SPE 124879 Rapid, Interactive Assessment of … 124879 Rapid, Interactive Assessment of Petrophysical and Geometrical Effects on Density and Neutron Logs Acquired in Vertical and Deviated

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Rapid, Interactive Assessment of Petrophysical and Geometrical Effects on Density and Neutron Logs Acquired in Vertical and Deviated Wells Zoya Heidari, SPE, Carlos Torres-Verdín, SPE, Alberto Mendoza, SPE, Olabode Ijasan, SPE, and Ben Voss, The University of Texas at Austin; Bill Preeg, SPE, Consultant

Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 4–7 October 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Borehole, geometrical, and petrophysical effects can significantly affect density and neutron logs. Tool location around the perimeter of the borehole, tool standoff, and wellbore deviation have a measurable effect on density and neutron logs, hence on the estimation of porosity and fluid density. It is difficult to diagnose and quantify these effects a priori without numerical modeling. The multiple-particle, radiation-transport Monte-Carlo code (MCNP) has traditionally been used by the logging industry to simulate borehole nuclear measurements acquired in complex rock formations. Despite its versatility and accuracy, MCNP is not numerically efficient for rapid simulation of nuclear logs and does not lend itself to interactive testing of multiple petrophysical/fluid hypotheses.

We describe the successful application of a new method for nuclear-log simulation based on linear iterative refinement of

nuclear sensitivity functions pre-calculated with MCNP. The procedure is fast, accurate, and efficient in most practical logging applications, including the simulation of nuclear logs acquired in invaded formations and in highly-deviated wells. Density and neutron logs acquired over a depth segment of 1000 ft can be accurately simulated within minutes of CPU time, compared to days with MCNP. Simulations are successfully verified against MCNP in a number of extreme cases of borehole, petrophysical, and fluid conditions wherein the error of the simulations does not exceed 2 porosity units.

We implement the new simulation method to reproduce several field examples where logs are affected by presence of clay

and invasion with water- and oil-base muds. Our rapid simulation procedure enables the interactivel quantification of the relative effect of clay, invasion, fluid density, and rock petrophysical properties on field logs. It also permits efficient integration with induction resistivity measurements for assessment of free and clay-bound water saturation, as well as for the assessment of residual hydrocarbon saturation. .Additional simulations of density and neutron logs are performed to quantify the influence of shale laminations on hydrocarbon-bearing and invaded thinly-bedded formations. Because of their longer radial length of investigation compared to density logs, neutron logs may exhibit false cross-over effects across thin beds.

Introduction

Monte Carlo algorithms are commonly used to numerically simulate borehole nuclear measurements because of their flexibility to include complex rock formation properties and arbitrary radioactive sources and sensors. However, while Monte Carlo methods are ideal for source/sensor design and for appraisal of borehole environmental effects, they are not sufficiently fast to be used for routine petrophysical interpretation and/or for integration with other borehole measurements. The widespread use of Logging-While-Drilling (LWD) measurements, for instance, emphasizes the need of real-time petrophysical interpretations of borehole measurements acquired in complex rock and geometrical conditions. Petrophysical interpretations also require that nuclear measurements be readily integrated with resistivity measurements to yield more precise and less uncertain calculations of hydrocarbon reserves. Additionally, the petrophysical interpretation of thinly-bedded rock formations requires that inversion techniques be used to reduce shoulder-bed effects on the measurements. In turn, inversion requires repeated numerical simulations of the measurements and it is therefore prohibitive with Monte Carlo methods.

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Watson (1984) for the first time introduced the idea of using linear sensitivity functions to accelerate nuclear log simulation. Quite recently, we introduced a new simulation algorithm that approximates nuclear measurements acquired in vertical and deviated wells (Mendoza et al., 2009a and 2009b). The algorithm described by Mendoza et al. (2009) is based on linear iterative refinements performed with spatial sensitivity functions (referred to as Flux Sensitivity Functions, or FSFs) precomputed with the nuclear-radiation, industry-standard MCNP code (X-5 Monte Carlo Team MCNP, 2003) via the adjoint formulation of Boltzmann’s equation and enhanced with importance statistical sampling. Compared to MCNP simulation results, the iterative FSF method is at least two orders of magnitude faster while delivering relative errors below 2-5% (Mendoza et al., 2009a). The FSFs are pre-calculated with MCNP by including all tool (geometrical and material properties, source-sensor separations, energy bands, etc.) and borehole properties (borehole size, borehole fluid, mudcake weight, mudcake thickness, tool standoff, etc.) while assuming (albeit not restrictively) a homogeneous and infinite rock formation. A pre-computed library of FSFs typically includes a large set of rock formation properties and a fixed set of tool and borehole environmental conditions. Borehole nuclear measurements acquired across non-homogeneous rock formations are approximated with local iterative (fixed-point) refinements of the FSFs until the final FSF is consistent with the simulated result (Mendoza et al., 2009). Convergence of the fixed-point iteration is usually achieved in no more than 4 iterations. Simulations and comparisons against MCNP for extreme cases of rock formation properties and shoulder beds conclusively indicate that the approximate method to simulate density and neutron measurements meets the necessary stability, speed, and accuracy conditions to be used routinely in well-log interpretation and inversion, including the challenging case of deviated wells (Mendoza et al., 2009b).

The objective of this paper is to document examples of application of the FSF-based simulation method for the interactive

and quantitative interpretation of borehole nuclear measurements. We consider several field examples as well as synthetic examples based on field measurements. The examples consider cases of rock formations invaded with both water-base (WBM) and oil-base (OBM) mud to quantify the effect of invasion on both density and neutron logs. In addition, the examples quantify the influence of invasion on borehole resistivity measurements for comparison purposes. Special emphasis is placed on the effect of both rock petrophysical properties (e.g. porosity, permeability, capillary pressure, and relative permeability) and fluid properties (e.g. density and viscosity) on the numerically simulated density and neutron logs. We also consider the important case of nuclear measurements acquired across thinly-bedded rock formations to quantify the relative effect of shoulder beds on both density and neutron logs. Our simulations confirm the need for fast and accurate simulation of nuclear logs for static and dynamic quantification of rock formation properties. Interactive nuclear modeling is also necessary to quantify multiple hypotheses concerning rock quality and for prediction of measurements in complex multi-layer sequences.

