CN115698467A - Method for matrix-acid stimulation design in limited entry liners - Google Patents

Abstract

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

Description

Method for matrix-acid stimulation design in limited entry liners

Technical Field

The present invention relates to fluid delivery in a system for stimulating an oil or gas well in a carbonate petrochemical reservoir.

Background

The goal of stimulation is to increase the productivity of an oil or gas well while minimizing the amount of stimulation fluid introduced into the oil or gas well. A common stimulation method for oil or gas wells in carbonate reservoirs is acid stimulation, by which a selected acid is allowed to chemically react with the reservoir rock (carbonate), which results in solubilization of the reservoir rock and increases the productivity of the oil or gas well.

For those completely "open hole" oil or gas wells, one complicating factor is the acid placement in the well, i.e., the ability of the acid to be distributed over the entire section of the reservoir. "pushing" acid from the surface typically results in a well stimulation because most of the acid reacts at the heel of the well.

To ensure proper placement and efficient use of the acid, so-called "limited inlet liner" (LEL) technology has been introduced. The LEL is a liner with a plurality of holes distributed along the length of the liner for transferring acid into the reservoir rock. The LEL technique was developed for acid distribution and acid stimulation of long horizontal wells and is also known as "controlled acid injection (CAJ)". The acidizing process of reservoir rock using LEL can be expressed in several different mathematical ways. The first mathematical approach is a completely transient approach in tracking the motion of the acid front over time. This full transient approach is also referred to as a "transient simulator" and is useful for matching (and/or reproducing) historical pressure data and flow rate data of existing acid stimulation processes. The first mathematical approach assumes that the pore size distribution of the pore sizes of the plurality of pores has been optimized and uses this pore size distribution as an input in the calculation. Transient simulators attempt to capture the physical characteristics of the chemical reaction between acid and rock that are required to increase the productivity of an oil or gas well. Transient simulation takes into account the dissolution pattern of the acid in the rock. These dissolution patterns are called "wormholes". These dissolution patterns depend on, for example, the injection rate of the acid, the rock type of the rock, the permeability, or the injection temperature of the acid. Transient simulators require a large amount of computing power and, therefore, transient simulations are time consuming.

The second mathematical approach for modeling the acidification process using LEL is the steady state approach. In this steady state approach, the pumping rate change of the acid in the LEL is neglected and only the final acid profile of the acid is evaluated in the steady state simulation. This steady state method is fast and the distribution of pore sizes can be varied using computer software to match the desired acid coverage of acid in each section of the LEL along the well.

The concept used for the acidification process is the distribution of multiple pores in the LEL. The holes may be of different sizes and/or may be spaced at intervals along the LEL and serve as flow restrictions. This size and positioning of the holes can result in mechanical changes in the flow of acid along the LEL. Proper design of the pore size distribution ensures that the reservoir section is treated with acid and that the acid is efficiently utilized in the acidification process. Aspects of the calculation of pore size distribution have been addressed in a number of references listed below.

Another complicating factor is ensuring maximum acid penetration into the reservoir rock. Acid is an expensive commodity and should not be used to dissolve all rock adjacent to the wellbore area, i.e. adjacent to the LEL. Conversely, the stimulation protocol should be designed so that the acid penetrates into the formation as far as possible, since this case would result in the highest negative skin and therefore the highest productivity index.

Laboratory experiments by many authors clearly show that for any given rock formation, acid penetration is dependent on the acid interstitial velocity. There is an optimum rate that minimizes the amount of acid required to create a deep dissolution pattern (i.e., wormholes). The optimal velocity depends on the rock and acid system (type, concentration, temperature). In addition to ensuring uniform acid coverage, the pore size distribution must also be designed to maximize the propagation of wormholes through the rock formation.

Another problem is associated with acid stimulation of both vertical and horizontal wells. The challenge is to achieve uniform stimulation throughout the well completion trajectory. Some operators choose not to stimulate the well, others press in from the wellhead, and others stimulate through the coiled tubing. Staged completions are employed that allow staged acidizing and use of diverters. A few operators use the Limited Entry Liner (LEL) concept, but do not describe a full workflow for hole sizing. Due to the variety of considerations, the design of LELs remains a challenging subject in terms of different pore sizes and frequencies.

In EP1184537B1, the authors describe the LEL concept for matrix-acid stimulation (known as controlled acid jetting) and develop a steady state model using polynomial approximation with orthogonal coordination. However, their models assume a constant friction factor and do not describe the workflow for designing an optimal pore size distribution. Their model does not estimate the maximum design rate, does not take into account the experimental wormhole curve, does not have an epidermal model and also does not estimate the required acid coverage and the optimal distance between wells.

US8,321,190 B2 discloses a system and method for stimulating fluid transport to increase the productivity of a well by introducing acid into the reservoir rock of the well using a stimulation liner. The stimulation liner is provided with a plurality of preformed holes which form flow channels, so-called "mudcakes", between the interior of the liner and the annular space around the liner. The us patent further describes a method for simulating and/or calculating the distribution of holes in a stimulation liner to ensure adequate acid coverage in the reservoir rock. The simulation of the hole locations was performed in a trial and error analysis by applying a transient model to distinguish the possible locations of the holes in the liner. The us patent also describes that the distribution of pores can be simulated using a steady state model.

It is further disclosed in the U.S. patent that the simulation includes the step of calculating the pressure drop along the stimulation liner as a dimensionless pressure function or using a polynomial approximation. However, the model disclosed in the us patent does not describe a workflow for designing an optimal pore size distribution. The model also does not estimate the maximum design rate and does not take into account the experimental wormhole curve. The disclosed method also does not contemplate physically segmenting the wellbore using swellable packers. Nor does this us patent application disclose a skin model.

U.S. patent application No. us 2016/2454049 A1 discloses an apparatus and method for simulating and/or controlling fluid flow during continuous injection of multiple fluids in a formation and/or wellbore. More specifically, this us patent application describes the adaptation of a commercial reservoir simulator (Eclipse from schlumberger) to handle transient displacements in a wellbore. Numerical modeling is used to determine the conditions and operating parameters required to ensure the best possible distribution of acid, effective control of the rate of wormhole growth in multiple sections of the well, displacement of mud along the entire reservoir section, and the management of significant formation pressure gradients along the reservoir section. Also disclosed is a matrix-acid stimulation using a controlled acid injection (CAJ) liner. The us patent application focuses on understanding the pressure response during well intervention and considers that this pressure response is a key requirement for designing and improving well intervention work. No workflow is disclosed for optimizing pore size distribution. The U.S. patent application is directed to capture friction reduction (as a function of rate, chemical concentration, etc.) as the acid front advances down through the wellbore.

