This paper shows how certain standard practices used in the investment evaluation process may be wrong in certain circumstances, thus leading to an incorrect decision. The main reason behind these mistakes can be attributed to the routinary and inflexible use of the standard techniques, without making a prior critical analysis of them and their applicability to the project under study. Such analysis should fully contemplate the environment within which the project will take place and the objectives pursued by the evaluation.
The evaluation of a project can be divided sequentially into the following six stages:
(1) Preparing the different forecasts involved: Investments, Productions, Prices, Expenses, Taxes, etc.
(2) Dividing the project into periods.
(3) Solving the Cash Flow Equation for each of the periods.
(4) Adopting the Form of Discounting best representing the physical reality. Calculation of the economic yardsticks.
(5) Sensitivity analysis.
(6) Submission to Management. Decision making.
In many cases, as evaluations become a routine task, some of the stages of the process receive only superficial treatment, without due careful consideration. Thus, it is possible that a division into periods that is recommendable for many projects, or a form of discounting that is usually correct, be used in circumstances where they do not represent the physical reality.
Routine is one of the main causes of conceptual errors that are sometimes made in project evaluations. Other important causes are the universal use of methods and equations originally derived for particular cases, the use of economic yardsticks with physical meanings not fully understood by the analyst, and the lack of a logical reasoning process contemplating not just the project but also its environment, the temporary circumstances of the company, and the objectives of the evaluation.
The erroneous situations that will be discussed in this paper are the following:
Case A. Failure to apply the incremental criteria to all of the elements in the evaluation.
Case B. Improper duration of the periods.
Case C. Improper form of discounting.
Case D. Erroneous treatment of the intermediate negative cash flows when calculating the Present Value Index.
Case E. Arbitrary elimination of the last periods or sequential repetition of the projects for comparison purposes. Case F. Improper tax treatment.
Case G. In projects where more than one result is possible, such as in an exploration scheme, replacing the systematical evaluation of all of the alternatives and weighing the results through the Expected Value, for the analysis of just a number of particular cases.
Case H. Failure to consider some inflationary effects.
Case A. Failure to apply the incremental criteria to all of the elements present in the evaluation. The basic rule for analyzing the convenience of a given investment is to consider only its incremental effects, that is, to evaluate the difference between the performance of the company with the project and without the project. Because of this, incomes or disbursements that are independent of the investments being analyzed must not be included in the evaluation. The following are some examples of elements that are at times erroneously considered in an evaluation:
The equation of the rheological model of Herschel & Bulkley and the relevant expressions of pressure drops, valid both for circular and annular sections, are applied to determine the three characteristic parameters of a drilling fluid, having yield pseudoplastic behaviour, and flowing in the drilling hydraulic circuit, starting from circulation tests. A typical standard drilling hydraulic circuit consists of the surface circuit (stand pipe, rotary hose, swivel and kelly), the circular section (inside the drill string with variable diameters), the bit and the annular section (the gap between the wall borehole or casing and the drill string). In this circuit the drilling mud enters the drill pipe, comes out from the bit, flows up to the annulus up to the surface, where it after a short time for cleaning is put back in the circuit.
The parameters to be solved are the yield point , the consistency index k and the flow behaviour index n.
By means at least three flow tests at a certain drilling depth, with the bit off bottom, the pump rates and the relative stand pipe pressures are recorded.
The obtained N couple of values of stand pipe pressure and pump rate, the geometry of the hydraulic circuit and the fluid density are the input data for a numerical procedure to determine the three parameters of the considered drilling fluid.
In this way, using this numerical process, a non linear system of N equations (with N 3) with three unknowns (the three parameters of the fluid: n, k and ) is solved determining the Herschel & Bulkley rheological parameters.
This procedure takes into account the more probable solution for each tentative value of the flow behaviour index np, considering the infinite couples of and k satisfying the input value of stand pipe pressure, and the mean square deviation is calculated for each tentative value of, np: the minimum value of the MSD gives the solution tern of the non linear system of N equations.
In this paper a brief description of the mathematical model and the numerical process used will he reported and a calculation using field data from circulation test carried out in a surface section of an ultradeep well located in the Po valley, will be done.
The results will be compared with the obtained results using the readings on the same drilling mud performed on Fann VG 35 viscometer and it can be seen that not always the rheological tern determined from the viscometer data coincides with the equivalent rheological tern found considering the drilling well as viscometer.
