 and g>0. f or g may be calibrated/correlated using experimental data aimed at measuring the rate of growth of the notch under various conditions describing v_jet,i⁰, Φ_acid, Φ_rock. f or g may also be determined by modeling the notching phenomenon and further calibrated/correlated using some experimental data aimed at determining the parameters of f or g. Using Equations 8-13, the time evolution of p_n(x) can be determined.

The pressure of the fluid in the formation rock 16 may be obtained by solving the classical mass and momentum balance equations related to flow in porous media. For instance, assuming radial flow around the wellbore 14, the fluid pressure into formation rock 16 may be obtained by solving the following equation:

∂ ∂ t ( ρ f ( p f ) ⁢ ϕ ⁡ ( t , MD , r ) ) + 1 r ⁢ ∂ ∂ r ( r ⁢ ρ f ( p f ) ⁢ k ⁡ ( t , MD , r ) μ f ( t , MD , r ) ⁢ ∂ p f ∂ r ⁢ ( t , MD , r ) ) = 0 ( 14 )

• • ρ_f: density of the fluid in the formation rock porosity (kg/m³)
  • ϕ: porosity of the formation rock 16 (volume fraction)
  • r: radial distance into the formation rock 16 from the center of the wellbore 14 (m)
  • k: permeability of the formation rock 16 (m²)
  • μ_f: viscosity of the fluid in the formation rock porosity (Pa·s)
  • p_f: pressure in the fluid in the formation rock porosity (Pa)

The mass rate of fluid exchanged between the annulus 52 and the formation rock 16 may be determined from the following equation:

q f ( t , MD ) = - 2 ⁢ π ⁢ k ⁡ ( t , MD , r w ( MD ) ) ⁢ r w ( MD ) ⁢ ρ f μ f ( t , MD , r w ( MD ) ) ⁢ ∂ p f ∂ r ⁢ ( t , MD , r w ( MD ) ) ( 15 )

• • q_f: mass flux of fluid exchanged between the annulus 52 and the formation rock 16 (kg/m/s).
  • r_w: borehole radius or radial distance to the rock face of the formation 16 from the center of the wellbore 14 (m)

Equation 15 may be integrated as follows:

q f ( t , MD ) = 2 ⁢ π ⁢ ( ρ f ⁢ k μ f ) _ ⁢ p a ( t , MD ) - p f ( t , MD , r ) ln ⁡ ( r / r w ) = 0 ( 16 )  ⁢ ( ρ f ⁢ k μ f ) _ : the ⁢ average ⁢ value ⁢ of ⁢ ( ρ f ⁢ k μ f ) _ ⁢ between ⁢ r w ⁢ and ⁢ r ( ρ f ⁢ k μ f ) _ = ln ⁡ ( r / r w ) ∫ r r w μ f ρ f ⁢ k ⁢   ( t , MD , ω ) ⁢ d ⁢ ln ⁡ ( ω ) ( 17 )

In front of the perforation #i, p_a in Equation 16 should be replaced by P_n,i.

q f , i ( t , MD p , i ) = 2 ⁢ π ⁢ ( ρ f ⁢ k μ f ) _ ⁢ p n , i ( t , MD p , i ) - p f ( t , MD p , i , r ) ln ⁡ ( r / r w ) = 0 ( 18 )

• • q_f,i: mass flux of fluid exchanged between the annulus 52 and the formation rock 16 close to perforation #i (m²/s)

q_f(t, MD) applies over a distance δMD* not affected by the jets. q_f,i(t, MD) applies over a distance δMD_wh,i affected by the jets. This is illustrated in FIG. 7.

To complete the set of equations required to determine all pressures and flow velocities along the LEL 12, along the liner annulus 52, and in the formation rock 16 around the wellbore 14, the mass balance describing mass exchange between the different parts of the wellbore 14 may be solved:

∂ ∂ t ( δ ⁢ V l ( ρ l ) ) + ( A l ⁢ ρ l ( p l ) ⁢ v l ) ⁢ ( MD + δ MD ) - ( A l ⁢ ρ l ( p l ) ⁢ v l ) ⁢ ( MD ) = - Q l , a ( 19 ) ∂ ∂ t ( δ ⁢ V a ⁢ ρ a ( p a ) ) + ( A a ⁢ ρ a ( p a ) ⁢ v a ) ⁢ ( MD + δ MD ) - ( A a ⁢ ρ a ( p a ) ⁢ v a ) ⁢ ( MD ) = Q l , a - Q a , f - Q q , f , i ( 20 )