The prior art teachings do not address the problems presented. For example, the assumption of a constant friction factor in EP1184537B1 does not predict the actual phenomenon well. The authors of EP1184537B1 also do not lay the design foundation for optimal pore size distribution. It cannot estimate the maximum design rate and the required acid coverage. This work did not combine experimental wormhole curves and epidermal models. US8,321,190 B2 uses a transient model. The teaching also does not describe an optimal pore size distribution.

Object of the Invention

The present invention, which will be described in further detail in subsequent paragraphs, includes a newly developed method and system for stimulating a well that at least partially addresses the above-described challenges in a novel and inventive manner.

Disclosure of Invention

It is therefore an object of the present invention to provide a system and method for determining the designed pumping rate of an acid to ensure uniform penetration of the acid into reservoir rock of an oil field while ensuring that the injection pressure remains below the fracturing pressure of the rock. The system and method also provide an accurate and efficient numerical solution strategy for providing an initial estimate of the number of holes per section that satisfies the acid coverage per section and the pressure drop (dp) over the last of the holes, particularly in the case of acid stimulation of a well completed in a carbonate reservoir with a limited entry liner or LEL liner.

According to a first aspect of the invention, the accuracy of the simulation of fluid transport of an acid in a system for stimulating a well in a material formation of a resource reservoir may be significantly improved by including a workflow for designing an optimal pore size distribution. Thus, an optimized pore size distribution in the liner of the LEL liner system is modeled, which results in improved modeling accuracy and provides improved construction and operation of the stimulation system. Well productivity and acid use are enhanced through improved construction and operation of the stimulation system. In particular, an initial estimate of the number of holes and the cross-sectional area of the holes for each section of the liner is provided. The cross-sectional area is based on the optimal velocity for minimizing the amount of acid required to create the dissolution pattern and the calculated designed pumping rate for ensuring that the injection pressure remains below the fracturing pressure of the well material formation. The number of apertures along the wall of the liner satisfies the acid coverage per section and the pressure drop over the last of the apertures (dp), where the pressure drop over the last of the apertures (dp) is linearly related to the cross-sectional area, such that an initial estimate can be found from the relationship between clearance velocity, pumping rate and total cross-sectional aperture area for a particular discharge coefficient and liner configuration.

According to some embodiments, it is another object to improve the accuracy of the simulation by ensuring that the annulus pressure remains below the fracture pressure. The maximum pumping rate allowed is determined by the permeability, fluid viscosity, length of completion interval, skin and difference between annulus pressure and reservoir pressure.

According to some embodiments, it is another object to improve the accuracy of the simulation by estimating wormhole characteristics in order to optimize the pore size distribution.

Wormhole estimation includes nodal analysis calculations performed to estimate the downhole temperature at the heel of the liner, and based on the selection of acid, permeability, and temperature, the optimal velocity for wormhole propagation and the expected volume of pores to penetrate are estimated.

According to some embodiments, model accuracy has been improved by providing a method that includes estimating the total number of holes and the pressure drop (dp) of the pressure on the last one of the holes. Based on the optimal speed and the calculated design pumping rate, a total cross-sectional area of the orifices is calculated, wherein the area is linearly related to the pressure drop (dp) across the last one of the orifices.

According to another aspect, a data processing system is configured to perform the steps of the methods 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 a pore size distribution that satisfies the acid coverage per section and the pressure drop (dp) over the last of the pores in case of acid stimulation of a well completed in a carbonate reservoir with an LEL liner. The method comprises the following steps:

-executing a series of algebraic equations for initial pore size distribution guessing;

-calculating the acid coverage and the pressure drop (dp) over the last of the wells;

-comparing the acid coverage and the pressure drop (dp) over the last one of the wells with the design variables in a first iteration;

-uniformly reducing the number of holes over the section in the next iteration until a pressure drop (dp) over the last one of the holes is met; or

-as a first step, uniformly increasing the number of holes over the segment in the next iteration until dp over the last one of the holes is met; and

-performing a second step, the second step comprising;

-as a first iteration, redistributing the existing number of holes between the segments, wherein;

-the zone where the calculated acid coverage is most different from the design value is exchanged for one well;

-performing a next iteration until the acid coverage is met; and

-performing the first and second steps until a pressure drop (dp) over the last one of the wells and an acid coverage are met.

According to yet another aspect, the invention relates to a method of stimulating a well by means of a workflow system for adjusting a pore size distribution that satisfies the acid coverage per zone and the pressure drop (dp) over the last one of the pores in case of acid stimulation of a well completed in a carbonate reservoir with an LEL liner. The method comprises the following steps:

-running a simulation to determine the wellhead pressure once the pressure drop (dp) and acid coverage on the last one of the holes is met;

-if the wellhead pressure exceeds the maximum pressure rating, adjusting the friction reducing agent concentration and re-running the simulation; and/or

-increasing the pipe inner diameter (pipe ID) in the presence of the existing friction reducer; and/or

-reducing the pumping rate such that the wellhead pressure rating remains below the 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 a pore size distribution that satisfies the acid coverage per zone and the pressure drop (dp) over the last one of the pores in case of acid stimulation of a well completed in a carbonate reservoir with an LEL liner. The method comprises the following steps:

-running a simulation to determine whether the distance between LEL holes, defined as the length of the stimulation reservoir section divided by the number of holes, should not exceed twice the expected final wormhole radius;

-if the distance between the LEL holes is too small, increasing the LEL hole size by 1mm and repeating the simulation; or

-if the distance between the LELs is too large, reducing the LEL pore size by 1mm and repeating the simulation; or

-if the LEL hole is close or equal to twice the wormhole radius, continue outputting the result.

According to yet another aspect, it relates to a method of stimulating a well by means of a workflow system for adjusting a pore size distribution that satisfies acid coverage per zone and pressure drop (dp) over the last of the pores in case of acid stimulation of a well completed in a carbonate reservoir with an LEL liner, wherein the constraint is:

-the annulus pressure exceeds the minimum reservoir pressure to avoid cross flow inside the wellbore;

-the annulus pressure does not exceed the fracturing pressure to avoid fracturing;

-wellhead pressure does not exceed a maximum design pressure rating;

the cross-sectional area of all LEL holes, taken together, may be equal to or exceed the minimum cross-sectional area to avoid additional pressure drop (dp) during normal production or injection of the well after stimulation;

the average distance between two adjacent LEL holes may be equal to twice the radius of the wormholes; and

-that the liner inner diameter (liner ID) does not exceed the borehole size.

