A method for diversion in smart liner completions via sequentially dissolving liner nozzle plugs

Abstract

The present invention relates inter alia to a liner (20) for engagement in a wellbore (12), the liner (20) comprising: a closed end (24) and an open end (22); a plurality of nozzles (28) for transferring fluid radially from the liner (20), wherein the nozzles (28) are plugged with degradable nozzle plugs (104, 204, 304); wherein at least some of the nozzles (28) comprise nozzle plugs (104, 204, 304) with different degradation characteristics, such that the nozzle plugs (104, 204, 304) take different times to degrade, in particular to dissolve, when subjected to an acid; and wherein the nozzle plugs (104, 204, 304) are arranged in the liner axially ordered depending on their degrading characteristic. The invention further relates to a method of stimulating a well. The invention also relates to a method, performed by a data-processing system, of simulating fluid transport in a system for stimulating a well in a material formation.

Description

[0001] October 14, 2024Abu Dhabi National Oil Company A174900WO MAJ / EssSMART LINER 1. Field of the invention5 This invention relates to a liner for engagement in a wellbore, and a method of stimulating a well with the liner. The invention also relates also to a method of stimulating a well with a liner by removing nozzle plugs and sealing open nozzles. The invention further relates to fluid transport in a system for0 stimulating a well in a material formation. 2. BackgroundIn the field of oil and gas extraction, well stimulation techniques such as hydraulic fracturing and acidizing are important for enhancing the permeability of rock formations and increasing the flow of5 hydrocarbons. Wellbore liners play an important role in theseoperations, serving as conduits for the controlled delivery of fluids into the rock formation. Conventional liners often face challenges in precisely managing the timing and location of fluid release, leading to inefficiencies and suboptimal stimulation results.0 In view of the foregoing, there is a need for an improved liner. It is thusan object to provide a liner with plugged nozzles that open at specific times allowing for a more precise and staged fluid release. Acidizing treatments are commonly used in the oil and gas industry to enhance the permeability of rock formations surrounding a wellbore.5 Acidizing is a common method used to dissolve minerals and materials that impede the flow of hydrocarbons, thereby improving the well's productivity. Traditional acidizing methods often involve uniformly applying acid to the entire wellbore, which can lead to inefficiencies and uneven stimulation of the rock formation. The invention focuses on providing a more controlled and targeted approach to acid distribution, optimizing the treatment process. In view of the foregoing, there is a need for an improved method of stimulating wells. It is thus an object to provide a method for more precise and controlled acid stimulation of a well by using a linerequipped with multiple plugged nozzles, wherein the nozzle plugs can becontrollably removed and the nozzles can be sealed again for precise acid targeting. The following paragraphs are based on the European patent application 20179814.7. The purpose of stimulation is to the enhance productivity of the well. A common stimulation method for carbonate reservoirs is acid stimulation whereby the selected acid is allowed to chemically react with the reservoir rock, which leads to dissolution and enhanced productivity. For wells completed open-hole, a complicating factor is the acid placement, i.e. the ability to distribute acid across the entire reservoir section. Bull-heading acid from the surface typically results in a mediocre stimulation treatment because the majority of the acid is spent reacting at the heel of the well. The solution to the acid placement challenge is addressed by the limited-entry liner (LEL) technique, also denoted as controlled-acid jetting (CAJ). The key concept is to distribute small holes of varying sizes and frequency in the liner. These holes act as flow restrictions, which leads to mechanical diversion of flow along the liner. An appropriate hole size distribution design is capable of ensuring that the entire reservoir section is treated with acid. Aspects of the hole size distribution has been addressed in a number of references. A further complicating factor is to ensure maximum acid penetration into the reservoir rock. Acid is an expensive commodity and should not be spent on dissolving all the rock in the near-wellbore area. Rather, the stimulation programme should be designed in such a way that acid penetrates as furthest as possible into the formation because this situation leads to the highest negative skin and hence the highest productivity index. Lab experiments by a number of authors clearly show that for any given rock, acid penetration depends on the interstitial velocity of acid. There exists an optimum velocity, which minimizes the amount of acid needed to generate dissolution patterns known as wormholes. This optimum velocity depends on the rock, and the acid system (type, concentration, temperature). In addition to ensuring uniform acid coverage, the hole- size distribution must also be designed in such a way that it maximizes the propagation of wormholes. Problem pertains to acid stimulation of both vertical and horizontal wells. The challenge is to achieve uniform stimulation throughout the completed well trajectory. Some operators choose not to stimulate the wells, others bullhead from the wellhead, others stimulate through a coiled tubing. Segmented completions which allow acidization in stages and use of diverters is employed. A few operators make use of the Limited Entry Liner (LEL) concept, but does not describe a comprehensive workflow for the hole size design. The design of LEL in terms of varying hole sizes and frequency remain a subject matter of challenge because of multiplicity of considerations. US 2009 / 294122 A1 discloses method of simulating fluid transport in a system for stimulating a well in a material formation of a resource reservoir, the system comprising a conduit element arranged in said well, the conduit element comprising a conduit wall including one or more openings for discharging a fluid into the material formation surrounding the conduit element. US 2016 / 245049 A1 discloses a controller for controlling an apparatus for performing a wellbore intervention or process, a corresponding processing device and method for simulating and / or controlling consecutive flow of a plurality of fluids in a wellbore of arbitrary geometry.EP 1184537 B1, the authors describe the LEL concept (called controlledacid jet) for matrix-acid stimulation and develop a very simple steady- state model using polynomial approximation with orthogonal collocation. However, their model assumes constant friction factor and does not describe a workflow for design of the optimum hole size distribution. Their model does not estimate the maximum design rate, does not take into account the experimental wormhole curve, does not have a skin model and is unable to estimate the required acid coverage and the optimum distance between holes.US 8,321,190 B2, the authors model a limited-entry liner with atransient model. The flow equations are the same as in the current work, but their numerical solution scheme uses the finite-difference method to solve for pressure and rate in the liner whereas as in this work, polynomial approximation with orthogonal collocation is used. Their model is transient, whereas the current work describes a steady-state solution. Their workflow does not allow design of the actual hole size distribution but is used to analyse data from an existing stimulation job.The teachings of the prior art do not solve the problem. For example, theassumption of constant friction factor in EP1184537B1 does not bode well for the actual phenomenon. They have also not laid out design basis for optimum hole size distribution. It cannot estimate maximum design rate and required acid coverage. The work did not incorporateexperimental wormhole curves and skin model. US 8,321,190B2 used atransient model. The teachings also did not describe optimum hole size distribution.SummaryThe above objects are at least partially achieved by the subject matter ofindependent claims 1 and 17. Preferred embodiments are the subject ofthe dependent claims, and the skilled person will find clues to other suitable aspects of the present invention in the overall disclosure of the present application.An aspect of the invention relates to a liner for engagement in a wellbore,the liner comprising: a closed end and an open end; a plurality of nozzlesfor transferring fluid radially from the liner, wherein the nozzles areplugged with degradable plugs; wherein at least some of the nozzles comprise plugs with different degradation characteristics, such that theplugs take different times to degrade, to dissolve, when subjected to anacid; and wherein the plugs are arranged in the liner axially ordereddepending on their degrading characteristic. Such a liner wellbore liner can be used in oil and gas extraction. The nozzles, being plugged with degradable materials, allow for controlled fluid release at different stages of the wellbore operation. For example, as the acid flows through the liner, it dissolves the plugs at different times, enabling a sequential and controlled fluid transfer. This orderly degradation helps in efficiently managing the pressure and flow within the wellbore, leading to optimized extraction processes and reducedoperational costs. In this document, nozzle plugs are also referred to asplugs. This can be improved when the degrading characteristics are based onthe permeability profile along the well. Permeability refers to the abilityof a porous geological formation (such as rock or sediment) surrounding the wellbore to allow fluids (like oil, gas, or water) to flow through it. It is a measure of how easily these fluids can move through the interconnectedpores or spaces within the rock or sediment. A tailored degradationapproach can ensures that the acid is applied precisely where it is needed most, optimizing the stimulation process and improving the overall efficiency of well production. By aligning the degradation of nozzle plugs with the well's permeability profile, this method enhances theeffectiveness of the stimulation treatment, which can lead to betterhydrocarbon recovery and reducing the risk of formation damage orinefficient acid use. Further improvement can be achieved when thenozzle plugs comprise an acid-degradable material. Using acid-degradable materials for the nozzle plugs can offer more precise control over when and where the acid is released. This ensures that acid is applied only where it is most needed, improving the efficiency and effectiveness of the well stimulation process. This method thus allows for a safer and more controlled stimulation process, leading to enhanced well productivity and better management of well integrity.Even further improvement is achieved when the nozzle plugs comprisewax, glass, ceramic, aluminum, chromium, rubber, elastomer and / or polylactic acid. Using different materials for the nozzle plugs allows for precise control over the degradation rates and characteristics of the plugs. Each material has distinct properties that make it suitable for different wellbore conditions and operational needs. For instance, wax and polylactic acid can be designed to degrade quickly when exposed to certain conditions, while materials like glass and ceramic may offer more controlled and slower degradation. This flexibility in material choice allows for tailored solutions to specific wellbore environments, enhancing operational efficiency, reducing downtime, and improving overall safety by ensuring that the plugs degrade in a predictable and controlled manner. Even further improvement is achieved when the nozzle plugs are adapted not to degrade when subjected to brine.Such a design ensures that the nozzle plugs remain intact in the presenceof brine, a common fluid in wellbore operations. By being resistant to brine, the plugs only degrade when exposed to specific substances like acids, allowing for precise control over when and how the plugs dissolve. This enhances the reliability and predictability of fluid release, especially in environments where brine is present. It improves the efficiency of operations by preventing premature degradation of the plugs, ensuringthat the desired fluid transfer sequences are maintained, and operationalintegrity is upheld.Further improvement is achieved when the nozzles have a diameter in therange of 1 to 10 mm, preferably in the range 2 mm to 6 mm, even morepreferably in the range 2 mm to 4 mm.Such a specifying of the nozzle diameters ensures consistent fluid flow and pressure management within the wellbore. Having nozzles within the optimal diameter range of 2 to 4 mm allows for precise control over the rate of fluid transfer, enhancing the efficiency of operations like hydraulic fracturing. Uniform nozzle diameters further simplify the design and manufacturing processes and ensure predictable performance across all nozzles. This consistency is important for maintaining balanced pressure distribution and achieving desired operational outcomes, such as effective fluid injection and extraction, reducing the risk of mechanicalfailures and improving overall wellbore performance. The purpose of theholes is two-fold: 1) to create a high-velocity jet of acid during a stimulation treatment, which promotes deep wormholing in the immediate vicinity of each hole, and 2) to promote an even distribution of acid along the entire completion interval.This embodiment is further improved when the liner comprises at leastone segment with a length of 100 m that comprises 5 to 40 holes, preferably 10 to 30 holes, more preferably 15 to 20 holes, most preferably about 16 holes. Such a specifying of the length and number of holes ensures consistentand optimal fluid distribution within the wellbore. A 100-meter segmentwith approximately 5 to 40 holes allows for a balanced and manageableflow rate, enhancing the control over fluid injection or extraction processes. This precise configuration supports efficient pressure management and fluid distribution, which is important in operations like hydraulic fracturing or acidizing. By having a defined number of holes, operators can better predict and manage the performance of the liner, leading to improved operational efficiency, reduced risks of uneven pressure distribution, and enhanced overall wellbore treatment effectiveness. This embodiment can be improved further when the liner comprises atleast one segment with a length of 100 m wherein the axial distancebetween the holes in each section is in the range of 3 to 35 m, preferably6 to 25 m, more preferably 8 to 20m.Such a setting of the axial distance between the holes