Influences of Well Test Techniques and Uncertainty in Petrophysics on Well Test Results

Abstract

In the present study, well logs and well test data of both conventional build-up tests and Mini-DST from different oil and gas fields are utilized to evaluate the effects of uncertainty in petrophysics and test techniques on well test results. This includes producing wells from the Nile Delta and Western Desert-Egypt together with published results from West Qurna oil field-Iraq. Results indicated that permeability is strongly dependent on petrophysics interpretation, particularly pay thickness, while the radius of investigation is significantly dependent on fluid properties, especially compressibility. The skin factor calculations showed great sensitivity towards the pressure measurements with medium influences on porosity and oil viscosity. The calculations of Mini DST and Build-up test are compared within the uncertainty context for effective permeability, radius of investigation, and skin factor, and the findings are discussed in detail. In all cases, error analysis indicated that well test results and interpretation of conventional build up data are largely stable and may reduce overall uncertainty to 30% of the corresponding Mini-DST results/interpretation. The results of this study not only characterize each input parameter involved in the interpretation of well test data but also confirms the superiority of conventional build-up on Mini-DST techniques.

1. Introduction

Being the primary source for subsurface formations and reservoir characterization, petrophysical analysis plays an important role in geological and reservoir studies. By definition, formation evaluation involves the process of using borehole measurements to evaluate subsurface formations [1]. Reservoir characterization aims usually at discretizing a petroleum reservoir into subunits, layers, or grid blocks, and assigning values for the pertinent physical properties [2]. Well test analysis became a valuable reservoir characterization tool to: recognize heterogeneous reservoir behaviors such as double and composite permeability, identify partial penetration or limited entry, analyze horizontal wells, and handle a wide range of boundary effects [3]. In addition, petrophysical cutoffs may significantly affect reservoir characterization for both static and dynamic models that considerably influence the realizations of the asset value [4]. Petrophysical properties are one great source of uncertainty that still constitutes a challenge for reservoir characterization, but accurate petrophysical analysis may involve a limited number of skeptical reservoir properties [5]. To quantitively evaluate uncertainty in petrophysical parameters, Monte Carlo simulation helps analyze the probable scenarios for each individual parameter [6]. It is first used in business [7] and with the advent of the computer, mathematical models are created to simulate complex projects including the petroleum industry [8].

In the Monte Carlo simulation, the output uncertainty is calculated using randomly selected values and their distribution with sensitive case analyses [9]. It could be suitable to quantify uncertainty, but only when an appropriate interpretation model is adopted. Adams [10] applied such a technique for porosity uncertainty analysis and recommended it as a good practice for other petrophysical parameters. In reserve analysis, Komlosi & Komlosi [11] applied Monte Carlo simulation and concluded that the accuracy of results is dependent on the reliability of the model, reservoir parameters, and the conditions of the forecast. The uncertainty analysis using probability density functions and Monte Carlo simulations could be essential in well test analysis. The pressure and rate of measurement represented the main errors that lead to major uncertainty in well test analysis [12] and the important input data can be constrained within an approximate uncertainty range. This includes the permeability-thickness product (kh, known within 15%), permeability (k, known within 20%), the wellbore storage constant (C, known within 20%), the skin factor (S, known within ±0.3), and distance to the boundary (known within 25%) [12]. Furthermore, the Monte Carlo method proved to be a powerful tool to determine risk in petroleum exploration and development projects, and also estimate the reliability of reservoir models [10]. However, uncertainty evaluation and quantification in well test analysis are not sufficiently addressed in the literatures, being an important issue, to reduce the risk and improve decision making in field management and the petroleum industry.

Prediction of porosity and permeability is important for identifying hydrocarbon accumulation in a reservoir and differentiates between pay zones and seals at different intervals. Well logging, core data analysis, and well testing methods are available to measure reservoir parameters in new drilled wells. Wireline formation testers (MDT, DST, or Mini DST) are powerful tools to estimate formation permeability (K), skin effect (S), and radius of investigation (Rinv). This is typically achieved by a drawdown pressure profile (generated by a known flow rate pulse) or the build-up of pressure transients [13]. Pressure and fluid sampling is accomplished by setting an elastic packer and a small diameter probe. The packer hydraulically segregates approximately one meter of the formation from the hydrostatic pressure, whereas the probe permits the tool communication with the tested zone [14]. The dual packer module is equipped with two gauges; the first gauge (typically a strain gauge) measures the pressure inside the packer component to monitor the packer setting pressure. The second pressure gauge (Crystal Quartz Gauge, CQG) is attached to the port located between the two packers and is developed specially to respond accurately and quickly to the flow line pressure and temperature transients [15]. The tool also enables to us collect down hole fluid samples from the test point and conveys the samples to the surface for identification and PVT analysis.

The most common well testing technique utilized in practice belongs to pressure-buildup. The pressure build-up test is accomplished by shutting-in a producing well to measure the pressure build-up versus time [16]. Different graphical techniques are developed for well test data interpretation and these techniques depend basically on pressure drop equations. Interpretation theory typically depends on the 2D analytical solutions of the radial diffusivity equation. The Perrine method, the most common method utilized to solve multiphase pressure-buildup tests, assumes uniform fluid distribution around the tested well. It also considers uniform fluid saturation and negligible capillary pressure [17]. Well test analysis has four main sources of uncertain parameters [18], with various ranges of importance and resolvability, including:

• Pressure data errors that have been developed by noise, drift, temperature effects, and time shifts.
• Errors in the flow rate measurements.
• Ambiguity in response interpretation (matching different models with apparently equal verisimilitude).
• Rock properties and fluid properties uncertainty.