According to yet another aspect, it relates to a method of stimulating a well by means of a workflow system for adjusting a pore size distribution that satisfies the acid coverage per section and the pressure drop (dp) over the last of the pores in case of acid stimulation of a well completed in a carbonate reservoir with an LEL liner. The method further comprises the following steps:

-inputting one or more of the following parameters: average reservoir pressure per zone, crack propagation pressure, permeability per zone, porosity, length of completion interval, wellbore radius, pipe inner diameter (pipe ID), liner inner diameter (liner ID), pipe roughness, acid properties, number of zones, desired acid coverage per zone, pore size per zone, and/or discharge coefficient in a series of algebraic equations for initial pore size distribution guessing.

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 steps of the method and to communicate the results of those steps to a user of the system or device. Such systems or devices include, but are not limited to, computers, notebooks, handheld electronic devices, or electronic workstations.

Drawings

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

FIG. 1: a schematic cross-sectional view of a wellbore and a limited entry liner is shown;

FIG. 2: showing a schematic cross-sectional view of a wellbore that is segmented into sections using packers;

FIG. 3: a flow chart is shown which depicts an implementation of the present invention in a step-by-step manner;

FIG. 4 is a schematic view of: a portion of figure 3 is shown in more detail;

FIG. 5: the effect of rate on dissolution of acid is graphically shown by the etch pattern in the Texas create chain:

FIG. 6: illustrating the effect of gap velocity on the volume of pores to be penetrated;

FIG. 7: illustrating the relationship between the temperature of the acid at the inlet of the liner at different pumping rates and wellhead temperatures;

FIG. 8: illustrating the volume of acid required to achieve a certain wormhole length based on the volume of pores to be penetrated from the core displacement data;

FIG. 9: illustrating the relationship between pumping rate, pressure drop (dp) across the last of the orifices, discharge coefficient and total cross-sectional orifice area;

FIG. 10: illustrating the effect of reynolds number on friction factor for different pipe roughness values;

FIG. 11: the effect of drag reduction on friction factor is illustrated in the Prandtl-Karman graph;

FIG. 12: illustrating the effect of drag reduction on friction pressure;

FIG. 13: the epidermal factor as a function of stimulation coverage is graphically illustrated; and

FIG. 14: the epidermal factor as a function of wormhole radius is illustrated.

Detailed Description

The limited inlet liner includes a plurality of unevenly spaced apertures, the purpose of which is to evenly distribute fluid, in this case acid, along the section of the reservoir to be stimulated. This concept was originally described by Shell (Shell) in 1963 for fracturing applications (Lagrone and Rasmussen, 1963) and is still widely used today. Later, it was adapted by Maersk Oil (maskski Oil) as a matrix-acid stimulation and patented (known as controlled acid injection or CAJ) and implemented on a large scale in the north sea chalk reservoir, 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) in addition to others (Mitchell et al, 2014; van Domelen et al, 2011, 2012). Rodrigues et al (2007) provide a good general overview of stimulation techniques for low permeability reservoirs, and Shokry (2010) describes acid stimulation practices for offshore reservoirs in ADNOC.

Fig. 1 shows a schematic cross-sectional view of a wellbore 12. The wellbore 12 is generally formed by techniques well known in the art and includes a wall 14 formed by the drilling process, a front end 16 extending into the formation 18, and a rear end 20 for accessing the wellbore.

A limited entry liner 20 is introduced into wellbore 12. The liner 20 has an open end 22 and an opposite sealed end 24. An annulus 22 is formed between the wall 14 and the exterior surface 26 of the liner.

Liner 20 is provided with a plurality of preformed apertures 28, which apertures 28 form flow passages between the interior of liner 20 and annulus 22. The holes 28 have a shape and position that conforms to a particular predefined specification.

Generally, the distance between adjacent apertures 28 along the liner 20 decreases toward the end 24 of the liner.

Acid is pumped into the liner in liner 20 and exits the apertures 28 at high velocity for injection into the formation 18. By limiting the number and size of the holes, a choking effect is obtained and during stimulation a significant pressure drop (dp) occurs between the inside and outside of the liner. The non-uniform geometric distribution of the holes is used to compensate for the friction pressure drop (dp) 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 inside the liner (due to choking above the holes) ensures that the acid eventually reaches the bottom of the liner and thus the well is stimulated along its entire length.

Acid is pressed from the surface and into liner 20 in the direction of arrow 30. The liner is not necessarily horizontal, but often horizontal. When the acid reaches the first hole 28, which has a size of 2mm to 7mm, the pressure drop (dp) over the hole is so high that only a small fraction of the acid leaves the liner through the hole; the remainder continues along the liner until it reaches the next hole where the same process is repeated. Appropriate pore size design can meet a specified acid coverage, which is defined as the number of barrels of acid per foot of reservoir section. Prior to stimulation, the mud may be circulated out so that only completion brines of the proper density are found in the wellbore 12.

The acid stimulation process is modeled by discretizing the wellbore 12 into a plurality of nodes 34, typically 100 to 400 nodes. The nodes need not be of the same size. From a practical design point of view, the wellbore is divided into a smaller number of sections 36. The zones may be physically separated from each other on the annulus side by hydraulic packers 32 (not shown), but need not be. As shown in fig. 2, the nodes may overlap between the two zones.

The displacement of the brine by the acid is believed to occur by a single phase plug flow with minimal dispersion. Negative excess mixing volumes were not considered. Prior to stimulation, the liner 20 is closed at the sealed end 24 and it is not cemented, which means that fluid can in principle flow in the annulus 22 along the wellbore trajectory before the packer 32 is set. In practice, the annular flow is mainly due to the injection of acid through the holes 28 perpendicular to the wellbore. For practical modeling purposes, the annular flow along the liner can be ignored.

The well completion design and associated modeling workflow included in this document allows for reservoir segmentation using packers and the resulting liner is therefore referred to as a segmented limited inlet liner. The desired acid coverage for each segment can be specified, taking into account the differences in porosity, permeability, initial water saturation, and reservoir pressure. The number of zones used to model the process may be greater than the number of intervals the packers are spaced apart.

The design of the pore size distribution is mainly dependent on the liner geometry and flow rate, which in turn is limited by the reservoir properties, i.e. the reservoir permeability. Acid stimulation is inherently transient in nature, in that once the acid reacts with the reservoir rock minerals, the skin factor at any given location along the well changes over time from an initially positive value (caused by the mud cake) towards a negative value. This reaction of the acid with the reservoir rock mineral results in the formation of highly conductive fluid flow paths in the reservoir rock. These fluid flow paths are commonly referred to as "wormholes". These wormholes are desirable in stimulation of reservoir rock because they allow further propagation of acid into the reservoir rock so that subsurface hydrocarbons can flow along them once the acid is depleted. If the skin evolution over time is uniform along the well, the flow distribution is not affected, which means that the whole process can be modeled based on steady-state principles.