within thesespecified ranges can ensure optimal fluid distribution and pressure management throughout the wellbore. The distances of 10 to 15 meters between holes can provide a balanced configuration that supports effective and controlled fluid flow. This precise spacing can enhance the ability to manage sequential fluid release, reducing the risk of pressurespikes or uneven fluid distribution. Such an arrangement is beneficial in processes like hydraulic fracturing or well stimulation, where maintaining consistent pressure and fluid flow is important to operational success. By adhering to these spacing guidelines, the liner improves overall efficiency, safety, and effectiveness of wellbore treatments.Further improvement is achieved when at least one packer is separatingtwo segments from one another.Incorporating at least one packer to separate the sections of the liner enhances control over fluid distribution and pressure management within the wellbore. Packers act as barriers that isolate each section, allowing for independent fluid control in each segment. This design is useful in multi-stage hydraulic fracturing, where precise isolation of different sections is necessary to optimize fluid injection and extraction processes. By effectively sealing and separating the sections, packers prevent crossflow between segments, ensuring that the desired pressure and fluid distribution are maintained. This leads to improved efficiency, better resource management, and enhanced overall performance of wellbore operations.Further improvement is achieved when the acid is HCl.Using hydrochloric acid (HCl) to degrade the plugs ensures a predictable and effective dissolution process. HCl is a commonly used acid in wellbore operations due to its strong corrosive properties and ability to rapidly break down specific materials. By utilizing HCl, the liner ensures that the degradation of the plugs occurs in a controlled and efficient manner, facilitating precise fluid release at desired stages. This enhances the overall efficiency of operations such as hydraulic fracturing, acidizing, or other well stimulation techniques by providing reliable and quick degradation of plugs, optimizing fluid flow, and improving the management of pressure and resources within the wellbore.Another aspect of the invention relates to a method of stimulating a wellcomprising the steps: providing a liner according to the above disclosureinto a wellbore; and subsequently applying the steps of: applying acid intothe liner, so that the acid dissolves nozzle plugs and flows through un-plugged nozzles; applying sealing material into the liner, configured anddedicated to seal un-plugged nozzles. Such a method can effectively stimulate a well by using a structured approach to fluid management. First, the insertion of the specially designed liner with degradable plugs ensures that fluid flow can be precisely controlled. Applying acid dissolves the plugs at specific nozzles, allowing for targeted fluid release and optimal pressure management within the wellbore. Following this, the application of a sealing material to the now-open nozzles prevents backflow and isolates different sections, ensuring that the stimulation process is contained and directed as intended. This method can enhance well efficiency by providing precise control over the stimulation process, reducing the risk of uncontrolled fluid migration, and improving overall resource management and operational safety.This embodiment can be further improved when the liner is provided fora completion interval with a length higher than 5000 feet, morepreferably higher than 10000 feet, even more preferably higher than20000 feet.Using this method for long completion intervals, such as those exceeding 20,000 feet, is beneficial for deep well operations. The extended liner ensures that fluid stimulation can be managed over a large section of the wellbore, allowing for comprehensive and effective treatment. This method is suited for deep wells requiring precise control over fluid release and pressure management across extensive lengths. The ability to use the liner in such long intervals enhances operational flexibility, improves the efficiency of resource extraction, and ensures thorough stimulation of the wellbore, leading to better overall performance and increased production rates in deep well operations.Further improvement is achieved when the acid comprises HCl.Using hydrochloric acid (HCl) for the stimulation method ensures efficient and predictable degradation of the nozzle plugs. HCl's strong corrosive properties make it highly effective in breaking down the materials used for the plugs, facilitating controlled fluid release through the nozzles. This method is especially advantageous for wellbores with long completion intervals, ensuring that the acid efficiently dissolves the plugs along the entire length of the liner. By using HCl, operators can achieve precise management of fluid distribution and pressure, enhancing the overall effectiveness of well stimulation processes such as hydraulic fracturing, acidizing, or other wellbore treatments, ultimately leading to improved production rates and operational efficiency.Another aspect of the invention relates to a method of stimulating a wellcomprising the steps: providing a liner with a closed end and an open end, the liner comprising a plurality of nozzles for transferring fluid radially from the liner, wherein at least some of the nozzles are pluggedwith nozzle plugs; applying acid into the liner, and thereafter: removingone or more nozzle plugs, so that acid can flow through the thus openednozzles radially from the liner; and sealing open nozzles. Such a method helps selectively removing the nozzle plugs, also referred to as plugs, so that the acid can be directed to precise locations, enhancing the efficiency of the well stimulation process. After the acid treatment, the open nozzles are sealed to prevent further fluid loss and maintain the integrity of the well. This method offers the advantage of controlled acid placement, leading to improved well performance and reduced operational costs. Further improvement is achieved when the removing and sealing steps are based on the permeability profile along the well. Permeability refers to how easily fluids can flow through the rock or sediment around the wellbore. Such an approach enables precise targeting of acid treatment, by tailoring the removal and sealing process to the permeability profile, this method optimizes the efficiency of well stimulation, enhances hydrocarbon recovery, and reduces the risk of damage to the well structure or formation. Further improvement is achieved when the sealing step comprises subsequently sealing from the closed end towards the open end. Such a sequential sealing towards the open end ensures that any remaining acid is pushed towards the open end, maximizing the treatment of the well. This approach provides an orderly and efficient sealing process, which helps to maintain well integrity and optimizes the distribution of the acid treatment. This method enhances the control over the stimulation process, ensuring consistent and effective results.The skilled person knows that the subsequent sealing from closed endtowards the open end, e.g. with sealing balls can also be not in a strictmathematical sense from closed end towards the open end but in ageneral sense, as the sealing balls might not seal the open nozzles in the strict order of the nozzles.The method can be further improved when the removing of nozzle plugsis done to control the pressure inside of the acid filled liner. Such a controlled removal allows for a gradual and precise release of acid through the nozzles, preventing sudden pressure changes that could compromise the well structure. By managing the pressure in this manner, the method ensures a more controlled and effective acid stimulation process, enhancing the well's productivity while minimizingthe risk of damage. This approach provides a safer and more efficientmeans of well stimulation, resulting in optimized well performance and extended operational life.The method can be further improved when the removing of nozzle plugsis done to control the fluid transfer from the liner, to even the acidcoverage in the wellbore.Such a strategic removal ensures that the acid is evenly spread,enhancing the treatment's effectiveness by targeting all necessary areas within the wellbore uniformly. By ensuring an even coverage, this method maximizes the contact between the acid and the wellbore surfaces, leading to a more efficient stimulation process. This technique improves the overall performance of the well, ensuring a more consistent and thorough acid treatment, thereby optimizing production and extending the well’s operational life. Even further improvement is achieved when the removing step comprises removing nozzle plugs subsequently from the closed end towards the open end. Such a sequential removal allows for a controlled and systematic release of acid, ensuring a gradual and uniform pressure drop within the liner. By managing the pressure and acid flow in this orderly manner, the method ensures that the acid treatment is evenly distributed along the wellbore. This approach enhances the effectiveness of the stimulationprocess by ensuring consistent acid contact with all targeted areas,leading to improved well productivity and stability. This method provides a precise and efficient technique for well stimulation, resulting in optimized acid distribution and enhanced well performance.Further improvement is achieved when the removing step involves adegradation, dissolving, breaking, melting and / or exploding of the nozzle plug(s).These techniques are chosen based on the specific requirements of thewell stimulation process, ensuring the most effective and efficient removal of the plugs. This flexibility allows for precise control over the timing and manner of acid release, leading to optimal distribution and pressure management within the wellbore. This method enhances the efficiency and effectiveness of the well stimulation process, resulting in improved well performance and longevity.The method can be further improved when the nozzle plugs compriseheating elements and / or explosives for removing the nozzle plugs. The components are activated to remove the plugs, allowing for the controlled release of acid. The heating elements can be used to melt the plugs, while the explosives can be precisely detonated to break or remove the plugs. This method ensures that the plugs are effectively and efficiently removed, providing a reliable means to manage the acid flow and pressure within the liner. By incorporating such advanced removal techniques, the method enhances the precision and control of the well stimulation process, leading to improved distribution of acid, optimal well performance, and increased operational safety.Further improvement can be achieved when the removing is triggeredby an acoustic signal and / or an electromagnetic signal. The nozzle plugs are removed by triggering mechanisms such as acoustic signals (sound waves) or electromagnetic signals (radio waves or other electromagnetic frequencies). These signals remotely activate the heating elements or explosives integrated within the nozzle plugs, causing them to degrade, dissolve, break, melt, or explode. This remote triggering allows for precise timing and control over the removal process, ensuring the plugs are removed in a sequential and controlled manner from the closed end towards the open end of the liner. By using acoustic or electromagnetic signals, the method enhances the efficiency, safety, and precision of the well stimulation process, leading to improved acid distribution, optimized well performance, and increased operational reliability.Further improvement is achieved when the triggering by anelectromagnetic signal comprises Radio-frequency identification (RFID).For instance, each nozzle plug can be equipped with an RFID tag thatcan be remotely activated. When an RFID signal is sent, it could trigger the heating elements or explosives within the nozzle plugs, causing them to for instance degrade, dissolve, break, melt, or explode. This RFID- based triggering allows for precise control over which plugs are removed and the timing of their removal, ensuring a controlled release of acid from the closed end towards the open end of the liner. By using RFID technology, the method provides a highly accurate and efficient way to manage the well stimulation process, resulting in optimal acid distribution, enhanced well performance, and improved operational safety.Even further improvement is achieved when the removing is triggeredwhen one or more pressure thresholds are exceeded inside the liner.The nozzle plugs can be designed to be removed when the internalpressure of the liner exceeds certain predefined thresholds. This pressure-sensitive triggering mechanism ensures that the plugs are removed in response to the build-up of pressure, allowing for the controlled release of acid through the nozzles. This method provides an automatic and responsive way to manage the acid flow and pressure within the liner, enhancing the precision and efficiency of the well stimulation process. By ensuring that the plugs are removed at optimal pressure levels, this approach improves the distribution of acid, leading to better well performance and increased operational safety. Further improvement can be achieved when the nozzle plugs comprise an acid-degradable material. Utilizing acid-degradable materials for the nozzle plugs provides greater precision in controlling the timing and location of acid release. This approach ensures that acid is applied specifically where it is most needed, thereby enhancing the efficiency and effectiveness of the wellstimulation process. Consequently, this method promotes a safer andmore controlled stimulation process, resulting in improved well productivity and more effective management of well integrity.Even further improvement is achieved when the nozzle plugs comprisewax, glass, ceramic, aluminum, chromium, rubber, elastomer and / or polylactic acid. These materials are chosen based on their properties, such as melting point, solubility, and mechanical strength, which are suitable for different triggering mechanisms and well conditions. For instance, wax plugs can be melted with heating elements, while ceramic and glass plugs might be broken using controlled explosions or pressure thresholds. The choice of material allows for flexibility in the design and functionality of the plugs, ensuring they can be effectively removed as needed during the well stimulation process. This method enhances the efficiency and precision of acid distribution, leading to improved well performance and operational reliability.Even further improvement can be achieved when the sealing stepcomprises the introduction of ball sealers into the liner. These ball sealers are designed to move through the liner and lodge into the open nozzles, effectively