Numerical Simulation Method

As indicated above, the numerical simulation method for borehole nuclear measurements adopted in this paper is based on pre-computed FSFs and local iterative refinement. Mendoza et al. (2009a) describe the basis of the method and numerical implementation. Below, we briefly explain this method.

Linear iterative refinement method. The principle behind the linear iterative approximation method is to simulate nuclear logs in non-homogeneous formations using multiple FSFs pre-computed in homogeneous formations (which also include the borehole). We have a pre-computed library of FSFs for a range of porosity-matrix-fluid mixtures in homogeneous formations. The linear iterative refinement method updates the FSF at each sample point based on the results of a previous pass. The variety of geometric factors contained in the library enables the selection of the most appropriate base-case FSFB for the simulation of nuclear borehole measurements across non-homogeneous formations. Moreover, the library of FSFs may incorporate several non-homogeneous base-case formations that include tool standoff and mudcake. (Mendoza et al., 2009)

The first step in the linear iterative refinement is to assume a matrix type and an average porosity value to select an initial FSF for nuclear log simulation. The second step consists of simulating the tool response (i.e., detector count rate) which correlates to porosity under specific assumptions of fluid and matrix type. In a third step, at each sample point a new FSF is calculated by linear interpolation between available functions corresponding to the closest neighbors of the simulated porosity. Finally, the updated FSF is used to simulate a new value of porosity. The procedure repeats itself until the simulated porosity matches the sensitivity function used for the simulation. The fast iterative refinement method incorporates the 3D spatial capabilities of the neutron and density FSFs, and can be used in both vertical and deviated well geometries. (Mendoza et al., 2009a)

For the purposes of this paper, we benchmark the reliability, accuracy, and efficiency of the method with a synthetic

model that includes invasion. The specific tool and source/sensor configuration of density and neutron tools assumed in the numerical simulations corresponds to the “Longhorn” design described by Mendoza et al. (2007). Figure 1 shows the

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corresponding model, constructed from a field example discussed in a subsequent section. In both of the iput models for MCNP and the fast iterative refinement method, we assume piston-like invasion. The presented multi-layer example contains gas and the drilling mud is WBM. Table 5 summarizes additional properties of this formation. Figure 1 shows a good agreement between the simulation results of MCNP and the fast linear iterative refinement method. The maximum error is 0.5 and 1.2 water-filled sandstone porosity units in neutron and density measurements, respectively. The CPU time for the MCNP simulations is about 20 hours for each well log. By comparison, it takes 2 minutes to complete the simulation with the fast linear iterative refinement method.

Figure 1: Comparison of fast linear iterative refinement method (solid lines) and Monte Carlo (MCNP) simulations (dashed lines) of neutron and density measurements. From left to right, panels show: formation bulk density model in [g/cm3], simulated neutron and density porosity logs in a vertical well across horizontal layers of different petrophysical properties, and the migration length (Lm) of the model used for the simulation of the neutron log. The assumed model is based on the field example of North Louisiana’s tight-gas sand. Table 5 summarizes the petrophysical properties assumed for this geometry.

The high accuracy and very low CPU time make the fast linear iterative refinement method an adequate option for forward

and inverse simulations of nuclear logs in complex geometries, including environmental effects and mud-filtrate invasion. An important application of such a fast and accurate modeling process is to improve the assessment of petrophysical properties. To estimate petrophysical properties, we use an iterative algorithm incorporating well-log interpretation and forward simulation of nuclear and resistivity logs. Iterative Algorithm to Simulate Well Logs and to Estimate Petrophysical Properties. The first step for interpretation consists of generating an initial guess of multi-layer petrophysical properties. It is based on initial log interpretation and petrophysical analysis. The second step includes simulating the process of mud-filtrate invasion to generate spatial distributions of fluid saturation from which we obtain spatial distributions of migration length, density, and electrical resistivity. In the third step, we numerically simulate the corresponding resistivity and nuclear logs. Finally, by comparing simulated to measured well logs, we update the petrophysical properties of the multi-layer model to decrease the difference between measured and numerically simulated well logs. Considering nuclear and resistivity logs at the same time in the iterative process reduces non-uniqueness of the results. Moreover, sensitivity of nuclear logs to petrophysical properties decreases in extreme cases when drilling mud is OBM in oil-bearing zones. In these extreme cases, resistivity logs improve the estimation. For simplicity, numerical simulations described below assume that measured logs have been corrected for borehole environmental effects. Available core data also decrease non-uniqueness of results by adjusting some of the unknown input properties. The flowchart in Figure 2 describes the simulation method and the iterative algorithm used to estimate multi-layer petrophysical properties. Subsequent sections briefly explain the process of mud-filtrate invasion and well-log simulation. Mud-Filtrate Invasion. Fluid flow simulation is performed with CMG (Computer Modeling Group LTD), a commercial reservoir simulator capable of solving three-dimensional (3D), three-phase fluid-flow equations of porous media. Among the various CMG simulators available, we use STARS® for WBM simulations and GEM®, a compositional simulator, for OBM simulation. Two-dimensional (2D) simulations are implemented in cylindrical coordinates to model invasion in vertical wells under the assumptions of cylindrical flow and permeability isotropy. The petrophysical model is the input for fluid-flow simulation; it consists of non-shale porosity, initial water saturation, absolute permeability, rock-fluid properties (relative permeability and capillary pressure curves), formation fluid properties, invasion parameters, and formation fluid and mud properties. These properties can be defined using core measurements or basic well-log interpretation; otherwise they are

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calculated using the above-described iterative process which simultaneously honors resistivity and nuclear logs. Rates of mud-filtrate invasion are calculated using the one-dimensional (1D) radial method described by Salazar and Torres-Verdín (2009), which considers the interplay between mudcake buildup and rock petrophysical properties.

Figure 2: Iterative workflow adopted in this paper to estimate unknown petrophysical properties. Iterations are intended to progressively improve the agreement between measured and numerically simulated resistivity and nuclear well logs by making adjustments to layer-by-layer petrophysical properties. Simulation of Well Logs. From the simulation of mud-filtrate invasion, we obtain new values for petrophysical properties such as water saturation and salt concentration. The new petrophysical properties update the values for density, neutron migration length, and resistivity for each simulation grid. We calculate bulk density via the equation

( )1r sh sh m sh s f sC Cρ ρ ρ φ ρ φ= + − − + , (1)

where ρsh is shale density, Csh is volumetric concentration of shale, ρm is matrix density, φs is non-shale porosity, and ρf is fluid density. Migration length is used for the simulation of neutron measurements. To calculate migration length for each simulation grid, we use Schlumberger’s commercial software, SNUPAR (McKeon and Scott, 1989). Chemical compositions for each grid and their corresponding volumetric concentrations are input to SNUPAR. The fast linear iterative refinement method incorporates the spatial distributions of density and migration length to simulate density and neutron logs (Mendoza et al., 2009a).