The effect of errors in flow rate data on well test interpretation is evaluated using simultaneous analysis and the results showed a great reduction in the degrees of interpretation freedom, the disappearance of the wellbore storage effect, and the complete vanishing of interlayer flows [19]. El-Hawary [20] emphasized the effect of data management to improve well test results and concluded that the main sources of error typically involve the improper design of the well test experiment and imprecise data entry. In addition, fractured reservoirs entail significant uncertainty in pressure and flow rate measurements as addressed by Archer et al. [21]. Khasanov et al. [22] evaluated the uncertainty in parameter estimation and erroneous input data for a specific reservoir model. In addition, the error bounds in well test analysis for the most common parameters involve permeability and skin factor with uncertainty ranges [3]. Jirjees and Abdulaziz [1] evaluated the effect of uncertainty in petrophysics and fluid properties on well test interpretation in West Al Qurna Oil Field, South Iraq and concluded that petrophysics uncertainty had marked influences on the interpretation of well test results (effective permeability, S and Rinv). The present study evaluates the effects of uncertainty in petrophysical parameters interpretation and well testing techniques, conventional build-up versus Mini DST, on the interpretation of well test results. It also characterizes the influences of individual input data on each interpretation parameter and analyzes the overall errors associating well testing techniques within the uncertainty context. This can be achieved through two basic steps; calculations of basic petrophysical parameters with error analysis in individual wells and well test data analysis using the interpreted petrophysical data under different probabilities. In addition, sensitivity analysis for petrophysical parameters involved in well test data interpretation is conducted for each interpreted well test parameter.

2. Materials and Methods

The impact of petrophysics uncertainty on pressure build-up test is evaluated using six wells (ND-1, ND-2, ND-3, ND-4, WD-1, and WD-2) from the Nile Delta (ND) and Western Desert (WD) of Egypt with complete records of well test and log data. All log data analysis and calculations are completed using the relationships presented in

Table 1. A list of equations for petrophysical calculations.

Petrophysical Parameter	Equations	Reference
Volume of clay	${\mathrm{V}}_{\mathrm{sh}}=\frac{0.5\times {\mathrm{I}}_{\mathrm{GR}}}{1.5-{\mathrm{I}}_{\mathrm{GR}}}$	Stieber, 1970 [25]
Porosity	Φ = ΦD1 + $\frac{\mathsf{\Phi }\mathrm{N}1-\mathsf{\Phi }\mathrm{D}1}{1-\frac{\mathsf{\Phi }\mathrm{N}1-\mathsf{\Phi }\mathrm{N}2}{\mathsf{\Phi }\mathrm{D}1-\mathsf{\Phi }\mathrm{D}2}}$	Bertozzi et al., 1981 [26]; Dewan, 1983 [27]
Water saturation	$\frac{1}{\sqrt{{\mathrm{R}}_{\mathrm{t}}}}=\left(\sqrt{\frac{{\varnothing }^{\mathrm{m}}}{\mathrm{a}\times {\mathrm{R}}_{\mathrm{w}}}}+\frac{{{\mathrm{V}}_{\mathrm{sh}}}^{\left(1-\frac{{\mathrm{V}}_{\mathrm{sh}}}{2}\right)}}{\sqrt{{\mathrm{R}}_{\mathrm{sh}}}}\right)\times {{\mathrm{S}}_{\mathrm{w}}}^{\frac{\mathrm{n}}{2}}$	Poete, 2012 [28]; Schlumberger, 2010 [23]

. After basic petrophysical parameters calculations (Average volume of shale: V_shavg, Average water saturation: S_wavg, Average porosity: Φ_avg, and Net-to-Gross ratio: N/G) in all pay zones, the possible error distributions of interpretation parameters are calculated. Figure 1 presents the Monte Carlo simulation workflow for uncertainty analysis to the calculated petrophysical parameters after constructing error distribution and randomization. For a particular parameter in any simulation, the shift is randomly calculated using an unsystematic pick to the time of the CPU clock [23]. Then, the shift characteristics including distribution, type, and value (high and low) are determined and subsequently, the results of each simulation are displayed as distributions. This eliminates any bias in the shift of individual parameters that significantly influence the error analysis for each parameter. The influence of each input parameter on the overall error results can be evaluated using sensitivity analysis and the relative importance of individual input parameters is graphically presented using the Tornado plot. Error calculations are executed using the technique described by Jarjis and Abdulaziz [1] and all petrophysical/error calculations are implemented using IP–V3.6 (Interactive Petrophysics; Lloyd’s Register, London, UK) software [24].

Well test data analysis typically includes four main steps; data preparation and processing, identification of the flow regime using diagnostic tools, model generation for both wellbore and reservoir, and finally post model investigations. In data preparation, The Pan System software, a powerful tool for preparing and editing well test data from conventional gauges and wireline formation testers, uses analytical and numerical methods to analyze and history match the available transient well test data. The skin factor (S), radius of investigation (Rinv), different permeability types (K), and initial pressure (Pi) are the key parameters of post modeling results. In the present analysis, the empirical correlations in (

Table 2. The PVT correlations used for well ND-1.