blocking them and preventing further fluid flow. This method ensures a reliable and efficient sealing process, maintaining the integrity of the well and preventing unwanted fluid loss. By using ball sealers, the method enhances the control and effectiveness of the well stimulation process, leading to improved well performance and extended operational life.Further improvement can be achieved when the sealing step comprisesthe introduction of a signal transmitter into the liner, configured and dedicated to trigger a removal of the nozzle plugs. Such a signal transmitter can be configured to send acoustic or electromagnetic signals, such as RFID signals, to trigger the removal of any remaining nozzle plugs. This ensures that the plugs can be removed in a controlled and sequential manner, from the closed end towards the open end, facilitating the even distribution of acid and maintaining optimal pressure conditions within the liner. The introduction of a signal transmitter enhances the precision and efficiency of the well stimulation process by providing an additional layer of control over the timing and sequence of plug removal. This method ensures effective acid coverage, improved well performance, and greater operational safety.This embodiment can be further improved when the signal transmitteris arranged inside a ball, in particular a ball sealer. Such a ball sealer, containing the signal transmitter, moves through theliner and can even lodge into open nozzles to seal them. The signaltransmitter inside the ball sealer is configured to send acoustic or electromagnetic signals, such as RFID signals, to trigger the removal of for instance any remaining nozzle plugs. This optionally dual-functionball sealer can not only seal the nozzles but also can ensure controlledand sequential plug removal from the closed end towards the open end. This approach enhances the precision and efficiency of the well stimulation process, ensuring optimal acid distribution, maintaining proper pressure conditions, and improving overall well performance and operational safety.Further improvement is achieved when the liner is provided for acompletion interval with a length higher than 5000 feet, morepreferably higher than 10000 feet, even more preferably higher than20000 feet.Such an extended length capability allows for the effective treatment oflong horizontal or vertical sections of the wellbore. The method's design accommodates the challenges associated with such extensive intervals, ensuring that the acid distribution, pressure management, and sealing processes remain efficient and effective over the entire length of the liner. By providing a liner suitable for these extensive completion intervals, the method enhances the overall stimulation process, leading to improved well productivity, optimized resource extraction, and increased operational efficiency.Even further improvement is achieved when the nozzle plugs are notdegrading when subjected to brine. For instance, nozzle plugs that are used in the well stimulation method are made from materials that resist degradation when exposed to brine. This can ensure that the plugs maintain their integrity and functionality during the initial stages of the well stimulation process, even in saline environments. The plugs will only degrade or be removed when subjected to specific triggering mechanisms, such as heating, chemical dissolution, mechanical breaking, melting, or controlled explosions. By ensuring that the nozzle plugs are brine-resistant, the method enhances the reliability and control of the stimulation process, ensuring that the acid distribution and pressure management remain effective throughout the procedure. This durability against brine exposure leads to more consistent well performance and improved operational efficiency.Further improvement is achieved when the nozzles comprise a diameterin the range 2 to 6 mm, preferably in the range 3 mm to 4 mm.Such a specific diameter range is chosen to balance the flow rate and control of acid distribution. Smaller nozzles provide more precisecontrol over the acid flow, which is necessary for targeted stimulationand even acid coverage within the wellbore. By ensuring the nozzles fall within this diameter range, the method enhances the efficiency and effectiveness of the well stimulation process, leading to improved well performance, optimal acid distribution, and better overall operationaloutcomes. The holes serve two main purposes: first, to generate a high-velocity jet of acid during stimulation treatment, promoting deepwormholing near each hole, and second, to ensure an even distribution of acid throughout the entire completion interval.Even further improvement is achieved when the liner in the completioninterval comprises 5 to 40 holes per 100 m of length, preferably 10 to 30 holes, more preferably 15 to 20 holes, most preferably 16 holes. Such a specific hole density ensures that the acid is evenly distributed along the length of the completion interval, enhancing the stimulation process. A higher density of holes allows for more targeted and efficient acid placement, while the preferred range optimizes the balance between sufficient coverage and structural integrity of the liner. By configuring the liner with this hole density, the method improves the efficiency and effectiveness of acid distribution, resulting in enhanced well performance, better resource extraction, and increased operational efficiency.Further improvement is achieved when the distance between the holesis in the range of 3 to 35 meter, preferably 6 to 25 meter, morepreferably 8 to 20 meter.Such a specific spacing ensures that the acid is distributed evenly along the completion interval, providing thorough coverage of the targeted wellbore areas. By optimizing the distance between the holes, the method balances efficient acid placement with maintaining the structural integrity of the liner. This configuration enhances the overall stimulation process, leading to improved well performance, more effective resource extraction, and increased operational efficiency.Even further improvement is achieved when the method furthercomprises the step of providing one or more packer(s) for separatingvarious sections of the liner. These packers are strategically placed within the liner to isolate specific intervals, allowing for targeted acid treatment of individual sections. This segmentation enables more precise control over the acid distribution, ensuring that each section receives the appropriate amountof stimulation. By using packers to separate the liner into various sections, the method enhances the efficiency and effectiveness of the stimulation process, leading to improved well performance, better resource extraction, and increased operational control. This approach also allows for customized treatment of different zones within the wellbore, optimizing overall production. Further improvement is achieved when at least some, preferably all, ofthe nozzle plugs comprise a self-destruct device for removal of the nozzle plug, in particular a detonator. These self-destruct devices are designed to activate and remove the plugs at precise moments, ensuring controlled and effective acid flow through the nozzles. By incorporating detonators or similar self-destruct mechanisms, the method allows for the reliable and timely removal of nozzle plugs, enhancing the control over the acid distribution process. This approach ensures that the acid is delivered to the targeted zones within the wellbore efficiently and safely. The use of self-destruct devices in the nozzle plugs improves the overall effectiveness of the well stimulation process, leading to optimized well performance, better resource extraction, and increased operational safety. This embodiment can be further improved when the self-destruct devicecomprises an explosive with a TNT equivalent of 10 to 100 g, preferably 15 to 80 g, more preferably 20 to 50 g. The following paragraphs are based on the European Patent Application20179814.7. It is thus an object of the present invention to provide an accurateand efficient numerical solution strategy for providing an initial estimate of the number of holes per segment which honours the acid coverage per segment and the drop in pressure (dp) across the last hole, in particular in the context of acid stimulation of wells completed in a carbonate reservoir with a Limited-Entry- Liner or LEL liner.According to an aspect of the invention the accuracy of simulations of fluidtransport in a system for stimulating a well in a material formation of a resource reservoir can significantly be improved by including a workflow for design of the optimum hole-size distribution. Therefore optimized hole-size distribution in the liner of a LEL liner system is modelled, which results in an improved modellingaccuracy and providing an improved construction and operation of thestimulation system. In particular, providing an initial estimate of the number of holes per segment which honours the acid coverage per segment and the dp across the last hole, such that the initial estimate can be found from the relationship between interstitial velocity, pump rate, and total cross-sectional hole area for a particular discharge coefficient and liner configuration. According to some embodiments, it is a further object to improve the accuracy of the simulation by ensuring that the annulus pressure remains below fracturing pressure. The maximum allowed pump rate is dictated by the permeability, thefluid viscosity, the length of the completed interval, the skin, and the differencebetween annulus pressure and reservoir pressure. According to some embodiments, it is a further objective to improve the accuracy of the simulation by estimating wormholing characteristics to facilitate an optimal hole-size distribution. The wormholing estimate includes a nodal analysis calculation performed to estimate the downhole temperature at the heel of the liner, and based on the choice of the acid, the permeability and the temperature, the optimum velocity for wormhole propagation is estimated together with the anticipated pore volume to breakthrough. According to some embodiments, the model accuracy has been improved by providing a method comprising estimation of the total number of holes and dp in pressure across the last hole. Based on the optimum velocity and the calculated design pump rate, the total cross-sectional area of the holes is calculated, wherein the area is linearly correlated with the dp across the last hole. According to another aspect, a data processing system is configured to perform the steps of the method described herein. According to yet another aspect, the invention relates to a method of stimulating a well by means of a workflow system for adjusting the hole-size distribution which honours the acid coverage per segment and the drop in pressure (dp) across the last hole in the context of acid stimulation of wells completed in a carbonate reservoir with a LEL liner. The method comprises:- performing a series of algebraic equations for an initial hole-size distributionguess;- calculating acid coverage and dp across the last hole;- comparing acid coverage and dp across the last hole against design variable ina first iteration;- evenly decreasing the number of holes across a segment for the next iterationuntil dp across the last hole is honoured; or- evenly increasing the number of holes across a segment for the next iterationuntil dp across the last hole is honoured, as a first step; and- performing a second step which includes;- redistributing existing number of holes between segments as a first iteration,wherein;- segments, where the calculated acid coverage is the furthest away from designvalues, exchange one hole;- performing the next iteration until acid coverage is honoured; and- performing the first step and the second step until dp across the last hole andacid coverage is honoured; and wherein the method further comprises:- operating the system for stimulating the well in the material formation usingthe optimized hole-size distribution; and wherein the method further compromises: a plurality of N sequential stages of simulating fluid transport, wherein the total number of holes to be taken into account for the hole-size distributionoptimization increases from stage 1 to stage N;initially specifying a desired average acid coverage and deduct a required total acid volume to be pumped; initially estimating the acid volume per stage to be the total acid volume divided by the number of stages N; optimizing the hole-size distribution for the first stage; optimizing the hole-size distribution for the next stage, while keeping the hole- size distribution of the previous stage constant. This method can be improved when the material formation around the wellcomprises permeabilities in the range of 0,1 millidarcy to 50 millidarcy.This method can further be improved when the material formation around the wellcomprises an average permeability of less than 2 millidarcy, wherein the ratio of highestpermeability to lowest permeability is higher than 3. This method can be subject to even further improvement when in the well thedrain length is higher than 20000 feet when the wellbore diameter is around 6inch or the drain length is higher than 30000 feet when the wellbore diameter isaround 8.5 inch. According to yet another aspect, the invention relates to a method of stimulating a well by means of a workflow system for adjusting the hole-size distribution which honours the acid coverage per segment and the drop in pressure (dp) across the last hole in the context of acid stimulation of wells completed in a carbonate reservoir with a LEL liner. The method comprises: -running a simulation once the dp across the last hole and acid coverage is honoured to determine the wellhead pressure;- adjusting the friction reducer concentration and re-running the simulation if thewellhead pressure exceeds the maximum pressure rating; and / or- increasing the tubing ID in presence of existing friction reducer; and / or- reducing the pump rate, such that the wellhead pressure rating is maintainedbelow a maximum pressure rating. According to yet another aspect, the invention relates to a method of stimulating a well by means of a workflow system for adjusting the hole-size distribution which honours the acid coverage per segment and the drop in pressure (dp) across the last hole in the context of acid stimulation of wells completed in a carbonate reservoir with a LEL liner. The method comprises:- running a simulation to determine whether the distance between LEL holes,defined as the length of the stimulate reservoir section divided by the number of holes, should not exceed twice the expected final wormhole radius;- increasing the LEL hole size by 1mm if the distance between LEL holes is toosmall, and repeating the simulation; or- decreasing the LEL hole size by 1mm if the distance between the LEL is toolarge, and repeating the simulation; or- proceeding with an output