Petrophysical parameters prediction and uncertainty analysis in tight sandstone reservoirs using Bayesian inversion method

Accepted Manuscript Petrophysical parameters prediction and uncertainty analysis in tight sandstone reservoirs using Bayesian inversion method Ruidong Qin, Heping Pan, Peiqiang Zhao, Chengxiang Deng, Ling Peng, Yaqian Liu, Mohamed Kouroura PII:

S1875-5100(18)30187-2

DOI:

10.1016/j.jngse.2018.04.031

Reference:

JNGSE 2553

To appear in:

Journal of Natural Gas Science and Engineering

Received Date: 13 September 2017 Revised Date:

3 April 2018

Accepted Date: 19 April 2018

Please cite this article as: Qin, R., Pan, H., Zhao, P., Deng, C., Peng, L., Liu, Y., Kouroura, M., Petrophysical parameters prediction and uncertainty analysis in tight sandstone reservoirs using Bayesian inversion method, Journal of Natural Gas Science & Engineering (2018), doi: 10.1016/ j.jngse.2018.04.031. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Petrophysical parameters prediction and uncertainty analysis in tight sandstone reservoirs using Bayesian inversion method

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Ruidong Qin, Heping Pan*, Peiqiang Zhao, Chengxiang Deng, Ling Peng, Yaqian Liu, Mohamed KOUROURA Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China

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Abstract: Petrophysical parameters are of great importance in the evaluation and characterization for reservoirs, especially for the unconventional reservoirs with complex properties. The geophysical inversion is an efficient and economic method to obtain the petrophysical parameters. In this paper, Bayesian inversion method is presented to predict petrophysical model with conventional well logs. Statistical analysis results of accepted Markov Chain Monte Carlo (MCMC) samples are used to study the uncertainty of forecasted parameters, since the MCMC is a powerful approach to obtain adequate samples obeying the posterior distribution of Bayesian inversion. The proposed method is applied to reservoirs of the Xiashihezi Formation which are typical tight sandstone layers in the Ordos Basin. Model prediction and corresponding uncertainty analysis are presented in detail at a specific depth. The interactive effects of multiple petrophysical parameters are investigated by correlation coefficients. Then, the accuracy and reliability of predicted model is validated by both forward log responses and core data of the whole depth interval. According to the results and discussions, it can be concluded that: (1) a reasonable prior information of model parameters will simplify the inversion problem, which provides much conveniences of statistical analysis of the MCMC samples; (2) the weak correlation between each two petrophysical parameters indicates that it is reasonable and feasible to disregard dependence of parameters; (3) synthetic logs calculated by predicted model are in good agreement with observed well logs, which implies the precision and credibility of Bayesian inversion; (4) the predicted porosity, permeability and minerals content are consistent with core data, verifying the effectiveness and reliability of proposed method and inversion results; (5) it is an advantage of Bayesian inversion to locate the most probable reservoirs with the extreme value. Keywords: Petrophysical parameters; Bayesian inversion; Markov Chain Monte Carlo; conventional well logs; tight sandstone reservoirs _______________________________________________________________________________ [email protected] (H. Pan) [email protected] (R. Qin)