PVT Correlations for PTA
Inputs	Type of Correlation	Outputs
GOR (Scf/bbl) = 1135	Oil correlation for PVT Standing	${\mathrm{P}}_{\mathrm{b}}$ & ${\mathrm{R}}_{\mathrm{s}}$
GOR (scf/bbl) = 1135, API = 52° & ${\mathsf{\gamma }}_{\mathrm{g}}$ = 0.626	Standing	${\mathrm{B}}_0$
API = 52°, Temp = 195 °F	Beal	${\mathsf{\mu }}_{\mathrm{o}}$
Water correlation for PVT
Salinity = 50 kppm	Spivey	${\mathrm{B}}_{\mathrm{w}}$
Salinity = 50 kppm	Dodson & Standing	${\mathrm{C}}_{\mathrm{w}}$
Salinity = 50 kppm	Van-wingen & Frick	$\mathsf{\mu }\mathrm{w}$
Salinity = 50 kppm	Katz	${\mathrm{R}}_{\mathrm{sw}}$

) are used to generate a PVT model to manage the real PVT data deficiency in the Nile Delta wells. The diagnostic plots to identify flow regimes, reservoir model, and boundary model are completed using line fitting techniques. The analytical model for well test data analysis comprises the mathematical formulas presented in Table 3 and all calculations in well test analysis and interpretation are accomplished using Pan System software (V 3.5) [29].

Uncertainty in buildup well test results and sensitivity analysis are addressed using similar approaches to the petrophysics uncertainty analysis (Figure 1). The input parameters responsible for uncertain well test results comprise two main categories. The first category includes the fluid parameters such as formation volume factor (B), viscosity (µ), fluid compressibility (K), and flow rate (Q) that usually impose fewer influences on the well test uncertainty results compared to the effect petrophysics dataset [1]. The second category involves the petrophysical parameters including net pay thickness, porosity, water saturation, and rock compressibility. Porosity typically has substantial effects on the radius of investigation and skin factor calculations and therefore porosity uncertainty effect is evaluated in the present analysis. Net pay and water saturation (S_w) are parameters in oil permeability (K_o) and gas permeability (K_g) calculations and consequently, uncertainty in these parameters may impact the calculations of all permeability types. The uncertainty simulation algorithm typically utilizes the distribution of individual parameters involved in calculations and therefore, a histogram for each petrophysical parameter is primarily constructed. The normal distribution has been used to fit the data points of porosity and water saturation while the triangle distribution fits the net pay data. Uncertainty and sensitivity analysis is completed using the VOSE plugin (Excel, Microsoft Office 2016; Microsoft Corporation, Redmond, WA, USA) [30] modified by Jirgis and Abdulaziz [1], to fit applications (Model risk VOSE, 2016; Vose Software, Gent, Belgium). This plugin is basically adopted to evaluate oil zones and, in this work, has been upgraded to include the gas zones as well. The upgraded code is presented in Appendix A. To evaluate the effect of the well test technique applied in pressure buildup experiments, the well test results under petrophysics uncertainty of the present study are compared to the corresponding results from the Mini DST measurements in the West Qurna oil field (WQOF) presented by Jirgis and Abdulaziz [1].

Table 3. A list of pressure transient equations used in the present calculations.

Parameters	Equation	Equation No.	Reference
Ko (mD)	${\mathrm{K}}_{\mathrm{o}}=\frac{162.6 \mathrm{QoBo} \mathsf{\mu }\mathrm{o}}{\mathrm{mh}}$	(1)	[17]
Kw (mD)	${\mathrm{k}}_{\mathrm{w} }=\frac{162.6 \mathrm{Qw} \mathrm{Bw}\mathsf{\mu }\mathrm{w}}{\mathrm{mh}}$	(2)	[17]
Kg (mD)	${\mathrm{K}}_{\mathrm{g}}=\frac{1637{ \mathrm{q}}_{\mathrm{sg}} \mathrm{T}}{\mathrm{mh}}$	(3)	[31]
Kabs (mD)	$\mathrm{Kabs}=\frac{\mathrm{Ko}}{\mathrm{Kro}} \mathrm{or} \mathrm{Kabs}=\frac{\mathrm{Kw}}{\mathrm{Krw}} \mathrm{or} \mathrm{Kabs}=\frac{\mathrm{Kg}}{\mathrm{Krg}}$	(4)	[17]
Skin factor for oil (inch)	$\mathrm{S}=1.151\left[\frac{\mathrm{P}1\mathrm{hr}-\mathrm{Pwf}\left(\Delta \mathrm{t}=0\right)}{\mathrm{m}}-\log \frac{\mathrm{K}}{\varnothing { \mathsf{\mu }\mathrm{o} \mathrm{ct} \mathrm{rw}}^2}+3.23\right]$	(5)	[17]
Skin factor for gas (inch)	$\mathrm{S}=1.151\left[\frac{\mathrm{m}\left(\mathrm{P}1\mathrm{hr}\right)-\mathrm{m}\left(\mathrm{Pwf}\left(\Delta \mathrm{t}=0\right)\right)}{\mathrm{m}}-\log \frac{\mathrm{K}}{\varnothing { \mathsf{\mu }\mathrm{g} \mathrm{ct} \mathrm{rw}}^2}+3.23\right]$	(6)	[31]
Rinv (ft)	$\left(\mathrm{Rinv}\right)=0.029\sqrt{\frac{\mathrm{K} \Delta \mathrm{t}}{\varnothing \mathsf{\mu }\mathrm{ct}}}$	(7)	[32]

Table 3. A list of pressure transient equations used in the present calculations.