Besides the stand pipe pressure relative to an 17 1/2" run (from 2900 m to 3060 m) will be monitored using this procedure: calculated SPP data using the equivalent rheological tern and the rheological parameters from viscometer readings, using different rheological models such as Bingham, Ostwald & de Waele and Herschel & Bulkley, will be compared to field stand pipe pressure data. It can be seen that the overall average error between measured and calculated SPP (using the Herschel & Bulkley equivalent tern) has been drastically reduced to very low error while the calculated SPP using viscometer readings with the most rheological models today used in practice could lead to large errors misleading an accurate evaluation of the SPP on the rig floor.
This method could be useful not only to calculate and predict exactly the SPP, but also to evaluate with accuracy the annular pressure drop and the corresponding ECD in order to have the maximum allowable pump rates without fracturing the crossed formation, besides could be used to monitor the SPP behaviour for potential occurring problem in the hydraulic circuit such as wash out, plugged nozzles and in the case of gas kicks in the well.
Also this method, if applied to different drilling depths, could give information on the influence of pressure and temperature, existing in the well, on the rheology of the drilling mud.
During drilling operations it is very important to know exactly the pressure drop along the hydraulic circuit for many reasons. The most important are the following:
The perturbation method provides approximate solutions of the well pressure for arbitrarily heterogeneous media. Although theoretically limited to small permeability variations, this approach has proved to be very useful, providing qualitative understanding and valuable quantitative results for many applications. The solution is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As presented in previous papers, deriving such permeability weighting functions appears as a complicated calculation, available only for special cases. This paper presents a simple and general method to calculate the permeability weighting function. In the Laplace domain, the permeability weighting function is easily related to the pressure solution of the background problem. Since Laplace pressure solutions are known for many situations (various boundary conditions, stratified and composite media etc), the associated permeability weighting function can be immediately derived. Among other examples, we calculate and discuss the well pressure solution for a horizontal well producing from a heterogeneous reservoir.
The trend for reservoir characterization has stimulated the study of well testing in more complex heterogeneous media.
Well testing in heterogeneous media has been studied by three approaches: exact analytical solutions, numerical simulations and approximate analytical solutions. Exact analytical solutions exist for a restricted class of problems, involving some simple symmetry: layered reservoir, single linear discontinuities, radially composite systems etc. Rosa and Horner computed the exact solution in the case of an infinite homogeneous reservoir containing a single circular permeability discontinuity. Most of these analytical solutions are written in the Laplace domain. Numerical methods can treat much more general situations, but have some disadvantages: their use is cumbersome, investigation is empirical and general insights are difficult to be extracted, results are inaccurate if the time and the spatial discretization were not carefully conducted. Approximate analytical solutions can be a practical way to understand the pressure behavior in geometrically complex heterogeneous media. Kuchuk et al. proposed one of these approximate methods. Another popular class of approximate analytical solutions is based on the first-order approximation obtained from perturbation methods.
This paper is related to these first-order approximate solutions of the well pressure in arbitrarily heterogeneous reservoirs. In particular, we propose an easy and general method to calculate the permeability weighting function in various flow geometries. In the next section, we define what the permeability weighting function is and review previous works in the domain. After that, we present our method to calculate the weighting permeability functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well.
The Permeability Weighting Function
The perturbation method is a well known technique to solve partial differential equations involving mathematical difficulties, like variable coefficients. According to this technique, we start from an easier problem, the background problem, to modify or perturb it. The full problem is approximated by the first few terms of a perturbation expansion, usually the first two terms.
In our context, we start from considering a background medium with permeability k0 and with specified boundary conditions. The permeability k0 may vary in the space, i.e k0 (xD) What is important is that the background problem has a known exact analytical solution, PDO (XD , tD).
The full problem has the same boundary conditions of the background problem but the permeability k(XD) differs from ko(XD) in arbitrary regions of the space. Rigorously speaking, k(XD) / k0 (XD) has to be close to 1 in order to obtain sound approximations. In practice, errors tend to be small, say less than 10%, even for relatively greater contrasts, say up to 10, between these permeabilities, depending on the specific problem.
It is very important to determine the economical feasibility of a fishing operation in order to know whether to continue or interrupt the fishing procedure. This decision, when erroneously taken, can often lead to unexpected losses of money and time.
In the past the decisions concerning whether to continue or interrupt a fishing operation were based primarily on the operator's previous experience. This procedure often led to wrong decisions and consequently unnecessary monetary losses.