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1. Introduction

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Oil and natural gas, with global consumption growth average of 1.6% and 1.5% respectively, are still the primary energy worldwide (BP, 2017). Due to the fast increase of energy demand and consistent decline of conventional hydrocarbon, unconventional reservoirs, such as tight or shale reservoirs, have been paid more and more attention by most countries and oil companies (Clarkson et al., 2012; Zou et al., 2012a; Hu et al., 2015; Zhao et al., 2017). The unconventional resources are becoming the most important contributor to annual production both in China and USA (Law and Curtis, 2002; Li et al., 2012; Jia et al., 2012; Sakhaee-Pour and Bryant, 2014). Therefore, there will be a rapid development of unconventional reservoirs in the future energy supply for its large reserves and production. Tight sandstone reservoirs are the major unconventional hydrocarbon resources and distributed all over the world (Ma et al., 2017). The rapid progress of exploration and development of tight sandstone reservoirs promotes related geological problems and associated technology after the great breakthrough in 1970s (Cui et al., 2017). The Ordos Basin in China is a famous hydrocarbon-bearing basin which contains many oil and gas fields such as Changqing oil field and Sulige gas field (Yang et al., 2008). The primary petroleum systems, Mesozoic oil system and Paleozoic gas system, are demonstrated to be tight reservoirs. (Lan et al., 2016; Guo and Xie, 2017). According to previous literatures and documents, tight sandstone reservoirs are characterized with porosity lower than 10%, permeability lower than 1 mD, pore throat diameter less than 1 µm and hydrocarbon saturation less than 60% (Zou et al., 2012b; Wang et al., 2017). Poor reservoir quality and strong heterogeneities, also, are typical features of tight sandstones (Lai and Wang, 2015). For that reason, tight sandstone reservoirs cannot be explored commercially to obtain economic flow rates and natural gas volume unless adopting novel technology such as hydraulic fracture treatment and horizontal drilling (Wu et al., 2017). As a result of poor reservoir quality and inconspicuous fluid characteristics of tight sandstone reservoirs, it is a challenging task to estimate porosity, permeability and water saturation precisely by conventional method. These petrophysical parameters, however, are necessary and essential in the process of tight sandstone reservoirs identification, interpretation and exploitation. Therefore, geophysical inversion provides another way to estimate the petrophysical parameters with conventional well logs. In the last 30 years, many different inversion approaches have been utilized to calculate the petrophysical parameters. Gradient decent method was firstly used in the GLOBAL program to interpret the well logs (Mayer and Sibbit, 1980). Later, some other linear or linearized optimization methods, including conjugate gradient method (Rodriguez et al., 1988; Peeters and Visser 1991), quasi-Newton method or variable metric method (Luo et al. 2005), are also adopted to search the minimum of a locally continuous objective function. With the improvement and development of inversion algorithm, non-linear inversion approaches were proposed to compute petrophysical parameters for their advantage of searching the global optimum rather than local optimum. Therefore, many non-linear inversion methods, such as Simulated Annealing (Zhou et al., 1992; Szucs and Civan, 1996), Genetic Algorithm (Dobróka and Szabó, 2012), Particle Swarm Optimization (Sun et al., 2016), Bacterial Foraging Algorithm (Pan et al., 2016) and machine learning method (Hall, 2016; Hall and Hall, 2017), are successfully employed in the well logs interpretation. The Bayesian inversion was firstly brought forward and 2

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developed based on the Bayes theorem by Tarantola and Valette (1982). Different with other inversion methods mentioned above, Bayesian inversion can provide the results of both best-fitting model parameters and their probability from a statistical perspective (Buland and Kolbjørnsen, 2012). Besides, Bayesian inversion is able to takes advantage of prior information and handle the multiple local minima which tend to be the challenge to other inversion methods. Consequently, it is a natural choice to apply the Bayesian inversion to many geophysical problems (Grana, 2016). The Bayesian inversion introduction can refer to, for example, Mosegaard and Tarantola (2002); Tarantola (2005); Tarantola and Mosegaard (2007). The MCMC is an efficient and reliable method to characterize the posterior probability distribution from complex, high-dimensional model space (Zhang et al., 2016). Based on the MCMC realizations, the proportion of exceeding values can be used to identify the zones exceeding a certain threshold of petrophysical parameter (Claprood et al., 2014). In this paper, the Bayesian inversion is introduced to predict petrophysical parameters and analyze the uncertainty in the tight sandstone reservoirs. The inversion results are represented by the posterior probability density from which confidence interval is available. The predicted models and probability of extreme values are utilized to characterize the tight sandstones and locate the hydrocarbon reservoirs.

Fig. 1. (A) Blue area showing the location of the Ordos Basin in China, (B) Study area and 3

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tectonic units of Ordos Basin (Modified after Zhao et al., 2008 and Mi et al., 2016).