The present invention includes a comprehensive algorithm for designing the hole size distribution of a limited inlet liner. The following section describes an algorithm for designing pore size distribution that achieves a specified (usually uniform) distribution of acid volume per interval length, also referred to as acid coverage.

This algorithm is schematically shown in fig. 3 and 4. Fig. 3 shows the overall algorithm, while fig. 4 shows a more detailed part of fig. 3. The algorithm will now be discussed with reference to FIG. 3 and the first box, input data, and constraints 1000.

Input data and constraints 1000:

as a starting point for implementing the algorithm, input data constraints are entered into the system. The input data includes rock properties, completion data, fluid properties and other data such as pumping rate, number of nodes used for numerical algorithms, pressure drop over the last of the holes of the liner (dp) and annulus pressure. These inputs are known or may be derived from historical data from the wellbore.

The algorithm defines certain constraints that must be observed in the operation of the system. These constraints form part of the input data and constraints 1000. Constraints include, but are not limited to: the annulus pressure must exceed the minimum reservoir pressure to avoid cross flow inside the wellbore; the annulus pressure must not exceed the fracturing pressure to avoid fracturing; the wellhead pressure must not exceed the maximum design pressure rating-which in turn affects the design rate and/or the amount of friction reducer to be added; the cross-sectional area of all the LEL holes, taken together, should be equal to or exceed the minimum cross-sectional area to avoid creating additional pressure drop (dp) after stimulation during normal production or injection of the well, which affects the number and size of the holes; the average distance between two adjacent LEL holes should be equal to twice the radius of the wormholes formed along the limited inlet liner-this will affect the pressure drop across the last LEL hole (dp), which is a design variable; and the liner inside diameter (liner ID) cannot exceed the wellbore size.

Moving to the next step, as shown in FIG. 3 at block 1002

Initial variable calculation 1002:

based on the input for each zone, the maximum rate for each zone is determined by applying the instantaneous inflow equation. It should be noted that although the well is horizontal, it acts as a vertical well in the early injection phase since the boundary is not yet felt. Thus, the reservoir section length L replaces the reservoir thickness H.

B is an acid forming volume factor, which is in the range of 1.0 to 1.1. In practice, B is assumed to be 1. Viscosity is the maximum of the oil or gas viscosity and the acid viscosity. In heavy oil reservoirs, the transient phase injection rate is initially controlled by the oil properties. Therefore, the temperature of the molten metal is controlled,

μ _{maximum of} ＝max(μ _o ,μ _Acid(s) ) Equation 2

The permeability will see contributions from both the horizontal as well as the vertical direction:

the vertical/horizontal permeability ratio may reach a value in the range of 0.01 to 1.0. For the present application, this value is close to 1, which makes the overall permeability equal to the horizontal permeability.

The diffusivity is given as

Where the total system compressibility is given as a contribution from the rock and the fluid phases present in the pore space.

c _{General assembly} ＝c _Rock +S _w ×c _w +(1-S _w )×c _o Equation 5

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

The maximum pumping rate allowed is then the sum of the individual segment rates:

however, any section that must remain unstimulated and therefore requires a void-free joint does not contribute to the calculation of the total rate. To start the design algorithm detailed later, the actual design rate is taken to be a value 10% to 30% lower than the maximum allowable rate. This value may be adjusted in subsequent iterations.

T is the total pumping time calculated by the acid coverage and length of all zones

It should be noted that T depends on Q, which depends on T.

In the 80's of the 20 th century, with the pioneering work of Fogler and colleagues at the university of Michigan (Hoefner et al, 1987; hoefner and Fogler, 1989; bernadiner et al, 1992; fredd and Fogler,1996, 1997, 1999; fredd et al, 1997), research began on the basic principles of matrix-acid stimulation, which demonstrated that the reaction of acids with rock can produce different etching patterns depending on the type and concentration of the acid and the rate and temperature. Key subsequent contributions in the literature to current understanding include work by Halliburton (Gdanski and Norman, 1986; gdanski and van domeen, 1999; gdanski, 1999), buijse and Glasbergen (2005) and Hill and colleagues of the university of texas a & M (Al-Ghamdi et Al, 2014; dong et Al, 2014, 2016; duz et Al, 2016; etten et Al, 2015; furui et Al, 2005, 2008, 2010, b; izgec et Al, 2008; ndonohong et Al, 2006, 2018; sasong ko et Al, 2011; schlbwaert et Al, 2018; shirley et Al, 2017; shukla et Al, 2016). Additional references to experimental and theoretical studies of wormhole growth are set forth in these references.

Fig. 5 shows the effect of rate on dissolution by a series of images 100. The low rate results in uniform dissolution and thus very inefficient use of the acid. The leftmost image 102 illustrates this. In this image, the acid 104 does not penetrate the formation 106 to any significant extent. At a slightly higher rate (i.e., moving from left to right in the image), the acid forms wormholes 108 through the rock. In fact, any acid preparation has an optimum rate at which the volume of acid required to etch the pattern from inlet to outlet is minimal. This volume is referred to as the volume of the pore 202 to be penetrated. It should be noted that 15% HCL corresponds to 4.4M, so the 0.5M concentration used in the experiment was rather low.

Fig. 6 illustrates the effect of gap velocity 200 on the volume of a hole 202 to be penetrated at two different temperatures 204A (depicted by a dashed line) and 204B (depicted by a solid line). The temperature increase (i.e., from a temperature 204A of 25 ℃ to a temperature 204B of 600C) results in a higher reaction rate and thus faster dissolution; therefore, optimal wormhole growth requires higher acid rates to avoid depleting all acid near the wellbore. It should also be clear that pumping at a rate slightly above the optimum rate is better than pumping at a rate below the optimum rate. In low permeability reservoirs, the maximum pumping rate is limited by the fracture pressure, which may prevent the operator from reaching the optimal speed. In this case, a different formulation of acid 104 needs to be selected to shift the curve to the left, and preferably also to the bottom.

Wormhole data can be reproduced using a model proposed by Buijse and Glasbergen (2005) containing two fitting constants α and β that can be reformulated with the lowest point on the curve (optimal gap velocity 200, optimal pore volume to penetrate 202)

Both the increased temperature 204 and the increased HCl concentration increase the optimal speed 200 of the wormholes. For low permeability rocks, where the optimum rate may be limited by crack propagation pressure, it may be beneficial to reduce the acid concentration, despite the increase in the pore volume 202 to be penetrated and thus the increase in the volume of acid solution required. If the acid concentration is halved, the volume must be doubled to maintain the same number of moles. Several authors have investigated the effect of weak acids, see Punnapala et al (2014) and Shirley et al (2014). Friction reducers can shift the PV curve upward, which means more acid is needed to achieve the same skin.