• • δ_MD: elementary variation around MD (m)
  • A_l: cross-sectional area of the LEL 12 (m²)
  • A_a: cross-sectional area of the annulus 52 (m²)
  • δV_l: elementary liner volume=A_l×δ_MD (m³)
  • δV_a: elementary liner annulus volume=A_a×δ_MD (m³).
  • Q_l,a: mass flow rate from the LEL 12 inside to the annulus 52 over δ_MD (kg/s).
  • Q_a,f: mass flow rate from the annulus 52 to the formation rock 16 away from the perforations 48 over δ_MD (kg/s).
  • Q_a,f,i: mass flow rate from the annulus 52 to the formation rock 16 near the perforations 48 over δ_MD (kg/s)

Q l , a = ∑ i ∈ Ω ⁡ ( MD , δ ⁢ MD )  π ⁢ d p , i 2 ⁢ v jet , i 0 × { ρ l if v jet , i 0 > 0 ρ q if v jet , i 0 < 0 ( 21 ) Ω ⁡ ( MD , δ ⁢ MD ) = { i ⁢ such ⁢ that ⁢ MD p , i ∈ [ MD , MD + δ ⁢ MD ] } ( 22 ) Q a , f = q f ( t , MD ) ⁢ δ ⁢ MD * × { ρ a if q f > 0 ρ f if q f < 0 ( 23 ) Q a , f , i = ∑ i ∈ Ω ⁡ ( MD , δ ⁢ MD )  q f , i ( t , MD ) ⁢ δ ⁢ MD wh , i × { ρ a if q f > 0 ρ f if q f < 0 ( 24 )

• • δMD*: the lateral extent within δMD over which Equation 16 applies (m). δMD_wh,i=δMD−δMD*: the lateral extent within δMD over which Equation 18 applies (m)

δMD* and MD_wh,i are illustrated in FIG. 8.

Equations 1-24 may be solved together to predict the flow velocities and pressures along the inner part of the LEL 12, along the liner annulus 52, and in the formation rock 16 at all measured depths and times.

It has been claimed that jetting acid improves wormhole growth rates at lower injection rates. It has been suggested that the excess pressure inside the notch (ρ_n,i) at the point of impingement promotes larger fluxes at the wormhole tips when compared to cases without jetting but with the same injection rate. FIG. 9 illustrates the Pore Volume to Break Through (PVBT, inversely proportional to wormhole propagation rate) measured experimentally with and without jetting¹. Much lower PVBT values are observed at lower interstitial velocity when jetting is used. ¹ Impact of Acid Jetting on Carbonate Stimulation. Beckham, R E, Shuchart, C E and Buechler, S R. Doha, Qatar: s.n., 2015. International Petroleum Technology Conference. IPTC-18360-MS. It is noted that FIGS. 3 and 9 of this disclosure are extracted from this reference, for clarification of certain aspects of the embodiments described herein.

With this in mind, it seems reasonable to imagine that the excess pressure P_n,i at the impingement point produces an excess rate into the wormholes originating from the notch tip 56, compared to the rate going into the wormholes originating at azimuths outside the notch. This is explained by comparing Equations 16 and 18 with P_n,i>p_a. Consequently, this excess local rate could push the limit of the face dissolution regime towards lower rates as seen in FIG. 9. In general, the depth of the wormholes may be determined using so-called wormholing models. The general form of such model is the following:

∂ r wh ∂ t ⁢ ( t , MD ) = f wh ( r wh , q f , Φ acid , Φ rock ) ( 25 ) ∂ r wh , i ∂ t ⁢ ( t , MD ) = f wh ( r wh , i , q f , i , Φ acid , Φ rock ) ( 26 )

• • r_wh: radial distance from the center of the wellbore 14 to wormhole tips for MDs away from any MD_p,i·(m).
  • r_wh,i: radial distance from the center of the wellbore 14 to wormhole tips for MDs close to MD_p.i·(m).
  • f_wh: wormholing model

The resolution of Equations 25 and 26 may lead to the situation represented in FIG. 10. The lateral extent δMD_wh,i around MD_p,i over which Equation 26 applies and beyond which Equation 25 applies is not known and should be calibrated using field or experimental data. FIG. 11 illustrates the lateral extent δMD_wh,it around perforation #I over which Equation 26 applies.

