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Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 26–29, 2020

Paper Number: SPE-204264-STU

... Pressure. reservoir surveillance drillstem/well testing compressibility factor drillstem testing specific gravity calculation production monitoring upstream oil & gas bottom hole pressure gravity profile numerical simulation depth interval assumption

**eqn**gradient data production...
Abstract

As the economic importance of natural gas continues to grow because of its relatively clean output, the exploitation of natural gas wells is of increasing importance in the energy industry. Monitoring of these gas wells is key and the Static Bottom Hole Pressure is an integral parameter when we talk about reservoir/well evaluation. The Static Bottom Hole Pressure is most often acquired through downhole gauge measurements. However, this method is disadvantaged by associated risk and cost of execution. This paper presents a new method for calculating Static Bottom Hole Pressure. This new model considers the changes in the critical properties in the well in a fine grid segmentation of the well and uses the different values of specific gravity in a numerical simulation that involves a modification to the Cullender and Smith’s Equation that accounts for the variation in gas gravity profile values, fine grid segmentation, and well geometry. Based on the results obtained, this model was seen to provide a more accurate estimate than the existing methods. The model was tested on 30 wells using the generated software and its results were benchmarked against other exiting models and compared with gauge measurements to validate the error reduction. Error estimation on the model showed that this method gives an average accuracy of 99% with an average absolute error of 0.304%. The results of this work showed that this method was effective in estimating Static Bottom Hole Pressure.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 30–October 2, 2019

Paper Number: SPE-195947-MS

... perturbation technique flow in porous media Upstream Oil & Gas pore pressure well PI Permeability reduction analytical solution Poisson production rate Reservoir Characterization coal seam gas approximation variation pi solution nucleus permeability reservoir

**Eqn**equation bulk modulus...
Abstract

Reservoir depletion can induce substantial changes in the stress state of the rock. The coupled interaction between the pore fluid pressure and rock stress will then alter the reservoir permeability, which in turn reversely affects the productivity index of the production well. A new nonlinear analytical solution is developed for the drawdown-dependent productivity index of reservoirs under steady-state flow. Biot's theory of poroelasticity is used to derive the depletion-induced changes in the reservoir rock porosity and permeability. The well-known Mindlin's solution for a Nucleus of Strain in a semi-infinite elastic medium is applied as Green's function and integrated over the depleted volume of reservoir rock to obtain the 3D distribution of stress and volumetric strain distributions. The fluid transport equation is nonlinearly coupled to the solid mechanics solution via the stress-dependent permeability coefficients. A perturbation technique is applied to mathematically treat the described nonlinearity to solve for the coupled equations of pore fluid flow and rock stress under steady-state flow. The good match between the obtained analytical approximations for productivity index and the numerical solutions verifies the correctness and robustness of the proposed model. Results indicate and confirm the expected strong dependency of the well productivity index to the drawdown magnitude as well as the poroelastic constitutive parameters of the reservoir rock, with the highest sensitivity to drained bulk modulus, followed by the reservoir depth and solid-grain modulus. The lowest PI sensitivity is to the pore fluid modulus and Poisson's ratio. The resulting productivity index is found out to be drawdown-dependent, which can render values substantially different than the productivity index estimate from the conventional flow-only analysis. The presented estimates for the related nonlinear productivity index can be readily used by the practicing engineers.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 26–28, 2016

Paper Number: SPE-184482-STU

... costs. Upstream Oil & Gas normalized relative permeability curve

**Eqn**relative permeability curve reservoir simulation best case saturation graph flow in porous media automatic hm rock type run 1 Modeling & Simulation Fluid Dynamics permeability curve normalized relative...
Abstract

A typical hydrocarbon reservoir consists of multiple rock types or facies. In reservoir modelling, capturing the properties of each rock type is essential. Relative permeability data, which describes rock-fluid and fluid-fluid interactions is normally developed for each rock types and is often a tuning parameter in History Match (HM); a very important element of reservoir simulation. The objective of this paper is to analyze the effect of applying normalized relative permeability data to model a reservoir. The effect is examined using both manual and automatic History Match, and then predicting the Estimated Ultimate Recovery (EUR) in two scenarios. Relative permeability curves for each rock type are prepared using corey saturation function equations. The normalization phase requires using separate sets of equations, after which the normalized curve will be denormalized for input into the simulation model. The resulting relative permeability curve is then used in HM by applying manual and automatic methods, after which the resulting matches are used to predict recovery. The reservoir simulation output using the normalized relative permeability curve is then compared to the base case scenario, in which case, individual rock types relative permeability curves were utilized. Reservoir simulation outputs from normalized relative permeability curves were found to compare very well with outputs using individual rock type relative permeability, with improved efficiency, for the reservoir under study. Normalized relative permeability data, when used with sound engineering judgement, can be very efficient. Mostly desired as an essential part of History Match, having a single representative set of relative permeability data for the reservoir can improve efficiency and save costs.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 30–October 2, 2013

Paper Number: SPE-166081-MS

... Integrating

**Eqn**. (2) , yields: (3) u = η G 1 r ∫ r w r rPdr + C 1 r 2 + C 2 r According to the theory of poroelasticity for the plane strain condition, the radial stress component is equal to: ( Wang 2000 ) (4) σ rr = E...
Abstract

In-situ reservoir stresses change during the production period due to the reduction in pore pressure. Consequently, the upper limit of the drilling mud window; i.e., formation fracture pressure, reduces corresponding to the pore pressure depletion. Since drilling operations in critical situations such as deep water and depleted reservoirs require precise prediction of the mud window, it's of vital importance to accurately assess the reservoir stress path; i.e., the change of in-situ stresses with pore pressure. Currently used models for reservoir stress path prediction can be divided into two main categories: space- and time-independent and spatio-temporal models. Space- and time-independent models have been used for several decades. According to industry demand of precise stress path prediction, spatio-temporal models have recently been developed for stress distribution determination within reservoirs. However, the developed stress path models are based on the transient flow regime and can not be utilized in depleted reservoirs wherein the pressure response usually reaches the reservoir boundary. The aim of this study is to provide an analytical model for predicting reservoir stress changes at different times and locations within reservoirs during the pseudo-steady-state flow regime. Constitutive equations from the theory of poroelasticity are combined with a force-balance equation to find the relationship between reservoir stresses and pore pressure. The application of the proposed model for reservoir stress path prediction is illustrated using a numerical example.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 30–October 2, 2013