This paper describes the implementation of a decision-making method based on risk analysis theory and previous operation results from the field under study. The method leads to more accurate decisions on a daily basis allowing the operator to verify each day of the operation if the decision being carried out is the one with the highest probability to conduct to the best economical result.
Fishing problems can occur during drilling, completion or production operations in oil and gas wells. These costly and undesirable remedial operations have been resulting in annual losses of US$ 500 million worldwide.
Recently a decision-making method to deal with this problem was developed. Based on risk analysis theory, probability distribution and previous operation results, the method conducts to more accurate decisions allowing the operator to verify if the decision being carried out is the one with the highest probability to conduct to the best economical result.
In Ref. 1, the basic concepts of the method were presented. Also derivations of all equations were presented and thoroughly discussed. Following, a review of the mainly concepts and equations.
- Economic fishing time (EFT): EFT represents, at a certain moment of the operation, the amount of time within the operation should be concluded in order to produce a satisfactory economic outcome.
- Probability of Success: This is the probability that the operation presents to be successfully concluded within the EFT.
- Expected cost of the operation: Value calculated using the principles of risk analysis theory. It involves not only the costs but also the operation probabilities of success and failure at a certain moment.
- Conditional probability of success at day i of the EFT:
- Conditional probability of failure at day i of the EFT:
This paper describes a simple procedure, based in the work of Vogel and Wiggins, to develop IPR curves at any stage of depletion or at any time, for solution gas-drive reservoirs, for two or three phase flow, from results of a Black-Oil reservoir simulator.
In many applications, the use of numerical simulators to represent reservoir behavior would require an excessive computational time. In such a case, one alternative is to use analytical Inflow Performance Relationships (IPR).
One application of IPR is a production optimization method implemented in this work to choose the best value of production parameters such as tubing diameter and choke size. In this procedure, only discrete values are considered and a economic factor is used to optimize an objective function which is the present value of cumulative oil production.
This procedure is compared with the results obtained from simultaneous simulation of reservoir and production facilities which requires much greater computational time.
This work shows: (1) a simple automatic procedure to generate IPR curves from reservoir simulators results, (2) an optimization procedure where only available values of each production parameter are considered; and (3) that dynamic IPR curves that vary with depletion can be used to represent reservoirs in optimization procedures. Problems related to the development of analytical IPR curves are also discussed.
The estimation of individual performance of oil wells can be used to determine, for instance, an optimum method of production, adequate design of artificial lift, success design stimulation, and treatments and forecast of production performance. In many of these cases, the utilization of numerical simulators results in a very high time consumption while IPR curves can be utilized to represent reservoir performance with low computation effort.
Gilbert utilized curves which related flow rate and pressure. He was the first who called them IPR curves.
Weller developed one method to calculate depletion performance in solution-gas drive reservoirs applicable for all saturation conditions considering steady state flow and variable gas-oil ratio.
Vogel presented an empirical method to estimate the pressure-production behavior of oil wells producing from solution-gas drive reservoirs based on reservoir simulator results. He used Weller method to calculate IPR curves with a variety of PVT properties and relative permeability data.
These methods considered only two-phase flow (oil and gas). Brown presented a solution developed in Petrobras to calculate three-phase flow IPR curves. This correlation uses a combination of Vogel type equation for the oil IPR and a constant productivity index (PI) for the water IPR. Oil and water fractions were held constant for all flowing bottom hole pressures.
Wiggins, Russell and Jennings developed analytical IPR curves based on the physical nature of the multiphase flow system. They extended these ideas from two to three-phase flow systems.
In this paper, it is described a simple procedure, based in the works of Vogel and Wiggins, to developed IPR curves at any stage of depletion for solution gas-drive reservoirs using results obtained from a Black-Oil simulator.
IPR curves are used here to optimize production parameters such as tubing diameter and choke sizes. Most of the published work about production optimization, for instance, Caroll, developed non-linear techniques to optimize production performance using some continuous variables. In this work, only discrete values are considered to optimize the objective function which is the present value of cumulative oil production. Therefore, this procedure can be used to optimize several parameters simultaneously with low computation effort.
The development of the mathematical model is based on the following assumptions: (1) homogeneous limited reservoir; (2) reservoir are initially above bubble point pressure; (3) radial flow, (4) Darcy's law for multiphase flow applies; (3) gravity and capillary effects are neglected; (6) isothermal conditions; (7) no gas solubility in water; and (8) fully penetrating wellbore.