Talbot and Gdanski (2008) proposed a general wormhole model in which they correlated two input parameters to the Buijse-Glasbergen model as a function of rock and acid properties and temperature. However, they do not specify the value of the constant in their correlation.

In the present invention, we use a concept that we move the default wormhole curve shown in fig. 6 up, down, left or right depending on the temperature 204, permeability and acid type. Table 1 shows some rough rules of thumb in adjusting the best (lowest) point on the wormhole curve. Based on the default curve, the optimal point is moved by the indicated amount. The optimum point cannot be lower than (0.1 ). The values in the tables are indicative only and are used to illustrate the concept.

Table 1: optimum wormhole growth parameters

The acid reactivity increases with temperature 204, which means that the optimal rate 200 of wormhole growth also increases. For low permeability reservoirs, it may be difficult to reach optimal velocities without fracturing the formation. Therefore, it is important to estimate the downhole temperature of the acid 104 as it reaches the formation 106.

As shown in fig. 7, which illustrates the relationship between the temperature of the acid at the inlet of the liner 300 at different pumping rates 302 and wellhead temperatures 304. It is advantageous to inject at a high rate and as low a wellhead temperature as possible to limit in situ acid reactivity. This is shown by line 304A. As the temperature increases, we can see an increase in acid reactivity at the inlet of the liner 300 through line 304B and line 304C. Furthermore, any brine used to remove the mud prior to acid stimulation should be injected at as low a temperature as possible.

The temperature used to adjust the wormhole curve is the temperature of the acid as it enters the reservoir, not the reservoir temperature.

Economides et al (1994) derived the following equation based on the volume of pore to be penetrated 202 from the core displacement data to determine the volume of acid required to achieve a certain wormhole length of 400:

this formula is plotted in fig. 8. The ratio V/L is referred to as acid coverage in bbl/ft 402.

The equivalent skin 404 is given by:

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

Alternatively, the acid stimulation may be fixed, which enables calculation of the maximum final wormhole length 400 and thus the final negative epidermis 404.

Fig. 9 illustrates the results of a larger sensitivity analysis involving a pumping rate 500, a pressure drop (dp) across the last one of the orifices 502 (502A-502E, respectively), a discharge Coefficient (CD) 504 (504A-504E, respectively), and a total orifice cross-sectional orifice area 506. The linear relationship between total orifice cross-sectional orifice area 506 and pumping rate 500 is derived from sensitivity analysis. Thus, the pressure drop (dp) 502 required to obtain a particular total orifice cross-sectional orifice area 506 may be predicted. This constrains the total pore cross-sectional pore area 506 to have to be equal to or greater than the minimum cross-sectional area to avoid imposing an additional pressure drop (dp) 502 during production/injection after stimulation, thus resulting in a constraint on the pressure drop (dp) 502 on the last of the pores, which can be estimated based on the relationship provided by the sensitivity analysis. This is a novel concept.

Wherein:

a = aQ + b equation 10

a = α dP + β equation 11

b = γ dP + δ equation 12

At this stage, we can estimate the initial pore size distribution for the starting point of the algorithm. This is achieved by

Depicted at block 1004 in fig. 3.

The next step, block 1006, requires the establishment of an equation. These equations are then solved as part of the following step, block 1008, mentioned later in this specification.

Equation 1004 is established:

the equations of motion for isothermal one-dimensional duct flow describe the pressure drop (dp) as contributions from friction, gravity and acceleration. The gravity term dominates in the vertical section of the wellbore, while friction losses become relatively more important in the horizontal section. The acceleration term is only required when there is a change in velocity, such as when fluid enters the liner from a pipe (change in internal diameter) or when fluid exits through an aperture in the liner. The contribution of the acceleration term to the total pressure drop (dp) is less than 5% and can generally be neglected.

θ is the angle relative to the Z-axis, and D is the pipe diameter. The acceleration term may be expressed in terms of volumetric flow Q instead of velocity v,

vannin friction factor f is defined in terms of wall shear stress

Thus, the frictional pressure of Newtonian flow decreases (dp) _{Friction by friction} ) Comprises the following steps:

for laminar flow, the fanning friction factor is related to the reynolds number,

reynolds number is given as

The pressure difference due to hydrostatic head is determined according to:

the fanning friction factor of the tube flow in a smooth tube is described by the Prandtl-Karman equation:

for a rough pipe, the friction factor depends on the relative roughness ε/D of the pipe and is given as

Fig. 10 illustrates the effect of reynolds number 600 on friction factor 602 for different pipe roughness values 604 (604A to 604H, respectively). Typical relative roughness of the new pipe is 10 ^-4 。

There is a potential discontinuity from laminar to turbulent flow because the flow regime is undefined in the 1000 to 2000 reynolds number region. This has no effect on the LEL pore design. Fig. 10 shows that the roughness only works if it exceeds 0.0001.

Typical pumping rates are 5 to 40bbl/min, depending on the reservoir permeability and liner length. Such rates can result in high surface pressures and therefore require proper design of the upper completion. Friction pressure losses are often required to be reduced to stay within safe operating limits and this requirement may require the use of Drag Reducing Agents (DRAs). Drag reducers are mostly dilute polymer solutions that, when added to a solvent, such as water or acid, can reduce the frictional resistance to flow under turbulent mechanisms. In some cases, very low concentrations (thousands of ppm) can reduce friction by as much as 70%. However, according to some studies, friction reducers may cause reservoir damage.

When a drag reducer is added, a region called an elastic sublayer is formed between the viscous sublayer and the newton's nucleus. The extent of the elastic sublayer will be determined by the amount and type of polymer and the flow rate.

Maximum drag reduction is achieved when the elastomeric sublayer extends to occupy the entire pipe cross-section. Drag reduction of dilute polymer solutions in turbulent pipe streams is defined between two general asymptotes and the maximum drag reduction asymptote described by newtonian turbulence. In between, is the so-called polymerization mechanism, in which, see fig. 11, the friction factor relationship is approximately linear in the Prandtl-Karman coordinate system. The aggregation mechanism can be described by two parameters: the initial wave number w and the slope increment δ, whereby the polymer solution slope exceeds the newton slope. The onset of drag reduction occurs at a well-defined starting wavenumber. For a given polymer solution, w is substantially the same for different pipe diameters. For a solution of a given polymer-solvent combination, w is substantially independent of polymer concentration.

In modeling the effect of drag reducers, it is assumed that the fluid friction factor is reduced and the fluid viscosity remains unchanged. As can be seen from fig. 10, via the reynolds number, the acid viscosity has little effect on the friction loss under typical operating conditions.

The following formula developed by Virk (1971, 1975) relates the friction factor to the concentration of drag reducing agent for the pipeline flow:

the parameters of the drag reduction model are

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

The maximum drag reduction asymptote for a pipeline flow is described by the following equation:

FIG. 11 shows the effect of drag reduction on friction factor in Prandtl-Karman plots.