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Fig. 2. Upper Paleozoic Stratigraphic section of Hangjinqi area (Modified after Yang et al., 2008).

2. Data

2.1 Study area

Bayesian inversion method is applied to tight sandstone reservoirs in northern Ordos Basin, China (Fig. 1(A)). The Ordos Basin, with area about 330,000 km2, is a large intracontinental sedimentary basin in the Western Block of the North China Craton (NCC). The basin is divided into 6 tectonic units including Yimeng Uplift, Weibei Uplift, Yishan Slope, Western Fold-Thrust Belt, Tianhuan Depression and Jinxi Fault-Fold Belt (Xiao et al., 2008; Yang et al., 2015). Based on former geological knowledges in the Ordos Basin, it is widely recognized that there are three main petroleum systems which are Lower Paleozoic marine carbonate gas system, Upper 4

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Paleozoic terrigenous clastic rocks coal-formed gas system, and Mesozoic terrigenous clastic rocks oil system, respectively (Yang et al., 2008; Gao and Wang, 2017). This paper mainly concerns about the Upper Paleozoic gas system. The study area, Hangjinqi, is located in the Yimeng Uplift of northern Ordos Basin extending from 43°36′N to 44°24′N, and from 192°00′E to 194°00′E (Fig. 1(B)). As is illustrated in Fig. 2, three most important formations in this area are the Lower Permian Xiashihezi Formation (P1x), the Lower Permian Shanxi Formation (P1s) and the Upper Carboniferous Taiyuan Formation (C3t). In the Upper Paleozoic gas reservoirs, lithology mainly consists of coarse or medium grained lithic sandstone, lithic quartz sandstone and quartz sandstone (Du et al., 2016; Yang et al., 2016). In these formations mentioned above, sandstones are mixtures of mineral grain, rock fragment, cement and pore space according to thin section results. Porosity and permeability are the most effective parameters to characterize the reservoirs.

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2.2 Well logs and core data

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Well logs of Well-J6 is chosen to assess petrophysical properties of the Upper Paleozoic gas reservoirs. As is known to all, well log is an effective method to evaluate the reservoirs accurately. Lithological logs, such as gamma ray (GR) and spontaneous potential (SP), are used to distinguish reservoirs from non-reservoirs. And resistivity logs are employed to identify hydrocarbon (oil and gas) reservoirs due to the higher resistivity characteristic compared with aquifers. Moreover, gas reservoirs are recognized based on its unique features of porosity logs. For instance, density log value (ρb) decreases obviously, or excavation effect can be found in the neutron log (ΦN) when the reservoirs contain natural gas (Segesman and Liu, 1971). And six kinds of well logs, including ∆t, ρb, ΦN, GR, Rs and Rd (symbols used in this paper are listed in Table 1), are applied to quantify the petrophysical parameters and then evaluate the gas reservoirs in this paper. Core data contains many crucial prior information including porosity (φ), water saturation in invaded (Sxo) and uninvaded (Sw) zone, mineral component and content. However, it is costly, sometimes even impossible, to measure these petrophysical parameters at every depth of the reservoirs. For that reason, geophysical inversion methods are applied to predict petrophysical parameters, then inversion results are restrained by prior information of core data. Core data is obtained from some laboratory experiments results, such as relative permeability curves, mercury injection capillary pressure curves, and thin section analysis results. As the statistical results of core samples shown in the Table 2, in the Upper Paleozoic gas reservoir, porosity value range varies from 0.0420 to 0.2390 with average of 0.0929, and permeability changes from 0.03mD to 9.09mD with average of 0.87mD, which indicates that the Upper Paleozoic stratum in Hangjinqi area is a typical tight sandstone reservoirs. According to thin section results of Well-J6, the most obvious mineral grains are quartz (about 0.6, v/v) and feldspar (<0.05, v/v). Phyllite is the chief component of rock fragment (0.18, v/v) compared with rest minor minerals. The cement consists of the chlorite and some other clay minerals. However, it is difficult to provide an exact percentage of chlorite because cement is a mixture of many kinds of clay minerals. Therefore, it should be noted that clay mineral contents are replaced by a total volume of the shale (Vsh) in this paper.