Parameters	Equation	Equation No.	Reference
Ko (mD)	${\mathrm{K}}_{\mathrm{o}}=\frac{162.6 \mathrm{QoBo} \mathsf{\mu }\mathrm{o}}{\mathrm{mh}}$	(1)	[17]
Kw (mD)	${\mathrm{k}}_{\mathrm{w} }=\frac{162.6 \mathrm{Qw} \mathrm{Bw}\mathsf{\mu }\mathrm{w}}{\mathrm{mh}}$	(2)	[17]
Kg (mD)	${\mathrm{K}}_{\mathrm{g}}=\frac{1637{ \mathrm{q}}_{\mathrm{sg}} \mathrm{T}}{\mathrm{mh}}$	(3)	[31]
Kabs (mD)	$\mathrm{Kabs}=\frac{\mathrm{Ko}}{\mathrm{Kro}} \mathrm{or} \mathrm{Kabs}=\frac{\mathrm{Kw}}{\mathrm{Krw}} \mathrm{or} \mathrm{Kabs}=\frac{\mathrm{Kg}}{\mathrm{Krg}}$	(4)	[17]
Skin factor for oil (inch)	$\mathrm{S}=1.151\left[\frac{\mathrm{P}1\mathrm{hr}-\mathrm{Pwf}\left(\Delta \mathrm{t}=0\right)}{\mathrm{m}}-\log \frac{\mathrm{K}}{\varnothing { \mathsf{\mu }\mathrm{o} \mathrm{ct} \mathrm{rw}}^2}+3.23\right]$	(5)	[17]
Skin factor for gas (inch)	$\mathrm{S}=1.151\left[\frac{\mathrm{m}\left(\mathrm{P}1\mathrm{hr}\right)-\mathrm{m}\left(\mathrm{Pwf}\left(\Delta \mathrm{t}=0\right)\right)}{\mathrm{m}}-\log \frac{\mathrm{K}}{\varnothing { \mathsf{\mu }\mathrm{g} \mathrm{ct} \mathrm{rw}}^2}+3.23\right]$	(6)	[31]
Rinv (ft)	$\left(\mathrm{Rinv}\right)=0.029\sqrt{\frac{\mathrm{K} \Delta \mathrm{t}}{\varnothing \mathsf{\mu }\mathrm{ct}}}$	(7)	[32]

3. Results and Discussion

The basic petrophysical parameters (Φ, V_sh, S_w, and N/G ratio) are calculated from the available well log data ((ND-1, ND-2, ND-3, ND-4, WD-1, and WD-2). The ND wells are penetrating Abu Madi and/or Qawasim Formations of the Nile Delta basin while WD wells penetrate the Baharyia Formation of the Western Desert basin. The results of petrophysical characterization, uncertainty analysis, and sensitivity analysis of the studied wells are discussed and subsequently compared to the corresponding results in West Al Qurna Oil Field, South Iraq presented in detail in Jirjis and Abdulaziz [1].

3.1. Petrophysical Characterization

The appropriate cut-off values suitable for the various reservoir conditions of the present study are 0.07 for porosity, 0.65 for S_w, and 0.2 for V_sh which appear comparable to the corresponding values used in the operating company. The results of the Nile Delta wells showed two hydrocarbon bearing intervals: Abu Madi and Qawasim Formations. The first is the gas bearing zones of Abu Madi Formation which have been seen in wells ND-2 and ND-3, while the oil zones are seen at Qawasim Formation in wells ND-1 and ND-4. Alternatively, the Western Desert showed oil bearing intervals with five pay zones seen in well WD-1 and eight pay zones in well WD-2. An example presenting the input data and the resulting reservoir characteristics is shown on a Triple Combo display for well ND-1 in Figure 2. Results showed that Abu Madi Formation has clean clastic facies with low shale content (12%) and relatively good calculated porosity (23%). While Qawasim Formation showed an average porosity of match-able values (21.1%) in wells ND-1 and ND-4, and the clay content reported only 9.8%. For all pay and/or reservoir intervals of the studied wells, the Monte Carlo Simulation results are presented in a set of outputs for each individual reservoir property in form of histograms, cross plots, and tabulated values. A simple summary of the uncertainty analysis with (P10, P50, and P90) of the key petrophysical characteristics is presented in

Table 4. The reservoir parameters derived from Monte Carlo simulation for all well zones at different probabilities.