This paper presents a new method for determining the effect of well location within any reservoir boundary on well performance. There are several types of analytical as well as numerical methods used to solve potential flow problems in bounded systems. However, these are limited in their treatment of the reservoirs geometrical shape. For two dimensional problems a powerful tool used is conformal mapping. In conformal mapping, a problem is transferred from a geometrical domain in which the solution is sought, to a domain in which the solution is known.
The transformation then provides the solution in the original domain. The practical limitation of conformal maps has always been that they must be computed numerically, except for simple domains where the exact conformal map is known. With improvements in computer technology the method can be used for fast, accurate and flexible computation of solutions to these problems. Traditionally, Dietz shape factors have been used to account for wellbore location within the drainage area. These have been presented for certain well locations in specific geometric domains. However, the technique described in this paper has been shown to be useful in determining solutions to flow problems in complex geometrical shapes, such as flow in fractured horizontal wells, under steady state flow conditions. In this study the basic techniques and their application to simple reservoir geometries will be presented. The results obtained compare closely with results obtained using the Dietz shape factors for certain limited wellbore and drainage area configuration. The application is presently limited to the steady state solution.
For several decades the majority of potential flow problems were solved using several simplifying assumptions and relatively simple geometrical domains. In recent times we have used numerical computing to solve these problems in virtually any domain as well as in three dimensional space. This computing power and expertise has not been wide spread and extensively harnessed. This is in part due to the fact, that most comprehensive reservoir simulators are not simple to use and the cost of such a resource can be prohibitively high, for small operators. Thus, the industry has continued to solve flow problems analytically by continued use of simplified flow domains, modified to account for non conformance from those domains. The objective of this paper is to highlight a technique used extensively in the past, with specific application to solving flow problems in reservoirs, with polygonal boundaries. The method provides some useful insights as wells as ease in solving flow problems for a variety of reservoir conditions, geometries and wellbore reservoir configurations. The method can be described as semi-analytical, since it requires both analytical as well as numerical computations.
Well Inflow Performance Model
The inflow performance model assumes that flow in the two dimensional plane can be extended into three dimensions by combining the two dimensional flow with the effects of flow in the vertical plane using an average reservoir thickness derived from structural contour maps or seismic surveys. The two dimensional flow is solved by use of conformal mapping theory, which guaranties the transformation of all boundary conditions in the physical plane to the mapped plane. A simple flow solution can then be applied to the mapped plane, to determine the flow in the physical plane. This theory has been shown to be accurate in the solving potential flow problems in other engineering disciplines. One of the key uses of this technique is that it is not necessary to have any geometric limitations on the reservoir and wellbore location.
Experience of Drilling the Horizontal Well VLD-1152 in Lagunillas Formation, Block IV, Lake Maracaibo Basin, Venezuela.
The main objective of the horizontal well VLD-1152, located in Block IV, was to improve recoverable reserves which was impaired by pressure depletion and reservoir heterogeneities. The well represents an important challenge because it is the first horizontal well drilled in a depleted pressure area and it was drilled within a small productive interval of 25 feet thick only.
A pilot area was selected after a detailed multi-disciplinary study by geologists. petrophysicists and reservoir engineers. New 3D seismic interpretation revealed a structural model that conformed well with pressure behavior of the area. New information from well VLD-1112 were utilized to update the petrophysical properties and the volumetrics These data were input to develop improved reservoir description and build a reservoir model for flow simulation.
The results indicated that Layer VII is the most important drainage target. The principal reasons for selecting this unit were, good mechanical stability of the rock, absence of a water front and a secondary gas cap and the presence of a regional shale ar the top that might be used to navigate drilling.
Despite some operational problems encountered in drilling, the results were mostly satisfactory. The entire pay was penetrated and the geology and petrophysics of the drilled area came in line with our model predictions.
A pilot area, containing approximately 30 wells located in Block IV in Lake Maracaibo Basin,was selected as the site for a horizontal well (Fig.1). The target reservoir, VLC-52/VLD-192 Lower Lagunillas, commenced production in 1957 with well VLD-192.
