Method for performing matrix-acid stimulation design in limited-entry liner

Abstract

A method for stimulating a well (12) in a material formation, the method comprising modeling a workflow for designing a hole size distribution in a liner for a LEL liner (20) system, wherein a solution strategy for providing an initial estimate of the number of holes (28) for each section (36) satisfies an acid coverage for each section and a pressure drop (dp) on a last hole of the holes (28), wherein the initial estimate can be found from a relationship between a leak-off velocity, a pumping rate and a total cross-sectional hole area for a particular drainage factor and a liner configuration.

Description

Technical Field

[0001] This invention relates to fluid transport in systems for enhancing production in oil or gas wells in carbonate petrochemical reservoirs. Background Technology

[0002] The goal of production enhancement is to increase the productivity of an oil or gas well while minimizing the amount of enhancement fluid introduced into the well. A common production enhancement method used for oil or gas wells in carbonate reservoirs is acid enhancement, which allows a selected acid to chemically react with the reservoir rock (carbonate), leading to the dissolution of the reservoir rock and increasing the productivity of the oil or gas well.

[0003] For fully "open-hole" oil or gas wells, a complicating factor is acid placement within the well, i.e., the ability of the acid to distribute throughout the reservoir. "Pushing" acid in from the surface typically results in limited well enhancement treatments because most of the acid reacts at the wellhead.

[0004] To ensure proper acid placement and efficient acid utilization, the so-called "Limited Inlet Liner (LEL)" technology has been introduced. An LEL is a liner with multiple orifices distributed along its length to transfer acid into the reservoir rock. LEL technology was developed for acid distribution and acid enhancement in long horizontal wells and is also known as "Controlled Acid Jetting (CAJ)." The acidizing process of reservoir rock using LELs can be represented in several different mathematical ways. The first mathematical way is a fully transient method that tracks the movement of the acid front over time. This fully transient method is also known as a "transient simulator" and is useful for matching (and / or reproducing) historical pressure and velocity data of existing acid enhancement processes. The first mathematical way assumes that the orifice size distribution of the multiple orifices has been optimized and uses this orifice size distribution as input in the calculations. The transient simulator attempts to capture the physical properties of the chemical reaction between acid and rock that are required to improve the productivity of oil or gas wells. The transient simulation considers the dissolution patterns of acid in the rock. These dissolution patterns are called "wormholes." These dissolution patterns depend on factors such as the acid injection rate, rock type, permeability, or acid injection temperature. Transient simulators require significant computational power, and therefore transient simulations are time-consuming.

[0005] A second mathematical approach for modeling acidizing processes using the LEL is the steady-state method. In this method, variations in the pumping rate of acid within the LEL are ignored, and only the final acid distribution is evaluated in the steady-state simulation. This steady-state method is fast and allows for the use of computer software to vary the distribution of borehole size to match the desired acid coverage in each segment of the LEL along the well.

[0006] The concept used in acidizing processes is the distribution of multiple pores within the reservoir layer (LEL). These pores can have different sizes and / or can be spaced apart along the LEL and act as flow restrictors. This pore size and positioning results in mechanical variations in acid flow along the LEL. Proper design of the pore size distribution ensures that reservoir sections are treated with acid and that the acid is used effectively during the acidizing process. Various aspects of pore size distribution calculations have been addressed in several references listed below.

[0007] Another complicating factor is ensuring that acid penetrates the reservoir rock to the maximum extent possible. Acid is an expensive commodity and should not be used to dissolve all the rock in the adjacent wellbore area, i.e., near the LEL. Instead, the enhancement program should be designed to allow acid to penetrate the formation as much as possible, as this results in the highest negative skin and therefore the highest productivity index.

[0008] Laboratory experiments by numerous authors have clearly demonstrated that, for any given rock formation, acid penetration depends on the interstitial velocity of the acid. There exists an optimal velocity that minimizes the amount of acid required to produce deep dissolution patterns (i.e., wormholes). This optimal velocity depends on the rock and the acid system (type, concentration, temperature). In addition to ensuring uniform acid coverage, the pore size distribution must also be designed to maximize the propagation of wormholes through the rock formation.

[0009] Another issue relates to acid enhancement for both vertical and horizontal wells. The challenge is achieving uniform production enhancement throughout the well's completion trajectory. Some operators choose not to enhance production at the wellhead, while others inject from the wellhead or enhance production via coiling. Staged completions that allow for phased acidizing and the use of steerables are employed. A minority of operators use the finite inlet liner (LEL) concept, but a comprehensive workflow for orifice size design is not described. Due to the diversity of considerations, LEL design remains a challenging subject at different orifice sizes and frequencies.

[0010] In EP 1 184 537B1, the authors described the LEL concept (referred to as controlled acid jetting) for matrix-acid enhancement and developed a steady-state model using a polynomial approximation with orthogonal coordination. However, their model assumes a constant friction factor and does not describe the workflow for designing the optimal pore size distribution. Their model does not estimate the maximum design rate, does not consider experimental pore profiles, does not include a skin model, and cannot estimate the required acid coverage or the optimal distance between pores.

[0011] US 8,321,190 B2 discloses a system and method for enhancing well productivity by introducing acid into the reservoir rock of a well using a production enhancement liner to improve fluid delivery. The production enhancement liner is provided with a plurality of pre-formed orifices that form flow channels, or so-called "mud cakes," between the interior of the liner and the annular space surrounding the liner. The US patent further describes a method for simulating and / or calculating the distribution of orifices in the production enhancement liner to ensure adequate acid coverage in the reservoir rock. Simulation of orifice locations is performed in experimental and error analysis by applying transient models to distinguish the possible locations of orifices in the liner. The US patent also describes the use of steady-state models to simulate the orifice distribution.

[0012] The U.S. patent further discloses steps in the simulation including calculating the pressure drop along the production-enhancing liner as a dimensionless function of pressure or using a polynomial approximation. However, the model disclosed in the U.S. patent does not describe a workflow for designing the optimal orifice size distribution. The model also does not estimate the maximum design rate and does not consider experimental wormhole curves. The disclosed method also does not consider the physical segmentation of the wellbore using an expandable packer. The U.S. patent application also does not disclose a skin model.

[0013] US Patent Application No. 2016 / 245049 A1 discloses an apparatus and method for simulating and / or controlling fluid flow during continuous injection of multiple fluids into a formation and / or wellbore. More specifically, this US patent application describes the adaptation of a commercial reservoir simulator (Schlenberger's Eclipse) to handle instantaneous displacement in the wellbore. Numerical modeling is used to determine the required conditions and operating parameters to ensure optimal possible acid distribution, effective control of pore growth rates in multiple sections of the well, mud displacement along the entire reservoir section, and handling of significant formation pressure gradients along the reservoir section. A matrix-acid enhancement method using a controlled acid jet (CAJ) liner is also disclosed. The US patent application focuses on understanding the pressure response during well intervention and considers this pressure response a key requirement for designing and improving well intervention operations. No workflow for optimizing pore size distribution is disclosed. The US patent application aims to capture friction reduction (as a function of rate, chemical concentration, etc.) as the acid front advances down through the wellbore.

[0014] The existing teachings do not address the problems raised. For example, the assumption of a constant friction factor in EP 1 184 537 B1 does not accurately predict actual phenomena. The authors of EP 1 184 537 B1 also did not establish a design basis for the optimal pore size distribution. It cannot estimate the maximum design rate and the required acid coverage. This work does not combine experimental pore size curves with an epidermal model. US 8,321,190 B2 uses a transient model. This teaching also does not describe the optimal pore size distribution.