It is also possible that the presence of the notch in front of a perforation 48 acts as a focus point for the flux of acid due to the notch geometry. In other words, the effective flux used to determine the rate of wormhole propagation in front of a perforation 48 in Equation 26 may have to be corrected as follows:

∂ r wh , i ∂ t ⁢ ( t , MD ) = f wh ( r w ⁢ h , i , α i ⁢ q f , i , Φ acid , Φ rock ) ( 27 )

• • α_i: acid flux focus factor for perforation #i (dimensionless)

The determination of α_i requires calibration using field or experimental data.

In the regions where the wormholes have propagated:

• • r<r_wh,i in front of the perforations 48
  • r<r_wh away from the perforations 48

The rock permeability should be increased by a certain factor α_k (typically very large) determined from laboratory experiments.

k ⁡ ( t , MD , r ) = α k ( MD ) ⁢ k ⁡ ( 0 , MD , r ) ( 28 )

FIG. 12 illustrates a flow diagram of a workflow 58 that enables engineers to assess the effect of acid jetting across liner perforation of wellbore simulation performance, and which may be at least partially performed by the data processing system 26 illustrated in FIG. 1, as described in greater detail herein. As illustrated, in certain embodiments, the workflow 58 begins with setting the wormhole propagations r_wh=r_wh,i=r_w at time=0 (block 60). Then, the flow problem may be solved using Equations 1-24, as described herein (block 62), for example, based on pressures in the LEL 12 (p_l), in the annulus 52 (p_a), and in the formation rock 16 (p_f); impingement pressure in front of (e.g., directly adjacent) the perforations 48 (p_n,i); flow velocities in the LEL 12 (v_l), in the annulus 52 (v_a), and across a perforation 48 (v_jet⁰); and flow rates into the rock formation 16 (q_f) away from the jets, and (q_f,i) near the jets. Then, the wormhole propagation r_wh around the wellbore 14, away from the perforations 48, may be calculated using p_a (block 64), for example, using Equation 25. Then, the wormhole propagation r_whⁿ around the wellbore 14, near the perforations 48, may be calculated using p_n (block 66), for example, using Equation 27. Then, near-wellbore permeability k(r<r_wh) away from the perforations 48 and k(r<r_wh,i) near the perforations 48, due to wormholing, may be updated (block 68), for example, using Equation 28. Then, the time may be updated from t=t+δ_t (block 70) and the workflow may be iteratively performed (e.g., continuously iteratively performed) again by returning to block 62.

Therefore, the workflow 58 illustrated in FIG. 12 defines a method for predicting wellbore stimulation performance of acid jetting through pre-perforated liners. For example, in certain embodiments, the method may include solving a flow problem associated with one or more perforations 48 created during a wellbore stimulation operation performed subsurface in a wellbore 14, for example, using Equations 1-24 (e.g., block 62 of the workflow 58). In addition, in certain embodiments, the method may include calculating a first amount of wormhole propagation r_wh around the wellbore 14 and away from the one or more perforations 48 created during the wellbore stimulation operation based at least in part on the solved flow problem, for example, using Equation 25 (e.g., block 64 of the workflow 58). In addition, in certain embodiments, the method may include calculating a second amount of wormhole propagation r_whⁿ around the wellbore 14 and near the one or more perforations 48 created during the wellbore stimulation operation based at least in part on the solved flow problem, for example, using Equation 27 (e.g., block 66 of the workflow 58). In addition, in certain embodiments, the method may include updating near-wellbore permeability away from the one or more perforations 48 created during the wellbore stimulation operation k(r<r_wh) and near the one or more perforations 48 created during the wellbore stimulation operation k(r<r_wh,i) associated with the first and second amounts of wormhole propagation, for example, using Equation 28 (e.g., block 68 of the workflow 58).

In addition, in certain embodiments, the method may include solving the flow problem utilizing a physics-based model to predict an effect of jetting on efficiency of a reactive fluid 50 during the wellbore stimulation operation based at least in part on data collected during the wellbore stimulation operation, as described herein. In certain embodiments, the method may include calibrating tuning parameters of the physics-based model utilizing experimental and field treatment data, real-time telemetry, production logs, flow quantification logs or distributed sensing results, as also described herein. In addition, in certain embodiments, the method may include adjusting the tuning parameters of the physics-based model based on data analytics and machine learning methods, as also described herein.