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Unit

∆t ρb ΦN GR Rs Rd Sxo Sw φ Vsh Vqz Vph Vfs K mmean mmp σ/σ- /σ+

Acoustic travel time Density Neutron porosity Natural gamma ray intensity Shallow electrical resistivity (invaded zone) Deep electrical resistivity (uninvaded zone) Water saturation in invaded zone Water saturation in uninvaded zone porosity Shale content Quartz content Phyllite content Feldspar content Permeability Mean model Most possible model Standard/lower/upper deviation

µs/m g/cm3 % API Ω.m. Ω.m. v/v v/v v/v v/v v/v v/v v/v mD

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Symbol

Table 2 The statistical result of porosity and permeability in Upper Paleozoic formation of Hangjinqi area. Permeability (mD)

mean

number

max

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mean

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max

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0.0825 0.0795 0.1010 0.1150 0.0865 0.0929

54 41 319 197 26 637

0.1160 0.1130 0.2060 0.2390 0.1310 0.2390

0.0490 0.0460 0.0560 0.0450 0.0420 0.0420

0.84 0.67 0.88 0.95 1.02 0.87

54 41 319 160 26 600

5.62 3.02 8.79 9.09 8.65 9.09

0.07 0.04 0.03 0.05 0.03 0.03

Vfs

Phyllite

Vph

Quartz

Vqz

Shale

Vsh

irreducible water free water

Shc,m

mobile hydrocarbon

Shc,r

residual hydrocarbon

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pore

Sxo

Sw

Feldspar

matrix

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P 1x P 1x 2 P 1x 1 P 1s C 3t Total

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Porosity (v/v)

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Fig. 3. Petrophysical rock model assumed in this paper. Volume of rock includes pore and solid matrix. Solid matrix consists shale and other three minerals. 6

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3. Petrophysics Modeling

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Equivalent petrophysical model is a conventional approximation to analyze the properties and characteristics of rocks (Schlumberger, 1991). Many different portions are consisted in the model based on the petrophysical diversities of each rock component. Then, the response of each component is investigated to calculate the total response of the whole rock volume. The petrophysical model of Heidari et al. (2012) includes solid matrix, wet shale and non-shale porosity. Zhao et al. (2016a, 2016b) took kerogen into consideration in the apparent porosity. Although there are various petrophysical models in different literatures, most models are mainly divided into pores, shale and matrix. The difference of these models is the constituent of the three chief parts according to the study area. The model chosen in this paper (in Fig. 3) is divided into pore (full of water and hydrocarbon) and rock matrix (including quartz, feldspar, phyllite and shale) based on the thin section analysis results in section 2.2. Then, the total volume is assumed as: V=ϕ+∑ =1 (1) Where Vmi is the volumetric fraction of i-th mineral component and φ is porosity; N denotes the total number of mineral categories. The mathematical relationship, which links petrophysical properties with observed well logs is termed response function. It is a conventional method to assume linear equations between rock components and well logs such as acoustic travel time, density, neutron porosity and gamma ray (Ellis and Singer, 2007). Multivarious elastic wave velocity models have been introduced to describe the categories of lithology and fluids in the porous rocks (Mavko et al, 2009). Voigt-Reuss-Hill equation is an average method, which provides a good approximation to estimate the effective elastic moduli of rock (Hill, 1952). In Raymer’s model, a quadratic equation is proposed to calculate the P-wave velocity in terms of the compressional wave velocity of rock matrix and pore fluids (Raymer, 1980). In view of sand-related and clay-related pores, Xu-White model is developed to estimate the elastic wave velocities in clay-sand mixtures (Xu and White, 1995). Even though many different nonlinear models have been introduced in the published literature, Wyllie’s time average equation also provide meaningful results that reveal the relation between velocity and fluid saturated rock (Wyllie, 1956). In the research of Wu and Grana (2017), Wyllie’s equation provides the similar prediction results with Raymer’s model in tight sandstone gas reservoir, and avoids the additional model parameters such as elastic modulus at the same time. For these reasons, Wyllie’s equation is chosen to compute the acoustic travel time in the stiff sandstone in this paper. ∆t = ϕ ∆t + ∆t 1 − + ∑ ∆t (2) Where ∆tmf, ∆thc, ∆tmi are the acoustic travel time of mud-filtrate, hydrocarbon and i-th mineral component separately. It is normally assumed to calculate the nuclear well logs including bulk density, neutron porosity and gamma ray intensity with the linear response equations below(Rabaute and Revil, 2003; Heidari et al, 2012). The linear bulk density response equation, which is related to the rock minerals and fluids is applied to the study of tight gas reservoirs by Woodruff et al. (2010). In addition, the linear relationship is also typical for neutron porosity and gamma ray intensity to evaluation the shaly sand formation (Dobróka and Szabó, 2012; Grana et al., 2012; Ajayi and 7