Paper Number: SPE-166074-MS

... volume (3) ζ = − ∅ ∇ . ( V f − V s ) (6) α = 1 − C s C b Drained bulk modulus, K b , is equal to the inverse of bulk compressibility. Substituting the aforementioned definitions back into

**Eqn**. (5) and defining S cr as storage capacity...
Abstract

Poisson's ratio is usually determined using well-logging, fracturing data and core samples. However, these methods provide us with a Poisson's ratio which is representative of only near-wellbore regions. In this paper, a technique is proposed by extending currently used pressure-transient testing concepts to include reservoir stresses. More specifically, the interference well test is generalized to find not only conventional flow parameters such as reservoir transmissivity and storage capacity, but also the average dynamic in-situ Poisson's ratio. This is accomplished by using the generalized diffusivity equation, which takes into account flow-induced stress changes. Firstly, a generalized diffusivity equation is formulated by considering a deformable porous medium. The main goal of the generalized diffusivity equation is to extend current well testing methods to include both fluid flow and rock mechanics issues, and to present a way to determine the rock mechanics-related property, Poisson's ratio, from interference well testing. The line source solution to the diffusivity equation is used to modify the current interference well testing technique. Field data is utilized to show the main steps of the proposed transient well testing analysis technique. An average in-situ value can be put in practice in different applications requiring accurate value of Poisson's ratio. Some examples of these include in-situ stress field determination, stress distribution and rock mass deformation, next generation of coupled fluid flow-geomechanical simulators, hydraulic fracturing design, wellbore stability analysis and sand production design. Using dynamic Poisson's ratio that could capture the flow-induced stress changes, we would be able to find the stress distribution due to production/injection within the reservoir more precisely. Accurate determination of stress distribution has a significant impact on wellbore stability issues in challenging wells, such as depleted and deep water reservoirs.

Proceedings Papers

#### What is the Characteristic Length Scale for Permeability? Direct Analysis From Microtomographic Data

Fabrice Bauget, Christoph Hermann Arns, Mohammad Saadatfar, Adrian Sheppard, Rob Sok, Michael Turner, Wolf Val Pinczewski, Mark Alexander Knackstedt

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 9–12, 2005

Paper Number: SPE-95950-MS

...

**Eqn**. 3 is defined by the percolation threshold of a non-wetting phase penetrating the pore space during a drainage (e.g., mercury porosimetry) experiment. This length scale is directly obtained from analysis of the digital image during the drainage process. Katz and Thompson21 argue that `c is given...
Abstract

Abstract Rock formation permeability is arguably the most important flow parameter associated with subsurface production and injection. Its importance is reected by the number of techniques (well-log evaluation and correlation, core measurement and well testing) used to estimate it. Clearly permeability should be linked to other porous media properties (e.g., surface area, porosity, pore/grain size). There have been numerous attempts over the last sixty years to establish a relationship between the permeability of a rock and other characteristic rock properties. Most empirical approaches for the prediction of permeability, which has units of length squared, propose a function of a characteristic length scale, formation factor (tortuosity) and porosity. The most widely used is the Carmen-Kozeny equation where the length scale is equated to the hydraulic radius (pore volume / pore surface area). Other length scales used include a critical pore radius associated with mercury injection experiments (Katz-Thompson), lengths associated with NMR relaxation (e.g., T2) and grain size and rock fabric measures. To uncover the relationship between permeability and other pore scale properties requires directly measuring the geometric and transport properties of the pore system. This is now possible with 3D microtomographic imaging (Knackstedt et.al. SPE 87009, Arns et.al. SPE 90368). In this paper we describe a comprehensive study of permeability correlation across a range of rock types. We directly compute permeability, formation factor, NMR response, hydraulic radius, rock fabric and texture, pore size and capillary pressure on 3D microtomographic images of 39 porous materials including over 30 clastic and carbonate samples from a wide range of reservoirs. Subsampling enables one to generate more than 6500 \independent" samples. Empirical correlations between permeability and various length scales are tested for a range of lithotypes including unconsolidated sands, homogeneous sands, consolidated reservoir sands, limestones and reservoir carbonates. We and that the most robust length scale correlation is based on the critical pore radius. All correlations which use the Formation factor as a measure of tortuosity give good predictions. Empirical correlations for permeability based on grain size perform well for permeabilities greater than one Darcy. Introduction A long standing and crucial problem in the study of ow in porous media is to relate the permeability k of a material saturated with a single uid to other petrophysical properties. Numerous correlations for permeability to a wide range of petrophysical properties (e.g., porosity, drainage capillary pressure, NMR response, grain fabric and texture, rock type and depositional environment) have been proposed. Testing of some correlations, where all necessary parameters are calculated, have been limited to a periodic array of spheres[1], model random sphere packs[2,3] and stochastic reconstructions of porous materials[4]. In the present paper our aim is to measure petrophysical properties directly from a large number of 3D digitized tomographic images and examine popular empirical correlations on realistic rock morphologies.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 9–12, 2005

Paper Number: SPE-97155-MS

... the deformations are the driving forcescausing changes in pressure.