[0015] Purpose of the invention

[0016] The present invention, which will be described in further detail in subsequent paragraphs, includes a newly developed method and system for increasing well production, which addresses at least in part the challenges described above in a novel and inventive manner. Summary of the Invention

[0017] Therefore, the object of this invention is to provide a system and method for determining the design pumping rate of acid to ensure uniform acid penetration into the reservoir rock of an oilfield, while ensuring that the injection pressure remains below the fracturing pressure of the rock. The system and method also provide an accurate and efficient numerical solution strategy for providing an initial estimate of the number of boreholes in each section that satisfy the acid coverage and pressure drop (dp) on the last borehole in each section, particularly in the case of acid enhancement in wells completed in carbonate reservoirs using finite inlet liners or LEL liners.

[0018] According to a first aspect of the invention, the accuracy of simulations of acid fluid transport in systems used for enhancing production in material formations of resource reservoirs can be significantly improved by including a workflow for designing optimal orifice size distribution. Therefore, modeling an optimized orifice size distribution in the liner of an LEL liner system leads to improved modeling accuracy and provides an improved construction and operation of the enhancement system. This improved construction and operation of the enhancement system increases well productivity and acid utilization. Specifically, initial estimates of the number of orifices and the cross-sectional area of ​​the orifices in each section of the liner are provided. The cross-sectional area is based on an optimal rate for minimizing the amount of acid required to generate a dissolution pattern and a calculated design pumping rate for ensuring the injection pressure remains below the fracturing pressure of the well material formation. The number of orifices along the liner wall satisfies the acid coverage of each section and the pressure drop (dp) on the last orifice, wherein the pressure drop (dp) on the last orifice is linearly related to the cross-sectional area, such that the initial estimate can be obtained from the relationship between the gap velocity, pumping rate, and total cross-sectional orifice area for a specific discharge factor and liner configuration.

[0019] According to some implementations, another objective is to improve the accuracy of the simulation by ensuring that the annular pressure remains below the fracturing pressure. The maximum permissible pumping rate is determined by permeability, fluid viscosity, completion interval length, skin pressure, and the difference between annular pressure and reservoir pressure.

[0020] According to some implementations, another objective is to improve the accuracy of the simulation by estimating the characteristics of the wormholes in order to achieve the optimal wormhole size distribution.

[0021] Wormhole estimation includes node analysis calculations performed to estimate the downhole temperature at the heel of the liner, and estimates the optimal velocity for wormhole propagation and the expected pore volume to be penetrated based on the selection of acid, permeability, and temperature.

[0022] According to some implementations, model accuracy has been improved by providing a method that includes estimating the total number of orifices and the pressure drop (dp) on the last orifice. Based on the optimal velocity and the calculated design pumping rate, the total cross-sectional area of ​​the orifices is calculated, where this area is linearly related to the pressure drop (dp) on the last orifice.

[0023] According to another aspect, the data processing system is configured to perform the steps of the methods described herein.

[0024] According to another aspect, the present invention relates to a method for well enhancement by means of a workflow system for adjusting the orifice size distribution, which satisfies the acid coverage of each section and the pressure drop (dp) on the last orifice in a well completed in a carbonate reservoir using LEL liners. The method includes:

[0025] -Execute a series of algebraic equations for guessing the initial hole size distribution;

[0026] - Calculate the acid coverage and the pressure drop (dp) on the last well in the well;

[0027] - In the first iteration, compare the acid coverage and the pressure drop (dp) on the last well in the well with the design variables;

[0028] - In the next iteration, uniformly reduce the number of holes in the segment until the pressure drop (dp) on the last hole is satisfied; or

[0029] - As a first step, in the next iteration, the number of holes on the segment is increased uniformly until dp on the last hole is satisfied; and

[0030] - Perform the second step, which includes;

[0031] - As the first iteration, the existing number of holes is redistributed among the segments, where;

[0032] - Exchange one hole in the section where the calculated acid coverage differs most from the design value;

[0033] -Execute the next iteration until the acid coverage requirement is met; and

[0034] - Perform steps one and two until the pressure drop (dp) and acid coverage on the last hole in the orifice are satisfied.

[0035] According to another aspect, the present invention relates to a method for well enhancement by means of a workflow system for adjusting the orifice size distribution, which satisfies the acid coverage of each section and the pressure drop (dp) on the last orifice in a well completed in a carbonate reservoir using LEL liners. The method includes:

[0036] - Once the pressure drop (dp) and acid coverage on the last borehole in the well are satisfied, run a simulation to determine the wellhead pressure;

[0037] - If the wellhead pressure exceeds the maximum pressure rating, adjust the friction reducer concentration and rerun the simulation; and / or

[0038] - Increase the inner diameter of the pipe (pipe ID) in the presence of existing friction reducers; and / or

[0039] - Reduce the pumping rate to keep the wellhead pressure rating below the maximum pressure rating.

[0040] According to another aspect, the present invention relates to a method for well enhancement by means of a workflow system for adjusting the orifice size distribution, which satisfies the acid coverage of each section and the pressure drop (dp) on the last orifice in a well completed in a carbonate reservoir using LEL liners. The method includes:

[0041] - Run simulations to determine whether the distance between LEL holes, defined as the length of the production-enhancing reservoir segment divided by the number of holes, should not exceed twice the expected final wormhole radius;

[0042] - If the distance between LEL holes is too small, increase the LEL hole size by 1mm and repeat the simulation; or

[0043] - If the distance between LELs is too large, reduce the LEL hole size by 1mm and repeat the simulation; or

[0044] - If the LEL hole is close to or equal to twice the radius of the wormhole, continue outputting the results.

[0045] According to another aspect, there is a method for well enhancement using a workflow system for adjusting the orifice size distribution, which satisfies acid coverage and pressure drop (dp) on the last orifice in a well completed in a carbonate reservoir using LEL liners, wherein the constraints are:

[0046] - Annular pressure exceeds minimum reservoir pressure to avoid crossflow within the wellbore;

[0047] - The annular pressure should not exceed the fracturing pressure to avoid fracturing;

[0048] - The wellhead pressure shall not exceed the maximum design pressure rating;

[0049] - The sum of the cross-sectional areas of all LEL holes can be equal to or greater than the minimum cross-sectional area to avoid additional pressure drop (dp) during normal production or injection of the well after boosting production;

[0050] - The average distance between two adjacent LEL holes can be equal to twice the radius of the worm hole; and

[0051] - The inner diameter of the liner (liner ID) must not exceed the borehole size.

[0052] According to another aspect, a method for well enhancement by means of a workflow system for adjusting the orifice size distribution, which satisfies the acid coverage of each section and the pressure drop (dp) on the last orifice in a well completed in a carbonate reservoir using LEL liners. The method further includes:

[0053] - Enter one or more of the following parameters: average reservoir pressure per segment, fracture propagation pressure, permeability per segment, porosity per segment, completion interval length, wellbore radius, pipe inner diameter (pipe ID), liner inner diameter (liner ID), pipe roughness, acid characteristics, number of segments, desired acid coverage per segment, pore size per segment, and / or emission coefficients in a series of algebraic equations used for initial pore size distribution estimation.

[0054] According to another aspect of the invention, the data processing system is configured to perform the steps of the method for increasing well production as described herein.