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=ϕ , + , 1− (3) + ∑ , Φ = ϕ Φ , +Φ , 1− + ∑ Φ , (4) =ϕ + 1− + ∑ (5) Where subscript in each equation indicates the logging response coefficients of mud-filtrate (mf), hydrocarbon (hc) and i-th mineral component (mi) respectively. These response values of different rock constitutions are usually obtained from laboratory experiments. The resistivity is usually estimated based on some nonlinear models. Therefore, the non-linear part of forward problem mainly involves the calculation of shallow and deep resistivity (Rs and Rd). Many different saturation models can be utilized to simulate the resistivity logs, such as clean sandstone formation model (Archie 1942), dispersed shale model (Poupon et al., 1954) and dual-water model (Waxman and Smits 1968). In the studies of Amiri et al. (2012, 2015), the Indonesia water saturation model (Poupon and Leveaux, 1971) is found to be the most reliable method to predict the water saturation in tight gas shaly sandstones. Consequently, the Indonesia formula is chosen to compute the resistivity logs in this paper.

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Where Rs and Rd are shallow and deep resistivity respectively, Rsh is shale resistivity. Sxo and Sw denote the water saturation in invaded and uninvaded zone respectively. And a, b are constants representing rock texture properties, m is the cementation exponent, n is the saturation exponent. These four empirical coefficients are determined from laboratory experiment results or related literatures. The resistivity of formation water (Rw) is derived by the formation water salinity data. The resistivity of mud-filtrate (Rmf) is based on the field record. As a general rule, forward modeling mathematical formula is written in the form of Eqs. (8), where m and d denote model vector and data vector respectively (Eqs. (9) and (10)), and the superscript T means transpose. Therefore, the synthetic log responses can be figured out after determining the petrophysical model and every log response coefficient. 9=: ; (8) D ; = [ , = , >, ? @A , B , ? ] (9)

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9 = [ΔF, , Φ , , ? , G ]D (10) Then Bayesian inversion is implemented to estimate the fluid saturation (Sxo, Sw), porosity (φ) and solid minerals’ content (Vsh, Vqz, Vph, Vfs) in the chosen tight sandstone reservoirs. The log response coefficients of different minerals, which have been studied a lot, are determined by the published literatures such as Ellis and Singer, 2007. While it is complex to decide the log response of shale due to the mixture of many shale minerals, we recommend to determine the shale response coefficients on the basis of well logs of nearby shale formations which are of high gamma ray intensity. Then it is readily to compute the total well log value if the response coefficients of each mineral and fluid are determinate. Further parameters such as mobile and residual hydrocarbon saturation (Shc, m and Shc, r) are calculated with the following equations: − = (11) , = (12) ,H = 1 − Moreover, based on core samples of the Upper Paleozoic formation in study area, the empirical 8

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K = 0.0346 N 4O.P QR- 4 = 0.6 Where K (mD) denotes the permeability, and φ (v/v) is the porosity.