**eqn**approximation effective stress stress change porosity depletion compressibility reservoir geomechanics reservoir simulator upstream oil & gas permeability assumption simulator modeling & simulation...
Abstract

Abstract Geomechanics is often represented in conventional reservoir simulators bypressure dependent treatment of porosity and/or permeability. The paper gives asystematic treatment of the subject and shows different methods forpressure-dependent approximations. Although there is no single method that would provide the best approximationunder all circumstances, reasonable approximations exist in several situations.For porosity coupling, the primary factor is the type of deformation. Accurateapproximations are given for the cases of uniform depletion with differentdeformation assumptions. For permeability coupling, new methods have beendeveloped and tested against coupled simulations. The results show clearly that large errors can result from simpleapproaches, in estimating both reservoir pressure decline and wellproductivity/injectivity. Introduction In coupled geomechanical and reservoir modelling, one can correctlyrepresent the dependence of reservoir porosity and permeability on effectivestress or deformation of porous media. However, because of the complexity andcost of coupled modelling, it is often necessary to approximate these effectsin conventional (uncoupled) reservoir simulators. Most commercial models offeroptions for pressure-dependent porosity (or rock compressibility) andpermeability. In spite of the popularity of the approach, confusion exists about how tobest translate the rock mechanics lab data (measured as a function of effectivestress) to functions of pressure only, and little is known about the errorsresulting from this simplification. This paper gives a systematic discussion ofthe subject and provides methods for the computation of rock compressibilityand pressure depemdent permeability for uncoupled modelling from the truegeomechanical data. Both aspects of the coupling (i.e., coupling via porosityand via permeability) will be treated. The methodology utilizes a combinationof analytical considerations and comparisons of coupled and uncoupledsimulations. In all reservoirs, the changes in pressure P and temperature T induced by recovery operations are accompanied by changes of stress state. Insome problems reservoir undergoes deformation caused by outside forces (e.g., from another reservoir or aquifer zone). In the first (more common) case the P and T changes are the driving forces causing deformation andstress changes, while in the second the deformations are the driving forcescausing changes in pressure.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 26–29, 2004

Paper Number: SPE-90370-MS

... is tested on experimental data from a laboratory set-up. reservoir surveillance measurement noise flow rate pressure measurement production monitoring soft sensor vector upstream oil & gas ekf model parameter spe 90370 gas-lift well unknown input

**eqn**ensemble kalman filter...
Abstract

Abstract This paper considers the use of extended Kalman Filtering as a soft-sensing technique for gas-lift wells. This technique is deployed for the estimation of dynamic variables that are not directly measured. Possible applications are the estimation of flow rates from pressure measurements or the estimation of parameters of a drift-flux model. By means of simulation examples different configurations of sensor systems are analyzed. The estimation of drift-flux model parameters is demonstrated on real data from a laboratory set-up. Introduction The smart well paradigm involves the instrumentation of wells with sensors and actuators, which can be used for monitoring and control purposes. From a monitoring point of view, the use of sensors that measure different properties at several locations is preferred. However, because of practical and economical reasons such demands are unrealistic. Some measurements, like pressure measurements, are more readily available than others (e.g. oil flow rate). To have access to the unmeasured variables, the concept of soft sensing is used in this paper: the unmeasured dynamic variables are estimated from the measured ones by fitting a model to the measurements using extended Kalman Filtering [6]. In this paper a gas-lift well is considered. Possible applications of soft sensing for gas-lift wells are the estimation of gas and oil flow rates from pressure measurements and the parameter estimation for models that describe the multiphase flow phenomena. The use of soft sensing for well operations has been described in e.g. [9], [10]. In [9], [10] ensemble Kalman Filtering is used as the soft sensing algorithm, whereas in this paper extended Kalman Filtering is used. The main difference between these two algorithms is the prediction of the state-covariance matrix: ensemble Kalman filtering uses an ensemble of nonlinear state predictions to construct the predicted state-covariance matrix, whereas extended Kalman Filtering uses a locally linearized model to predict the state-covariance matrix. For models that are moderately nonlinear, in the sense that the change of the dynamics is small within two subsequent sampling times, extended Kalman filtering works well since in such cases the linear approximation between two sampling times is accurate. For highly nonlinear models this approximation is no longer accurate, and the use of an ensemble of nonlinear predictions may improve the predicted state covariance matrix [4]. However, the nonlinearity of the gas-lift model considered in this paper proved to be modest in the investigated operating region, which justifies the use of local linearizations. According to [9] a disadvantage of the extended Kalman Filter is the large computational demand of the numerical linearizations, which require a number of model evaluations that is of the same order as the number of state variables. However, in the ensemble Kalman Filter in [9] and [10] the number of nonlinear model predictions is set to 100, which is also of the same order as the number of state variables (159 in [10]: a discretization of 20 meters for a 1000 meter well results in 50 sections and each section consists of 3 states, additionally 9 states are used for the estimation of model parameters). Drawing the ensemble members randomly from a distribution introduces a stochastical component in the prediction of the state-covariance matrix of an ensemble Kalman Filter. This stochastic dependency on the random realization of the ensemble can be circumvented by choosing the realizations as suggested in [4] and [5], but this results in a number of ensemble members that is twice the number of state variables. Thus with respect to the computational load the extended Kalman Filter may be preferred over the ensemble Kalman Filter in the case of the application for gas-lift wells. The organization of the paper is as follows: first brief descriptions are given of both the model for the gas-lift well and of the extended Kalman Filter. Next, different measurement configurations are analyzed by means of simulations: the use of pressure measurements along the tubing, and the use of topside measurements from the annulus and the tubing. These configurations can be used for the on-line estimation of the gas and oil flow rates in the tubing, acting as a multiphase-flow soft-sensor. Besides, unknown model parameters can be estimated on-line in order to keep the model on track. For the estimation of unknown parameters of a drift-flux model, the soft sensor is tested on experimental data from a laboratory set-up.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 5–8, 2003

Paper Number: SPE-84181-MS

... single phase region phase boundary phase region transient simulation simulation calculation pipeline simulation computation time stability analysis initial estimate shadow phase region coefficient critical point timestep