of results if the LEL holes is close or equal to twicethe wormhole radius. In one embodiment the model describes a first order non-linear boundary value problem and consists of two first-order coupled non-linear differential equations with boundary conditions at each end of the interval of the independent variable,coupled with a set of algebraic equations, wherein polynomial approximation isused to solve the numerical model. According to yet another aspect, the method of stimulating a well by means of a workflow system for adjusting the hole-size distribution which honours the acid coverage per segment and the drop in pressure (dp) across the last hole in the context of acid stimulation of wells completed in a carbonate reservoir with a LEL liner in which the constraints of;- annulus pressure exceeding minimum reservoir pressure to avoid cross-flowinside wellbore;- annulus pressure does not exceed fracturing pressure to avoid fracturing;- wellhead pressure does not exceed maximum design pressure rating;- cross-sectional area of all LEL holes combined may be equal to or exceed aminimum cross-sectional area to avoid creating an additional pressure drop during normal production or injection of the well after stimulation;- average distance between two neighbouring LEL holes may be equal to twicethe wormhole radius; and- liner ID not exceeding the wellbore size, are honoured.According to yet another aspect, the method of stimulating a well by means of a workflow system for adjusting the hole-size distribution which honours the acid coverage per segment and the drop in pressure (dp) across the last hole in the context of acid stimulation of wells completed in a carbonate reservoir with a LEL liner. The method comprises further:- input of one or more of the following parameters; average reservoir pressure persegment, fracture propagation pressure, permeability per segment, porosity, length of the completed interval, wellbore radius, tubing ID, liner ID, pipe roughness, acid properties, number of segments, desired acid coverage per segment, hole size per segment and / or discharge coefficient in a series of algebraic equations for an initial hole-size distribution guess. According to yet another aspect of the invention, a data processing system is configured to perform the steps of the method of stimulating a well as described herein. The term data processing system includes any electronic system or device having a processor configured to perform the step of the method, and to communicate the outcome of those steps to a user of the system or device. Such system or device includes, but is not limited to, a computer, a laptop, a handheld electronic device, or electronic workstation.Brief description of the figuresIn the following, preferred embodiments of the disclosure are disclosed by reference to the accompanying figures.Figure 1: shows a schematic cross-sectional view of a well-bore andlimited entry liner.Figure 2: shows a schematic cross-sectional view of a well-bore which issectionalized into segments by the use of packers.Figure 3: shows a flow diagram, depicting the implementation of thecurrent invention in a step-wise fashion.Figure 4: shows a portion of figure 3 in greater detail.Figure 5: shows, by way of images, the effect of rate on dissolution of anacid, by etching patterns in Texas cream chalk.Figure 6: illustrates the impact of interstitial velocity to pore volume tobreakthrough.Figure 7: illustrates the relation between the temperature of the acid atthe entrance of the liner under different pump rates and wellhead temperatures.Figure 8: illustrates the volume of acid required to achieve a certainwormhole length based on the pore volume to breakthrough from core flood data.Figure 9: illustrates the relationship between the pump rate, the dpacross the last hole, the discharge coefficient and the total cross- sectional hole area.Figure 10: illustrates the effect of Reynold’s number on friction factor fordifferent values of pipe roughness. Figure 11: illustrates the impact of drag reduction on the friction factor, in a Prandtl-Karman plot.Figure 12: illustrates the influence of drag reduction on friction pressure.Figure 13: illustrates the zero’s of orthogonal polynomials (100 in total) against the length of the liner. Figure 14: illustrates skin factor as a function of stimulation coverage.Figure 15: illustrates skin factor as a function of wormhole radius.Figure 16: illustrates a liner according to the invention in a wellbore in a side view. Figure 17: shows another liner in a well bore in a side view.Figure 18: depicts a liner according to the invention duringstimulating in a well bore in a side view. Figure 19: shows the liner of Figure 18 at a different stage ofstimulating in a side view.5. Detailed description of the figuresThe subsequent sections provide a detailed description of the invention, referencing the accompanying illustrations for clarity. The descriptions represent examples only and are not intended to limit the invention's scope. Identical reference numerals across the figures and text denote the same components. The illustrations may not reflect actual size or scale; their dimensions, proportions, and depictions of elements might be enhanced for better understanding and visual convenience. The following paragraphs are based on the European Patent Application20179814.7. The limited-entry liner comprises a number of unevenly spaced holeswith the purpose to distribute fluid, in this case acid, evenly along the reservoir section to be stimulated. The concept was initially described in 1963 by Shell for fracturing applications (Lagrone and Rasmussen, 1963) and is still widely applied. It was later adapted for matrix-acid stimulation and patented by Maersk Oil (known as controlled acid jetting or CAJ) and implemented in North Sea chalk reservoirs on a large scale, see Hansen (2001) and Hansen and Nederveen (2002). Since then, this novel stimulation concept has been tested by various operators such as ConocoPhilips (Furui et al., 2010a,b), Petrobras (Fernandes et al., 2006), ExxonMobil (Sau et al., 2014; Troshko et al., 2015), ZADCO (Issa et al., 2014) among others (Mitchell et al., 2014; van Domelen et al., 2011, 2012). Rodrigues et al. (2007) provided a good general overview of stimulation techniques for low-permeability reservoirs and Shokry (2010) described the acid stimulation practice in ADNOC for offshore reservoirs. Figure 1 shows a schematic cross-sectional view of a well-bore 12. The well-bore 12 is conventionally formed by techniques commonly known in the art, and includes a wall 14 created by the drilling process, a leading end 16, which extends into the formation 18, and a trailing end 20 for accessing the well-bore. A limited-entry liner 20 is introduced into the well-bore 12. The liner 20 has an open end 22 and opposed sealed end 24. An annulus 22 is formed between the wall 14 and outer surface 26 of the liner. The liner 20 is provided with a number of pre-formed holes 28 that form flow passages between the interior of the liner 20 and the annular space 22. The holes 28 have a shape and location that comply with particular, pre-defined specifications. Typically, the distances between adjacent holes 28 along the liner 20 decrease towards the end 24 of the liner. The acid is pumped into the liner in the liner 20 and exits holes 28 at high velocities resulting in jetting into the formation 18. By limiting the number and size of holes, a choke effect is obtained and a significant pressure drop occurs between the inside and the outside of the liner during stimulation. A non-uniform geometric distribution of the holes is used to compensate for the friction pressure drop along the liner section. This means that the average hole spacing decreases towards the bottom of the liner. The open annulus 22 outside the liner in combination with the overpressure on the inside of the liner (due to the choking over the holes) ensures that the acid eventually reaches the bottom of liner, and the well is thus stimulated along its full length. Acid is bull-headed from the surface and enters the liner 20 in the direction of arrows 30. The liner does not have to be horizontal but very often is. When acid reaches the first hole 28, which has a size of 4-7 mm, the pressure drop across the hole is so high that only a small portion of the acid exits the liner through thehole; the remaining portion continues along the liner until it reaches the next holewhere the same process is repeated. An appropriate hole-size design makes it possible to honour a specified acid coverage, defined as barrels of acid per feet of reservoir section. Prior to the stimulation, the mud can be circulated out so that only completion brine with the right density is found in the wellbore 12. The acid stimulation process is modelled by discretizing the wellbore 12 into a number of nodes 34, typically 100-400. The nodes do not need to have the same size. From a practical design point of view, the wellbore is split into a smaller number of segments 36. These segments may be physically isolated from each other on the annulus side by hydraulic packers 32 (not shown) but do not have to. Nodes can overlap between two segments, as shown in Figure 2. Displacement of brine by acid is considered to occur by single-phase plug flow with minimal dispersion. The negative excess mixing volume is not taken into account. The liner 20 is closed at sealed end 24 before stimulation and it is not cemented, which means that fluid can in principle flow in the annulus 22 along the well-bore trajectory before packers 32 are set. In practice, annulus flow occurs predominantly due to jetting of acid through the holes 28, perpendicular to the wellbore. Annulus flow along the liner can be ignored for practical modelling purposes. The completion design, and the associated modelling workflow covered in this document, allows for reservoir segmentation using packers and the resulting liner is hence referred to as a segmented limited-entry liner. The desired acid coverage can be specified per segment to take into account differences in porosity, permeability, initial water saturation, and reservoir pressure. The number of segments for modelling the process can be larger than the number of packer- isolated intervals. Design of the hole-size distribution depends primarily on liner geometry and flow rate, which in turn is governed by reservoir properties, i.e. reservoir permeability. Acid stimulation is inherently transient in nature because the skin factor at any given position along the well changes with time from a positive value initially (caused by a mud filter cake) towards a negative value once the acid has reacted with the reservoir rock minerals. If the skin evolution over time is uniform along the well, it will not affect the flow distribution, which means that the overall process can be modelled based on steady-state principles. The invention consists of a comprehensive algorithm for designing the hole-size distribution for limited-entry liners. The next sections describe the algorithm for designing a hole-size distribution which achieves a specified (often uniform) distribution of acid volume per interval length, also known as acid coverage. The algorithm is shown schematically in Figures 3 and 4. Figure 3 shows the overall algorithm whereas Figure 4 shows a more detailed part of Figure 3. The algorithm is discussed by reference now to Figure 3 and the first block, input data and constraints 1000. Input data and constraints 1000: As a starting point for implementation of the algorithm, input data constraints are entered into the system. The input data is made up of rock properties, completion data, fluid properties and other data, such as pump rate, number of nodes for the numerical algorithm, pressure drop across the last hole of the liner and annulus pressure. These inputs are either known, or may be sourced from historical data from the wellbore. The algorithm defines certain constraints which must be adhered to in the functioning of the system. These constraints form part of the input data and constraints 1000. The constraints includes, but are not limited to; annulus pressure must exceed minimum reservoir pressure to avoid cross-flow inside wellbore; annulus pressure must not exceed fracturing pressure to avoid fracturing; wellhead pressure must not exceed maximum design pressure rating– in turn this impacts the design rate and / or the amount of friction reducer to beadded; cross-sectional area of all LEL holes combined should be equal to or exceed a minimum cross-sectional area to avoid creating an additional pressuredrop during normal production or injection of the well after stimulation - thisimpacts the number and size of the holes; average distance between twoneighbouring LEL holes should equal twice the wormhole radius - this impactsthe pressure drop across the last LEL hole which is a design variable; and, the liner ID cannot exceed the wellbore size. Moving on to the next step, as shown in Figure 3 block 1002. Initial variable calculations 1002: Based on the input per segment, the maximum rate per segment is found by applying the transient inflow equation. Note that although the well is horizontal, it acts as a vertical well in the early injection phase because the boundaries have not been felt. Hence, the reservoir section length, L, replaces the reservoir thickness, H. Equation 1 B is the acid formation volume factor, which is in the range 1.0 to 1.1. In practice, it is assumed to be 1. The viscosity is the maximum value of the oil or gas viscosity and the acid viscosity. In heavy oil reservoirs, the transient phase injectivity is initially controlled by the oil properties. Thus,^^^^ = max (^^ , ^^^^^) Equation 2The permeability will see a contribution from the two horizontal directions as well as the vertical direction:^^ =^^^^,^^^,^^^,^Equation 3 The vertical / horizontal permeability ratio may attain values in the range 0.01 to 1.0. For the current application, the value is close to 1, which makes the overall permeability equal to the horizontal permeability. The diffusivity is given as Equation 4 Where the total system compressibility is given as a contribution from the rock and the fluid phases present in the pore space. Equation 5rw refers to the wellbore radius. In gas reservoirs, co equals gas compressibility. The maximum pump rate allowed is then the sum of the individual segment rates:^ = ∑^ ^^^Equation 6 However, any segments which must be left unstimulated and therefore require joints without holes, do not contribute to the calculation of the total rate. To start the design algorithm detailed later, the actual design rate is taken as a value 10- 30% lower than the maximum allowed rate. This value may be adjusted in a subsequent iteration. T is the total pump time calculated from the acid coverage and length of all the segments Equation 7 It is noted that T depends on Q, which depends on T. Research into matrix-acid stimulation fundamentals took off in the 1980’s