Well	Zone	Depth		Output %	Pay Thickness (ft)	N/G	ØAVG	SwAVG	VCLAVG
		From	To
ND-1	Qawasim	8266	8401.5	10	112.5	0.83	0.168	0.194	0.175
				50	125	0.923	0.211	0.164	0.098
				90	129.5	0.956	0.245	0.134	0.025
ND-2	Abu Madi	9968.5	10,080	10	75.75	0.679	0.212	0.396	0.139
				50	93.5	0.839	0.23	0.357	0.075
				90	102.5	0.919	0.243	0.309	0.032
ND-3	Abu Madi	9921	10,054	10	91	0.684	0.22	0.397	0.176
				50	113	0.85	0.234	0.354	0.122
				90	122.5	0.921	0.246	0.315	0.067
ND-4	Qawasim	9154	9234	10	40	0.5	0.159	0.279	0.142
				50	55.5	0.694	0.215	0.238	0.086
				90	64.75	0.809	0.238	0.202	0.04
WD-1	Zone1	6361.5	6375	10	2.25	0.167	0.164	0.279	0.288
				50	7.25	0.537	0.197	0.237	0.174
				90	11.75	0.87	0.225	0.198	0.053
	Zone 2	6390	6405	10	1	0.067	0.122	0.376	0.297
				50	3	0.2	0.197	0.295	0.169
				90	6	0.4	0.233	0.23	0.019
	Zone3	6561	6620.5	10	6	0.101	0.162	0.46	0.259
				50	16.75	0.282	0.193	0.394	0.131
				90	26.25	0.441	0.216	0.343	0.033
	Zone 4	6793	6857.5	10	37.5	0.581	0.194	0.117	0.212
				50	56	0.868	0.208	0.091	0.164
				90	62.75	0.973	0.223	0.06	0.101
	Zone 5	6959.5	7105.5	10	20.5	0.14	0.178	0.192	0.268
				50	50	0.342	0.2	0.151	0.175
				90	68	0.466	0.221	0.113	0.08
WD-2	Zone 1	6739.5	6752.5	10	10.5	0.808	0.185	0.236	0.208
				50	12.75	0.981	0.208	0.205	0.123
				90	12.75	0.981	0.228	0.174	0.052
	Zone 2	6771.5	6791.5	10	2	0.1	0.141	0.213	0.316
				50	8.5	0.425	0.216	0.171	0.134
				90	11	0.55	0.249	0.142	0.026
	Zone 3	6951.5	6962	10	8.5	0.81	0.209	0.256	0.143
				50	9.5	0.905	0.224	0.224	0.083
				90	10.25	0.976	0.239	0.195	0.037
	Zone 5	7006	7028.5	10	10.5	0.467	0.151	0.39	0.184
				50	19.5	0.867	0.169	0.341	0.112
				90	22.25	0.989	0.188	0.299	0.043
	Zone 6	7055	7061.5	10	1	0.3	0.16	1	0.119
				50	4.5	0.692	0.184	0.419	0.049
				90	4.5	0.846	0.203	0.37	0
	Zone 7	7132	7143	10	0.1	0.02	0.15	1	0.192
				50	0.5	0.045	0.152	0.505	0.104
				90	6.5	0.591	0.201	0.417	0
	Zone 8	7217.5	7278	10	44.5	0.736	0.197	0.144	0.166
				50	49.5	0.818	0.211	0.122	0.114
				90	54.25	0.897	0.224	0.099	0.069
	Zone 9	7404.5	7514	10	35	0.32	0.193	0.31	0.154
				50	47.25	0.432	0.205	0.268	0.106
				90	53.25	0.486	0.219	0.231	0.066

.

Results in all wells showed essentially consistent values in most petrophysical characteristics, but such uniformity is markedly changed in the pay results. Among various reservoir parameters, the average porosity and clay content are moderately consistent under the different probabilities (Table 4). However, an apparent variation in pay thickness and N/G calculations is clearly perceived under the different probabilities.

Commonly, most studied wells showed stabilized facies, as indicated by the consistent values of the mean net pay thickness in both conventional log interpretation and uncertainty analysis. Instead, N/G showed an obvious instability in uncertainty calculations of the net pay results of the Qawasim Formation in wells ND-1 and ND-4. For example, at the Qawasim zone of well ND-1, the N/G calculations varied from 83% at P10 to 95.6% at P90, whereas in well ND-4 the N/G falls between 50% at P10 and 80.9% at P90 (Table 4). Compared to pay thickness in the Qawasim Formation, the N/G uncertainty results showed relatively moderate changes in Abu Madi pay zones at the various percentiles. The N/G of the Abu Madi Formation goes as low as 67.9% at P10 and increases to 92.1% at P90 (Table 4). Such instability in probability calculation could be attributed to the gradational change in sedimentary facies at the formation boundaries. The porosity analysis of all pay results (Table 4) provides a relatively consistent average porosity values that typically fall between 15 and 24%. Compared to the conventional log analysis, uncertainty analysis of porosity calculation usually shows slightly fewer estimates in the average porosity (P50) except for wells WD-1 and WD-2 which tend to provide slightly higher estimates. The discrepancy in the mean porosity values typically falls within 0.8% in Qawasim and Bahariya pay zones, but report approximately 0.3% in Abu Madi Formation (Table 4). Alternatively, S_w calculations under uncertainty showed different trends compared to the conventional log analysis. The log calculated S_w is dominantly slightly overestimated in Qawasim and Abu Madi Formations but in the pay zones of WD-1 well (zones 1 and 4) and WD-2 well (zones 1, 3, 5, 8, and 9) showed various trends, yet behaves similar to Qawasim and Abu Madi pay zones. Generally, the log-derived average S_w is similar to the uncertainty analysis with a difference between 0.1 (Qawasim and Bahariya pay zone) and 0.9% as reported in Abu Madi pay zone (Table 4). The volume of shale (V_sh) calculations indicate that the conventional log average inclines towards overestimation that may approach 6.4% as calculated in zone 1 and zone 2 of WD-1 well, but the common values typically fall between 0.2% and 0.9%.