In order to simulate resistivity logs, we calculate the spatial distribution of formation resistivity using the dual-water

model, based on the new petrophysical properties after mud-filtrate invasion. Then we numerically simulate array-induction resistivity measurement (AIT1). Heidari et al. (2009) provide further technical details about the iterative approach of nuclear and resistivity simulations as well as of the simulation of resistivity logs. Results and Discussions

We describe four examples of the interactive assessment of petrophysical and geometrical effects on density and neutron

logs. The first three examples are challenging field cases that emphasize how modeling makes the estimation of petrophysical properties more accurate. After obtaining the final multi-layer model based on the iterative approach, we examine the effect of volumetric shale concentration, residual hydrocarbon saturation, and mud-filtrate invasion on the simulated nuclear logs. The first field example is a tight-gas sand zone, which was drilled with WBM. This zone consists of a large portion of residual hydrocarbon saturation causing cross-over between density and neutron logs even after deep mud-filtrate invasion. The second case is an oil-bearing zone, which was invaded with OBM. Although oil saturation is above 0.70, high concentration of shale causes a prominent separation of density and neutron logs. In these two cases, we investigate the effect of mud-filtrate invasion and petrophysical parameters such as volumetric concentration of shale and residual hydrocarbon saturation on the simulated density and neutron logs. The third field example is another tight-gas sand zone, which was drilled with WBM. In this field example, it is not possible to estimate gas saturation based on conventional interpretation methods or using only resistivity logs. The reason is the deep mud-filtrate invasion in extremely salty connate water. Nuclear log simulation is expected to improve the assessment of hydrocarbon saturation. The fourth example is a multi-layer synthetic case. We simulate density and neutron logs for both a vertical an a high-angle well across the same multi-layer petrophysical model to investigate the effect of deviation angle on nuclear measurements.

Field Example No. 1: North Louisiana’s Tight-Gas Sand. In this field example, we analyze a North Louisiana’s tight-gas sand. The objectives are first to investigate the effect of different petrophysical properties on the simulated nuclear logs, and second, to obtain petrophysical properties based on nuclear and resistivity well-log simulation. The formation was drilled with WBM. The sedimentary rock contains very fine- to fine-grained sandstone and some fossiliferous oolitic limestone.

1 Mark of Schlumberger

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Texturally, it is mature quartz arenite and subarkose sand (Salazar et al., 2006) originated from fluvial deposition during early Cretaceous times. Large separation between resistivity logs with different radial lengths of investigation is an indication of deeply invaded beds (Figure 3). Despite the fact that invasion is deep, neutron and density logs exhibit a substantial cross-over which is due to residual gas saturation. Quantifying this separation with the proposed simulation method enables the estimation of residual gas saturation.

Tables 1 and 2 summarize the Archie’s parameters, matrix, mud, fluid and formation properties, and rock-fluid properties

reported for the tight-gas sand. We perform the petrophysical analysis and associated numerical simulations in a 22 ft thick sand interval. Table 3 describes the average values for porosity, initial water saturation, and permeability. These values are calculated by averaging the properties obtained from standard well-log interpretation. We assume that there is no free water in the formation; hence initial water saturation is equal to irreducible water saturation. The time-constant average rate of mud-filtrate invasion is 0.93 [ft3/day/ft] for five days of invasion.

The iterative simulation process begins with an initial guess obtained from the layer-by-layer averages of porosity,

permeability, and volumetric concentration of shale. Table 4 describes the petrophysical properties of the initial multi-layer model. There are no core data available to initialize residual gas saturation. With the initial values listed in Table 4, we iteratively adjust residual gas saturation and volumetric concentration of shale to match the cross-over between simulated and measured neutron and density logs. This process yields 0.1-0.3 residual gas saturation for the tight-gas sand formation. In the next step, we iteratively adjust the remaining parameters based on the procedure described in Figure 2. Figure 3 shows the final simulation results. Table 5 summarizes the final petrophysical properties obtained at the end of the iterative process. The average difference between simulated and measured logs in the cleanest sands is 2% for neutron porosity, 3% for density, and 10% for resistivity.

Figure 3: Comparison of numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (center panel), and neutron and density logs (right-hand panel) for the multi-layer model of North Louisiana’s tight-gas sand. The left-hand panel shows the spatial distribution (radial and vertical directions) of water saturation. Time of mud-filtrate invasion (tinv) is five days. Petrophysical properties were obtained from the matching of field logs with numerical simulations (Table 5).

In the next step, we adjust the petrophysical properties of the model based on the the values reported in Table 5 and investigate the effect of residual hydrocarbon saturation, mud-filtrate invasion, and volumetric concentration of shale on the simulated well logs. The effect of residual gas saturation is investigated in Figure 4. We assumed 0.1 residual gas saturation in the numerical simulations. Neutron and density logs do not match the measured logs anymore, with their cross-over decreased from 8.0 to 0.5 sandstone porosity units. This decrease in cross-over is due to lower gas saturation in the invaded formation caused by a decrease of residual gas saturation. Moreover, the shallow measured and simulated resistivity logs are no longer in good agreement due to lower gas saturation in the radially shallow part of the reservoir. It should be mentioned that combining nuclear logs with resistivity logs makes it possible to obtain accurate estimations of residual gas saturation, which was impossible with resistivity logs alone. In turn, more accurate estimations for residual hydrocarbon saturation lead to improved estimations of other petrophysical properties. The maximum permeability obtained with our interpretation method is 12% higher than the maximum permeability reported by Salazar et al. (2006) based solely on the inversion of resistivity logs.

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Figure 5a shows the numerically simulated nuclear and resistivity logs before mud-filtrate invasion. Simulated density and neutron porosity logs exhibit a larger cross-over compared to the one measured after invasion. This behavior is expected because of higher gas saturation in the pre-invaded formation. Resistivity curves do not exhibit separation before invasion, thus verifying that the separation of apparent resistivity curves is not due to post-processing artifacts.