**eqn**computational speed composition trivial solution pvt...
Abstract

Abstract Approaches to reducing the computation time spent on flash calculations in compositional, transient simulations are presented. In a conventional flash calculation the majority of the simulation time is spent on stability analysis even for systems far into the single phase region. A criterion has been implemented for deciding when it is justified to skip the stability analysis. With the implementation of the developed time saving initiatives, it has been shown for a number of compositional, transient pipeline simulations, that it is possible to reduce computation time spent on flash calculations by between approximately 85–90%. Introduction Oil production involves a series of transient flow scenarios. Examples are miscible gas displacement in petroleum reservoirs and multiphase flow in pipelines including slug formation and start-up scenarios. In a typical transient flow simulation the system is discretizised into a number of cells or sections. Phase amounts and phase properties are needed for each cell or section to solve the conservation equations in the model, and specifically for transient pipeline simulations to calculate e.g. heat loss to the surroundings, liquid holdup, and pressure drop. If the overall composition is constant during the simulation, the phase properties can be stored in precalculated tables listing the needed properties as a function of pressure and temperature [1],[2] . This is in the following referred to as a non-compositional table based simulation. When simulating e.g. miscible gas displacement in reservoirs, the assumption of a constant overall composition with time and location is not adequate, since the injection gas will dissolve in the reservoir fluid and vice versa. Similarly in many typical multiphase pipelines, the fluid composition will vary due to velocity differences between phases, interfacial mass transfer, and merging networks. In these situations a compositional approach is useful. A compositional model has the drawback that the computation time is higher than that of a non-compositional, table based approach. The phase amounts and properties must be evaluated in each cell or section in each timestep. Furthermore phase compositions are required to calculate the interfacial mass transfer. Nevertheless the increased accuracy in the fluid description makes the compositional approach to represent multiphase pipelines and reservoirs with large compositional variations attractive. Flash Equilibrium Calculations in Compositional, Transient Simulations The physical properties of a fluid in a cell or section depend on whether the fluid is present as a single phase or splits into several equilibrium phases. A flash calculation is therefore required in each timestep in order to determine the number of equilibrium phases and their amounts and compositions. Even with relatively few mixture components the computation time of a compositional, transient simulation far exceeds that of a similar table based transient simulation and the computation time increases with the number of components. Analyses show that even with an optimized algorithm the flash computation time will typically constitute 70–80% of the total computation time required for a compositional, transient pipeline simulation. It is therefore highly desirable to be able to reduce the computation time associated with the flash calculations. Consider, as an example, a typical transient pipeline simulation. The total number of discretizised sections depends on the total length of the pipeline and the desired degree of detail in the simulation, but a typical number of sections is around 100. The total number of timesteps depends on slug formation, liquid holdup, and varying boundary conditions, but is often of the order of 50,000. That is, a compositional, transient pipeline simulation comprising 100 sections and 50,000 timesteps requires 5 million flash calculations.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 29–October 2, 2002

Paper Number: SPE-77688-MS

... provided a careful choice of process parameters is made, our method produces reliable estimates. regularization

**eqn**spe 77688 square problem production control production monitoring flow period constraint error level pressure match upstream oil & gas pressure signal error weight...
Abstract

Abstract Current trends towards permanent downhole instrumentation allow the acquisition of large sets of well test data ranging over much longer periods of time than previously imaginable. Such data sets can contain information about the reservoir at a substantially larger radius of investigation than that accessible to conventional derivative analysis, which is limited to the interpretation of single flow periods at constant rate. By contrast, deconvolution methods do not suffer from this constraint as they are designed to perform well test analysis at variable flow rate. Recently we presented a new method for the deconvolution of well test data in which the problem is reformulated as a separable nonlinear Total Least Squares problem which accounts for uncertainties in the measurement of both rate and pressure data. 8 In this paper we report a number of improvements to our algorithm, and derive error bounds for rate and response estimates in the presence of uncertainties in the data, for which we assume simple Gaussian models. We illustrate our method by applying it to a small simulated example and two large sets of field data with up to 6000 hours of pressure data and up to 450 flow periods. Introduction With current trends towards permanent downhole instrumentation, continuous bottomhole well pressure monitoring is becoming the norm in new field developments. The resulting well test data sets, recorded mainly during production, consist of hundreds of flow periods and millions of pressure data points stretched over thousands of hours of elapsed time. Such data sets contain information about the reservoir at distances from the well which can be several orders of magnitude larger than the radius of investigation of a single flow period. Conventional derivative analysis is therefore ill equipped to access the full potential information content. What is required is an analysis method which can estimate the response which the reservoir would exhibit when subjected to a single drawdown at constant rate over the entire production period. In mathematical terms, this is a deconvolution problem. Over the last 40 years, this problem has received sporadic, but recurring attention in the Petroleum and Ground Water Engineering literature. At last year's SPE Annual Technical Conference, we presented a new approach which is based on a nonlinear Total Least Squares formulation and accounts for errors in the measured rates as well as in the pressure signal 8 (henceforth referred to as "paper I"). We tested our method with simulated well test data as well as a small-scale field example with 31 hours of usable duration. Since then we have successfully applied our method to a number of substantially larger data sets from production wells with permanent downhole gauges. In this follow-up we give a brief summary of the current state of our approach, including some minor modifications made since the first paper appeared, and a list of unresolved issues. We also derive analytic expressions for the expected bias vector and covariance matrix of the estimated parameter set based on simple Gaussian models for the measurement errors in pressure and rate signals. We then illustrate our method with a small simulated data set, showing the effect of varying levels of regularization on bias and variance. The second part focuses on practical aspects and includes results from some of the larger field examples mentioned above, including one which allows a direct comparison of our method with derivative analysis. Our experience to date suggests that, within reasonable limits of data quality, and provided a careful choice of process parameters is made, our method produces reliable estimates.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 3–6, 1999

Paper Number: SPE-56757-MS

... present optimal mud weight selection criteria to drill multi-lateral junctions. directional drilling oval hole upstream oil & gas critical collapse pressure ellipse junction