The discrepancy in pay S_w among the various percentiles falls within 3–4% while the (V_sh) shows a wider range (5–9%). The minimum change in average porosity calculations at different percentiles is found in well WD-2 of tje Bahariya zone with a 2.6% difference, but the maximum variation may approach 12% (Table 4). Sensitivity analysis normally uses Tornado plots to evaluate the effect of input parameters and interpretation models on the system performance. This is achieved by assigning a weight for each input parameter in the error analysis of the interpretation results. For example, the cutoffs applied for pay zone identification are classically the most effective parameter in N/G error analysis [1]. the Tornado plot for average porosity analysis in the pay zones of well ND-1 (Figure 3) illustrates the significant effect of the Gamma ray records on porosity calculations, while Neutron/Density measurements and V_sh cut off entailed secondary effects. On the other hand, the typical Archie parameters (m and n), deep resistivity measurements, and Gamma ray clean are the most sensitive parameters in average water saturation calculations but S_w and V_sh cutoffs had only intermediate effects (Figure 4). The Gamma ray records are the typical sensitive parameter in the volume of clay calculations however, the V_sh cutoff and Gamma ray clay represent the secondary parameters in these calculations. Nevertheless, the N/G error analysis showed a major influence induced by the V_sh cutoffs while Gamma ray clean and S_w cutoffs had an intermediate sensitivity. Limited effects on N/G calculation can be related to porosity cutoffs, and the Archie parameter (m).

3.2. Well Test Interpretation

Many reservoir parameters and well conditions have been evaluated for the subsequent well test calculations to estimate initial pressure, various types of permeability, skin factor, and radius of investigation [33]. Figure 5 presents a log-log match of the buildup data in well ND-1 while testing the Qawasim Formation. The results of the well test and uncertainty analysis of all pay zones in the studied wells are presented in

Table 5. The results of well test uncertainty analysis (mean value and output percentiles: 10, 50, and 90%) for all pay zones of the studied wells.

Wells	Formations	Probability	Ko (mD)	Kw (mD)	Kg (mD)	Kabs (mD)	Rinv (ft)	Skin (inch)
ND-1	Qawasim	Base Case	4752.03	20.70	-	4906.00	1566.00	3.80
		Mean	4752.99	20.73	-	4911.12	1570.7	3.81
		P10	4253.96	19.51	-	4417.40	1463.87	3.56
		P50	4752.03	20.68	-	4906.08	1565.38	3.80
		P90	5252.98	21.99	-	5409.87	1682.86	4.08
ND-4	Qawasim	Base Case	2665.91	1.63	-	3060.00	2231.00	3.50
		Mean	2718.10	1.67	-	3117.43	2238.86	3.50
		P10	2239.35	1.40	-	2579.6	2074.96	3.12
		P50	2668.98	1.64	-	3060.06	2230.20	3.57
		P90	3245.97	1.97	-	3712.65	2411.42	3.93
ND-2	Abu Madi	Base Case	0.000052	-	279.99	316.00	1203.00	15.00
		Mean	0.000053	-	281.98	318.86	1216.26	15.20
		P10	0.000048	-	253.79	289.97	1110.48	15.15
		P50	0.000053	-	280.05	316.00	1201.11	15.28
		P90	0.000058	-	312.12	350.60	1337.19	15.46
ND-3	Abu Madi	Base Case	2.10 × 10−6		356.88	390.98	1556.00	3.50
		Mean	2.10 × 10−6	-	360.85	395.20	1599.11	3.51
		P10	1.90 × 10−6	-	327.98	363.08	1385.69	3.47
		P50	2.10 × 10−6	-	357.70	391.00	1559.01	3.51
		P90	2.30 × 10−6	-	396.88	431.70	1852.65	3.57
WD1	Bahariya	Base Case	1046.03	0.621	-	1079.00	1677.00	3.38
		Mean	1063.19	0.606	-	1099.29	1685.45	3.46
		P10	861.36	0.530	-	894.21	1537.87	3.28
		P50	1045.19	0.600	-	1080.37	1678.2	3.47
		P90	1283.02	0.690	-	1323.29	1840.27	3.63
WD-2	Bahariya	Base Case	198.24	0.100	-	215.00	970.00	10.00
		Mean	201.58	0.100	-	217.70	972.94	10.00
		P10	171.64	0.090	-	186.33	913.68	9.92
		P50	199.01	0.100	-	214.70	970.28	10.00
		P90	234.11	0.110	-	252.08	1034.86	10.08

. In Qawasim pay zones, the absolute permeability (Kabs) reports a range of 2579.6–5409.87 mD in all wells. The radius of investigation showed generally the greatest calculated radius among the studied zones and varied between 2075 ft at P10 in ND-4 and 2411 ft in P90 of well ND-4 (Table 5). The minimum calculated differences in probabilities for the skin effect of the Qawasim zone is encountered in wells ND-1 (P50 = 3.8 and P90 = 4.08) and ND-4 (P50 = 3.6 and P90 = 3.9). In the Abu Madi zone, all calculated parameters under uncertainty showed slightly higher estimates compared to conventional results (

Table 6. A comparison of Monte Carlo uncertainty analysis (UA) results with the corresponding parameters from the conventional pressure transient analysis (PTA) for all well zones.