Volumetric concentration of shale is another parameter that serously affects neutron and density logs. Figure 5b illustrates

the numerically simulated nuclear and resistivity logs before mud-filtrate invasion where we neglect shale in the formation. Compared to the results shown in Figure 5a, the effect of volumetric concentration of shale on the cross-over of nuclear logs is 50% of the initial cross-over. In other words, the cross-over between neutron and density logs increases by 5 sandstone porosity units when we eliminate the effect of shale.

Figure 4: Comparison of the numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (center panel), and neutron and density logs (right-hand panel) for the multi-layer model of North Louisiana’s tight-gas sand. Residual gas saturation is assumed equal to 0.1. Remaining layer petrophysical properties are equal to those of estimated values. The left-hand panel shows the spatial distribution (radial and vertical directions) of water saturation. Time of mud-filtrate invasion (tinv) is five days.

(a) (b) Figure 5: Comparison of numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (center panel), and neutron and density logs (right-hand panel) for the multi-layer model of North Louisiana’s tight-gas sand before invasion. The left-hand panel shows the spatial distribution (radial and vertical directions) of water saturation. We observe no separation between apparent resistivity curves. (a) In this model, we consider the effect of volumetric concentration of shale. (b) The sand is assumed to be clean and the effect of clay is neglected. Remaining layer petrophysical properties are equal to those of estimated values in Table 5. The cross-over between neutron and density porosity logs in the sand zone increases by 5 sandstone porosity units compared to Figure 5a.

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In the next step, we simplify the invasion profile as a piston-like front to define the geometry for input to MCNP simulations. Figure 6 compares the rapid approximation of neutron measurements against MCNP simulations for the North Louisiana’s tight-gas sand field example penetrated by a vertical well. The maximum difference between the two simulation methods is equal to 2.6 sandstone porosity units for neutron porosity, and 2.3 for density porosity. A reason explaining this difference is the effect of piston-like invasion assumed in the input geometry defined for MCNP versus the actual invasion distribution defined for the fast iterative refinement method. This simplifying assumption makes the MCNP-simulated density and neutron porosity higher than those simulated with the fast linear iterative refinement method. We conclude that the fast linear iterative refinement method is a better choice for simulating nuclear logs. The reason is not only the more efficient CPU time, but also the possibility of including invasion and environmental effects in the numerical simulation.

Figure 6: Comparison of Monte Carlo (MCNP) simulations (dashed lines) and rapid approximation results (solid lines) of density and neutron measurements for the multi-layer model of North Louisiana’s tight-gas sand. Final petrophysical properties were obtained from the matching of field logs with numerical simulations (Table 5). Time of mud-filtrate invasion (tinv) is five days for the rapid approximation simulations. We assume a piston-like invasion profile to construct the MCNP geometry. Figure 3 shows the spatial distribution of water saturation and corresponding resistivity logs. Field Example No. 2: Oil-Bearing Zone. The second field example considers an oil-bearing sand saturated with high viscosity oil. Based on well-log interpretation, it follows that volumetric concentration of shale is very high in this formation due to high gamma-ray values and large separation between density and neutron porosity logs. The formation of interest was drilled with OBM. Separation of resistivity curves indicates null free water saturation in the top zone (Figure 7), increasing to 0.3 ten feet below. Even though resistivity logs suggest high water saturation in the bottom zone, production tests conclusively indicate null water production. Based on pressure measurements, it is known that all the layers in this sand are in hydraulic communication. A hypothesis to explain this behavior is the presence of high residual oil saturation in the bottom zone. In addition, it is likely that overbalance and production pressures are lower than water capillary trapping forces. Another explanation is the presence of high irreducible water saturation due to variable particle-size distribution and associated mixed wettability conditions. The lack of neutron-density cross-over renders the analysis even more complicated. Simulation of mud-filtrate invasion seems the only viable interpretation alternative to diagnose residual oil saturation in the bottom depth zone and distinguish the effect of residual hydrocarbon and capillary pressure on the distribution of vertical fluid saturation.

Tables 6 and 7 summarize the Archie’s parameters, matrix, formation properties, and rock-fluid properties for this field

example. Core data indicate values of absolute permeability and irreducible water saturation of 50 [md] and 0.25, respectively. The time-constant average rate of mud-filtrate invasion is 1.35 [ft3/day/ft] for one day of invasion. Even though the well was drilled with a deviation angle of 27° we approximate it with a vertical well. Reported hydrocarbon components from PVT analysis are lumped into five pseudo-components for the in-situ oil and three for OBM (Angeles et al, 2009). Table 8 summarizes the corresponding mud-filtrate and fluid properties. Initial values for petrophysical properties were calculated by averaging log interpretation results across the desired depths. Table 9 describes the average petrophysical properties assumed for the oil-bearing sand in this field example. Table 10 lists the initial values of multi-layer petrophysical properties.

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We iteratively match numerically simulated and measured well logs by adjusting residual oil saturation, relative permeability end points, non-shale porosity, and water saturation. Figure 7 shows the final match between numerically simulated and measured array-induction resistivity and nuclear logs. The average matching error is 6% for neutron porosity logs, 1% for density logs, and 5% for resistivity logs. Table 11 describes the final values of porosity, volumetric concentration of shale, water saturation, residual oil saturation, and wetting-phase relative permeability obtained after securing an acceptable match between simulated and measured well logs. This fast numerical method enables us to infer a high value of residual oil saturation at the bottom zone, which was impossible to quantify with classical well-log interpretation.

Figure 7: Comparison of numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (second panel from right), and neutron and density logs (right-hand panel) for the multi-layer, oil-bearing field example model. Final petrophysical properties were obtained from the matching of field logs with numerical simulations (Table 11). The left-hand panel shows the gamma-ray log. Time of mud-filtrate invasion (tinv) is one day.

Figure 8a shows the numerically simulated well logs before mud-filtrate invasion. We observe that, in the absence of

mud-filtrate invasion, a cross-over develops between neutron and density logs. The effect of invasion is about 1% on density logs and 13% on neutron logs in the oil-bearing sand. Comparison of these results against those obtained for the gas-bearing zone of Field Example No. 1, indicates that the density log is less sensitive to mud-filtrate invasion in the oil-bearing than in the gas-bearing sand. This behavior can be explained by the small density difference of approximately 0.25 [g/cm3] between oil and water. However, neutron logs remain sensitive to oil saturation since the migration length of in-situ hydrocarbon is approximately 5 [cm] longer than that of water.