**eqn**reservoir characterization society of petroleum engineers drilling operation borehole pressure stress...
Abstract

Abstract The number of multi-lateral wells are increasing in the North Sea area. Potential for improved accessibility and recovery has led to a significant interest both from oil companies and the service industry. With improved technology, these multi-lateral branches will be key elements in the development of the so called "smart" wells. Several multi-lateral wells recently drilled in the North Sea have experienced problems. One well had severe problems as the casing inside the branch deformed into the main wellbore, making both branches unavailable for reentry. Clearly, the formation at the junction had collapsed, forcing the casing to deform. A rock mechanics study was undertaken. The hole geometry above the junction was a circular hole, which became oval at the junction, and then split into two adjacent boreholes below the junction. Since the solution to this problem was not found in the literature, a new mathematical model had to be developed to model these geometries. This model was coupled to a stress model and a failure model for the rock. The paper will present the complete models. It was found that both the critical fracturing pressure and the collapse pressure changed as the hole geometry changed from a circular hole. The paper present a field case using the new models. The study showed that the stress concentration increased as the hole became oval at the junction. It was found that the oval geometry created more severe conditions both for mechanical hole collapse and fracturing. It was also found that at certain borehole pressures, this geometric effect diminished. One main conclusion of the study is that the allowable mud weight window is smaller at the junction itself compared to the holes above and below. The fracturing pressure is lower, and, the critical collapse pressure is higher in the junction than outside the junction. Therefore, the optimal mud weight is the most important condition for trouble-free drilling of a junction. The optimal stress conditions for the geometries will be defined in the paper, and based on these, the paper will present optimal mud weight selection criteria to drill multi-lateral junctions.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 3–6, 1999

Paper Number: SPE-56628-MS

... design andersen pipe

**eqn**pipe strength coefficient differential sticking pull force buoyancy Copyright 1999, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 3 6 October 1999. This...
Abstract

Abstract The complexity of the wells have increased significantly in later years. Reach has more than doubled, and high inclination and fully 3-D well paths are common. However, statistics shows that sidetracking the boreholes due to stuck pipe has also shown a significant increase, and is presently a high cost factor. The margins between success and failure are now much smaller. A larger study was initiated to understand the stuck pipe situation better, and to develop improved procedures. A mechanistic approach was chosen. The following elements were analyzed: the forces developed during differential sticking, pipe strength under combined loads; tension, torque and pressure, effects of buoyancy under various conditions like equal or different mud densities in drillpipe and annulus, wellbore friction as related to torque and drag. This paper presents new equations to determine depth to the stuck point in deviated wellbores, based on pulling tests and torsion tests. In particular it is shown that bends in the wellbore leads to more friction, which with the new equations results in a deeper stuck point in a deviated well compared to a similar vertical well. Knowing all the forces involved in a stuck pipe case, another analysis was performed to determine the action that has largest impact to free the pipe. One of the main conclusions is that the most important element to free the string is to keep the bottomhole pressure as low as possible. The paper will present three methods to free the pipe, which where developed from the analysis: maximum mechanical force method, minimum density method, and maximum buoyancy method. A detailed field case from the Yme field in the North Sea will demonstrate these methods, and show the effect on the stuck point using each method. In addition to the field case, the paper will in the Appendices present the complete equations for pipe stress and strength under 3-dimensional loading, and define the effects of buoyancy in deviated wellbores. In particular is the buoyancy issue resolved, showing the differences and similarities between "the piston force approach" and the law of Archimedes, as applied to deviated wellbores.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1998

Paper Number: SPE-49313-MS

... probability estimation variance artificial intelligence upstream oil & gas standard deviation correlation matrix reservoir simulation probabilistic reserve estimation model outcome

**eqn**uncertain variable spe 49313 uncertain input modeling & simulation point estimate method...
Abstract

Abstract This paper describes three alternatives to Monte-Carlo simulation: the first-order second moment method, the point estimate method, and the first-order reliability method, for the assessment of uncertainty in reservoir engineering calculations. These techniques arc proposed as more suitable than Monte-Carlo simulation when information regarding parameter uncertainty is limited, the forward model is computation intensive, and/or the probability associated with only a limited number of model outcomes is of interest. P. 785

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1998

Paper Number: SPE-48964-MS

... current methods give only parts of the information. P. 91^ extrapolation laplace transform upstream oil & gas artificial intelligence kernel function initial pressure estimation pressure history drillstem testing interpretation drillstem/well testing

**eqn**coefficient machine...
Abstract

Abstract This study presents a new way to integrate flow rate history and pressure history data in well-test analysis. The method used is an extension of the deconvolution principle to a chosen period of time before the well-test. It is based on the recovery of missing parts of the pressure history by taking into account the flow rate history and the available sections of measured pressure history. The method has the advantage of working without making any assumption about the reservoir by using the data itself to define the "model". A real example with 71 pressure points and 70 corresponding flow rate points shows how it is possible to recover the first 20 pressure points correctly considering that only the flow rate history and the last 51 pressure points are known. The main purpose of this procedure is to have an alternative to treat the problem of flow rate variations both before and during the well-test. In such a case, neither the deconvolution (only applicable for flow rate variations during the well-test, when flow rate and pressure are both known), nor the multirate superposition plots (too constraining on the shape of the flow rate history and on the late time reservoir behavior) can be applied properly. In addition, the principle of the method enables us to extend its use to the analysis of well-tests with pressure recording errors or missing pressure recording. Examples show that it is possible to make a good analysis of a well-test with missing pressure regions when current methods give only parts of the information. P. 91^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 23–26, 1990

Paper Number: SPE-20658-MS

... the fracture begins to screen-out the pressure increases (positive ap ) and the rate of fracture propagation(aL and aH) decreases toward zero. The dimensionless pressure slope for the rate case (IlIa) is about 2.5 to 1 (aH = 2.5 for two active wings). From