For the particular Drag Reducer (DRA) 606 model constant used, the maximum asymptotic line 608 is reached only when the DRA concentration exceeds 2000 ppm. The case where no DRA is added is shown by 610. The amount of DRA that is incrementally increased is shown by lines 612, 614, and 616, respectively.

In a 6 "Inside Diameter (ID) liner, a pumping rate of 25bbl/min equivalent to 36000bbl/d results in a Reynolds number of about 321635, which is well within the turbulent regime.

Fig. 12 illustrates the effect of drag reduction on friction pressure 620 in a 10000ft long 4.5 od top completion line as a function of pumping rate 500. By adding 1000ppm of DRA, the friction is reduced to 1/3. The concentration of DRA is similar to that illustrated in fig. 11.

The limited entry liner includes a plurality of apertures that allow fluid to exit the liner and enter the annulus and then the reservoir. The holes are small compared to the liner size, both in length and diameter, and can therefore be considered apertures. Pressure drop across N holes in the liner (dp) _Hole(s) ) Can be calculated as:

Q _hole(s) Is the flow rate through the orifice in bbl/min. The positive direction is the direction from the liner into the annulus. D _Hole(s) Is the inside diameter of the hole in the liner in inches. N is the total number of pores. C _D Is a dimensionless emission coefficient that takes into account the fact that: due to the short length of the hole (equal to the pipe thickness), the pressure loss can only be partially recovered. Based on the work of Crump and Conway (1988), the flow of water and gelled fluid in sharp-edged round boreholes used lower values of 0.56; depending on the type of fluid and the actual drilling mode, this value can also reach 0.90, see El-Rabba et al (1997) and McLemore et al (2013). During the first LEL design run, C _D Should be considered as a sensitivity variable. Drilling at a small angle may reduce the splash back of unconsumed acid thereby adversely affecting the formation and improving the injection process.

The model for the friction factor in the presence of drag reducer is combined with the model for the friction factor of newtonian turbulent pipe flow in a rough pipe.

If no drag reducer is used, δ =0. If a drag reducer is used, the roughness is set to zero.

Insert expression for Reynolds number:

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

The algorithm proceeds to inner loop 1100. This is guided by block 1008, solving the equations.

The final step of the inner loop 1100 is to determine if the solution vector is constant, block 1012.

Whether the solution vector is constant 1012:

typically, the Newton-Raphson technique will converge within 5 iterations using carefully selected relaxation parameters to guide convergence during the first iteration. This method ensures that the final convergence rate is quadratic.

The intra-iteration loop will repeat by following arrow 1014 and restarting the solution of the equations, as set forth at block 1008.

The iterative inner loop 1100 is completed when the absolute change of the solution vector is below a certain threshold, typically 1E-12. To avoid the possibility of an infinite loop, the program stops after a pre-specified number of iterations, typically in the range of 10 to 20, has been reached.

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

Calculate acid coverage 1018:

once the stimulation flow rate is calculated from the solution, the acid coverage of each liner section is the product of the section flow rate and the pumping time. If the total pumping rate changes during operation, the yield increase for each zone will change.

C _{Acid, i} ＝Q _{Increase in production i} Equation 97 of x T

The transient period of time during which the acid front moves through the liner while displacing brine must also be considered. However, this is compensated for when the water displaces the acid at the end of the job. The time required for the front to reach a given position i is called the hold time, which is calculated in a recursive manner:

since the liner flow rate gradually decreases towards zero at the heel, it is clear that the acid front takes longer to displace the brine out of the liner. In other words, the inner portion sees the acid longer than the outer portion. The pore size distribution should compensate for this. Thus, the hold time is also a measure of the minimum time required for the water to displace acid from the liner at the end of stimulation.

The next step to determine whether the pressure drop (dp) across the last of the orifices matches is shown in block 1020. This step is combined with the following block 1022 to determine whether the design acid coverage matches.

Is the pressure drop (dp) across the last of the orifices matched 1020? :

the pressure drop (dp) across the last of the holes is calculated as the difference between the pressure in the last node of the liner and the annulus stimulation pressure (annulus stimulation pressure is constant and user specified):

dP _{last hole} ＝P _{Lining tube, n} -P _{Increase production} Equation 99

Is the acid coverage designed to match 1022? :

the difference between the calculated acid coverage and the specified target acid coverage is given as

This equation ensures that the dCOV (acid coverage distribution) function is always positive. Therefore, it must be minimized to obtain the best possible match. Relative acid coverage was determined as follows:

turning to FIG. 4, the above two steps are combined into block 1050.

While the inner loop 1100 includes material balances that account for a given combination of LEL holes, pumping rates, and other variables, the first portion of the outer loop 1200 includes adjusting the LEL hole size distribution to match the desired pressure drop (dp) across the last of the holes, both pressure drops (dp) 1052, and the desired acid coverage for each section 1054. The outer loop 1200 is used to satisfy both constraints.

Therefore, the pore size distribution must be satisfied, as shown in block 1024

Update pore size distribution 1024:

if the pressure drop (dp) is too small 1056, there are too many LEL pores, and then one LEL pore 1058 is subtracted from the segment with the highest relative acid coverage, and then the material balance inner loop 1100 is recalled via block 1006.

If the pressure drop (dp) is too great (arrow 1060), there are too few holes, and then one hole is added to the zone 1062 with the lowest non-zero relative acid coverage and the material balance inner loop 1100 is recalled via block 1006.

No adjustment was made to the segment with zero acid coverage.

If the pressure drop (dp) approaches the target value within a certain tolerance, the acid coverage distribution dCOV 1054 is calculated. At this point, the total number of LEL holes is correct, but the holes need only be redistributed between the segments. One LEL hole is added to the zone with the lowest non-zero relative acid coverage and one LEL hole 1064 is subtracted from the zone with the highest relative acid coverage. The inner loop 1100 is then re-enabled via block 1006 and the procedure is repeated until the dCOV function reaches a minimum value. Since the algorithm adjusts the integer value, i.e., the number of LEL holes, the dCOV function cannot be exactly zero.

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

Is the calculated WHP below the maximum constraint? :

each wellhead has a maximum pressure rating such as 5000psia, 6500psia, and higher. Similarly, each tube has a maximum pressure rating. Thus, if the reservoir pressure is high, the design rate may result in a wellhead pressure that exceeds the pressure rating.

If the calculated wellhead pressure exceeds the maximum rating of the pipe (indicated by arrow 1028), then adjusting the design (block 1030) requires the following steps:

step 1: if the previous design is based on zero friction reduction, 2000ppm of friction reducer is added. The simulation was re-run.