[0055] The term "data processing system" includes any electronic system or device having a processor configured to perform the steps of a method and transmit the results of those steps to a user of the system or device. Such systems or devices include, but are not limited to, computers, laptops, handheld electronic devices, or electronic workstations. Attached Figure Description

[0056] These and other features of the invention will become more apparent from the following description of embodiments by way of example, with reference to the accompanying drawings, in which:

[0057] Figure 1 A schematic cross-sectional view of the wellbore and the finite inlet liner is shown.

[0058] Figure 2 A schematic cross-sectional view of a wellbore is shown, which has been divided into sections using packers.

[0059] Figure 3 The flowchart shown depicts the implementation of the invention step by step.

[0060] Figure 4 : Showed in more detail Figure 3 Part of;

[0061] Figure 5 The effect of rate on acid dissolution is illustrated graphically using etching patterns in Texas cream chalk.

[0062] Figure 6 The diagram illustrates the effect of gap velocity on the volume of the hole to be penetrated.

[0063] Figure 7 The figure illustrates the relationship between the acid temperature at the inlet of the liner under different pumping rates and wellhead temperatures.

[0064] Figure 8 The illustration shows the volume of acid required to achieve a certain wormhole length based on the pore volume to be penetrated, using core displacement data.

[0065] Figure 9 The diagram illustrates the relationship between pumping rate, pressure drop (dp) at the last orifice in the orifice, discharge factor, and total cross-sectional orifice area.

[0066] Figure 10 The figure illustrates the effect of Reynolds number on friction factor for different pipe roughness values.

[0067] Figure 11 The Prandtl-Karman curve illustrates the effect of drag reduction on the friction factor.

[0068] Figure 12 The diagram illustrates the effect of drag reduction on frictional pressure.

[0069] Figure 13 The diagram illustrates the epidermal factor as a function of yield-increasing coverage; and

[0070] Figure 14The figure illustrates the epidermal factor as a function of the pore radius. Detailed Implementation

[0071] A limited inlet liner comprises multiple non-uniformly spaced orifices designed to uniformly distribute fluid—in this case, acid—along the reservoir section to be enhanced. This concept was originally described by Shell in 1963 for fracturing applications (Lagrone and Rasmussen, 1963) and remains widely used today. Later, it was adapted by Maersk Oil for matrix-acid enhancement and patented (known as controlled acid injection or CAJ) and implemented on a large scale in the North Sea Cretaceous reservoirs; see Hansen (2001) and Hansen and Nederveen (2002). Since then, this novel production enhancement concept has been tested by various operators, including ConocoPhilips (Furui et al., 2010a, b), Petrobras (Fernandes et al., 2006), ExxonMobil (Sau et al., 2014; Troshko et al., 2015), and ZADCO (Issa et al., 2014), in addition to others (Mitchell et al., 2014; van Domelen et al., 2011, 2012). Rodrigues et al. (2007) provided a good overall overview of production enhancement techniques for low-permeability reservoirs, and Shokry (2010) described acid enhancement practices for offshore reservoirs in ADNOC.

[0072] Figure 1 A schematic cross-sectional view of borehole 12 is shown. Borehole 12 is typically formed using techniques known in the art and includes a wall 14 formed by the drilling process, a front end 16 extending into the formation 18, and a rear end 20 for access to the borehole.

[0073] A limited inlet liner 20 is introduced into the borehole 12. The liner 20 has an open end 22 and a opposite sealed end 24. An annular space 22 is formed between the wall 14 and the outer surface 26 of the liner.

[0074] The liner 20 is provided with a plurality of pre-formed holes 28, which form flow channels between the interior of the liner 20 and the annular space 22. The holes 28 have a shape and position that conform to specific predefined specifications.

[0075] Typically, the distance between adjacent holes 28 along the liner 20 decreases toward the end 24 of the liner.

[0076] Acid is pumped into the liner 20 and exits at high speed from the orifice 28, thus being injected into the formation 18. A choke effect is achieved by limiting the number and size of the orifices, and a significant pressure drop (dp) occurs between the inside and outside of the liner during production enhancement. The non-uniform geometry of the orifices compensates for the frictional pressure drop (dp) along the liner section. This means that the average orifice spacing decreases towards the bottom of the liner. The open annulus 22 outside the liner, combined with the overpressure inside the liner (caused by the choke above the orifices), ensures that the acid ultimately reaches the bottom of the liner, and thus the well is enhanced along its entire length.

[0077] Acid is forced in from the surface and enters liner 20 in the direction of arrow 30. The liner is not necessarily horizontal, but is often horizontal. When the acid reaches the first borehole 28, which has a size of 2 mm to 7 mm, the pressure drop (dp) on the borehole is so high that only a small portion of the acid leaves the liner through the borehole; the rest continues along the liner until it reaches the next borehole, where the same process is repeated. Appropriate borehole sizing can satisfy a specified acid coverage, defined as the number of buckets of acid per foot of reservoir section. Before boosting production, the mud can be circulated out so that only completion brine of appropriate density is found in borehole 12.

[0078] The acid enhancement process is modeled by discretizing the wellbore 12 into multiple nodes 34, typically 100 to 400 nodes. The nodes do not need to have the same size. From a practical design perspective, the wellbore is divided into a smaller number of segments 36. These segments can be physically separated from each other on the annular side by hydraulic packers 32 (not shown), but this is not always necessary. Figure 2 As shown, nodes can overlap between two segments.

[0079] The displacement of brine by acid is assumed to occur via a single-phase plug flow with minimal dispersion. Negative excess mixing volume is disregarded. Prior to production enhancement, the liner 20 is sealed at the sealing end 24 and is not cemented, meaning that fluid can, in principle, flow along the wellbore trajectory in the annulus 22 before the packer 32 is installed. In practice, the annulus flow is primarily due to acid injection through the orifice 28 perpendicular to the wellbore. For practical modeling purposes, the annulus flow along the liner can be neglected.

[0080] The well completion design and associated modeling workflow included in this document allow for reservoir segmentation using packers, and the resulting liner is referred to as a segmented finite inlet liner. The desired acid coverage for each segment can be specified, taking into account differences in porosity, permeability, initial water saturation, and reservoir pressure. The number of segments used to model this process can be greater than the number of intervals separated by the packers.

[0081] The design of the pore size distribution depends primarily on the liner geometry and flow velocity, which in turn are constrained by reservoir characteristics, namely reservoir permeability. Acid enhancement is inherently transient because once the acid reacts with the reservoir rock minerals, the skin factor at any given location along the well will change over time from an initial positive value (caused by the mud cake) towards a negative value. This reaction between the acid and the reservoir rock minerals results in the formation of highly conductive fluid flow paths within the reservoir rock. These fluid flow paths are commonly referred to as "wormholes." These wormholes are desirable during reservoir enhancement because they allow acid to propagate further into the reservoir rock, enabling subsurface hydrocarbons to flow along these wormholes once the acid is depleted. If the skin evolution over time is uniform along the well, it will not affect the flow distribution, meaning the entire process can be modeled based on steady-state principles.

[0082] This invention includes a comprehensive algorithm for designing the orifice size distribution of a finite inlet liner. The following section describes the algorithm for designing the orifice size distribution, which achieves a specified (typically uniform) distribution of acid volume per interval length, also known as acid coverage.

[0083] exist Figure 3 and Figure 4 The algorithm is illustrated schematically in the figure. Figure 3 The overall algorithm is shown, while Figure 4 It shows Figure 3 More details. Now refer to... Figure 3 Let's discuss the algorithm in conjunction with the first box, the input data, and constraint 1000.