with the pioneering work of Fogler and co-workers from 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) who demonstrated that the acid reaction with the rock gives rise to different etching patterns depending on the type and concentration of acid as well as the velocity and the temperature. Key subsequent contributions in the literature to the current understanding includes work by Halliburton (Gdanski and Norman, 1986; Gdanski and van Domelen, 1999; Gdanski, 1999), Buijse and Glasbergen (2005), and Hill and coworkers from 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). Further references to experimental and theoretical studies on wormhole growth are listed within these references. Figure 5 shows the effect of rate on dissolution through a series of images 100. A low rate leads to uniform dissolution and hence a very inefficient usage of the acid. This is shown by the image to the far left 102. In the image the acid 104 has not permeated the formation 106 to any appreciable extent. At slightly higher rates (i.e. moving from left to right in the images), the acid creates wormholes 108 through the rock. In fact, any acid formulation has an optimum velocity at which the least volume of acid is required to etch a pattern from inlet to outlet. This volume is called the pore volume to breakthrough 202. Note that 15% HCl corresponds to 4.4M, hence the 0.5M concentration used in the experiment is quite low. Figure 6 illustrates the impact of interstitial velocity 200 on pore volume to breakthrough 202 at two different temperatures 204A (depicted by the dot-dash line) and 204B (depicted by the solid line). A temperature increase (i.e. from temperature 204A at 25oC to temperature 204B at 600C) leads to higher reaction rate and hence faster dissolution; optimum wormhole growth therefore requires a higher acid velocity to avoid spending all the acid near the wellbore. It is also clear that it is better to pump at a rate which is slightly above the optimal than below. In a low-permeability reservoir, the maximum pump rate is limited by the fracturing pressure, which may prevent the operator from reaching the optimum velocity. In such situations, it is necessary to select a different acid 104 formulation to shift the curve to the left and preferably also down. The wormhole data can be reproduced with a model proposed by Buijse andGlasbergen (2005) containing two fitting constants, ^and ^, which can bereformulated in terms of the lowest point on the curve (optimum interstitial velocity 200, optimum pore volume to breakthrough 202): ^ ^ ^ ^^ = ^^^,^ ^^^ ×^^^^^^ (^^^^^Equation 8 ^^^,^)^ Increased temperature 204 and increased HCl concentration both increase the optimum velocity 200 for wormholing. For low-permeability rocks where the optimum rate may be limited by the fracture propagation pressure, it may be beneficial to reduce the acid concentration, although the pore volume to breakthrough 202 increases and hence the volume of acid solution needed. If the acid concentration is halved then the volume must double to maintain the same number of moles. Several authors have investigated the effect of weaker acids, see Punnapala et al. (2014) and Shirley et al. (2014). A friction reducer may shift the PV curve upwards, which means that more acid is required to achieve the same skin. Talbot and Gdanski (2008) proposed a general wormhole model where they correlate the two input parameters to the Buijse-Glasbergen model as a function of rock and acid properties as well as temperature. However, they do not specify the values of the constants in their correlation. In this invention, we make use of a concept whereby we shift the default wormhole curve shown in Figure 6 up, down, left, or right as a function of the temperature 204, the permeability, and the acid type. Table 1 shows some rough rules-of-thumb when adjusting the optimum (lowest) point on the wormhole curve. Based on the default curve, the optimum point is shifted with the amount indicated. The optimum point cannot be lower than (0.1, 0.1). Values in the table are only indicative and serve to illustrate a concept. Table 1: Optimum Wormhole growth parameters Acid reactivity increases with temperature 204, which means that the optimum velocity 200 for wormhole growth also increases. For low-permeability reservoirs, it can be difficult to reach the optimum velocity without fracturing the formation. Therefore, it is important to evaluate the downhole temperature of the acid 104 when it reaches the formation 106. As shown in Figure 7, which illustrates the relation between the temperature of the acid at the entrance of the liner 300 under different pump rates 302 and wellhead temperatures 304. It is an advantage to inject at high rate and at the lowest possible wellhead temperature to limit the in-situ acid reactivity. This is shown by line 304A. As the temperature increases, lines 304B, and 304C we can see an increase in the acid reactivity at the entrance of the liner 300. Furthermore, any brine used to clean out the mud prior to the acid stimulation should be injected at the lowest possible temperature. The temperature to be used for adjusting the wormhole curve is the temperature of the acid when it enters the reservoir, not the reservoir temperature. Economides et al. (1994) derived a formula to determine the volume of acid required to achieve a certain wormhole length 400 based on the pore volume to breakthrough 202 from core flood data: Equation 9 The formula is plotted in Figure 8. The ratio V / L is known as the acid coverage in bbl / ft 402. The equivalent skin 404 is given as: The algorithm aims to achieve a given final skin factor and then calculates the equivalent wormhole radius and then the required acid coverage. However, for economic considerations, the maximum acid coverage is limited by the acid volume which can be pumped. For instance, in offshore wells, the volume islimited by acid boat capacity. In the current application, the acid coverage shouldnot exceed 1.5 bbl / ft. Alternatively, the acid stimulation can be fixed, which enables calculation of the maximum, final wormhole length 400 and consequently the final, negative skin 404. Figure 9 illustrates the outcome of a larger sensitivity analysis involving the pump rate 500, the dP across the last hole 502 (502A to 502E, respectively), the discharge coefficient 504 (504A to 504E, respectively) and the total hole cross-sectional hole area 506. The linear relationship is very clear, and it is thereforepossible to predict the dP 502 required to obtain a certain cross-sectional hole area 506. This constraint that the cross-sectional LEL hole area 506 must be equal to or larger than a minimum cross-sectional area to avoid imposing anaddition pressure drop during production / injection after stimulation thereforeresults in a constraint on the dP across the last hole, which can be estimated based on the relationship provided by the sensitivity analysis. This is a novel concept, where:^ = ^^ + ^ Equation 10^ = ^^^ + ^ Equation 11^ = ^^^ + ^ Equation 12At this stage, we are able to estimate the initial hole-size distribution for the starting point of the algorithm. This is depicted by block 1004 in Figure 3. The next step, block 1006, requires that the equations are set up. These are then solved as part of the following step, block 1008, dealt with later in this specification. Set up equations 1004: The equation of motion for isothermal one-dimensional pipe flow describes the pressure drop as a contribution from friction, gravity, and acceleration. The gravity term dominates in the vertical section of the wellbore, whereas friction losses become relatively more important in the horizontal section. The acceleration term is only required when velocity changes occur, such as when fluid enters the liner from the tubing (change in inner diameter), or whenever fluid exits through a hole in the liner. The contribution of the acceleration term to the total pressure drop is less than 5% and can often be neglected. ^^= −^^^^^^^^ ^ −^^^^ ^^ −^^^^^^ ^^ ^^ Equation 13 Equation 14^ is the angle relative to the z-axis and D is the pipe diameter. The accelerationterm can be expressed in terms of volumetric flow Q instead of velocity v, Equation In field units, this becomes Equation 16 f, is defined in terms of the wall shear stress Equation 17 Hence the friction pressure drop for Newtonian flow is: Friction pressure loss, in psi / ft, for pipe flow of Newtonian fluids becomes: For laminar flow the Fanning friction factor is linked to the Reynolds number: Equation 20 The Reynolds number is given as^^ = 15916^^ ^^ Equation 21 Friction pressure loss for annulus flow of Newtonian fluids in concentric pipes: Equation 22 For laminar flow in the annulus, the Fanning friction factor is defined as:^ =^^ ^^ Equation 23 The Reynolds number is given as Equation 24 The pressure difference due to the static head is found from: Equation 25Where h is the depth in ft TVD. ^^= ^^^^ Equation 26Thus, Equation 27Annular flow can occur if acid is first injected through a coiled tubing inside the production string followed by the main acid treatment injected into the production tubing while the coil is still in hole. The Fanning friction factor for pipe flow in smooth pipes is described by the Prandtl-Karman equation: ^ ^.^^ ^ ^= 4^^^^^^^^^^^ − 0.4 = −4^^^^^^^^^ Equation 28 For rough pipes, the friction factor depends on the relative pipe roughness, c / D, and is given as Equation 29 Figure 10 illustrates the effect of Reynold’s number 600 on friction factor 602 for different values of pipe roughness 604 (604A to 604H, respectively). A typical relative roughness for a new pipe is 10-5. Similarly, for annular flow in smooth pipes, the friction factor becomes Equation 30 Finally, for annular flow in roughness pipes, the friction factor is given as Equation 31 There is a potential discontinuity going from laminar to turbulent flow because the flow regime is poorly defined in the 1000-2000 Reynolds number region. This has no impact on the LEL hole design. Figure 10 shows that roughness plays a role only if it exceeds 0.0001. Typical pumping rates are 5-40 bbl / min, depending on reservoir permeability and liner length. Such rates may lead to high surface pressures and thus require the upper completion to be designed appropriately. There is often a need to reduce the friction pressure loss to stay within safe operating limits and this requirement may necessitate the use of drag reducing agents (DRA). Drag reducers are mostly dilute polymer solutions, which lower the frictional resistance to flow in the turbulent regime when added to a solvent, for instance water or acid. Very low concentrations (a few thousand ppm) may in some instances reduce friction by as much as 70%. Friction reducers, may, however, cause reservoir damage, according to some studies. When adding drag reducing agents a zone named the elastic sub-layer is formed between the viscous sub-layer and the Newtonian core. The extent of the elastic sub-layer will be governed by the amount and type of polymer, and by the flow rate. Within the elastic sub-layer the turbulence structure is significantly different from the Newtonian plug. The turbulent eddies are broken up, whereby the flow characteristics approach those of laminar flow. Therefore, it follows that the friction factor is independent of the relative roughness of the pipe in the presence of drag reducing agents. Maximum drag reduction is achieved when the elastic sub-layer extends to occupy the entire pipe cross-section. Drag reduction by dilute polymer solutions in turbulent pipe flow is bounded between the two universal asymptotes described by the Prandtl-Karman law for Newtonian turbulent flow and a maximum drag reduction asymptote. In between is the so-called polymeric regime in which the friction factor relations are approximately linear in Prandtl- Karman coordinates, see Figure 11. The polymeric regime may be described by two parameters: The wall shear stress at the onset of the drag reduction, ^w*, (or equivalent the onset wave number w*) and the slope increment,^, by which the polymer solution slope exceeds Newtonian slope. The onset of drag reduction occurs at a well defined onset wave number. For a given polymer solution w* is essentially the same for different pipe diameters. For solutions of a given polymer-solvent combination w* is approximately independent of polymer concentration. When modelling the effect of the drag reducer, it is assumed that the fluid friction factor is reduced and that the fluid viscosity remains the same. Acid viscosity, via the Reynolds number, has a minor impact on friction losses at typical operating conditions, as seen from Figure 10. The following formula, developed by Virk (1971, 1975) relates the friction factor to the concentration of the drag reducer for pipe flow: 0.4 − ^^^^^^^√2^^∗^ Equation 32 The drag reduction model parameters are^ = ^^^^^^Equation 33K and α are constants. The parameters are specific to the chemical used and mustbe fitted based on flow loop test data provided by the vendor. The maximum drag reduction asymptote for pipe flow is described by:Equation 34Figure 11 shows the impact of drag reduction on the friction factor, in a Prandtl- Karman plot. For the particular Drag Reducing Agent (DRA) 606 model constants used, the maximum asymptote 608 is only reached if the DRA concentration exceeds 2000 ppm. Without the addition of a DRA is shown by 610. Incrementally increasing the amount of DRA is shown by lines 612, 614 and 616 respectively. In a 6” ID liner, a pumping rate of 25 bbl / min, equivalent of 36000 bbl / d, leads to a Reynolds number of approximately 321635, which is well inside the turbulent flow regime. Figure 12 illustrates the influence of drag reduction on friction pressure 620 in a 10000 ft long 4.5” OD top completion string as a function of pump rate 500. Friction is reduced to 1 / 3 by adding 1000 ppm DRA. The concentration of DRA is similar to that as illustrated in Figure 11. The limited-entry liner consists of a number of holes allowing fluid to exit the liner and enter the annulus and subsequently the reservoir. The holes are small compared to the liner dimensions, both in terms of length and diameter and can therefore be considered as an orifice. The pressure drop across N holes in the liner may be calculated as: Equation 35 Qhole is the flow rate in bbl / min through the holes. The positive direction is from the liner and into the annulus. Dholeis the inner diameter, in inches, of the holes in the liner. N is the total number of holes. CD is the dimensionless discharge coefficient, which accounts for the fact that the pressure loss is only partially recovered due to the short length of the hole (equal to the pipe thickness). Based on the work by Crump and Conway (1988), a lower value of 0.56 is used for flow of water and gelled fluids in round sharp-edged drilled holes; values up to 0.80 are also possible, depending on fluid type and how the hole was actually drilled, see