Formations	Ko (mD)		Kw (mD)		Kg (mD)		Kabs (mD)		Rinv (ft)		Skin (Inch)
	PTA	UA	PTA	UA	PTA	UA	PTA	UA	PTA	UA	PTA	UA
Qawasim ND-1	4752.03	4752.99	20.7	20.73	-	-	4906.00	4911.12	1566.00	1570.70	3.80	3.81
Qawasim ND-4	2665.91	2718.1	1.63	1.67	-	-	3060.00	3117.43	2231.00	2238.86	3.50	3.50
Abu Madi ND-2	5.2 × 10−5	5.3 × 10−5	-	-	279.99	281.98	316.00	318.86	1203.00	1216.26	15.0	15.20
Abu Madi ND-3	2.10 × 10−6	2.10 × 10−6	-	-	356.88	360.85	390.98	395.20	1556.00	1599.11	3.50	3.50
Bahariya WD-1	1046.03	1063.19	0.621	0.606	-	-	1079.00	1099.29	1677.00	1685.45	3.38	3.46
Bahariya WD-2	198.24	201.58	0.1	0.1	-	-	215.00	217.70	970.00	972.74	10.00	10.00

). For instance, the Kabs calculations varied between 289.9 mD at P10 and 350.6 mD at P90 in well ND-2, whereas in well ND-3 varied between 391 mD at P10 and 431.7 mD at P90 (Table 5). In contrast, the calculated P90 for Rinv showed significant changes compared to the conventional calculations. For example, the conventional calculations in well ND-3 reported only 1559 ft while the P90 was 1852 ft indicating a 16% increase in Rinv values. This effect is decreased to approximately a 10% difference between the P50 and P90 in well ND-2 (P50 = 1201 ft and P90 = 1337 ft). The maximum changes in skin factor of the Abu Madi zone are recorded in well ND-2 (15.46 in. at P90), compared to the corresponding parameter in well ND-3 (3.57 in. at P90) indicating high damage in the producing zone of well ND-2. In the Bahariya zones, the calculated P90 of Kabs showed significant changes compared to the conventional calculations. For example, the conventional calculations in well WD-1 reports Kabs of 1079 mD while the P90 provides 1323 mD indicating an 18% increase. This effect decreased to approximately 14% in well WD-2, (P50 = 214.7 mD, P90 = 252 mD). These marked changes in Kabs of these wells can be attributed to the high uncertainty in pay thickness calculations of WD-1 well, (P50 = 50 ft versus P90 = 68 ft; Table 4).

Sensitivity analyses are implemented for the entire probability region (P0 to P100) to evaluate the performance of the calculation models of different permeabilities (K_o, K_w, K_g), Rinv, and skin factors in all the studied wells. The Tornado plots of well ND-1, as an example, are presented in Figure 6. Generally, for oil wells, K_o, and K_w calculations are significantly affected by the uncertainty in pay thickness with the minimal effect observed in well ND-1 and maximum effect reported in well WD-1 (

Table 7. The error analysis results for all wells under full probability region (P0 to P100).

Wells	Formations	Parameters	Pay Thick.	Φ	µo	Bo	Ct	Flow Rate	Pressure
ND-1	Qawasim	Ko	8.9%	0.4%	5.9%	4.8%	0.5%	0.5%	7.6%
		Kw	5.03%	0.17%	0.2%	0.3%	0.2%	0.2%	5.8%
		Rinv	0.3%	6.4%	4.3%	0.2%	3.4%	0.2%	0.4%
		Skin	0.2%	2%	1.2%	0.3%	1.1%	0.3%	8.3%
ND-4	Qawasim	Ko	16.6%	0.8%	7.8%	5.5%	0.5%	2.7%	14.6%
		Kw	17.8%	6.6%	6.4%	6.2%	6.3%	6.3%	15.8%
		Rinv	0.3%	5.9%	4.1%	0.5%	10.3%	0.4%	0.3%
		Skin	0.5%	1.4%	0.9%	0.3%	2.4%	0.4%	14.9%
WD-1	Bahariya	Ko	17.2%	0.9%	11.2%	2.8%	0.8%	16.9%	3.7%
		Kw	17%	0.4%	0.3%	0.6%	0.6%	0.5%	3.4%
		Rinv	0.5%	3.8%	5.3%	0.2%	6.7%	0.3%	0.1%
		Skin	0.2%	1.2%	1.5%	0.1%	1.9%	0.2%	5.7%
WD-2	Bahariya	Ko	14.2%	0.6%	10.7%	2.6%	0.8%	4.5%	8.2%
		Kw	14.1%	1.0%	0.9%	1.1%	0.9%	0.8%	7.9%
		Rinv	0.2%	1.8%	5.3%	0.3%	2.7%	0.2%	0.3%
		Skin	0.02%	0.2%	0.7%	0.1%	0.3%	0.02%	0.2%
ND-2	Abu Madi	Kg	10.4%	0.4%	0.2%	0.4%	0.3%	2.8%	7.6%
		Kw	7.2%	0.5%	0.5%	0.4%	0.5%	0.5%	6.2%
		Rinv	0.50%	2.8%	9.8%	0.1%	3.8%	0.2%	0.3%
		Skin	0.05%	0.3%	1.1%	0.02%	0.4%	0.01%	0.03%
ND-3	Abu Madi	Kg	10.2%	0.2%	0.5%	0.4%	0.4%	2.2%	6.5%
		Kw	10.2%	0.3%	0.5%	0.3%	0.4%	0.3%	6.4%
		Rinv	0.6%	2.0%	8.2%	0.3%	14.9%	0.6%	0.4%
		Skin	0.05%	0.2%	0.9%	0.03%	1.7%	0.05%	0.04%