[0084] Input data and constraints 1000:

[0085] As a starting point for implementing the algorithm, input data constraints are entered into the system. Input data includes rock properties, completion data, fluid properties, and other data such as pumping rate, the number of nodes used in the numerical algorithm, the pressure drop (dp) on the last hole in the liner, and annular pressure. These inputs are known or can be derived from historical data from the wellbore.

[0086] This algorithm defines certain constraints that must be followed during the operation of the system. These constraints form part of the input data and constraint 1000. Constraints include, but are not limited to: the annular pressure must exceed the minimum reservoir pressure to avoid cross-flow within the wellbore; the annular pressure must not exceed the fracturing pressure to avoid fracturing; the wellhead pressure must not exceed the maximum design pressure rating—which in turn affects the design rate and / or the amount of friction reducer to be added; the sum of the cross-sectional areas of all LEL holes should be equal to or greater than the minimum cross-sectional area to avoid additional pressure drop (dp) during normal production or injection of the well after enhancement—this affects the number and size of the holes; the average distance between two adjacent LEL holes should be equal to twice the radius of the wormhole formed along the finite inlet liner—this affects the pressure drop (dp) on the last LEL hole, which is a design variable; and the liner inner diameter (liner ID) cannot exceed the wellbore size.

[0087] Move to the next step, such as Figure 3 As shown in the middle frame 1002

[0088] Initial variable calculation 1002:

[0089] Based on the input for each segment, the maximum rate for each segment is determined by applying the instantaneous inflow equation. It should be noted that although the well is horizontal, it is used as a vertical well in the early injection phase because the boundary has not yet been sensed. Therefore, the reservoir segment length L is used instead of the reservoir thickness H.

[0090]

[0091] B is the acid formation volume factor, which ranges from 1.0 to 1.1. In practice, B is assumed to be 1. Viscosity is the maximum of the oil or gas viscosity and the acid viscosity. In heavy oil reservoirs, the initial injection rate is initially controlled by oil characteristics. Therefore,

[0092] μ 最大 =max(μ o ,μ 酸 Equation 2

[0093] Penetration will see contributions from both horizontal and vertical directions:

[0094]

[0095] The vertical / horizontal permeability ratio can reach values ​​ranging from 0.01 to 1.0. For this application, the value is close to 1, which makes the total permeability equal to the horizontal permeability.

[0096] The diffusion rate is given as

[0097]

[0098] The overall system compressibility is given as a contribution from the rock and the fluid phase present in the pore space.

[0099] c 总 =c 岩石 +S w ×c w +(1-S w )×c o Equation 5

[0100] rw refers to the wellbore radius. In gas reservoirs, co equals the gas compressibility.

[0101] Then, the maximum allowable pumping rate is the sum of the rates of each section:

[0102]

[0103] However, any sections that must be kept from increasing production and therefore require porous joints do not contribute to the overall rate calculation. To initiate the design algorithm, which will be detailed later, the actual design rate is taken as 10% to 30% lower than the maximum permissible rate. This value can be adjusted in subsequent iterations.

[0104] T is the total pumping time calculated using the acid coverage and length of all sections.

[0105]

[0106] It should be noted that T depends on Q, and Q depends on T.

[0107] In the 1980s, research into the fundamental principles of matrix-acid enhanced production began with the pioneering work of Fogler and colleagues at the University of Michigan (Hoefner et al., 1987; Hoefner and Fogler, 1989; Bernadiner et al., 1992; Fredd and Fogler, 1996, 1997, 1999; Fredd et al., 1997). They demonstrated that the reaction of acid with rock produces different etching patterns depending on the type and concentration of acid, as well as the rate and temperature. Key subsequent contributions to the current understanding in the literature include work by Halliburton (Gdanski and Norman, 1986; Gdanski and van Domelen, 1999; Gdanski, 1999), Buijse and Glasbergen (2005), and Hill and colleagues at Texas A&M University (Al-Ghamdi et al., 2014; Dong et al., 2014, 2016; Dubetz et al., 2016; Etten et al., 2015; Furui et al., 2005, 2008, 2010a, b; Izgec et al., 2008; Ndonhong et al., 2016, 2018; Sasongko et al., 2011; Schwalbert et al., 2018; Shirley et al., 2017; Shukla et al., 2006). Other references on experimental and theoretical studies of wormhole growth are listed within these references.

[0108] Figure 5 The effect of rate on dissolution is illustrated by a series of images 100. Low rates result in uniform dissolution and therefore very inefficient use of acid. This is shown in the leftmost image 102. In this image, acid 104 has not penetrated the formation 106 to any noticeable extent. At slightly higher rates (i.e., moving from left to right in the image), acid forms wormholes 108 through the rock. In fact, any acid formulation has an optimal rate at which the volume of acid required to corrode the pattern from the inlet to the outlet is minimized. This volume is referred to as the pore volume to be penetrated 202. It should be noted that 15% HCl corresponds to 4.4 M, therefore the 0.5 M concentration used in the experiment is rather low.

[0109] Figure 6The effect of the interstitial velocity 200 on the volume of pore 202 to be penetrated is illustrated at two different temperatures 204A (depicted by a dashed line) and 204B (depicted by a solid line). Increasing the temperature (i.e., from 25°C at temperature 204A to 60°C at temperature 204B) results in a higher reaction rate and therefore faster dissolution; thus, optimal pore growth requires a higher acid rate to avoid exhausting all the acid near the wellbore. It should also be clear that pumping at a rate slightly above the optimal rate is better than pumping at a rate below the optimal rate. In low-permeability reservoirs, the maximum pumping rate is limited by fracturing pressure, which may prevent the operator from reaching the optimal rate. In this case, a different acid 104 formulation needs to be selected to shift the curve to the left, and preferably also downwards.

[0110] The wormhole data can be reproduced using a model proposed by Buijse and Glasbergen (2005), which includes two fitting constants α and β, and can be reinterpreted using the minimum point on the curve (optimal gap velocity 200, optimal pore volume to be penetrated 202).

[0111]

[0112] Both increased temperature (204) and increased HCl concentration improved the optimal pore penetration rate (200). For low-permeability rocks where the optimal rate may be limited by crack propagation pressure, it may be beneficial to reduce the acid concentration, although the pore volume to be penetrated (202) increases and therefore the volume of acid solution required increases. If the acid concentration is halved, the volume must be doubled to maintain the same molar number. Several authors have investigated the effects of weak acids; see Punnapala et al. (2014) and Shirley et al. (2014). Friction reducers can shift the PV curve upward, meaning more acid is needed to achieve the same skin.

[0113] Talbot and Gdanski (2008) proposed a general wormhole model in which they correlated two input parameters as functions of rock and acid properties, as well as temperature, to the Buijse-Glasbergen model. However, they did not specify the values ​​of the constants in their correlation.

[0114] In this invention, we utilize a concept that, based on temperature 204°C, permeability, and acid type, Figure 6 The default wormhole curve shown shifts up, down, left, or right. Table 1 shows some rough rules of thumb for adjusting the optimal (lowest) point on the wormhole curve. Based on the default curve, the optimal point shifts by the indicated amount. The optimal point cannot be lower than (0.1, 0.1). The values ​​in the table are indicative only and are used to illustrate concepts.

[0115] Table 1: Optimal Parameters for Wormhole Growth

[0116]

[0117] Acid reactivity increases with temperature 204, which means the optimal rate of pore growth 200 also increases. For low-permeability reservoirs, reaching the optimal rate without fracturing the formation can be difficult. Therefore, it is important to assess the downhole temperature at which acid 104 reaches the formation 106.