El-Rabba et al. (1997) and McLemore et al. (2013). CD should be considered a sensitivity variable during the first LEL design jobs. Drilling the holes at a slight angle may reduce the splash-back of unspent acid hitting the formation and improve the jetting process. The model for the friction factor in the presence of a drag reducer is combined with the model for the friction factor for Newtonian turbulent pipe flow in rough pipes. Equation 36 If no drag reducers are used then ^ = 0. If drag reducers are used the roughnessis set to zero. Inserting the expression for Reynolds number: Equation 37 The hole distribution function,^^∗^^^^, is defined as number of holes per foot along the liner. For the present application, it is a step function reflecting different geometrical distribution of holes in sections along the liner. It can also be specified on the basis of the actual liner tally in which case the length of the individual sections equals the length of the joints. A mass balance across a unit liner section states that the rate lost across a liner segment equals the rate through the holes along the same segment: ^∗ = −^^ ^^^^^^^^^^^ ^^^^^^^ Equation 38 Rearrangement of terms gives Equation For steady state applications the annulus pressure is a user-provided input, which should not exceed the fracture propagation pressure. The model for the liner without an inner string includes two differential equations to describe the pressure and flow rate profile within the liner. In addition, a set of algebraic equations defines the Fanning friction factor for Newtonian turbulent flow with and without drag reducer. The inner diameter, the hole diameter and the well inclination may change along the liner. The flow rate change inside the liner, which is equal to the flow through the holes, is described by: Equation It is clear that the liner pressure must exceed the annulus pressure at all times to avoid unwanted cross-flow from a segment with higher reservoir pressure. Zones with substantially higher or lower pressures should be isolated with packers to improve acid coverage control. Transformation into the dimensionless independent variable u=x / L, where L is the total length of the liner and introducing position-dependent constants gives: ^^ = 0.052^^ ^^^^(^) Equation 46^^ ^^∗=^^^^(^)^^^^^^^(^)^^^ ^^.^^^^^ Equation 47 ^^ ^^= ^^^^^^ Equation 48 ^ ^^= ^.^^ Equation 49 Equation 50 ^.^^^^^ ^^=^^(^)Equation 51 Boundary conditions:^^^^^^(^ = 0) = ^^ Equation 52^^^^^^(^ = 1) = 0 Equation 53The stimulation rates in bbl / min / ft are found from:^^^^^^^^^^^^ = −^^ Equation 54 The flow rate through the holes in bbl / min / hole is found from: Equation 55 The algorithm enters into the inner loop 1100. This lead by block 1008, Solve equations. Solve equations 1008: The model describes a first order non-linear boundary value problem and consists of two first-order coupled non-linear differential equations with boundary conditions at each end of the interval of the independent variable, coupled with a set of algebraic equations. The problem can be solved in at least two ways, either using a finite-difference scheme, see Mogensen and Hansen (2007), or with polynomial approximation. Following the approach outlined by Hansen and Nederveen (2002), polynomial approximation is used to solve the numerical model. The liner is spatially discretised with N nodes, as will be described in the subsequent paragraphs. The solution vector, Y, is defined as an array containing the pressure, P, in each of the N nodes, the flow rate, Q, in each of the N nodesand the friction factor, f, in each of the N nodes. Equation 56The iteration scheme to be outlined in the subsequent paragraphs is somewhat sensitive to the guess of the initial solution vector. A safe starting guess is to assume that the pressure in the liner decreases linearly with position, while remaining larger than the specified annulus pressure. The same approach is taken for the rate inside each liner segment, which must equal the total flowrate at theinlet and zero at the end. The initial estimate of the friction factor should be 0.01or lower. Note that since the flow rate is zero at the end of the liner, the flow regime will go from being turbulent around the heel towards laminar at the toe. The N collocation points are found as zeros of N’th order orthogonal polynomials 700, against the length of the liner 702; in total there are N-2 interior points and 2 interval endpoints. Location of the zeros is shown in Figure 13 using 100 collocation points. Use of polynomial approximation of order N gives:^^(^) = (^ − ^^)(^ − ^^) … (^ − ^^) Equation 57Construction of Lagrange polynomials are defined by: Equation 58 The nodes of the polynomial are found from the orthogonality condition: Equation 59Expansion of the four dependent variables into an N’th degree Legendreinterpolation polynomial ^ ^ ^ 0 ) gives: Based on experience, the number of collocation points is set to 100 and is independent of the number of physical holes in the liner. Rewriting the equations gives the discretised model: Equation 65 5 The maximum drag reduction asymptote is not included in the model formulation. In practice, use of drag reduction concentrations in the order of 1000-2000 ppm does not exceed the asymptotic line. Equation EquationEquation 15 Equation Equation Equation Equation Boundary conditions: 25^^ = ^^^^^^,^ − ^^ = 0 Equation 75^^ = ^^^^^^,^ = 0 Equation 76Q0 is the specified pump rate. The stimulation rates in bbl / min / ft are found from: 77 78 The next step as part of the inner loop 1100, is to update the solution vectors, as shown by block 1010. Update solution vectors 1010: The 3N set of non-linear algebraic equations is solved using a Newton-Raphson iteration procedure. Given an initial estimate of the solution vector an improved estimate is obtained from: Equation 79 The Jacobian matrix is defined with the following elements:^ =Equation 80 The elements of the Jacobian matrix are set up as described below. The collocation matrix is defined as: ^^^^^ ^^^^^^^,^ = ^^^ = ^^^^Equation 86Equation 87 Function H (j = 1..N-1, i = j): ^^^^^^^^^^,^ = 0 Equation 88 ^^^ = ^1 ^ = 1^^^^^^^,^ 0 ^ = 2.. ^Equation 93 Equation 94 Equation 95 Equation 96The final step of the inner loop 1100 is to determine whether the solution vector is constant, block 1012. Is solution vector constant 1012: Typically, the Newton-Raphson technique will converge within 5 iterations using carefully selected relaxation parameters to guide the convergence during the first iterations. The method ensures that final convergence speed is quadratic. The iterative inner loop will repeat by following arrow 1014, and restating resolving the equations, as set forth from block 1008. This iterative inner loop 1100 finishes when the absolute change to the solution vector is below a certain threshold value, typically 1E-12. To avoid the possibility of an infinite loop, the procedure stops after a pre-specified number of iterations has been reached, typically in the range 10-20. Once the solution vector is deemed constant, the next step is to follow arrow 1016 to calculate the acid coverage, depicted by block 1018. Calculate acid coverage 1018: Once the stimulation flow rates are calculated from the solution, the acid coverage per liner segment is the product of segment flow rate and pumping time. If the overall pump rate changes during the job, the stimulation rate for each segment changes.^^^^^,^ = ^^^^^,^ × ^ Equation 97 The transient period where the acid front moves through along the liner while displacing the brine must also be taken into account. However, this is compensated for when water displaces acid at the end of the job. The time it takesfor the front to reach a given position i, is called the retention time, which iscalculated recursively: Equation 98 Since the liner flow rate gradually decreases towards zero at the heel, it is clear that it takes gradually longer time for the acid front to displace the brine out of the liner. In other words, the inner part sees acid for longer time than the outer part. The hole-size distribution should compensate for this. The retention time is therefore also a measure of the minimum time needed for water to displace acid from the liner at the end of the stimulation. The next step is shown in block 1020, to determine if the dp across the last hole is matched. This step goes in combination with the following block 1022 which is to determine if design acid coverage matched: Is the dp across the last hole matched 1020? The dP across the last hole is calculated as the difference between the pressure in the last node of the liner and the annulus stimulation pressure (which is constant and user-specified):^^^^^^ ^^^^ = ^^^^^^,^ − ^^^^^ Equation 99Is design acid coverage matched 1022? The difference between calculated and specified target acid coverage is given as Equation 100 This formulation ensures that the dCOV (acid coverage distribution) function is always positive. Hence, it must be minimized to obtain the best possible match. The relative acid coverage is determined as follows: Equation 101 Turning to Figure 4, the above two steps are combined into block 1050. Whereas the inner loop 1100 consists of solving the material balance for a given combination of LEL holes, pump rate and other variables, the first part of the outer loop 1200 consists of adjusting the LEL hole size distribution to match both the desired pressure drop across the last hole, dP 1052, and the desired acid coverage for each segment 1054. The outer loop 1200 serves to honour both constraints at the same time. Therefore, the hole size-distribution must be satisfied, as shown in block 1024. Update hole-size distribution 1024: If the dP is too small 1056, then there are too many LEL holes and one LEL hole is then subtracted from the segment with the highest relative acid coverage 1058 and the material balance inner loop 1100, via block 1006, is then reinvoked. If the dP is too large (arrow 1060), there are too few holes, and one hole is then added to the segment with the lowest, non-zero relative acid coverage 1062 and the material balance inner loop 1100, via block 1006, is reinvoked. Segments with zero acid coverage are not adjusted. If the dP is close to the target value within a certain tolerance, then the acid coverage distribution dCOV is calculated 1054. At this point, the total number of LEL holes is correct but the holes just need to be redistributed among segments. One LEL hole is added to the segment with the lowest, non-zero relative acid coverage, whereas one LEL hole is subtracted from the segment with the highest relative acid coverage 1064. Then the inner loop 1100, via block 1006, is reinvoked and the procedure is repeated until the dCOV function reaches a minimum. Since the algorithm adjusts integer values, i.e. number of LEL holes, it is not possible for the dCOV function to be exactly zero. Once the minimum dCOV function is reached, it must be determined if the calculated wellhead pressure (WHP) is below the wellhead pressure maximum constraint, as shown in block 1026. Is calculated WHP below max. constraint? Every wellhead has a maximum pressure rating, such as 5000 psia, 6500 psia and higher. Similarly, every tubing has a maximum pressure rating. Therefore, if the reservoir pressure is high, the design rate may give rise to a wellhead pressure, which exceeds the pressure rating. If the calculated wellhead pressure exceeds the maximum rating of the tubing (shown by arrow 1028), adjustment to the design (block 1030) requires the following steps: Step 1: If the previous design was based on zero friction reduction, then add 2000 ppm friction reducer. Re-run the simulation.Step 2: If friction reducer is already present, investigate the possibility toincrease the tubing ID. Re-run the simulation.Step 3: If step 2 is not possible, reduce the rate, re-run the simulation, andloop until the calculated WHP is below the maximum pressure rating of the tubing. Next, the average hole distance constraint must be met, block 1032. Is average hole constraint met? As described earlier, Economides et al. (1994) derived a formula to determine the volume of acid required to achieve a certain wormhole length based on the pore volume to breakthrough from core flood data: Equation 102 The ratio V / L is known as the acid coverage in bbl / ft. The equivalent skin is given as Equation 103 Schwalbert et al. (2018) defined the stimulation coverage as twice the wormhole radius relative to the length of the perforated interval, which for LEL completions equals the distance between LEL holes. Equation 104 Turing to Figure 14 which illustrates skin factor as a function of stimulation coverage. Therefore the skin factor 800 becomes constant when the stimulation coverage 802 reaches 50%. Figure 15 shows that an effective wormhole radius 804 of 20 ft would result in an equivalent negative skin factor 800 of -4, assuming that all the wormholes generated along the well have the same radius. Combining the two plots shows that the maximum distance between wormholes should not exceed twice the wormhole length. A skin of -3 for the entire well, for instance, means that the holes should be drilled with a maximum distance of 30 ft. This means that the average distance between LEL holes, defined as the length of the stimulate reservoir section divided by total number of holes, should not exceed twice the expected final wormhole radius. The following check is therefore performed: ∆^ ^^^^^= ^^^^^^ ≤ 2^^^ Equation 105If the average hole constraint is not met it becomes necessary to adjust the hole size, see block 1034: Adjust hole size 1034: Based on the evaluation of the above equation, the following possible actions are taken: If the distance between LEL holes is too small, the LEL hole size can be increased by 1 mm and the entire simulation is then repeated. If the distance between LEL holes is too large, the LEL hole size can be decreased by 1 mm and the entire simulation is then repeated. If the average distance between LEL holes is close to or equal to twice the wormhole radius, the algorithm has converged with a final design and proceeds with output of results, block 1036. Output results 1036: Output results consist of the following items: Node properties, including position, pressure, rate, friction factor, number of holes per foot, velocity, retention time, stimulation rate, cumulative volume of acid leaving the node through holes. Segment properties, including segment number, segment interval, number of holes in segment, distance between holes, calculated and design acid coverage, acid coverage ratio, acid stimulation rate, acid velocity at the exit point of the holes, pore volume to breakthrough, final wormhole radius, and final skin factor. Actual versus specified pressure drop across last holes. Average overall distance between LEL holes. Total number of LEL holes, total cross-section area of LEL holes, equivalent ID of total number of LEL holes. Wellhead pressure and bottom-hole pressure