**Eqn**. (B-5) the slope for an efficiency of...
Abstract

SPE Members Abstract This paper presents a foundation and methodology for real time three-dimensional hydraulic fracturing simulation and analysis. The equations governing fracture propagation are summarized for both rate and net pressure boundary conditions. A new dimensionless pressure slope parameter Is introduced which prevents using chaotic measured or calculated pressures. This parameter also helps identify near wellbore restrictions. The real time fracturing simulator utilizes the same numerical modules and routines as the design program. This will insure design, real time and program. This will insure design, real time and post-design simulation compatibility. Simulated post-design simulation compatibility. Simulated results for the rate and net pressure driven models can be displayed concurrently during the hydraulic stimulation. Comparative studies of the rate and net pressure driven numerical results are included. A posttreatment analysis of a real time field case study posttreatment analysis of a real time field case study is presented to illustrate the application of this real time fracturing system. The importance of using a dimensionless pressure slope is also discussed. Introduction During the 1980's on-site monitoring of the hydraulic fracturing process evolved to new levels of sophistication through the use of on-site computers. This enabled the operator to perform numerous analyses and fracture diagnostics during stimulation The operator no longer had to rely on strictly digital data but was now able to view the fracturing process graphically. When such monitoring systems were coupled with skid technology detailed pumping and blending quality control analyses were possible. The information provided by such monitoring systems began to impact provided by such monitoring systems began to impact the decision making process on-site and on future treatments. The process monitoring capabilities of on-site computers with skid units has proven to be a useful and cost effective application of technology. Currently on-site computers are used for monitoring, quality control analyses, pressure diagnostics and real time fracture simulation using rate and pressure driven 3-D models. A description of the hardware/software and real time hydraulic fracturing simulator is discussed below. A number of comparative studies between the rate and pressure driven solutions are presented and illustrate the similarity of numerical results. Finally, a real time field example is shown for the rate and pressure driven solutions. DESCRIPTION OF HARDWARE/SOFTWARE AND FRACTURE MODEL Hardware and Software The on-site monitoring and fracture software is designed to run on a 80386/80486 IBM compatible computer. The system requires four megabytes of memory, a math co-processor, hard disk, VGA graphics and an RS-232 communications port. This hardware configuration is commercially available on many laptop, portable and desktop models. A printer can also be used to allow real time printer can also be used to allow real time digital and graphical output to be down-loaded during monitoring. The system is portable and can be easily interfaced by a common RS-232 interface. The system can be interfaced to another computer such as a service company monitoring system or be interfaced to a data acquisition system for independent transducer monitoring. P. 417

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 2–5, 1988

Paper Number: SPE-18020-MS

... drilling fluids and materials diameter reservoir surveillance gas concentration transition upstream oil & gas production control annular pressure drilling transportation rise velocity gas velocity society of petroleum engineers

**eqn**migration flow pattern well control...
Abstract

Abstract This paper presents a new method to predict both gas velocity and gas concentration for processes of well control operations in vertical or processes of well control operations in vertical or near vertical wells. The effect of gas concentration on gas velocity is taken into account. The dispersion of the gas-contaminated region of drilling fluid with time can be determined for both static and circulating well conditions. The method was integrated into an existing well control simulator computer program. The improved numerical mode closely predicted measured casing pressure profiles that were obtained in a 6000 ft research well. Introduction The control of high pressure gas encountered while drilling for hydrocarbon reservoirs is an expensive and dangerous problem of the oil producing industry. When unexpected high producing industry. When unexpected high pressures are encountered, flow of gas from the pressures are encountered, flow of gas from the formation to the wellbore occurs. Once detected, gas influx is stopped by shutting-in the well. This is the first measure in a series of corrective actions taken to bring the well under control. Models of well control operations are useful for training of drilling personnel, for analysis of past blowouts, and for evaluating alternative pressure control procedures. Computer advancements have made possible the development of well control simulators. Early well control simulators assumed that the gas enters a wellbore as a continuous slug. Recent simulators consider more than one two-phase flow pattern, but some flow-pattern boundary conditions are arbitrary. A common feature in previously published well control models is a constant slip velocity of the gas contaminated region. The slip velocity is defined to be either zero early models, or the terminal velocity of a single bubble. Unfortunately, these assumptions do not account for the dispersion of the gas contaminated region of drilling fluid with time and often lead to inaccurate results. The main objective of this paper is to present a model that will permit a determination of both the velocity and concentration of gas contaminated regions of drilling fluid during pressure control operations for complex flow geometry such as that present on floating drilling vessels in deep water. present on floating drilling vessels in deep water. Ultimately, it is hoped that a more complete understanding of the behavior of gas contaminated regions of drilling fluid will lead to improved blowout prevention practices. The model presented permits improved estimates of down-hole permits improved estimates of down-hole conditions in the field as well as the development of improved computer software for simulating well control operations. The work is limited to vertical or near vertical wells. It is important to point out that the problem of gas influx in the wellbore during conventional well drilling operations is not continuous and thus differs from the classical concurrent upwards two-phase flow applied to oil and gas production. Concurrent steady-state flow is generally approached only in the bottom portion of the borehole during the time prior to the detection of the gas influx. Upon detection of the gas influx, the gas is shut-in. At this point, the gas contaminated zone will migrate upwards due to its buoyancy. Finally, the procedures of well control operations call for the transportation of the gas contaminated region. Fig. 1 shows the different processes of the well control operations which must be modelled. These processes will be referred to as (1) generation, (2) migration, and (3) transportation of the two-phase region. P. 19

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 22–26, 1985

Paper Number: SPE-14207-MS

... upstream oil & gas flow in porous media society of petroleum engineers gas well inertial resistance

**eqn**permeability wellbore inertial coefficient thickness equation machine learning porosity fluid dynamics coefficient artificial intelligence spe 14207 correlation core plug...
Abstract