[0118] like Figure 7 As shown, this diagram illustrates the relationship between the acid temperature at the inlet of the liner 300 and different pumping rates 302 and wellhead temperatures 304. It is advantageous to inject at a high rate and the lowest possible wellhead temperature to limit in-situ acid reactivity. This is shown by line 304A. As the temperature increases, we can see the increase in acid reactivity at the inlet of the liner 300 through lines 304B and 304C. Furthermore, any brine used to purge the mud before acid production enhancement should be injected at the lowest possible temperature.

[0119] The temperature used to adjust the pore profile is the temperature at which the acid enters the reservoir, not the reservoir temperature.

[0120] Economides et al. (1994) derived the following formula based on the volume of pores to be penetrated from core displacement data 202 to determine the volume of acid required to achieve a certain pore length of 400:

[0121]

[0122] exist Figure 8 The formula is plotted in [the image]. The ratio V / L is called the acid coverage ratio, and the unit is bbl / ft 402.

[0123] The equivalent epidermis 404 is given as:

[0124]

[0125] The algorithm aims to achieve a given final epidermal factor and then calculate the equivalent wormhole radius and subsequently the desired acid coverage. However, for economic reasons, the maximum acid coverage is limited by the volume of acid that can be pumped. For example, in offshore wells, the volume is limited by the capacity of the acid vessel. In this application, the acid coverage should not exceed 1.5 bbl / ft.

[0126] Alternatively, acid production can be fixed, which makes it possible to calculate the maximum final wormhole length 400 and thus the final negative epidermis 404.

[0127] Figure 9The diagram illustrates the results of a larger sensitivity analysis, involving pumping rate 500, pressure drop (dp) on the last of the orifices 502 (502A to 502E, respectively), discharge coefficient (CD) 504 (504A to 504E, respectively), and total orifice cross-sectional area 506. The linear relationship between total orifice cross-sectional area 506 and pumping rate 500 was derived from the sensitivity analysis. Therefore, the pressure drop (dp) 502 required to achieve a specific total orifice cross-sectional area 506 can be predicted. This constrains the total orifice cross-sectional area 506 to be equal to or greater than the minimum cross-sectional area to avoid imposing additional pressure drop (dp) 502 during production / injection after boosting production, thus leading to a constraint on the pressure drop (dp) 502 on the last orifice, which can be estimated based on the relationship provided by the sensitivity analysis. This is a novel concept.

[0128] in:

[0129] A=aQ+b Equation 10

[0130] a=αdP+β Equation 11

[0131] b=γdP+δ Equation 12

[0132] At this stage, we are able to estimate the initial hole size distribution used as the starting point for the algorithm. This is due to...

[0133] Figure 3 Box 1004 in the text depicts this.

[0134] The next step, box 1006, requires establishing equations. These equations are then solved as part of the following steps, box 1008, which will be mentioned later in this specification.

[0135] Equation 1004 is established:

[0136] The equations of motion for isothermal one-dimensional pipe flow describe the pressure drop (dp) as contributions from friction, gravity, and acceleration. The gravity term dominates in the vertical section of the wellbore, while frictional losses become relatively more significant in the horizontal section. The acceleration term is only needed when the velocity changes, such as when fluid enters the liner from the pipe (change in inner diameter) or when fluid exits through orifices in the liner. The acceleration term contributes less than 5% to the total pressure drop (dp) and is generally negligible.

[0137]

[0138]

[0139] θ is the angle relative to the Z-axis, and D is the pipe diameter. The acceleration term can be represented by the volumetric flow rate Q instead of the velocity v.

[0140]

[0141] Fanning friction factor f is defined based on wall shear stress.

[0142]

[0143] Therefore, the frictional pressure of the Newtonian flow decreases (dp) 摩擦 )for:

[0144]

[0145] For laminar flow, the Fanning friction factor is related to the Reynolds number.

[0146]

[0147] The Reynolds number is given as

[0148]

[0149] The pressure difference caused by the static head is determined as follows:

[0150]

[0151] The Fanning friction factor of pipe flow in a smooth pipe is described by the Prandtl-Karman equation:

[0152]

[0153] For rough pipes, the friction factor depends on the relative roughness ε / D of the pipe and is given as...

[0154]

[0155] Figure 10 The figure illustrates the effect of Reynolds number 600 on friction factor 602 for different pipe roughness values ​​604 (604A to 604H). A typical relative roughness of the new pipe is 10. -4 .

[0156] There is a potential discontinuity between laminar and turbulent flow because the flow mechanism is not well-defined in the 1000 to 2000 Reynolds number region. This has no impact on LEL orifice design. Figure 10 It is shown that roughness only works when it exceeds 0.0001.

[0157] Typical pumping rates range from 5 bbl / min to 40 bbl / min, depending on reservoir permeability and liner length. Such rates can lead to high surface pressures and therefore require proper top-end completion design. It is often necessary to reduce frictional pressure losses to stay within safe operating limits, and this may necessitate the use of drag-reducing agents (DRAs). DRAs are mostly diluted polymer solutions that, when added to a solvent such as water or acid, reduce frictional resistance in turbulent flow. In some cases, very low concentrations (a few thousand ppm) can reduce friction by up to 70%. However, according to some studies, drag-reducing agents may cause reservoir damage.

[0158] When drag-reducing agents are added, a region called the elastomeric sublayer is formed between the viscous sublayer and the Newton's core. The extent of the elastomeric sublayer is determined by the amount and type of polymer and the flow rate.

[0159] Maximum drag reduction is achieved when the elastomeric sublayer extends to occupy the entire pipe cross-section. The drag reduction of a diluted polymer solution in turbulent pipe flow is defined between the two general asymptotes described by Newtonian turbulence and the maximum drag reduction asymptote. Between these two lies the so-called polymerization mechanism, in which see [further details omitted]. Figure 11 The friction factor relationship is approximately linear in the Prandtl-Karman coordinate system. The polymerization mechanism can be described by two parameters: the initial wavenumber w* and the slope increment δ, whereby the slope of the polymer solution exceeds the Newtonian slope. Drag reduction begins at a well-defined initial wavenumber. For a given polymer solution, w* is essentially the same for different pipe diameters. For a given polymer-solvent combination solution, w* is approximately independent of the polymer concentration.

[0160] When modeling the effect of drag-reducing agents, it is assumed that the fluid friction factor decreases while the fluid viscosity remains constant. From Figure 10 As can be seen from the Reynolds number, the acid viscosity has little effect on friction loss under typical operating conditions.

[0161] The following formula, developed by Virk (1971, 1975), correlates the friction factor with the concentration of drag-reducing agent used for pipe flow:

[0162]

[0163] The parameters of the drag reduction model are

[0164]

[0165] K and α are constants. These parameters are specific to the chemicals used and must be fitted based on flow loop test data provided by the supplier.

[0166] The maximum drag-reducing asymptote for pipe flow is described by the following equation:

[0167]

[0168] Figure 11 The Prandtl-Karman curve illustrates the effect of drag reduction on the friction factor.

[0169] For the specific drag reducer (DRA) used, the model constant 606 is reached only at DRA concentrations exceeding 2000 ppm, with the maximum asymptote 608. The case without added DRA is shown by 610. The incrementally increased amounts of DRA are shown by lines 612, 614, and 616, respectively.