during pumping. Wellhead pressure and bottom-hole pressure immediately after shut-in, known as instantaneous shut-in pressure (ISIP). Acid volume required, total pumping time, assuming pumping occurs at design rate. Total liner volume, total tubing volume, displacement volume, retention time. A detailed tally list containing the number and size of LEL holes for each joint to be run in hole, as well as the order in which the joints must be run in hole. Furthermore, the total number of joints with a particular number and size of holes is summarised, such as number of joints with 0, 1, 2 or 3 LEL holes of size 4 mm, 5 mm, or 6 mm etc. Wellhead pressure and bottom-hole pressure during pumping are calculated from the pressure at the first node and then subtracting hydrostatic pressure and adding friction up to the given gauge depth. Wellhead and bottom-hole instantaneous shut-in pressures ISIP are calculated from the pressure at the first node and then subtracting hydrostatic pressure up to the given gauge depth. The friction is zero because the rate is zero during an ISIP. Calculation procedure: Input to the numerical design model includes:- Average reservoir pressure- Fracture propagation pressure- Permeability- Porosity- Length of the completed interval- Wellbore radius- Tubing ID, liner ID, pipe roughness- Acid properties (type, concentration, density, viscosity)- Number of segments- Acid coverage per segment- Hole size per segment- Discharge Coefficient Step 1. Estimate the Pump Rate The software will then estimate the design pump rate based on the standard transient inflow model (not the Darcy model, which is a steady-state assumption) while ensuring that the injection pressure remains below fracturing pressure. Key parameters include the permeability, the length of the completed interval, and the difference between annulus pressure and reservoir pressure. For the calculation, it is assumed that the skin can be reduced to zero. Thus, note that because the stimulation job typically takes less than 24 hours, the injectivity is higher than predicted by the Darcy formulation. The reason is that the boundaries are not yet felt by the pressure pulse emitted during stimulation. So, even though the flow inside the liner is a steady-state formulation, the inflow model used for design of the pump rate is transient. Step 2. Estimate Wormholing Characteristics A nodal analysis calculation must be performed to estimate the downhole temperature at the heel of the liner. Based on the choice of acid system, the permeability, and the temperature, the optimum velocity for wormhole propagation is estimated, together with the anticipated pore volume to breakthrough based on published literature data. The Buijse-Glasbergen model is used to characterise the wormholing at different velocities. Step 3. Estimate Total Number of Holes and the dP Across Last Hole Based on the optimum velocity and the calculated design pump rate, it is straightforward to calculate the total cross-sectional area of the holes. This area is linearly correlated with the dP across the last hole, which is a key design parameter. Step 4. Estimate Acid Coverage The stimulation design aims for a negative skin of -3 or better, which requires the holes to be not more than 30-40 ft apart. The model by Economides et al. (1994) is used to calculate the acid coverage required to achieve this skin. A higher acidcoverage requires more acid and longer pumping time and hence higher cost. Thismust be weighed against aiming for a more negative skin. Step 5. Calculate the Optimized Hole Distribution Provide an initial estimate of the number of holes per segment and let the software find the solution which honours the acid coverage per segment and the dP across the last hole. The initial estimate can be found from the relationship between interstitial velocity, pump rate, and total cross-sectional hole area for a particular discharge coefficient and liner configuration. Example: To illustrate the design concept in more detail, an example is shown below. The well in question will have an approximate reservoir length of some 7000 ft. Since a stand (3 drill pipe lengths) is approximately 91 ft, the well is numerically split into 8 segments, each with a length of 910 ft, corresponding to 10 stands. The initial design coverage is set to 1 bbl / ft. The transient inflow equation predicts that the maximum rate without fracturing the formation is 20 bpm, assuming that the skin is zero. As the stimulation progresses, the rate can be increased further. The resulting pumping time will be 6 hours, which leads to a slight adjustment of the design rate, but not much. Although the reservoir temperature is 250 F or more, nodal analysis based on the design rate of 20 bpm predicts a BHT of 140 F at the first hole. This temperature is used for estimating the position of the optimum velocity for wormhole propagation based on a measured curve and the Buijse-Glasbergen model. The final skin is initially assumed to be -3, which yields a maximum distance between adjacent holes of 30 ft. This corresponds to a pressure drop across the last hole of about 30 psia, which is then used as input for the design model. The discharge coefficient is assumed to be 0.70, which is mid-way between the theoretical minimum of 0.56 and a high value of 0.80. Post-job analysis will help identify the pressure drop across the holes and hence the actual discharge coefficient. A first estimate of the hole size distribution makes use of a linear relationship between hole cross-section area and pressure drop across the last hole. Based on this initial input, the actual optimum hole 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 holes is adjusted to match the pressure drop across the last hole as well as the acid coverage for each segment. The results from the calculations are show in the four plots above. The distance between adjacent holes is in the range 20-35 ft, which yields optimum stimulation coverage (wormholes cover the entire well length). The distance is not uniform because the hole size is chosen to be constant at 4 mm to avoid complicating the pilot design. Based on the wormhole growth model of PVbt versus interstitial rate, the minimum PVbt to be inserted into the skin model by Economides and based on the specified acid coverage of 1.0 bbl / ft. This yields a skin factor of -2.5, which is considered close enough to the initial estimate of -3. If a skin of -3 is desired, we would need to increase the acid coverage, recalculate the pumping time, recalculate the flow rate, redesign the hole sizes and then check the resulting skin. While the embodiment of the invention has been described above and discussed in detail, the invention is not deemed to be restricted to this particular embodiment. A person skilled in the art will appreciate that a number of variations may be made to the described embodiment or features thereof, without departing from the scope of the present invention. In particular, the invention is not deemed to be limited to use in LEL-liners, as has been described. Other systems involving material flow through conduits and / or material formations may benefit from the implementation of the current invention, and embodiment, as described above. References 1. Al-Ghamdi, A.H., Mahmoud, M.A., Wang, G., Hill, A.D., Nasr-El-Din, H.A.(2014). 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[0002] Figure 16 illustrates the liner according to the invention in a well bore ina side view. The liner 100 is positioned in a wellbore 12 in a rockformation 10 and comprises multiple sections, visible in the figure is afirst section 1001, a second section 1002 and a third section 1003.Packers 110 are used to separate the sections from one another. Nozzles are arranged all around the liner 100 extending in radial direction. Open nozzles 102 are arranged at the end of the liner, so that acid can flow through the open nozzles 102 into the rock formation 10, thus creatingworm holes 14. The nozzles in the first section 1001, the second section1002 and the third section 1003 remain plugged by plugs 104, as theplugs are not yet degraded. Such a design of the liner 100 enables precise control over the stimulation of the wellbore 12. By having multiple sections1001,1002,1003 separated by packers 110, fluid injection can be stagedand managed effectively. The placement of open nozzles 102 at the end of the liner 100 allows for initial acid flow to create wormholes 14 in the rock formation, enhancing permeability and facilitating the extraction process. The plugs 104 in the nozzles of the first, second, and thirdsections 1001, 1002, 1003 ensure that these sections remain isolateduntil the plugs degrade, allowing for sequential and controlledstimulation. This can improve the efficiency of well stimulation, reducethe risk of uncontrolled fluid migration, and optimize the overall extraction process by targeting specific areas within the wellbore 12 for treatment.Figure 17 shows another liner in a well bore in a side view. The liner 200is positioned in a wellbore 22 in a rock formation 20 and comprises multiple sections. Visible in the figure is a first section 2001, a secondsection 2002 and a third section 2003. Packers 210 are used to separatethe sections from one another. Nozzles are arranged all around the liner 200 extending in radial direction. In the second section 2002 and the third section 2003 nozzles are plugged by plugs 204. The plugs in the first section 2001 and the end of the liner are already degraded and / or destroyed, such that the nozzles are open. One nozzle in the first sectionand all nozzles at the end of the liner 100 are sealed nozzles 206. The sealed nozzles 206 are sealed by balls 208, after the acid created worm holes 24, so that stimulation in this area is inhibited.Such a design for the liner 200 allows for selective and controlledstimulation of the wellbore 22. The use of packers 210 to separatesections 2001, 2002, 2003 ensures that fluid injection can be precisely managed within each section. The degradation of plugs 204 in the firstsection 2001 and at the end of the liner 200 enables initial acid flow tocreate wormholes 24 in the rock formation 20, enhancing permeability.The subsequent sealing of these nozzles 206 with balls 208 preventsfurther fluid flow in these areas, allowing operators to focus stimulation efforts on other sections. This method optimizes the stimulation process by ensuring that only targeted areas are treated, improving the efficiency of resource extraction, and reducing the risk of over-stimulation or damage to the wellbore 22.Figure 18 depicts a liner according to the invention during stimulatingin a well bore in a side view. The liner 300 is positioned in a wellbore 32in a rock formation 30 and comprises multiple sections. Visible in thefigure is a first section 3001 and a second section 3002. Packers 310 areused to separate the sections from one another. Nozzles are arranged all around the liner extending in radial direction. Acid 350 flows in axial direction through the liner towards the closed end of the liner 3000.Open nozzles 302 are arranged at the end of the liner, so that acid canflow through the open nozzles into the rock formation 30, to stimulatethe wellbore. The nozzles in the first section 3001 are all open, thenozzles in the second section 3002 are partially plugged by nozzle plugs304. During the stimulation process, the liner 300 is strategically positioned in the wellbore 32, which penetrates the rock formation 30. The liner is divided into sections by packers 310, which help in isolating differenttreatment zones. As acid 350 is pumped through the liner, it flowstowards the closed end 3000. In the first section 3001, where all nozzles are open, the acid exits through these nozzles 302, effectively treating the surrounding rock formation 30. In the second section 3002, the presence of nozzle plugs 304 in some nozzles allows for controlled and selective acid release, ensuring targeted stimulation of specific areas. This configuration enhances the precision of the stimulation process, optimizes acid usage, and improves the overall efficiency of well production by ensuring thorough treatment of the rock formation while minimizing wastage. Figure 19 shows the liner of Figure 3 at a different stage of stimulatingin a side view. The liner 300 is still positioned in a wellbore 32 in a rockformation 30 and comprises multiple sections, visible in the figure is afirst section 3001, a second section 3002. Packers 310 are still used toseparate the sections from one another. Nozzles are arranged all around the liner extending in radial direction. Acid 350 flows in axial direction through the liner towards the closed end of the liner 3000. Now, some of the nozzle plugs 304 in the second section 3002 have been removed to create open nozzles 302, while some nozzles remain plugged by nozzles plugs 304. Balls 308 are sealing some of the nozzles in the first section 3001, so that acid 350 can no longer flow through the sealed nozzles 306. This way, the area to be stimulated can be specifically targeted by controlled removal of the plugs and the controlled introduction of sealing balls. Acid 350 flows towards the closed end 3000, and at this stage, selective removal of nozzle plugs 304 in the second section 3002 creates additional open nozzles 302, allowing the acid to exit and stimulate specific areas. Simultaneously, sealing balls 308 are introduced into the first section 3001 to block certain nozzles, forming sealed nozzles 306 and stopping the flow of acid through them. This selective plugging and sealing method allows for precise control over the acid distribution, ensuring that only the intended zones within the rock formation receive the treatment. This approach enhances the efficiency of the well stimulation process by focusing the acid on the most productive areas, improving resource extraction and overall well performance. The sealing can be achieved by subsequently sealing from the closed endtowards the open end. The removing of nozzle plugs can be done tocontrol the pressure inside of the acid filled liner. The removing of nozzle plugs is furthermore done to control the fluid transfer from the liner, to even the acid coverage in the wellbore. The removing of nozzle plugs can also be subsequently done from the closed end towards the open end. Removing of the nozzle plugs 304 can involve a degradation, dissolving, breaking, melting and / or exploding. The nozzle plugs 304 can also comprise heating elements and / or explosives for removing thenozzle plugs. The removing could as well be triggered by an acousticsignal and / or an electromagnetic signal, while the triggering could be achieved by an electromagnetic signal comprises Radio-frequencyidentification (RFID). The removing can be triggered when one or morepressure thresholds are exceeded inside the liner 300. The nozzle plugs 304 the nozzle plugs can comprise wax, glass, ceramic, aluminum, chromium, rubber, elastomer and / or polylactic acid.