The lack of neutron-density cross-over is due to high density of in-situ oil, high volumetric concentration of shale, and

deep radial invasion. Fast simulation of nuclear logs makes it possible to seperately quantify the effect of different petrophysical properties such as residual hydrocarbon saturation, porosity and volumetric concentration of shale, on nuclear logs, which definitely improves log interpretation. By neglecting the presence of clay and mud-filtrate invasion, simulated density and neutron porosity measurements decrease by 2% and 40% of the initial value, respectively, and a density-neutron cross-over develops in the sand zone (Figure 8b).

Figure 9 compares the rapid approximation of neutron measurements against MCNP simulations for the field example of

the oil-bearing zone. The maximum difference between the two simulation methods in the sand zone is equal to 1.6 and 1.8 water-filled sandstone porosity units for neutron porosity and density porosity, respectively. The assumption of piston-like invasion in defining the geometry for MCNP simulations is the reason for this difference. We observe that the difference between the fast iterative refinement method and MCNP simulations decreases in the oil-bearing zone compared to the tight-gas sand, originating from the corresponding decrease of the differences between density and migration length of mud-filtrate and in-situ hydrocarbon. Therefore, the effect of mud-filtrate invasion on nuclear logs decreases, thereby decreasing the difference with MCNP results (which is mostly due to the effect of invasion).

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(a) (b)

Figure 8: Comparison of numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (center panel), and neutron and density logs (right-hand panel) for the multi-layer model of the oil-bearing field example before invasion. We observe no separation between apparent resistivity curves. The left-hand panel shows the gamma-ray log. (a) In this model we consider the effect of volumetric concentration of shale. Neutron and density porosity logs overlap in the sand zone. The effect of invasion is a 4-porosity-unit decrease on the neutron log. (b) The sand is assumed to be clean and the effect of clay is neglected. Remaining layer petrophysical properties are equal to those of estimated values in Table 11. Neutron porosity decreases by 6 sandstone porosity units and the cross-over between neutron and density porosity logs increases by 6 sandstone porosity units, compared to results shown in Figure 8a.

Figure 9 compares the rapid approximation of neutron measurements against MCNP simulations for the field example of the oil-bearing zone. The maximum difference between the two simulation methods in the sand zone is equal to 1.6 and 1.8 water-filled sandstone porosity units for neutron porosity and density porosity, respectively. The assumption of piston-like invasion in defining the geometry in MCNP is the reason for this difference. We observe that the difference between the fast iterative refinement method and MCNP simulations decreases in the oil-bearing zone compared to the tight-gas sand.

Figure 9: Comparison of Monte Carlo (MCNP) simulations (dashed lines) and rapid approximation results (solid lines) of density and neutron measurements for the multi-layer, oil-bearing field example model. Final petrophysical properties were obtained from the matching of field logs with numerical simulations (Table 11). Time of mud-filtrate invasion (tinv) is one day for the rapid approximation simulations. We assume a piston-like invasion profile to generate the MCNP geometry. Figure 7 shows the corresponding resistivity and gamma-ray logs.

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Field Example No. 3: Upper Fort Riley’s Tight-Gas Sand. In this field example, we analyze the Upper Fort Riley’s tight-gas sand formation in the Hugoton field. The formation was drilled with WBM. Reservoir flow units consist of relatively thin marine carbonates separated by thin, low-permeability nonmarine siltstones. These layers are stacked in series of marine-nonmarine sedimentary cycles. Due to low porosity of about 0.12 in this formation, mud-filtrate invasion is very deep. Accordingly, estimating gas saturation from resistivity logs in the Hugoton field is difficult to impossible because of deep mud-filtrate invasion of freshwater mud and extremely salty connate water (Torres-Verdín et al., 2006). Adding nuclear logs to the modeling process improves the estimation of gas saturation in the Hugoton field. The main objective in this part is to achieve a fast petrophysical analysis to estimate movable and residual hydrocarbon saturation based on the simulation of nuclear and resistivity logs. Although invasion is deep, neutron and density logs exhibit a large cross-over due to residual gas saturation. Quantifying this separation with the proposed simulation method enables the estimation of residual gas saturation along with porosity, permeability and other petrophysical properties. The results of this analysis are estimations of petrophysical properties with accuracy difficult to achieve via conventional log interpretation.

Tables 12 and 13 summarize the corresponding Archie’s parameters, matrix, mud, fluid and formation properties, and

rock-fluid properties. We perform the petrophysical analysis and associated numerical simulations across a 31 ft-thick sand interval. Table 14 describes the average values for porosity, initial water saturation, and permeability for the sand under consideration. These values are calculated by averaging the properties obtained from standard well-log interpretation. The time-constant average rate of mud-filtrate invasion is 0.98 [ft3/day/ft] for one day of invasion.

After obtaining an initial guess from the layer-by-layer averages of porosity, permeability, and volumetric concentration of

shale, we iteratively adjust petrophysical propertries to match the simulated resistivity and nuclear logs to the measured ones. Figure 10 shows the final simulated resistivity and nuclear logs on top of measured ones. Table 15 summarizes the final petrophysical properties achieved at the end of the iterative process. The average difference between simulated and measured logs in the cleanest sands is 12% for neutron porosity, 2% for density, and 16% for resistivity. Figure 11 shows the numerically simulated nuclear and resistivity logs before mud-filtrate invasion. Simulated density and neutron porosity logs exhibit a larger cross-over compared to the one measured after invasion. This behavior is expected because of higher gas saturation in the pre-invaded formation.

Figure 10: Comparison of numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (center panel), and neutron and density logs (right-hand panel) for the multi-layer model of Upper Fort Riley’s tight-gas sand. Neutron porosity is displayed in limestone porosity units. The left-hand panel shows the spatial distribution (radial and vertical directions) of water saturation. Time of mud-filtrate invasion (tinv) is one day. Petrophysical properties were obtained from the matching of field logs with numerical simulations (Table 15).

Interpretation of this depth interval, including log simulation and 15 iterations to obtain final petrophysical parameters, took less than three hours. The sand interval includes gas saturation above 0.8 on average. In the middle zone, we estimated

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large residual gas saturation, which reduced the estimation of producible gas. The low CPU time for nuclear simulation, is the key to fast an accurate petrophysical interpretation.