[0170] In a 6” inner diameter (ID) liner, a pumping rate of 25 bbl / min, equivalent to 36,000 bbl / d, results in a Reynolds number of approximately 321,635, which is well within the turbulence mechanism.

[0171] Figure 12 The figure illustrates the effect of drag reduction on the frictional pressure of 620 in a 10,000 ft long 4.5” OD top completion line as a function of a pump rate of 500. Friction was reduced to one-third by adding 1000 ppm of DRA. The DRA concentration is similar to... Figure 11 The concentration shown in the figure.

[0172] The finite inlet liner comprises multiple orifices that allow fluid to exit the liner and enter the annulus and subsequently the reservoir. The orifices are small in both length and diameter compared to the liner size and can therefore be considered as orifices. The pressure drop (dp) at the N orifices in the liner... 孔 ) can be calculated as:

[0173]

[0174] Q 孔 This is the flow velocity through the orifice, measured in bbl / min. The positive direction is from the liner into the annulus. D 孔 This is the inner diameter of the holes in the liner, in inches. N is the total number of holes. C D This is a dimensionless emission factor that takes into account the fact that pressure loss can only be partially recovered due to the short length of the orifice (equal to the pipe thickness). Based on the work of Crump and Conway (1988), a lower value of 0.56 was used for the flow of water and gelling fluids in sharply edged circular boreholes; this value can also reach 0.90 depending on the fluid type and the actual borehole method, see El-Rabba et al. (1997) and McLemore et al. (2013). During the first LEL design work, C DThis should be considered a sensitivity variable. Drilling at a small angle can reduce unconsumed acid splashback, which can have adverse effects on the formation, and improve the jetting process.

[0175] The model for the friction factor in the presence of drag-reducing agents is combined with the model for the friction factor of Newtonian turbulent pipe flow in rough pipes.

[0176]

[0177] If no drag-reducing agent is used, then δ = 0. If a drag-reducing agent is used, then the roughness is set to zero.

[0178] Insert the expression for the Reynolds number:

[0179]

[0180] The flow between adjacent cells in the LEL is now fully described, and a set of nonlinear equations is generated that can be solved using standard mathematical techniques, such as finite difference and other techniques.

[0181] The algorithm enters the inner loop 1100. This is guided by box 1008, solving the equation.

[0182] The final step of the inner loop 1100 is to determine whether the solution vector is constant, box 1012.

[0183] Is the solution vector constant 1012?

[0184] Typically, the Newton-Raphson technique converges within 5 iterations using carefully chosen relaxation parameters, guiding convergence during the first iteration. This method ensures that the final convergence rate is quadratic.

[0185] As illustrated in box 1008, the iterative inner loop will repeat by following arrow 1014 and restarting the solution of the equation.

[0186] The inner loop 1100 completes when the absolute change in the solution vector falls below a certain threshold, typically 1E-12. To avoid the possibility of an infinite loop, the program stops after reaching a pre-specified number of iterations—usually in the range of 10 to 20.

[0187] Once the solution vector is considered constant, the next step is to calculate the acid coverage following arrow 1016, as depicted by box 1018.

[0188] Calculate acid coverage 1018:

[0189] Once the booster flow rate is calculated from the solution, the acid coverage of each liner section is the product of the section flow rate and the pumping time. If the total pumping rate changes during operation, the booster rate of each section will change.

[0190] C 酸,i =Q 增产,i ×T Equation 97

[0191] The instantaneous period during which the acid front moves along the liner and simultaneously displaces the brine must also be considered. However, this is compensated for when the water displaces the acid at the end of the operation. The time required for the front to reach a given position i is called the hold time, which is calculated recursively:

[0192]

[0193] Since the flow velocity in the liner gradually decreases to zero at the base, it is clear that the time it takes for the acid front to displace the brine out of the liner gradually increases. In other words, the inner portion sees the acid for a longer period compared to the outer portion. The orifice size distribution should compensate for this. Therefore, the retention time is also a measure of the shortest time required for water to displace the acid from the liner at the end of the production ramp-up.

[0194] Box 1020 shows the next step for determining whether the pressure drop (dp) on the last hole in the orifice matches. This step is combined with the following box 1022 for determining whether the design acid coverage matches.

[0195] Does the pressure drop (dp) on the last hole in the orifice match 1020?

[0196] The pressure drop (dp) at the last hole in the orifice is calculated as the difference between the pressure at the last node of the liner and the annular boost pressure (which is constant and user-specified):

[0197] dP 最后一个孔 =P 衬管,n -P 增产 Equation 99

[0198] Does the designed acid coverage match 1022?

[0199] The difference between the calculated acid coverage and the specified target acid coverage is given as

[0200]

[0201] This formula ensures that the dCOV (acid coverage distribution) function is always positive. Therefore, it must be minimized to obtain the best possible match. The relative acid coverage is determined as follows:

[0202]

[0203] Go to Figure 4 The above two steps are combined into box 1050.

[0204] While the inner circulation 1100 includes addressing material balance for a given combination of LEL orifices, pumping rates, and other variables, the first part of the outer circulation 1200 includes adjusting the LEL orifice size distribution to match the desired pressure drop (dp) on the last orifice in the orifice, pressure drop (dp) 1052, and the desired acid coverage for each segment 1054. The outer circulation 1200 is used to satisfy both of these constraints simultaneously.

[0205] Therefore, the hole size distribution must be satisfied, as shown in box 1024.

[0206] Update hole size distribution 1024:

[0207] If the pressure drop (dp) is too small 1056, there are too many LEL pores, and then one LEL pore is subtracted from the segment with the highest relative acid coverage 1058, and then the material balance internal circulation 1100 is reactivated via frame 1006.

[0208] If the pressure drop (dp) is too large (arrow 1060), there are too few pores, and then a pore is added to segment 1062 with the lowest non-zero relative acid coverage and the material balance inner circulation 1100 is reactivated via box 1006.

[0209] No adjustments are made to sections with zero acid coverage.

[0210] If the pressure drop (dp) is close to the target value within a certain tolerance, the acid coverage distribution dCOV 1054 is calculated. At this point, the total number of LEL orifices is correct, but the orifices only need to be redistributed between segments. One LEL orifice is added to the segment with the lowest non-zero relative acid coverage, while one LEL orifice 1064 is subtracted from the segment with the highest relative acid coverage. Then, the inner loop 1100 is reactivated via block 1006, and this procedure is repeated until the dCOV function reaches its minimum. Because the algorithm adjusts for integer values, i.e., the number of LEL orifices, the dCOV function cannot be exactly zero.

[0211] Once the minimum dCOV function is reached, it must be determined whether the calculated wellhead pressure (WHP) is below the maximum wellhead pressure constraint, as shown in box 1026.

[0212] Is the calculated WHP lower than the maximum constraint?

[0213] Each wellhead has a maximum pressure rating, such as 5000 psia, 6500 psia, and higher. Similarly, each pipe has a maximum pressure rating. Therefore, if the reservoir pressure is high, the design rate may result in wellhead pressures exceeding the pressure ratings.

[0214] If the calculated wellhead pressure exceeds the pipe's maximum rating (indicated by arrow 1028), adjusting the design (box 1030) requires the following steps:

[0215] Step 1: If the previous design was based on zero friction reduction, add 2000 ppm of friction reducer. Rerun the simulation.

[0216] Step 2: If a friction reducer is already present, investigate the possibility of increasing the pipe inner diameter (pipe ID). Rerun the simulation.