The reservoir, which is stratigraphically divided into L, M and N sands, has not been uniformly drained. Since 1960, most of the wells have been completed in the L and N sands; therefore, the M sand has been less depleted. (Fig. 2) Production declination was very intense and was partially controlled by a gas injection program in 1967. Dropping pressure however continued until getting 1000 psi (Fig. 3)
The drilling of VLD-1112 well at south of the selected area, contributed with valuable information to validate the petrophysical parameters and calculate a new OOIP number which was found to be 20 % greater than the initial estimate of 264 MMSTB.
Once the new geological model and petrophysical parameters were defined, the more prospective area with less operational risk was selected. The output was used as information to develop a dynamic model for the simulator. The results provided well defined boundaries conditions and indicated the absence of an independent aquifer.
The selection of the zone was based on a combined evaluation of different criteria of orientations, lengths and restrictions for the horizontal well. This zone showed low values of porosity and permeability and a depleted reservoir character, justifying drilling of horizontal wells in order to improve oil recovery and maximize the production rate. The recommended location was GOF-3.
The main target is Unit VII of M Sand Lower Lagunillas Member. This sand has an potential of 1200 STBOD, and is expected to have a water cut of 5% and a GOR of less than 1000 SCF/STB.
This paper presents an examination of the beneficial effects of aligned-interest alliances on overall project risk. Case studies in drilling and completion services demonstrate how alliances mitigate the risk of project cost overruns through joint planning, superior communication, and a clear understanding of work processes. Successful alliances emphasize common project goals and allow management responsibility for specific well operations and their inherent risks to be passed to the party with the most experience and knowledge.
A distinction is made between project risks (cost, schedule, safety, wellbore integrity) which may be assumed or shared by suppliers and project uncertainties (reserves, oil prices, weather) which should remain with the operator. As with any commercial transaction, prices for services must be linked to the terms under which they are offered. Typically, alliance contracts include bonus and penalty mechanisms. Other alternatives are also discussed. The risk- and reward-sharing characteristics of typical bonus/penalty contract provisions are examined and presented in detail. Introduction
An alliance as defined in this paper is a long-term relationship, founded on mutual trust and commitment, between a service company and an operating company. The focus is on providing benefits for both parties and may include sharing risks and rewards. The cornerstone of any strategic alliance is open communication, which helps lead to a spirit of teamwork among participants.
The term "alliance" is used broadly to describe a variety of relationships. In the context of this paper. however, the term "alliance" refers to substantially increased involvement of contractors in project management roles, such as (1) lead-services contractors or (2) project-well engineering coordinators, during well construction (including drilling and completion), production enhancement, and production maintenance. Such alliances are based on the belief that by combining resources during these phases, operators and service companies can both be more successful.
Formation of aligned interests and achieving a better understanding of the factors that drive additional costs into the suppliers' system are two ways to enhance a relationship beyond that of a traditional arrangement. An essential component of these arrangements is to ensure that all involved parties have a complete understanding of the total system cost as opposed to the price of individual products and services. In addition, a variety of incentive options can be implemented to further optimize revenue, and performance should be measured to help ensure continuous improvement.
The most common types of commercial options available within an alliance approach include the following:
- Lump sum-a fixed price is paid to the contractor. This option ensures project cost certainty but does little to align contractor and operator interests in long-term operating efficiency.
- Incentive/penalty payments-in addition to the price of the job, an incentive is paid to the contractor if the contractor finishes ahead of the original goal, or the contractor pays the operator a penalty amount if the contractor fails to achieve the original goal. This option aligns operator and contractor interests and encourages a team effort. Project success depends on the activities of all parties.
- Payment-related-to-production - the contractor is paid a per-barrel sum based on incremental production. This option aligns operator and contractor interests and provides a form of contractor financing.
Unreliability with drill stem testing (DST) tools is often experienced by operators in the development of high pressure/high temperature (HP/HT) wells. These environments are normally defined as having hydrostatic pressures of at least 10,000 psi and bottomhole temperatures of at least 300 F. Factors affecting reliability may include deteriorating mud conditions caused by increased depth and temperature ranges, high pressures. which hinder tool operation, and reduced sealing capability resulting from the high temperatures. Solutions to these issues are available, but in many cases, are not used because of insufficient pre-job planning.
This paper discusses a joint project undertaken by eleven major operators and service/testing companies in the Norway, UK area to address these problems and to determine methods that would increase the reliability of DST operations in HP/HT well testing. Emphasis was placed on the selection of the downhole equipment used for the HP/HT DST's and design of the full scale testing of the North-Sea-area field tools in use. Additionally, an investigation was to be made to establish methods that could be used to verify and increase equipment reliability. The project also included design parameters for contingency conditions that were not normally present during equipment operations but could exist in emergency situations.