And 2, step: the possibility of increasing the pipe inner diameter (pipe ID) is investigated if friction reducers are already present. The simulation was re-run.

And step 3: if step 2 is not feasible, the rate is reduced, the simulation is re-run, and a loop is made until the calculated WHP is below the maximum pressure rating of the pipe.

Next, the average hole distance constraint must satisfy block 1032.

Is the average pore distance constraint satisfied? :

as previously described, economides et al (1994) derived the following formula based on the volume of pore to be penetrated from the core displacement data to determine the volume of acid required to achieve a certain wormhole length:

the ratio V/L is called acid coverage in bbl/ft. The equivalent epidermis is given as

Schwalbert et al (2018) defined stimulation coverage as twice the radius of wormholes relative to the length of perforation intervals, which for LEL completions is equal to the distance between LEL holes.

Turning to fig. 13, the epidermal factor as a function of stimulation coverage is graphically illustrated.

Thus, when the stimulation coverage 802 reaches 50%, the epidermal factor 800 becomes constant.

FIG. 14 shows that assuming all wormholes produced along a well have the same radius, an effective wormhole radius 804 of 20ft will result in an equivalent negative skin factor 800 of-4. In connection with the two graphs it is shown that the maximum distance between the wormholes should not exceed twice the wormhole length. For example, a skin of-3 for the entire well means that the hole 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 stimulation reservoir section divided by the total number of holes, should not exceed twice the desired final wormhole radius. Thus, the following checks are performed:

if the average hole constraint is not met, then the hole size needs to be adjusted, block 1034

Adjustment of hole size 1034:

based on the evaluation of the above equations, the following possible actions are taken:

if the distance between the LEL apertures is too small, the LEL aperture size can be increased by 1mm and then the entire simulation repeated.

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

If the average distance between LEL apertures is close to or equal to twice the radius of the wormholes, the algorithm has converged to the final design and continues to output the result, block 1036.

Output result 1036:

the output results include the following items:

node characteristics including location, pressure, velocity, friction factor, number of holes per foot, velocity, retention time, stimulation rate, cumulative volume of acid exiting the node through the hole.

Segment characteristics including number of segments, segment spacing, number of pores in a segment, distance between pores, calculated and designed acid coverage, acid coverage ratio, acid stimulation rate, acid velocity at exit point of pores, pore volume to penetrate, final wormhole radius, and final skin factor

Actual pressure drop over last orifice with specified pressure drop (dp)

Average total distance between LEL holes

Total number of LEL holes, total cross-sectional area of LEL holes, equivalent Inner Diameter (ID) of total number of LEL holes

Wellhead pressure and bottomhole pressure during pumping

Wellhead pressure and bottom hole pressure just after shut-in, referred to as instantaneous shut-in pressure (ISIP)

The volume of acid required, total pumping time, assuming pumping occurs at the designed rate.

Total liner volume, total pipe volume, displacement volume, hold time

A detailed statistical list containing the number and size of LEL holes for each joint to be run in the hole and the order of joints that must be run in the hole. Furthermore, the total number of engaging members having a certain number and size of holes, such as the number of engaging members having 0, 1, 2 or 3 LEL holes of 3mm, 4mm, 5mm or 6mm size etc. are summarized.

The wellhead pressure and bottom hole pressure during pumping are calculated from the pressure at the first node, and then the hydrostatic pressure is subtracted and friction is added up to a given gauge depth.

The wellhead and bottom hole instantaneous shut-in pressures ISIP are calculated from the pressure at the first node, and then the hydrostatic pressure is subtracted up to a given gauge depth. The friction is zero because the velocity is zero during ISIP.

And (3) calculating a program:

inputs to the numerical design model include:

average reservoir pressure

Crack propagation pressure

Permeability of

Porosity of

Length of completion interval

Radius of borehole

Pipe inner diameter (pipe ID), liner inner diameter (liner ID), pipe roughness

Acid characteristics (type, concentration, density, viscosity)

Number of sections

Acid coverage per segment

-the pore size of each section

-coefficient of emission

Step 1. Estimating the pumping rate

The software will then estimate the design pumping rate based on a standard instantaneous inflow model (rather than the Darcy model as a steady state assumption) while ensuring that the injection pressure remains below the fracture pressure. Key parameters include permeability, length of completion interval, and difference between annulus pressure and reservoir pressure. For the calculation it is assumed that the epidermis can be reduced to zero. Therefore, it should be noted that the injection rate is higher than that predicted by Darcy's formula, since stimulation operations typically require less than 24 hours. The reason is that the boundary has not been felt by the pressure pulses emitted during stimulation. Thus, even if the flow inside the liner is a steady state formulation, the inflow model for the pumping rate design is transient.

Step 2, estimating characteristics of wormholes

Nodal analysis calculations must be performed to estimate the downhole temperature at the liner heel. Based on the selection of acid system, permeability and temperature, the optimal speed for wormhole propagation and the expected pore volume to penetrate is estimated from published literature data. The Buijse-Glasbergen model was used to characterize wormholes at different speeds.

Step 3. Estimating the total number of holes and the pressure drop (dp) over the last hole of the holes

Based on the optimal velocity and the calculated design pumping rate, the total cross-sectional area of the holes is directly calculated. The cross-sectional area is linearly related to the pressure drop (dp) over the last of the holes, which is a key design parameter.

Step 4, estimating acid coverage rate

Stimulation designs aim to have a negative skin of-3 or better, which requires that the spacing of the holes does not exceed 30ft to 60ft on average. The model of Econoids et al (1994) was used to calculate the acid coverage required to achieve this epidermis. Higher acid coverage requires more acid and longer pumping time and is therefore more costly. This must be balanced against the goal of pursuing more negative epidermis.

Step 5. Calculating the optimized pore distribution

An initial estimate of the number of wells per segment is provided and the software finds a solution that satisfies the acid coverage per segment and the pressure drop (dp) across the last one of the wells. The initial estimate can be found from the relationship between the gap velocity, pumping rate and total cross-sectional hole area for a particular discharge coefficient and liner configuration.

Example (c):

to illustrate the design concept in more detail, examples are shown below. The well in question will have a reservoir length of about 7000 ft. Since the legs (3 drill pipe lengths) are about 91ft, the well is numerically divided into 8 sections each having a length of 910ft, corresponding to 10 legs.

The initial design coverage was set to 1bbl/ft. The instantaneous inflow equation predicts that the maximum rate is 20bpm without fracturing the formation, assuming the skin is zero. The rate may be further increased as stimulation progresses. The resulting pumping time will be 6 hours, which results in a slight adjustment of the design rate, but not too much.