Abstract The pressure drop due to non-Darcy flow in a gas well producing at high rates can be determined from analysis producing at high rates can be determined from analysis of multi-rate pressure test data. In the absence of such data calculation of the non-Darcy term requires a knowledge of the coefficient of inertial resistance. This paper presents correlations for estimating the coefficient of inertial resistance from reservoir porosity, permeability and gas saturation. These porosity, permeability and gas saturation. These correlations were derived from analysis of multi-rate pressure tests conducted in over one hundred gas wells pressure tests conducted in over one hundred gas wells and are supported by experimental data using core plugs. They are found to be better than other plugs. They are found to be better than other published relationships in predicting observed published relationships in predicting observed performance. performance Introduction It is well known that deviation from Darcy's law occurs for fluid flow through porous media at high velocity. This phenomena is especially significant in the region surrounding the wellbore of a gas well producing at high rate. In this situation the Forchheimer equation can provide a better description of flow than Darcy's law. The Forchheimer equation may be represented by: (1) where a is the coefficient of viscous flow resistance (=1/k). The term beta is generally known as the coefficient of inertial resistance although other names have been proposed in the literature. The second term in this equation represents the extra pressure drop caused by the high flow rate. pressure drop caused by the high flow rate. The value of beta can be estimated from laboratory measurements on core plugs and from multi-rate pressure tests conducted in a well. Correlations pressure tests conducted in a well. Correlations published in the literature have all made use of published in the literature have all made use of the first method, and are often found to predict flow behaviour with unacceptable accuracy. In this work the second method has been used to calculate beta and experimental measurements on core plugs were also carried out for comparison. The plugs were also carried out for comparison. The beta-factors obtained have then been correlated with available reservoir parameters. The empirical correlations obtained by this method could be used in predicting the non-Darcy pressure drop for other wells where, due to reservoir or operational constraints it may be difficult to conduct any extended multi-rate flow test. INERTIAL AND TURBULENCE EFFECT In the early literature it was supposed that fluid flowing at high velocity through porous media would behave like a fluid flowing through a conduit and suffer energy loss from turbulence. The coefficient beta was therefore often termed "turbulence factor". More recently it has become clear that turbulence does not play any significant part on flow behaviour of interest in reservoir engineering, and that the deviation from Darcy's law is the result of: the convective acceleration and deceleration of fluid particles as they travel through the pores, and the displacement of fluid particles from a straight line due to the tortuous path in a porous media.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 16–19, 1984

Paper Number: SPE-13057-MS

... mathematical solution of

**Eqn**. 1, subject to the appropiate boundary conditions, is given in Appendix A. The final form of the solution of the pressure in the fracture, in dimensionless form, is written as: I J PfD(xD,zD) = PwD + L: L: A.· (xD' 2 D)qfD' · i=l j=1 lJ lJ The coefficients Aij(x0,zD) are given by...
Abstract

SPE Members Abstract A semianalytical solution was developed for the transient flow behavior of a reservoir with a well intersecting a partially-penetrating vertical fracture of finite conductivity. The transient pressure behavior of a well in this kind of system consists mainly of three flow periods: the early time, the infinite acting and the pseudoradial flow periods. The results of this study show that the flow behavior of the partially-penetrating fracture during the early time period is equivalent to that of a totally- penetrating fracture. This period consists of a penetrating fracture. This period consists of a bilinear flow period for low conductivity fractures, and of a linear reservoir flow period for moderate to highly-conductive fractures. The onset of the infinite-acting flow period is directly proportional to the square of the dimensionless fracture height, which is defined as the ratio between the fracture height and the fracture half length. The results show that as the value of this ratio becomes small, the infinite-acting flow period starts at very early times, such that the bilinear and the linear reservoir flow periods might not appear in the well response for practical values of time. The approximate start of the pseudo-radial flow period does not depend significantly on the fracture period does not depend significantly on the fracture conductivity, an the fracture penetration ratio, or on the dimensionless fracture height, for moderate to highly conductive fractures, and for fracture penetration ratios larger than about 0.2. The effect penetration ratios larger than about 0.2. The effect of the dimensionless fracture height on the pressure response of a partially-penetrating fractured well becomes negligible for penetration ratios larger than about 0.8. The vertical location of the fracture affects the behavior of the well only after the upper and/or lower boundaries of the reservoir become noticeable in the pressure response of the well. The same solutions are pressure response of the well. The same solutions are found for the early time and for the infinite-acting flow periods, until the boundary effects become evident. Introduction The effectiveness of hydraulic fracturing in increasing the productivity of damaged wells and wells located in low-permeability reservoirs has been recognized for many years. It has been known for some time that data obtained from tests of fractured wells reflects the characteristics of the fractured well-reservoir system. Hence many studies have been undertaken to provide the means to evaluate the benefits of fracturing operations. The effect of the conductivity of the fracture on the behavior of a fractured well-reservoir system was recognized early, and is reported in the works by van Poollen et al., Dyes et al:, McGuire and Sikora and Poollen et al., Dyes et al:, McGuire and Sikora and Prats. Steady-state results concerning the increase Prats. Steady-state results concerning the increase in productivity that a well would experience after fracturing were obtained by van Poolen et al., using a potentiometric model, by McGuire and Sikora, using an electric analog, and later by Prats through analytical procedures. Dyes et al. were mainly interested in the effect of fracturing on waterflooding operations. Their study was conducted using an electric analog. Among subsequent works that contemplated the transient behavior of a vertically-fractured well were the work by Scott who used heat flow analogy, and a later work by Russell and Truitt who used a finite-difference formulation of the problem. Totally-penetrating fractures of infinite conductivity were considered in these studies. Gringarten et al. obtained and analytical solution to the problem of transient flow of fluids towards fractured wells. Results were presented for the cases of wells with infinite conductivity and uniform flux vertical fractures. The method applied to solve these problems was based on the use of Green's and Source problems was based on the use of Green's and Source Functions whose usefulness in solving transient reservoir flow problems had been documented in a previous study. Type curves of the transient pressure previous study. Type curves of the transient pressure behavior of fractured wells were provided for use in type-curve matching procedures of well test data.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 16–19, 1984