Eleven tools including tester valves, multiple-cycle circulating valves, bypass valves and safety/circulating valves from different service company/suppliers were rigorously tested. The results of the tests, which compare the susceptibility of various tool design types to reliability problems, will be provided as well as the types of tool modifications that were subsequently made to increase reliability of the equipment for the HP/HT conditions. The data presented will also include results of testing conducted on the equipment modifications.
The method of testing helped determine changes needed to address reliability in HP/HT conditions and has demonstrated that DST tools could maintain integrity in this type of environment.
For over 60 years, drill stem testing has been used to determine reservoir parameters of potential producing zones. However, the early DST applications were traditionally limited to open hole at low to moderate temperatures and pressures. These conditions would include temperatures between 150 and 300 F and pressures ranging from 1,000 to 10,000 psi. With the expansion of offshore DST to cased-hole applications with increased temperature ranges of up to 410 F and increased hydrostatic pressures of up to 15,000 psi, traditional equipment experienced operational problems. These problems included the following in descending order of relative frequency:
1. Mud deterioration resulting from mud remaining static at temperature for several days, either in water-based systems in which the settling of solids caused the string to become stuck in the hole and to create plugs of solids that hindered pressure transmission to the tools, or in oil-based systems that increased in viscosity and gel strength, reducing capability to transmit operating pressures to downhole tools.
2. Tool design that was not fully suited for the conditions. This included insufficient pressure capabilities, especially in the area of air chambers, shear pin-operated tools that were susceptible to pressure and shock from the perforating guns, and string designs that were too complicated for the conditions. For example, several annulus pressure-operated tools run with operating pressures that are too close together because of an insufficiently low pressure test of the casing or liner lap.
Both of the above problems can be minimized with the application of prejob planning.
HoSim is a flow network solver which can determine rate and pressure at a given point in a flow system based on terminal and inflow relationships.
An analytic model for fractured horizontal wells has been integrated in conjunction with inflow through perforations and different geometries in the horizontal completion. The fractured well model used is based on the premise that a boundary value problem may be conformally mapped onto a complex plane. The model divides each fracture and the wellbore into a number of flow divisions. Flow into a section is based on the length of each section and the associated pressure distribution. The model is pseudo steady state, but has the capability of being integrated with a gridded reservoir simulator the see the reservoir behavior which may include water or gas coning over time.
This paper gives the theoretical development as well as a North Sea field case simulation and results. The simulator is a new tool that gives the completion/reservoir engineer a way to better optimize single or multiple fractured horizontal well designs for maximum recovery.
Evaluation of multiple completion and stimulation alternatives has become more difficult in recent years due to the many options available on the market today. One of the most challenging scenarios for simulation is the multiple fractured horizontal well. The potential for enhanced recovery coupled with large completion and stimulation expenditures creates a need for better understanding of these complex flow environments. The model presented here attempts to model a system composed of a reservoir drainage block, multiple orthogonal fractures intercepting a horizontal wellbore, and a completion within that horizontal wellbore. This model is constructed using the concept of a flow network, where each component of the system is further subdivided into multiple flow divisions.
Description of the System
A horizontal wellbore is exists within a rectangular drainage block which has dimensions of axbxh. The wellbore azimuth is coincident with the local least principal stress direction within the drainage block. Hydraulic fractures which are introduced in this well will therefore lie in a plane orthogonal to the wellbore. It is assumed that induced fractures are spaced evenly along the length of the wellbore, making it possible to divide the drainage block into subsections, with one fracture lying within each block. These subsections are indexed as b1,b2,...,bn. Each fracture is assumed to be rectangular, having dimensions ci, Lfi, hfi. A diagram of this system is shown in Figure 1. This subdivision of the problem into multiple subproblems assumes that there is no flow between adjacent subdivisions. The size of the end blocks is adjusted to account for the effect of the 'greedy' end fractures, but this assumption requires that each fracture under consideration be given the same conductivity profile when discretized in the network model.
The lengths of the subdivisions of the drainage block can be derived based on the geometry of the system described above by assuming that each equally spaced fracture is geometrically centered in its drainage block, with the exception of the first and last fracture. It can then be stated that the length of the first and last block is given by,