[0217] Step 3: If step 2 is not feasible, reduce the rate, rerun the simulation, and repeat the cycle until the calculated WHP is lower than the pipe's maximum pressure rating.

[0218] Next, the average hole distance constraint must satisfy box 1032.

[0219] Does the average hole distance constraint meet the requirements?

[0220] As previously described, Economides et al. (1994) derived the following formula based on the volume of pores to be penetrated from core displacement data to determine the volume of acid required to achieve a certain pore length:

[0221]

[0222] The ratio V / L is called acid coverage, and the unit is bbl / ft. The equivalent epidermis is given as...

[0223]

[0224] Schwalbert et al. (2018) defined the production coverage as twice the wormhole radius relative to the length of the perforation interval, which for LEL completions is equal to the distance between LEL holes.

[0225]

[0226] Turn Figure 13 The figure illustrates the epidermal factor as a function of yield coverage.

[0227] Therefore, when the yield coverage 802 reaches 50%, the epidermal factor 800 becomes constant.

[0228] Figure 14The diagram illustrates that, assuming all wormholes produced along the well have the same radius, an effective wormhole radius of 20 ft (804) would result in an equivalent negative skin factor of -4 (800). Combining the two graphs, it is shown that the maximum distance between wormholes should not exceed twice the wormhole length. For example, for the entire well, a skin factor of -3 means that wormholes should be drilled with a maximum distance of 30 ft.

[0229] This means that the average distance between LEL holes, defined as the length of the production-enhancing reservoir segment divided by the total number of holes, should not exceed twice the expected final borehole radius. Therefore, the following check should be performed:

[0230]

[0231] If the average hole constraint is not met, the hole size and frame 1034 need to be adjusted.

[0232] Adjustment hole size 1034:

[0233] Based on the evaluation of the above equations, the following possible actions may be taken:

[0234] If the distance between LEL holes is too small, the LEL hole size can be increased by 1 mm, and then the entire simulation can be repeated.

[0235] If the distance between LEL holes is too large, the LEL hole size can be reduced by 1 mm and then the entire simulation can be repeated.

[0236] If the average distance between LEL holes is close to or equal to twice the radius of the wormhole, the algorithm has converged to the final design and continues to output the results, box 1036.

[0237] Output result: 1036

[0238] The output includes the following items:

[0239] Node characteristics include location, pressure, rate, friction factor, number of holes per foot, velocity, holding time, production rate, and cumulative volume of acid leaving the node through the holes.

[0240] Segment characteristics, including the number of segments, segment intervals, number of pores in a segment, distance between pores, calculated and designed acid coverage, acid coverage ratio, acid production rate, acid velocity at the pore exit point, pore volume to be penetrated, final pore radius, and final epidermal factor.

[0241] The actual pressure drop at the last orifice versus the specified pressure drop (dp)

[0242] Average total distance between LEL holes

[0243] Total number of LEL holes, total cross-sectional area of ​​LEL holes, total equivalent inner diameter (ID) of LEL holes

[0244] Wellhead pressure and bottom hole pressure during pumping

[0245] The pressure at the wellhead and bottom hole immediately after shutting in the well is called the instantaneous shut-in pressure (ISIP).

[0246] Assuming pumping occurs at the design rate, the required acid volume and total pumping time are given.

[0247] Total liner volume, main pipe volume, replacement volume, holding time

[0248] A detailed statistical list is provided, which includes the number and size of LEL holes for each joint to run in the holes, as well as the order in which the joints must run in the holes. In addition, the total number of joints with specific numbers and sizes of holes is summarized, such as the number of joints with 0, 1, 2 or 3 LEL holes of sizes of 3mm, 4mm, 5mm or 6mm.

[0249] The wellhead and bottomhole pressures during pumping are calculated based on the pressure at the first node, and then the hydrostatic pressure is subtracted and friction is added until the given gauge depth is reached.

[0250] The instantaneous shut-in pressure (ISIP) at the wellhead and bottom is calculated based on the pressure at the first node and then subtracted from the hydrostatic pressure up to the given gauge depth. Friction is zero because the velocity is zero during ISIP.

[0251] Calculation procedure:

[0252] The inputs to the numerical design model include:

[0253] -Mean reservoir pressure

[0254] - Crack propagation pressure

[0255] -Penetration

[0256] -Porosity

[0257] - Length of the completion interval

[0258] -Bore hole radius

[0259] - Pipe inner diameter (pipe ID), liner inner diameter (liner ID), pipe roughness

[0260] - Acid properties (type, concentration, density, viscosity)

[0261] - Number of segments

[0262] - Acid coverage of each section

[0263] - Hole size for each section

[0264] - Emission coefficient

[0265] Step 1. Estimate the pumping rate

[0266] The software then estimates the design pumping rate based on a standard transient inflow model (rather than the Darcy model, which is used as a steady-state assumption), while ensuring that the injection pressure remains below the fracturing pressure. Key parameters include permeability, completion interval length, and the difference between annular pressure and reservoir pressure. For calculation purposes, it is assumed that the skin can be reduced to zero. Therefore, it should be noted that since production enhancement operations typically last less than 24 hours, the injection rate will be higher than that predicted by the Darcy formula. This is because the boundary has not yet been sensed by the pressure pulses emitted during production enhancement. Thus, even if the flow inside the liner is a steady-state formulation, the inflow model used for pumping rate design is transient.

[0267] Step 2. Estimate wormhole features

[0268] Node analysis calculations are necessary to estimate the downhole temperature at the liner heel. Based on the choice of acid system, permeability, and temperature, the optimal velocity for wormhole propagation and the expected pore volume to be penetrated are estimated using published literature data. The Buijse-Glasbergen model is used to characterize wormholes at different velocities.

[0269] Step 3. Estimate the total number of holes and the pressure drop (dp) on the last hole.

[0270] Based on the optimal velocity and the calculated design pumping rate, the total cross-sectional area of ​​the orifice is directly calculated. This cross-sectional area is linearly related to the pressure drop (dp) on the last orifice in the orifice, which is a key design parameter.

[0271] Step 4. Estimate acid coverage

[0272] Increased production designs aim to achieve a negative skin of -3 or better, which requires orifice spacing to average no more than 30 to 60 feet. A model by Economides et al. (1994) was used to calculate the acid coverage required to achieve this skin. Higher acid coverage requires more acid and longer pumping times, and therefore higher costs. This must be weighed against the goal of achieving a greater negative skin.

[0273] Step 5. Calculate the optimized pore distribution

[0274] Provide an initial estimate of the number of orifices in each section and allow the software to find a solution that satisfies the acid coverage and pressure drop (dp) on the last orifice in each section. The initial estimate can be obtained from the relationship between gap velocity, pumping rate, and total cross-sectional orifice area for a specific emission factor and liner configuration.

[0275] Example:

[0276] To illustrate the design concept in more detail, an example is shown below. The well under discussion will have a reservoir length of approximately 7,000 ft. Since the supports (3 drill pipe lengths) are approximately 91 ft each, the well is numerically divided into 8 sections, each 910 ft long—corresponding to 10 supports.

[0277] The initial design coverage rate was set at 1 bbl / ft. The instantaneous inflow equation predicts a maximum rate of 20 bpm, assuming zero skin coverage, without fracturing the formation. As production ramps up, the rate can be further increased. The resulting pumping time will be 6 hours, leading to a slight, but not significant, adjustment to the design rate.