Figure 11: Comparison of numerically simulated (solid lines) and measured (dashed lines) array-induction apparent resistivity logs (center panel), and neutron and density logs (right-hand panel) for the multi-layer model of Upper Fort Riley’s tight-gas sand before invasion. Neutron porosity is displayed in limestone porosity units. The left-hand panel shows the spatial distribution (radial and vertical directions) of water saturation.

Effect of Well Deviation Angle on Density and Neutron Logs: Synthetic gas-bearing zone. The objective of this synthetic case is to compare the simulated neutron porosity and density porosity logs across layers penetrated by vertical and HA/HZ wells. To this end, we consider a multi-layer case with large contrasts of petrophysical properties. The model consists of sandstone layers of 0.05 and 0.20 porosity bounded by shale shoulder beds. Sand beds are water-saturated in the lower zone and gas-saturated in the upper zone. Figure 12 compares simulated neutron and density logs for the cases of vertical and highly-deviated wells. Simulations shown in Figure 12 would require several days of CPU time with MCNP in the same depth interval as opposed to approximately 15 minutes with iterative refinement approximations for the vertical well and 18 minutes for the deviated well.

A neutron-density gas cross-over occurs near the top, where the model includes gas-saturated sands. In the thin beds where the thickness is shorter than the vertical resolution of the neutron tool, increasing the angle of well inclination decreases shoulder-bed effects in the low-porosity water sand at the bottom. We also observe a small cross-over across the thin layers near the bottom, across the water-saturated beds. The false gas effect in the bottom thin beds is due to the difference in spatial resolution between neutron and density measurements (neutron logs exhibit lower vertical resolution than density logs) in both the vertical and deviated well cases (Ellis and Chiaramonte, 2000). Moreover, radial length of investigation of neutron measurements is larger than that of density measurements. It leads to a difference between estimated neutron porosity values and those estimated from density measurements across the water-saturated thinly-laminated interval. The difference becomes smaller in the highly-deviated well compared to the vertical well.

This example emphasizes that simulating nuclear logs in different geometries helps to diagnose misleading effects

observed in the well logs due to deviation angle, thin beds, and shoulder beds. Diagnosing these misleading effects improves the reliability of well-log interpretations. On the other hand, the linear iterative refinement method for the numerical simulation of nuclear measurements makes it possible to assess petrophysical parameters quantitatively across different geometries and bed thicknesses.

12 SPE 124879

Figure 12: Neutron and density logs simulated with the linear iterative refinement approximation. From left to right, panels show: formation bulk density model in g/cm3, the migration length (Lm) of the model, simulated neutron and density porosity logs in a vertical well across horizontal layers of different petrophysical properties, simulated neutron and density porosity logs in a well deviated 85 degrees from the vertical, the migration length (Lm) of the model used for the simulation of the neutron log, and the bulk density model [g/cm3] for the deviated well. The vertical axis in shown on the left panel describes true vertical depth (TVD) of the vertical well and the right panel shows the measured depth (MD) depth across the same interval penetrated by a well deviated 85 degrees. A neutron-density gas cross-over occurs near the top, where the model includes gas-saturated sands. An interesting observation is a small false apparent gas cross-over effect across the thin layers near the bottom. This bottom section is a laminated sequence of 0.20 and 0.05 porosity water-saturated beds. The false gas effect is caused by the difference in spatial resolution between neutron and density measurements in both the vertical and deviated well cases (Ellis and Chiaramonte, 2000). Conclusions

Synthetic and field examples of the simulation of density and nuclear measurements confirm the need of fast simulation methods for quantitative and interactive assessment of petrophysical properties. The great variety of geometrical and petrophysical properties of rock formations make it necessary to quantify a multitude of hypotheses for reliable appraisal of static and dynamic petrophysical properties. Because density and neutron logs do not exhibit the same volume of investigation, they respond in different ways to invasion and shoulder beds, for instance. Consequently, petrophysical properties could be misinterpreted without reliable quantitative appraisal of shoulder, invasion, and well-inclination effects on nuclear logs.

Simulation of borehole nuclear measurements has been elusive in the past because of extremely time consuming and

cumbersome Monte Carlo methods. By contrast, simulation of borehole resistivity measurements is routinely performed by oil companies to appraise (and correct) shoulder-bed and invasion effects on processed well logs. This paper introduced a computer efficient, yet accurate algorithm for nuclear-log simulation as well as for the interactive assessment of petrophysical and geometrical properties of rock formations on measurements. The algorithm permits, among other things, quantitative assessment of multi-layer petrophysical properties in combination with resistivity logs. We considered examples of the assessment of matrix, fluid, and saturation properties on both density and neutron logs for the accurate calculation of porosity and fluid saturation. In addition, we showed that mud-filtrate invasion may have unequal effects on density and neutron logs depending on the radial length of invasion, type of mud, bed thickness, and dip. Such unequal effects may give rise to biased estimations of porosity and fluid saturation if not accounted for in the interpretation.

Experience with the interpretation of field data sets indicates that numerical simulation is a valuable procedure to verify

the accuracy, reliability, and internal consistency of measurements acquired with borehole nuclear tools. The numerical simulation method described in this paper could be readily interfaced with inversion procedures for the automatic estimation of static/dynamic multi-layer petrophysical properties in the presence of invasion and in combination with resistivity logs.