Although the reservoir temperature was 250F or higher, nodal analysis based on a design rate of 20bpm predicted 140F for BHT at the first hole. This temperature is used to estimate the location of the optimal velocity for wormhole propagation based on the measurement curve and the Buijse-Glasbergen model.

The final skin was initially assumed to be-3, which resulted in a maximum distance of 30ft between adjacent holes. This corresponds to a pressure drop (dp) of about 30psia over the last of the holes, which is then used as an input to the design model.

The emission coefficient is assumed to be 0.70, which is between the theoretical minimum of 0.56 and the high value of 0.85-0.90. Post-operation analysis will help determine the pressure drop (dp) across the orifice and hence the actual discharge coefficient.

The first estimate of the pore size distribution utilizes a linear relationship between the pore cross-sectional area and the pressure drop (dp) across the last of the pores. Based on this initial input, the actual optimal pore size distribution is calculated using the numerical algorithm outlined. In the inner loop, the flow equation is solved. In the outer loop, the number of holes is adjusted to match the pressure drop (dp) over the last of the holes and the acid coverage for each segment.

The results of the calculations are shown in the four graphs above. The distance between adjacent holes is in the range of 20ft to 35ft, which results in optimal stimulation coverage (wormholes covering the entire well length). The distance is not uniform as the hole size is chosen to be a constant 4mm to avoid complicating the pilot design.

Based on the wormhole growth model of PVbt with gap rate, ecolomides inserts the minimum PVbt into the epidermis model and is based on a specific acid coverage of 1.0 bbl/ft. This produced an epidermal factor of-2.5, which is considered to be close enough to the initial estimate of-3. If a skin of-3 is desired, we need to increase the acid coverage, recalculate the pumping time, recalculate the flow rate, redesign the orifice size and then inspect the resulting skin.

While embodiments of the present invention have been described and discussed in detail above, the present invention is not to be considered limited to the particular embodiments. It will be appreciated by those skilled in the art that various modifications could be made to the described embodiments or features thereof without departing from the scope of the present invention.

In particular, the invention is not to be considered as limited to use in the LEL liner already described. Other systems involving materials flowing through a conduit and/or material formation may benefit from implementations of the present invention and embodiments described above.

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Claims (11)

1.A method of simulating fluid transport of an acid in a system for stimulating an oil or gas well in a material formation, the system comprising a limited inlet liner, wherein the limited inlet liner is divided into a plurality of sections having a length less than the total length of the limited inlet liner and comprising one or more apertures along the wall of the limited inlet liner for discharging fluid into the material formation, wherein the method comprises: calculating an initial estimate of the number of apertures along the wall of the limited-inlet liner and an estimated cross-sectional area of the apertures, wherein the estimated cross-sectional area is based on a rate for providing the amount of the acid required to generate a dissolution pattern and a pumping rate for maintaining an injection pressure below a fracturing pressure of the material formation of the oil or gas well, wherein the initial estimate of the number of apertures along the wall of the limited-inlet liner is calculated to achieve sufficient acid coverage of the acid per zone and sufficient pressure on a last one of the apertures, wherein a pressure drop (dp) across the last one of the apertures is linearly related to the estimated cross-sectional area; and adjusting the initial estimate of the number of apertures along the wall of the limited-entry liner if the measured acid coverage of each section and the pressure drop across the last of the apertures are not satisfied.

2. The method of claim 1, wherein the system comprises: executing a series of algebraic equations for initial pore size distribution guessing; calculating the acid coverage and the pressure drop (dp) over the last well; comparing the acid coverage and the pressure drop (dp) across the last of the wells to design variables in a first iteration; uniformly reducing the number of holes on one or more of the sections in a next iteration until a pressure drop (dp) over the last hole is met; or as a first step, uniformly increasing the number of holes on said one or more of said sections in a next iteration until a pressure drop (dp) over said last hole is met; and performing a second step, the second step comprising: as a first iteration, redistributing an existing number of the holes between different ones of the one or more sections; swapping one well for the segment where the calculated acid coverage is furthest from the design value; performing a next iteration until the acid coverage is satisfied; and performing the first step and the second step until a pressure drop (dp) over the last well and the acid coverage are met.

3. The method of claim 1, wherein the system comprises: running a simulation to determine a wellhead pressure once a pressure drop (dp) across said last of said holes and said acid coverage are met; adjusting a friction reducer and rerunning the simulation if the wellhead pressure exceeds a maximum wellhead pressure rating; and/or increasing the pipe inner diameter (pipe ID) in the presence of existing friction reducers; and/or reducing the pumping rate such that the maximum wellhead pressure rating remains below the maximum wellhead pressure rating.

4. The method of claim 1, wherein the system comprises: running a simulation to determine whether a distance between adjacent ones of the apertures along the wall of the limited inlet liner does not exceed twice an expected final radius of wormholes formed along the limited inlet liner; increasing the size of the apertures along the wall of the limited inlet liner by an amount if the distance between the adjacent ones of the apertures along the wall of the limited inlet liner is too small, and repeating the simulation; or if the distance between the adjacent ones of the apertures along the wall of the limited inlet liner is too large, reducing the size of the apertures along the wall of the limited inlet liner by an amount, and repeating the simulation; or if the distance between adjacent ones of the holes along the wall of the limited inlet liner hole is close to or equal to twice the wormhole radius, continuing to output the result.

5. The method of claim 1, wherein the constraint is: the annulus pressure exceeds the minimum reservoir pressure to avoid cross flow inside the wellbore; the annular pressure does not exceed the fracturing pressure to avoid fracturing; the wellhead pressure does not exceed a maximum design pressure rating;

the sum of the cross-sectional areas of all of the holes along the wall of the limited inlet liner can equal or exceed a minimum cross-sectional area to avoid additional pressure drop (dp) after stimulation during normal production or injection of the acid into the well; the average distance between two adjacent apertures along the wall of the limited inlet liner can be equal to twice the wormhole radius; and that the liner inner diameter (liner ID) does not exceed the wellbore size.

6. A method according to claim 1, wherein the method provides an initial estimate of the number of holes over the section of the limited inlet liner along the wall of the limited inlet liner that meets the acid coverage of each said section and the pressure drop over the last one of the holes of the section (dp) for a hole size distribution in the construction and operation of the system, and adjusts the initial estimate of the number of holes over the section along the wall of the limited inlet liner if the acid coverage of each section and the pressure drop over the last one of the holes of the section (dp) are not met.

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

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

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

10. A data processing system configured to perform the steps of the method as described in any one of claims 1 to 9 for stimulating a well.

11. The data processing system of claim 10, comprising any electronic system or device having a processor configured to perform the steps of the method and communicate the results of those steps to a user of the system or device, including but not limited to a computer, notebook, handheld electronic device, or electronic workstation.

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