In addition, in certain embodiments, solving the flow problem may include determining one or more pressures (p_l) in LEL 12 used to perform the wellbore stimulation operation, one or more pressures (p_a) in an annulus 52 formed between the wellbore 14 and the LEL 12, and one or more pressures (p_f) in a formation 16 through which the wellbore 14 extends (e.g., as part of block 62 of the workflow 58). In addition, in certain embodiments, solving the flow problem may include determining an impingement pressure (p_n,i) directly adjacent the one or more perforations 48 created during the wellbore stimulation operation (e.g., as part of block 62 of the workflow 58). In addition, in certain embodiments, solving the flow problem may include determining one or more flow velocities (v_l) in an LEL 12 used to perform the wellbore stimulation operation, one or more flow velocities (v_a) in an annulus 52 formed between the wellbore 14 and the LEL 12, and one or more flow velocities (v_jet⁰) across the one or more perforations 48 created during the wellbore stimulation operation (e.g., as part of block 62 of the workflow 58). In addition, in certain embodiments, solving the flow problem may include determining one or more flow rates (q_f) into a formation 16 through which the wellbore 14 extends and away from jets that form the one or more perforations 48 created during the wellbore stimulation operation, and one or more flow rates (q_f,i) into the formation 16 through which the wellbore 14 extends and near the jets that form the one or more perforations 48 created during the wellbore stimulation operation (e.g., as part of block 62 of the workflow 58).

In addition, in certain embodiments, the method may include calculating the first amount of wormhole propagation r_wh around the wellbore 14 and away from the one or more perforations 48 created during the wellbore stimulation operation based at least in part on a pressure p_a in an annulus 52 formed between the wellbore 14 and an LEL 12 used to perform the wellbore stimulation operation that is determined as part of the solved flow problem (e.g., as part of block 64 of the workflow 58). In addition, in certain embodiments, the method may include calculating the second amount of wormhole propagation r_whⁿ around the wellbore 14 and near the one or more perforations 48 created during the wellbore stimulation operation based at least in part on an impingement pressure pn directly adjacent the one or more perforations 48 created during the wellbore stimulation operation that is determined as part of the solved flow problem (e.g., as part of block 66 of the workflow 58).

In addition, in certain embodiments, the method steps may be performed iteratively over time as a plurality of iterative loops. In certain embodiments, the method may further include adjusting one or more operational parameters of the wellbore stimulation operation during each iterative loop of the plurality of iterative loops. For example, certain parameters (e.g., concentrations, flow rates, pressures, temperatures, and so forth) of the reactive fluid 50 used during the wellbore stimulation operation may be adjusted based on the analysis described herein. In addition, in certain embodiments, the method steps may be performed in substantially real-time during performance of the wellbore stimulation operation. In other words, the analysis described herein may be performed while the wellbore stimulation operation is being performed, enabling relatively fast adjustments to operational parameters of the wellbore stimulation operation.

As such, the embodiments described herein include systems and methods for predicting wellbore stimulation performance of acid jetting through pre-perforated liners (e.g., LELs 12). In particular, the methods described include collecting data relating to a wellbore stimulation operation performed subsurface in a wellbore 14, and utilizing a physics-based model to predict the effect of jetting on efficiency of a reactive fluid 50 during the wellbore stimulation operation based at least in part on the collected data. In addition, experimental and field treatment data, real-time telemetry, production logs, flow quantification logs or distributed sensing (e.g., temperature, acoustic, strain, and so forth) results may be used to calibrate tuning parameters of the physics-based model, which may be adjusted based on data analytics and machine learning methods. For example, the methods described herein may include using jetting acids during the wellbore stimulation operation to create deep tunnels in geothermal wells to improve steam/heat recovery from geothermal reservoirs.

While only certain features of the invention have been illustrated and described herein, many modifications and changes will occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.

The techniques presented and claimed herein are referenced and applied to material objects and concrete examples of a practical nature that demonstrably improve the present technical field and, as such, are not abstract, intangible or purely theoretical. Further, if any claims appended to the end of this specification contain one or more elements designated as “means for (perform)ing (a function) . . . ” or “step for (perform)ing (a function) . . . ”, it is intended that such elements are to be interpreted under 35 U.S.C. § 112(f). However, for any claims containing elements designated in any other manner, it is intended that such elements are not to be interpreted under 35 U.S.C. § 112(f).