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Nomenclature a : Archie’s tortuosity factor Snwr : Non-wetting phase residual saturation, [frac.] Csh : Volumetric concentration of shale, [frac.] Sor : Residual oil saturation, [frac.] ep : Pore-size distribution exponent Sw : Wetting-phase saturation, [frac.] enw : Experimental exponent for krnw equation Swr : Residual wetting-phase saturation, [frac.] ew : Experimental exponent for krw equation Swt : Total water saturation, [frac.] k : Permeability, [md] tinv : Time of mud-filtrate invasion, [hours] krnw : Non-wetting-phase relative permeability φ : Porosity, [frac.] k0

mw : krnw end point φD : Density porosity, [frac.] krw : Wetting-phase relative permeability φN : Neutron porosity, [frac.] k0

rw : krw end point φs : Non-shale porosity, [frac.] Lm : Migration length [cm] φt : Total porosity, [frac.] m : Archie’s cementation exponent ρb : Bulk density, [g/cm3] n : Archie’s saturation exponent ρf : Fluid density, [g/cm3] Pc : Capillary pressure, [psi] ρm : Matrix density, [g/cm3] Pc

0 : Coefficient for Pc equation, [psi.darcy1/2] ρsh : Shale density, [g/cm3] Sgr : Residual gas saturation, [frac.] Acronyms AIT® : Array-Induction Tool OBM : Oil-Based Mud CMG : Computer Modeling Group Ltd ppm Parts Per Million FSF : Flux Sensitivity Function [1/cm3-eV] SNUPAR : Schlumberger Nuclear Parameter Code FSFB : Flux Sensitivity Function for an assumed base-case

background formation [1/cm3-eV] STARS® : Multi-Component Thermal Reservoir Simulator

from CMG GAPI : Gamma-ray American Petroleum Institute Units TVD : True Vertical Depth GEM® : Compositional Simulator from CMG WBM : Water-Based Mud GR : Gamma Ray 3D : Three Dimensional HA/HZ : High-angle/Horizontal 2D : Two Dimensional MCNP : Monte Carlo N-Particle Program 1D : One Dimensional MD : Measured Depth

Acknowledgements The work reported in this paper was funded by the University of Texas at Austin’s Research Consortium on Formation Evaluation, jointly sponsored by Anadarko, Aramco, Baker Hughes, BG, BHP Billiton, BP, ConocoPhillips, Chevron, ENI, ExxonMobil, Halliburton, Hess, Marathon, Mexican Institute for Petroleum, Nexen, Petrobras, RWE, Schlumberger, StatoilHydro, TOTAL, and Weatherford.

References Angeles, R., Torres-Verdín, C., and Elshahawi, H., 2009, History matching of multi-phase flow formation-tester measurements acquired

with focused sampling probes in deviated wells: 50th Annual Logging Symposium Transactions: Society of Professional Well Log Analysts, The Woodland, TX, June 21-24.

Computer Modeling Group Ltd., 2008, User’s guide: STARS: Computer Modeling Group Ltd., Calgary, Alberta, Canada. Ellis, D. V., and Chiaramonte, J. M., 2000, Interpreting neutron logs in horizontal wells: a forward modeling tutorial, Petrophysics, vol. 41,

no. 1, pp. 23 -32. Heidari, Z., Torres-Verdín, C., Mendoza, A., and Wang, G. L., 2009, Improving the assessment of residual hydrocarbon saturation with the

combined quantitative interpretation of resistivity and nuclear logs, paper FFF, in Transactions of the SPWLA 50th Annual Logging Symposium, The Woodlands, TX, June 21 – 24.

McKeon, D. C., and Scott, H. D., 1989, SNUPAR – a nuclear parameter code for nuclear geophysics applications: IEEE Transactions on Nuclear Geoscience, vol. 36, no. 1, February, pp. 1215–1219.

Mendoza, A., Preeg, W. E., and Torres-Verdín, C., 2007, Monte Carlo modeling of borehole nuclear measurements in vertical and horizontal wells in the presence of mud-filtrate invasion and salt mixing, Petrophysics, vol. 48, no. 1, pp. 28 - 44.

Mendoza, A., Torres-Verdín, C., and Preeg, W. E., 2009, Linear iterative refinement method for the rapid simulation of borehole nuclear measurements, Part I: vertical wells, in press, Geophysics.

Mendoza, A., Torres-Verdín, C., and Preeg, W. E., 2009, Linear iterative refinement method for the rapid simulation of borehole nuclear measurements, Part II: high-angle and horizontal wells, submitted to Geophysics.

Salazar, J. M., Torres-Verdín, C., Alpak, F. O., Habashy, T. M., and Klein, J. D., 2006, Estimation of permeability from array induction measurements: applications to the petrophysical assessment of tight-gas sands: Petrophysics, vol. 47, no. 6, pp. 527–544.

Salazar, J. M. and Torres-Verdín, C., 2009, Quantitative comparison of processes of oil- and water-base mud-filtrate invasion and corresponding effects on borehole resistivity measurements: Geophysics, vol. 74, no. 1, pp. E57-E73.

Torres-Verdín, C., Alpak, F. O., and Habashy, T. M., 2006, Petrophysical inversion of borehole array-induction log: Part II - Field data examples. Geophysics, v. 71, no. 5, pp. G261-G268.

Watson, C. C., 1984, Monte Carlo computation of differential sensitivity functions, in Transaction of The American Nuclear Society, vol. 46, pp. 655.

X-5 Monte Carlo Team MCNP, 2003, A general Monte Carlo N-Particle transport code, version 5, volume III, developer’s guide: University of California, Los Alamos National Laboratory, Los Alamos, New Mexico.

14 SPE 124879

Table 1: Archie’s parameters, matrix, mud, fluid, and formation properties assumed for North Louisiana’s tight-gas sand. Variable Value Units Archie’s tortuosity factor, a 1.00 - Archie’s cementation exponent, m 1.95 - Archie’s saturation exponent, n 1.75 - Connate water salt concentration 160 kppm Mud-filtrate salt concentration 3.60 kppm Shale density 2.68 g/cm3 Matrix density 2.65 g/cm3 Water density 1.00 g/cm3 Water viscosity 1.00 cp Gas density 0.016 g/cm3 Gas viscosity 0.02 cp Mud-filtrate density 1.00 g/cm3 Mud-filtrate viscosity 1.00 cp Formation temperature 210 °F Initial formation pressure 5000 psi Mud hydrostatic pressure 5825 psi Mudcake reference permeability 0.03 md Mudcake reference porosity 0.30 fraction Mud solid fraction 0.06 fraction Mudcake maximum thickness 1.02 cm Mudcake compressibility exponent 0.40 fraction Mudcake exponent multiplier 0.10 fraction Wellbore radius 10.26 cm Formation maximum invasion time 5.00 Days Irreducible water saturation 0.40 fraction

Table 2: Rock-fluid properties assumed for North Louisiana’s tight-gas sand. Pc

0 [psi.darcy1/2] ep k0rnw enw k0

rw ew Swr 1.4 2 0.92 2.5 0.85 4.5 0.4

Table 3: Averaged petrophysical properties assumed for North Louisiana’s tight-gas sand. Variable Value Units Thickness 22 ft Absolute permeability 210 md Non-shale porosity 0.121 fraction Total water saturation 0.631 fraction Volumetric concentration of shale 0.176 fraction Shale porosity 0.06 fraction