[0003] Reference list: 1610, 1720: rock formation 1612, 1722: wellbore 1614, 1724: worm hole 1600, 1700: liner 1602, 1702: open nozzle 1604, 1704: plug 1610, 1710: packer 1706: sealed nozzle 1708: ball 16001, 2001: first section 16002, 2002: second section 16003, 2003: third section 1830: rock formation1832: wellbore 1800: liner 1802: open nozzle 1804: nozzle plug 1806: sealed nozzle 1808: sealing ball 1810: packer 1850: acid flow 18000: end of liner 18001: first section 18002: second section

Claims

1October 14, 2024Abu Dhabi National Oil Company A174900WO MAJ / EssClaims 1. A liner for engagement in a wellbore, the liner comprising:a closed end and an open end;5 a plurality of nozzles for transferring fluid radially from the liner,wherein the nozzles are plugged with degradable nozzle plugs;wherein at least some of the nozzles comprise nozzle plugs with different degradation characteristics, such that the nozzle plugs takedifferent times to degrade, in particular to dissolve, when subjected to an0 acid; and wherein the nozzle plugs are arranged in the liner axially ordered depending on their degrading characteristic.

2. A liner according to claim 1, wherein the degrading characteristics are based on5 the permeability profile along the well.

3. A liner according to any one of the preceding claims, wherein the nozzle plugscomprise an acid-degradable material.0 4. A liner according to any one of the preceding claims 1 or 2, wherein the nozzleplugs comprise wax, glass, ceramic, aluminum, chromium, rubber, elastomer and / or polylactic acid.

5. A liner according to any one of the preceding claims, wherein the nozzle plugsare adapted not to degrade when subjected to brine.

6. A liner according to any one of the preceding claims, wherein the nozzles have adiameter in the range of 1 to 10 mm, preferably in the range 2 mm to 6 mm,even more preferably in the range 2 mm to 4 mm.0 7. A liner according to any of the preceding claims, wherein the liner comprises atleast one segment with a length of 100 m that comprises 5 to 40 holes,2 preferably 10 to 30 holes, more preferably 15 to 20 holes, most preferably about 16 holes.

8. A liner according to claim 7, wherein the liner comprises at least one segmentwith a length of 100 m wherein the axial distance between the holes in said segment is in the range of 3 to 35 m, preferably 6 to 25 m, more preferably 8 to20m.

9. A liner according to any of the preceding claims 7 or 8, wherein at least onepacker is separating two segments from one another.

10. A liner according to any one of the preceding claims, wherein the acid is HCl.

11. A method of stimulating a well comprising the steps:providing a liner according to any of the preceding claims into awellbore; andsubsequently applying the steps of: applying acid into the liner, so that the acid dissolves nozzle plugsand flows through un-plugged nozzles; applying sealing material into the liner, configured and dedicated to seal un-plugged nozzles.

12. A method according to preceding claim 11, wherein the liner is provided for acompletion interval with a length higher than 5000 feet, more preferably higherthan 10000 feet, even more preferably higher than 20000 feet.

13. A method according to preceding claims 11 or 12, wherein the acid comprisesHCl.

14. A method of stimulating a well comprising the steps:providing a liner with a closed end and an open end, the liner comprising a plurality of nozzles for transferring fluid radially from the liner, wherein at least some of the nozzles are plugged with nozzle plugs; applying acid into the liner, and thereafter:3 removing one or more nozzle plugs, so that acid can flow through thethus opened nozzles radially from the liner; and sealing open nozzles.

15. The method according to preceding claim 14, wherein the removing and sealingsteps are based on the permeability profile along the well.

16. The method according to preceding claim 14, wherein the sealing step comprisessubsequently sealing from the closed end towards the open end.

17. The method according to any one of the preceding claims 14 to 16, wherein theremoving of nozzle plugs is done to control the pressure inside of the acid filled liner.

18. The method according to any one of preceding claims 14 to 17, wherein theremoving of nozzle plugs is done to control the fluid transfer from the liner, in particular to even the acid coverage in the wellbore.

19. The method according to any one of preceding claims 14 to 18, wherein theremoving step comprises removing nozzle plugs subsequently from the closedend towards the open end.

20. The method according to any one of preceding claims 14 to 19, wherein theremoving step involves a degradation, dissolving, breaking, melting and / orexploding of the nozzle plug(s).

21. The method according to any one of preceding claims 14 to 20, wherein thenozzle plugs comprise heating elements and / or explosives for removing thenozzle plugs.

22. The method according to any one of preceding claims 14 to 21, wherein theremoving is triggered by an acoustic signal and / or an electromagnetic signal.

423. The method according to preceding claim 22, wherein the triggering by anelectromagnetic signal comprises Radio-frequency identification (RFID).

24. The method according to any one of preceding claims 14 to 23, wherein theremoving is triggered when one or more pressure thresholds are exceeded insidethe liner.

25. The method according to any one of preceding claims 14 to 24, wherein thenozzle plugs comprise an acid-degradable material.

26. The method according to any one of preceding claims 14 to 25, wherein thenozzle plugs comprise wax, glass, ceramic, aluminum, chromium, rubber, elastomer and / or polylactic acid.

27. The method according to any one of preceding claims 14 to 26, wherein thesealing step comprises the introduction of ball sealers into the liner.

28. The method according to any of the preceding claims 14 to 27, wherein thesealing step comprises the introduction of a signal transmitter into the liner,configured and dedicated to trigger a removal of the nozzle plugs.

29. The method according to the preceding claim 28, wherein the signal transmitteris arranged inside a ball, in particular a ball sealer.

30. The method according to any of the preceding claims 14 to 29, wherein the lineris provided for a completion interval with a length higher than 5000 feet, morepreferably higher than 10000 feet, even more preferably higher than 20000feet.

31. The method according to any of the preceding claims 14 to 30, wherein thenozzle plugs are not degrading when subjected to brine.

532. The method according to any of the preceding claims 14 to 31, wherein thenozzles comprise a diameter in the range of 2 to 6 mm, preferably in the range3 mm to 4 mm.

33. The method according to any of the preceding claims 14 to 32, wherein the linerin the completion interval comprises 5 to 40 holes per 100 m of length,preferably 10 to 30 holes, more preferably 15 to 20 holes, most preferably 16 holes.

34. The method according to any of the preceding claim 33, wherein the axialdistance between the holes is in the range of 3 to 35 m, preferably 6 to 25 m,more preferably 8 to 20 m.

35. The method according to any of the preceding claims 14 to 34, furthercomprising the step of providing one or more packer(s) for separating varioussections of the liner.

36. The method according to any of the preceding claims 14 to 35, wherein at leastsome, preferably all, of the nozzle plugs comprise a self-destruct device forremoval of the nozzle plug, in particular a detonator.

37. The method according to the preceding claim 36, wherein the self-destructdevice comprises an explosive with a TNT equivalent of 10 to 100 g, preferably 15 to 80 g, more preferably 20 to 50 g.

38. A method, performed by a data-processing system, of simulating fluid transportin a system for stimulating a well in a material formation, which system comprises a limited entry liner, LEL, wherein the liner is divided into segments, each of which having a length less than that of the total length of the liner, and which includes one or more holes along a wall of the liner for discharging a fluidinto the formation, characterized in that the method provides an initial estimate of the number of holes along the wall ofthe liner which honours the acid coverage per segment and the drop in pressure6 across the last hole, for an optimized hole-size distribution in construction and operation of the system,adjusting the initial estimate of the number of holes along the wall of the liner ifthe acid coverage per segment and the drop in pressure across the last hole is not honoured; and wherein the method further comprises: performing a series of algebraic equations for an initial hole-size distribution guess; calculating acid coverage and dp across the last hole; comparing acid coverage and dp across the last hole against a design variable in a first iteration;evenly decreasing the number of holes across a segment for a next iteration untildp across the last hole is honoured; orevenly increasing the number of holes across a segment for a next iteration untildp across the last hole is honoured, as a first step; and performing a second step which includes: redistributing existing number of holes between various segments as a first iteration, wherein segments, where the calculated acid coverage is the furthest away from design values, exchange one hole; performing the next iteration until acid coverage is honoured; and performing the first step and the second step until dp across the last hole and acid coverage is honoured; and wherein the method further comprises:operating the system for stimulating the well in the material formation using theoptimized hole-size distribution; and wherein the method further compromises:7 a plurality of N sequential stages of simulating fluid transport, wherein the total number of holes to be taken into account for the hole-size distribution optimzation increases from stage 1 to stage N; initially specifying a desired average acid coverage and deduct a required total acid volume to be pumped; initially estimating the acid volume per stage to be the total acid volume divided by the number of stages N; optimizing the hole-size distribution for the first stage; optimizing the hole-size distribution for the next stage, while keeping the hole- size distribution of the previous stage constant.

39. The method according to claim 38, wherein the material formation around thewell comprises permeabilities in the range of 0,1 millidarcy to 50 millidarcy.

40. The method according to claim 38 or claim 39, wherein the material formationaround the well comprises an average permeability of less than 2 millidarcy,wherein the ratio of highest permeability to lowest permeability is higher than 3.

41. The method according to any of the claims 38 to 40, wherein in the well the drainlength is higher than 20000 feet when the wellbore diameter is around 6 inch orthe drain length is higher than 30000 feet when the wellbore diameter is around8.5 inch.

42. The method according to any of the claims 38 to 41, wherein the methodcomprises: running a simulation once the dp across the last hole and acid coverage is honoured to determine a wellhead pressure;adjusting a friction reducer andre-running the simulation if the wellhead pressure exceeds a maximum pressurerating; and / or increasing a tubing ID in presence of existing friction reducer; and / or8 reducing a pump rate, such that the wellhead pressure rating is maintainedbelow a maximum pressure rating.

43. The method according any of the claims 38 to 42, wherein the methodcomprises: running a simulation to determine whether a distance between LEL holes doesnot exceed twice an expected final wormhole radius;increasing a LEL hole size by 1 mm if the distance between LEL holes is toosmall, and repeating the simulation; or decreasing the LEL hole size by 1 mm if the distance between the LEL is too large, and repeating the simulation; or proceeding with an output of results if the LEL holes is dose or equal to twice the wormhole radius.

44. The method according any of the claims 38 to 43, wherein the followingconstraints are honoured: annulus pressure exceeding minimum reservoir pressure to avoid cross-flow inside wellbore; annulus pressure does not exceed fracturing pressure to avoid fracturing; wellhead pressure does not exceed maximum design pressure rating; cross-sectional area of all LEL holes combined may be equal to or exceed a minimum cross-sectional area to avoid creating an additional pressure drop during normal production or injection of the well after stimulation; average distance between two neighbouring LEL holes may be equal twice the wormhole radius; and liner ID not exceeding a wellbore size, are honoured.

945. The method according any of the claims 38 to 44, wherein the methodcomprises: describing a first order non-linear boundary value problem consisting of twofirst-order coupled non-linear differential equations with boundary conditions at each end of an interval of the independent variable, coupled with a set ofalgebraic equations, wherein polynomial approximation is used to solve the numerical model.

46. The method according to any of the claims 38 to 45, wherein the fluid transportis a fluid transport in a LEL-liner.

47. The method according to any of the claims 38 to 46, wherein the simulation isperformed in discrete steps, each step required to be completed before a following step may take place.

48. A method according to any one of claims 38 through 47, wherein a dataprocessing system is configured to perform the steps of the method.

49. A data processing system configured to perform the steps of the method asdescribed in any one of the claims 38 to 48 for stimulating a well.

50. The data processing system as claimed in claim 49 which includes any electronicsystem or device having a processor configured to perform the step of the method, and to communicate an outcome of those steps to a user of the system or device,which system or device includes, but is not limited to, a computer, a laptop, a handheld electronic device, or electronic workstation.

51. A computer program comprising instructions, which when executed by acomputer, causing the computer to perform the method according to claims 38