Paper Number: SPE-13231-MS

... al, whose major concern was to study the sensitivity of fracture design parameters on ultimate well performance. dimensionless ipr curve sandface pressure reservoir calculation correlation

**eqn**upstream oil & gas objective well performance psia gas deliverability calculation...
Abstract

Abstract A new method for deliverability calculations of gas wells, which eliminates the need for conventional multipoint tests, is presented. Using the analytical solution for real gas flow under stabilized conditions, and a broad range of rock and fluid properties, an empirical relation for calculating gas well deliverability in an unfractured reservoir is developed. This relationship is similar to Vogel's dimensionless Inflow Performance Relation (IPR) for solution-gas drive reservoirs. Also developed in this study is a second empirical relation, which estimates future deliverability from current flow test data. A simple procedure for gas deliverability calculations, utilising procedure for gas deliverability calculations, utilising these two relations, is suggested. Introduction Deliverability testing is a commonly technique for predicting short-term and long-term behavior of gas wells. Typically, a well is flowed at different rates, and the pressure-rate-time response recorded. From an analysis pressure-rate-time response recorded. From an analysis of these data, information is obtained regarding the deliverability of the well, i.e. the ability to produce against a given back-pressure at, a given stage of reservoir depletion. Such forecasting is often the required input for designing production facilities, planning field development, estimating payout time, setting allowables etc. Traditionally, deliverability testing has been done using elaborate multi-point flow tests, e.g. Flow-after-flow tests, Isochronal tests, and Modified isochronal tests. Details of these method,; are well described in the literature, and will not be discussed here. Inflow performance refers to the relationship between now rate and sandface pressure at any given average reservoir pressure. By graphing flow rate as a fraction of the theoretical maximum rate, against sandface pressure as a fraction of the average reservoir pressure, Vogel introduced the concept of pressure, Vogel introduced the concept of dimensionless Inflow Performance Relation (IPR). From a detailed study of the inflow performance of wells producing from solution gas drive reservoirs, he concluded that a general dimensionless IPR curve could properly predict the flow behavior of such wells. SCOPE OF WORK Fig. 1, redrawn from Vogel, shows the nature of dimensionless IPR curves for single phase oil, two) phase gas-oil and single phase gas flow. As sketched schematically in fig. 1, the IPR curves for gas flow and two phase gas-oil flow not only lie close to each other, but also follow similar (non-linear) trends. Fence, intuitively at the least, it should be possible to develop a Vogel-type dimensionless IPR to predict the production performance, or the deliverability, of gas wells. This approach has been investigated and reported only once before, by Meng et al. They used the dimensionless IPR concept Lo develop transient inflow performance curves for production systems analysis in performance curves for production systems analysis in vertically fractured gas reservoirs under darcy flow conditions. This study expands upon the work of Meng et al to develop dimensionless IPR curves for stabilized nondarcy flow in unfractured gas reservoirs. The objective is to use these curves for current and future gas deliverability predictions, and thus eliminate the need for conventional multi-point tests. The basic idea is to offer a simple alternative to the elaborate multi-point testing methods current used. IL is in this aspect that the present study differs from that of Meng et al, whose present study differs from that of Meng et al, whose major concern was to study the sensitivity of fracture design parameters on ultimate well performance.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 26–29, 1982

Paper Number: SPE-11164-MS

... retention time

**eqn**SPE 11164 Sizing Separators for CO2 Flood Operations by J.E. Roye Jr., Amoco Production Co. Member SPE Copyright 1982, Society of Petroleum Engineers of AIME SPE Society of PetroltlIrTl Englrteers of AIME This paper was presented at the 57th Annual Fall Technical Conference and...
Abstract

Abstract The design of field separation equipment under CO2 flood operations requires that many variables be addressed. During a CO2, flood, separators will be required to effectively separate large volumes of multi-component gas streams from produced oil and water. Many design assumptions which served well in non CO2, hydrocarbon service cannot be successfully applied to the design of field separation equipment in CO2, flood operations. To adequately size separators for CO2 service, specific attention must be given to design parameters such as K factors, gas density, liquid density, retention time for gas-liquid separation CO2, and produced gas rates, and the type of vessel best suited for CO2 flood conditions. After a detailed study of the parameters involved in the design of field separation equipment for CO2 floods, several conclusions were reached. Conventional sizing charts are not applicable in CO2 operations. One to two minute retention time is not sufficient for gas-liquid separation in CO2 operations. K factors normally used for sizing separators should be derated for CO2 operations. Horizontal separators are better suited for CO2, flood operations than spherical or vertical separators. Computer sizing of separators is necessary to determine the optimum operating settings. Introduction Amoco Production Company has been evaluating the feasibility of CO2, flood operations. Engineering studies were performed to determine the operational effectiveness of existing field separation installations in Amoco Production Company's West Texas operations. After these studies were completed the information was compiled and performance noted. In addition to these field studies, several outside firms undertook various research projects directed toward describing the most efficient means of field projects directed toward describing the most efficient means of field separation for CO2 rich produced fluid streams. Gas-liquid separation is of critical importance in CO2 floods because of the high gas production associated with CO2 breakthrough. In some cases, rates at 160 MMSCFD for a single satellite battery serving 20 wells were predicted. In addition to the high rates, other separation factors considered were maximum operating pressure of 25 psig, at a temperature of 80F. Under these, conditions, CO2 exhibits a density of .295 lbs/ft and a compressibility of .987. A .65 gravity hydrocarbon gas under these 3 conditions would exhibit a density of .128 lbs/ft and a compressibility of .98. As seen from the density values, the CO2, is more than twice as heavy as the hydrocarbon gas stream. These density differences necessitate modification of sizing charts conventionally used in sizing field separation equipment. VESSEL DESIGN REQUIREMENTS In oil field operations, either the horizontal or vertical type separator account for almost all field separation installations. To determine the type vessel best suited for CO2, flood applications, both the horizontal and vertical types were compared for gas capacity, cost, gas surge capacity, and maintenance.