Table 4: Initial values of model properties assumed for North Louisiana’s tight-gas sand in different petrophysical layers. Layer thickness [ft] k [md] φs Csh Swt 3 58 0.043 0.358 0.926 2 312 0.170 0.117 0.481 5 257 0.153 0.069 0.448 5 334 0.176 0.100 0.452 3 259 0.154 0.126 0.574 3 30 0.033 0.278 0.976 1 0.0004 0.0001 0.515 1.000

Table 5: Final multi-layer petrophysical properties obtained after matching the measured logs with numerical simulations for North Louisiana’s tight-gas sand. Layer thickness [ft] k [md] φs Csh Swt Sgr 3 1.50 0.15 0.40 0.45 0.30 2 7.50 0.18 0.09 0.45 0.30 5 45.80 0.14 0.06 0.45 0.30 5 25.05 0.159 0.09 0.48 0.30 3 0.30 0.16 0.04 0.60 0.20 3 0.10 0.07 0.10 0.90 0.10 1 0.015 0.145 0.08 0.80 0.10 Table 6: Archie’s parameters, matrix, and formation properties assumed for the oil-bearing field example. Variable Value Units Archie’s tortuosity factor, a 1.00 - Archie’s cementation exponent, m 1.9 - Archie’s saturation exponent, n 2.1 - Connate water salt concentration 105 kppm Bound water resistivity 0.038 Ohm.m Shale density 2.76 g/cm3 Matrix density 2.65 g/cm3 Formation temperature 254 °F Initial formation pressure 18949 psi Mud hydrostatic pressure 19500 psi Mudcake reference permeability 0.03 md Mudcake reference porosity 0.30 fraction Mud solid fraction 0.06 fraction Mudcake maximum thickness 1.02 cm Mudcake compressibility exponent 0.40 fraction Mudcake exponent multiplier 0.10 fraction Wellbore radius 10.79 cm Formation maximum invasion time 1.00 Days

Table 7: Rock-fluid properties assumed for the oil-bearing field example.

Pc0 [psi.darcy1/2] ep k0

rnw enw ew 20 4.5 0.9 2.48 2

Table 8: Mud and fluid properties assumed for the oil-bearing field example. Variable Value Units Water density 1.00 g/cm3 Water viscosity 1.00 cp In-situ oil density 0.87 g/cm3 In-situ oil viscosity 5.50 cp Mud-filtrate density 0.8 g/cm3 Mud-filtrate viscosity 2.00 cp

Table 9: Averaged petrophysical properties calculated for the oil-bearing field example. Variable Value Units Thickness 15 ft Absolute permeability 35 md Non-shale porosity 0.21 fraction Total water saturation 0.29 fraction Volumetric concentration of shale 0.25 fraction Shale porosity 0.12 fraction

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Table 10: Initial multi-layer petrophysical properties for the oil-bearing field example. Layer thickness [ft] k [md] φs Csh Swt 2 31 0.22 0.24 0.23 3 53 0.22 0.18 0.19 2 30 0.20 0.21 0.25 3 71 0.22 0.24 0.26 2 14 0.18 0.26 0.30 1 2 0.17 0.19 0.46 1 1 0.14 0.49 0.56 1 0.001 0.20 0.33 0.38

Table 11: Final multi-layer petrophysical properties obtained from the matching of measured logs with numerical simulations for the oil-bearing field example. Layer thickness [ft] φs Csh Sw Sor krw 2 0.230 0.15 0.28 0.00 0.6 3 0.220 0.05 0.25 0.00 0.6 2 0.220 0.10 0.35 0.00 0.6 3 0.220 0.08 0.33 0.30 0.3 2 0.190 0.26 0.42 0.35 0.2 1 0.150 0.45 0.50 0.30 0.3 1 0.145 0.53 0.75 0.30 0.3 1 0.230 0.29 0.33 0.30 0.3 Table 12: Archie’s parameters, matrix, mud, fluid and formation properties assumed for Upper Fort Riley’s tight-gas sand. Variable Value Units Archie’s tortuosity factor, a 1.0 - Archie’s cementation exponent, m 1.89 - Archie’s saturation exponent, n 1.51 - Connate water salt concentration 200 kppm Mud-filtrate salt concentration 0.3 kppm Shale density 2.76 g/cm3 Matrix density 2.69 g/cm3 Water density 1.00 g/cm3 Water viscosity 1.00 cp Gas density 0.00098 g/cm3 Gas viscosity 0.01087 cp Mud-filtrate density 1.186 g/cm3 Mud-filtrate viscosity 1.0 cp Formation temperature 96 °F Initial formation pressure 92 psi Mud hydrostatic pressure 1153 psi Mudcake reference permeability 0.03 md Mudcake reference porosity 0.30 fraction Mud solid fraction 0.06 fraction Mudcake maximum thickness 1.02 cm Mudcake compressibility exponent 0.40 fraction Mudcake exponent multiplier 0.10 fraction Wellbore radius 10.00 cm Formation maximum invasion time 1.00 Days Irreducible water saturation 0.08 fraction

Table 13: Rock-fluid properties assumed for Upper Fort Riley’s tight-gas sand. Pc

0 [psi.darcy1/2] ep k0rnw enw k0

rw ew Swr 1.0 10 0.4 2.5 0.3 4.5 0.08

Table 14: Averaged petrophysical properties assumed for Upper Fort Riley’s tight-gas sand. Variable Value Units Thickness 31 ft Absolute permeability 3 md Non-shale porosity 0.12 fraction Total water saturation 0.19 fraction Volumetric concentration of shale 0.12 fraction Shale porosity 0.08 fraction

Table 15: Final multi-layer petrophysical properties obtained from the matching of measured logs with numerical simulations for Upper Fort Riley’s tight-gas sand field example. Layer thickness [ft] φs Csh Swt Sgr k[md]3 0.12 0.10 0.19 0.20 3 8 0.12 0.10 0.18 0.30 3 2 0.12 0.10 0.18 0.35 3 4 0.11 0.13 0.12 0.40 3 3 0.12 0.14 0.25 0.20 3 5 0.11 0.12 0.17 0.30 3 4 0.12 0.12 0.25 0.20 3