and Figure 6a). Alternatively, oil viscosity and pressure measurements had medium effects on K_o and K_w analysis (Table 7). On the other hand, the Rinv calculation is affected mainly by porosity and oil viscosity calculations (Figure 6b), while the uncertainty in pressure measurements and total compressibility showed secondary effects on Rinv calculations. Finally, the skin factor calculations are greatly influenced by the pressure measurements, while the total compressibility and porosity uncertainty had secondary effects (Table 7 and Figure 6c). For gas wells, K_g and K_w calculations are also strongly affected by pay thickness uncertainty in wells ND-2 and ND-3, with medium influences tempted by the reservoir pressure and flow rate (Table 7). Similarly, the skin factor calculations showed great sensitivity towards the gas viscosity and total compressibility, but the uncertainty in porosity had a secondary influence on skin factor calculations. The radius of investigation calculations is affected mainly by uncertainty in gas viscosity and total compressibility, with minimal impacts by porosity and pay thickness (Table 7).

3.3. Influences of Well Test Techniques

To evaluate the effect well test technique (methods of measurements) on the results of well test uncertainty analysis, the results of the present conventional pressure transient analysis (K, S, and Rinv) are compared to the results obtained from the modular dynamic test tool in WQOF presented by Jirgis and Abdulaziz [1]. WQOF involves three oil-bearing zones that fall within the Cretaceous succession including Saadi, Mishrif, and Yamama Formations. Well test results under uncertainty of these formations have been compared to the corresponding parameters in Qawasim, Abu Madi, and Bahariya Formations. Furthermore, the sensitivity analysis applied to well test results helped to evaluate the impact of each input parameter on the final output result using only two runs: one for the low shift value and the other for the high shift value. Under various uncertainties, the calculated parameters in the Saadi pay zone showed a limited change that could be attributed to the prevailing tight conditions induced by the sedimentary facies, while Mishrif pay zones showed remarkable changes [1]. Error and sensitivity analysis shows that the shift in results compared to the original values for all the parameters has an obvious influence on calculations of a specific parameter, such as permeability. Generally, in WQOF the permeability calculations of Mini-DST measurements are strongly affected by the pay thickness uncertainty with influences varied between 14.5% and 47%, with medium influences by pressure (10%), µ_o and B_o (9%), Φ (6%), and S_w (5%)). Similarly, the sensitivity analysis of the build-up test showed that the permeability calculations are affected by pay thickness (the reported influences fell between 5.03% and 17.8%), and pressure (14.6%), with medium influences by µ_o (11%) and B_o (5.5%). It is obvious that the pay thickness uncertainty in Mini DST is rigorous compared to its effect in the build-up test due to the long period and stability in the build-up test, despite the similar porosity uncertainty (6%) in both tests. Comparatively, the Rinv calculations in Mini DST are affected mainly by S_w (9%) and porosity (5%) uncertainties, with minimal influence by B_o, µ_o, and pressure, while the Rinv calculations in the build-up test are affected mainly by the total compressibility (10.3%) and porosity (6.4%), with minimal influence by pressure and oil flow rate. Alternatively, the skin factor calculations in Mini DST showed great sensitivity toward the pressure measurements (35%), but in the build-up test, showed great sensitivity towards the pressure measurements (14%) with medium influences on porosity and oil viscosity (Table 7). Such results indicate that the conventional build-up test provides comparatively stable measurements relative to Mini DST, and may decrease error analysis of most well test results down to approximately 30%. Moreover, permeability and Rinv calculations are significantly sensitive to petrophysical interpretation while skin factor is primarily dependent on pressure measurements. This manifests the importance of considering the uncertainty analysis of petrophysics and measurement techniques on well test results and interpretation. However, further applications of the proposed methodology should be extended to cover other well testing techniques at various reservoirs to confirm the influence of the depicted parameters on well test results.

4. Conclusions

The present study aimed to evaluate the effect of uncertainty in petrophysical and field measurement techniques on well test results and interpretation. The conventional MDT test (Mini DST) in wells from WQOF and the traditional DST test in wells from the Nile Delta and Western Desert basins, Egypt are evaluated using the conventional interpretation techniques and error analysis. Results showed that permeability calculations are strongly affected by uncertainty in pay thickness interpretation while Rinv calculations are affected by S_w and porosity uncertainty. Alternatively, the skin factor is significantly sensitive to the uncertainty in pressure measurements with moderate effects by porosity and oil viscosity. Error analysis of both Mini-DST and conventional pressure build-up test indicated that the well test results and interpretation of conventional build-up data are considerably stable and may reduce uncertainty by 30% relative to Mini-DST results. Also, tight formation typically, as demonstrated by the Saadi Formation, shows limited changes in the error analysis of well test interpretation.