[0278] Despite reservoir temperatures of 250°F or higher, nodal analysis based on a design rate of 20 bpm predicts a BHT of 140°F at the first pore. This temperature is used to estimate the location of the optimal velocity for wormhole propagation based on measured curves and the Buijse-Glasbergen model.

[0279] The final skin depth was initially assumed to be -3, which resulted in a maximum distance of 30 ft between adjacent holes. This corresponds to a pressure drop (dp) of approximately 30 psia on the last hole in the series, and was then used as input to the design model.

[0280] The emission factor is assumed to be 0.70, which falls between the theoretical minimum of 0.56 and the high of 0.85-0.90. Post-work analysis will help determine the pressure drop (dp) at the orifice and thus the actual emission factor.

[0281] The initial estimation of the orifice size distribution utilizes the linear relationship between the orifice cross-sectional area and the pressure drop (dp) at the last orifice in the orifice. Based on this initial input, the actual optimal orifice size distribution is calculated using the numerical algorithm outlined. In the inner loop, the flow equations are solved. In the outer loop, the number of orifices is adjusted to match the pressure drop (dp) at the last orifice in the orifice and the acid coverage for each segment.

[0282] The four graphs above illustrate the calculated results. The distance between adjacent holes ranges from 20 ft to 35 ft, which yields the optimal production coverage (wormholes covering the entire well length). This distance is not uniform because the hole size was chosen to be a constant 4 mm to avoid complicating the pilot design.

[0283] Based on the pore growth model using PVbt and gap ratio, Economides inserted the minimum PVbt into the epidermal model and based it on a specific acid coverage of 1.0 bbl / ft. This yielded an epidermal factor of -2.5, which was considered sufficiently close to the initial estimate of -3. If an epidermal factor of -3 is desired, we need to increase the acid coverage, recalculate the pumping time, recalculate the flow rate, redesign the pore size, and then examine the resulting epidermal model.

[0284] Although embodiments of the invention have been described and discussed in detail above, the invention is not intended to be limited to that particular embodiment. Those skilled in the art will understand that various modifications can be made to the described embodiments or features thereof without departing from the scope of the invention.

[0285] In particular, the invention is not intended to be limited to use in the LEL liner already described. Other systems involving material flow through conduits and / or material formations can benefit from the implementation of the invention and the embodiments described above.

[0286] References

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[0300] 14. Fredd, CN, Fogler, HS (1997). Chelating Agents as Effective Matrix Stimulation Fluids for Carbonate Formations. SPE-37212-MS, SPE-38167-MS, presented at the SPE International Symposium held in Houston, Texas, USA, from February 18 to 21, 1997.

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Claims

1. A method of simulating fluid transport in a system for stimulating an oil or gas well in a material formation, the system comprising a limited-entry liner, wherein, The limited-entry liner is divided into segments having lengths less than the overall length of the limited-entry liner and including one or more holes along the wall of the limited-entry liner for discharging fluid into the material formation, wherein the method includes: calculating an initial estimate of the number of holes along the wall of the limited-entry liner and an estimated cross-sectional area of the holes, wherein the estimated cross-sectional area is based on a velocity for providing an amount of the acid needed to create a dissolution pattern and a pumping rate for maintaining an injection pressure below a fracturing pressure of the material formation of the oil or gas well, wherein the initial estimate of the number of holes along the wall of the limited-entry liner is calculated to achieve sufficient acid coverage of the acid per segment and sufficient pressure drop across a last hole of the holes, wherein the pressure drop across the last hole of the holes is linearly related to the estimated cross-sectional area; and adjusting the initial estimate of the number of holes along the wall of the limited-entry liner if the measured acid coverage per segment and the pressure drop across the last hole of the holes are not satisfied.

2. The method of claim 1, wherein, The system includes: executing a series of algebraic equations for an initial hole size distribution guess; calculating the acid coverage and the pressure drop across the last hole; comparing the acid coverage and the pressure drop across the last hole to design variables in a first iteration; uniformly reducing the number of holes across one or more of the segments in a next iteration until the pressure drop across the last hole is satisfied; or as a first step, uniformly increasing the number of holes across the one or more of the segments in a next iteration until the pressure drop across the last hole is satisfied; and performing a second step including: as a first iteration, redistributing an existing number of the holes among different ones of the one or more segments; swapping one hole for a segment for which the calculated acid coverage differs the most from a design value; performing a next iteration until the acid coverage is satisfied; and performing the first step and the second step until the pressure drop across the last hole and the acid coverage are satisfied.

3. The method of claim 1, wherein, The system includes: running a simulation to determine a wellhead pressure once the pressure drop across the last hole of the holes and the acid coverage are satisfied; adjusting a friction reducer and rerunning the simulation if the wellhead pressure exceeds a maximum wellhead pressure rating; and / or increasing an inside pipe diameter in the presence of an existing friction reducer; and / or decreasing a pumping rate such that the wellhead pressure remains below the maximum wellhead pressure rating. The system includes: running a simulation to determine a wellhead pressure once the pressure drop across the last hole of the holes and the acid coverage are satisfied; adjusting a friction reducer and rerunning the simulation if the wellhead pressure exceeds a maximum wellhead pressure rating; and / or increasing an inside pipe diameter in the presence of an existing friction reducer; and / or decreasing a pumping rate such that the wellhead pressure remains below the maximum wellhead pressure rating.

4. The method of claim 1, wherein, The system includes running a simulation to determine if the distance between adjacent ones of the holes along the wall of the limited-entry liner is no more than twice the expected final radius of a wormhole formed along the limited-entry liner; if the distance between the adjacent ones of the holes along the wall of the limited-entry liner is too small, increasing the size of the holes along the wall of the limited-entry liner by an amount and repeating the simulation; or if the distance between the adjacent ones of the holes along the wall of the limited-entry liner is too great, decreasing the size of the holes along the wall of the limited-entry liner by an amount and repeating the simulation; or if the distance between the adjacent ones of the holes along the wall of the limited-entry liner is close to or equal to twice the wormhole radius, continuing to output results.

5. The method of claim 1, wherein, The constraints are that the annulus pressure exceeds the minimum reservoir pressure to avoid cross flow inside the wellbore; the annulus pressure does not exceed the fracturing pressure to avoid fracturing; The wellhead pressure does not exceed the maximum design pressure rating; The cross-sectional area of all of the holes along the wall of the limited-entry liner can equal or exceed a minimum cross-sectional area to avoid excessive pressure drop during normal production or injection of the acid into the well after stimulation; The average distance between two adjacent holes along the wall of the limited-entry liner can equal twice the wormhole radius; and the liner internal diameter does not exceed the wellbore size is satisfied.

6. The method of claim 1, wherein, The method provides an initial estimate of the number of holes along the wall of the limited-entry liner on the section of the limited-entry liner that satisfies the acid coverage of each of the sections and the pressure drop on the last hole of the section of the holes if the acid coverage of each section and the pressure drop on the last hole of the section of the holes are not satisfied, then adjusting the initial estimate of the number of holes along the wall of the limited-entry liner on the section.

7. The method of claim 1, wherein, The fluid delivery is a fluid delivery in a limited-entry liner.

8. The method of claim 1, wherein, The simulation is performed in discrete steps and each step can be completed before the next step.

9. The method of any one of claims 1 to 8, wherein, A data processing system is configured to perform the steps of the method.

10. A data processing system configured to perform the steps of the method according to any one of claims 1 to 9 for stimulation of a well.

11. The data processing system according to claim 10, comprising an electronic system or device having a processor configured to perform the steps of the method and to communicate the results of those steps to a user of the system or device.

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