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4. Methodology

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4.1 Bayesian inversion

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The classical geophysical inversion methods can provide only one certain solution since these methods do not take into consideration the uncertainties of both observed data and model parameters. Actually, it is difficult to obtain a definite inversion solution for two reasons: firstly, non-uniqueness of the inverse problem; secondly, data errors are ineluctable when measuring some physical parameters (Backus 1970; Backus and Gilbert, 1970). Bayesian inversion method can provide the probabilities of different inverse solutions using different prior information as constrains. The prior information of data and model is usually described by the probability density ρD(d) and ρM(m). In the simplest situation, the distribution of data and model are assumed to be Gaussian type and represented by covariance CD and CM respectively. Therefore, ρD(d) and ρM(m) can be denoted by the Gaussian prior probability density centered at dobs and mprior. ZP.Q

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Where j and k are the dimensions of data vector d and model vector m. Inverse problem is on the basis of forward problem, which predicts the error-free data d according to the given model m, and g represents the mathematical relationship between d and m (Eqs. (2)-(7)). When the theoretical uncertainties are not negligible due to the imperfect modeling or lack of some important information, the error-free Eq. (8) is replaced with a probabilistic relationship in Eq. (16) between model m and data d. Θ 9, ; = θ 9|; d] ; (16) Where θ(d|m) denotes probability density of d for every given model m, and µM(m) is a homogeneous probability density. If the theoretical uncertainties are disregarded, the theoretical probability density is: Θ 9, ; = δ 9 − : ; d] ; (17) In this paper, probabilistic relationship is described by the joint probability density Θ(d, m) shown in Eq. (17). Then the general solution of the inverse problem is written in the form of posterior probability density as in Eq. (18). f] ; = g ] ; h ; (18) Where k is a constant and L(m) is the likelihood function represented in Eq. (19). h ; =i

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n9

(19)

Where µD(d) is the homogeneous probability density in the data manifold. Moreover, σM(m) makes it possible for us to obtain more statistical information of model parameters, such as mean model, maximum likelihood model and uncertainty interval (Tarantola, 2005). Also, we can compute the possibility of a given model by integrating the probability density. 9

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It is feasible to compute the posterior probability density of every possible model only when the probability distribution is defined over a low (less than 5, more precisely) dimensional space (Tarantola, 2005). Whereas representing the probability density with an analytic method will become difficult when it comes to a large dimensional number of model space. Therefore, it is of great necessary to find some equivalent approaches to describe high dimensional probability distribution efficiently. Some sampling methods such as Gibbs sampler (Geman and Geman; 1984) and MCMC method (Wang et al., 2017; Sambridge and Mosegaard 2002), fortunately, can generate sufficient samples to approximate the probability density. MCMC method is selected in this paper for the swift and effective sampling and ability to search the multiple local minima in nonlinear inverse problem. The Metropolis-Hastings algorithm is adopted when using MCMC to obtain adequate samples of the posterior probability density. After obtaining enough samples which obey the prior distribution (like Gaussian or uniform), it is easy to accept or reject the samples with MCMC method. For instance, at an arbitrary step i, suppose mi is the current random model, and Metropolis-Hastings algorithm will decide whether accept the transition from mi to next model mj. The rejection rule is as follows: (1) Calculate the likelihood function L(m) according to Eq. (19) at mi and mj respectively; (2) If L(mj) ≥ L(mi), then transition from mi to mj is accepted; (3) If L(mj) < L(mi), then generate a random number, r, that distributes homogeneously between 0 and 1; if L(mj) / L(mi) > r, then accept the transition to mj, else stay at mi. It should be mentioned that sampling efficiency is of high significance when using MCMC approach to acquire enough independent samples (Mosegaard, 1995). It is a known fact that if the models are easily to be accepted, the random walk will spend more time to generate efficient models to fit the distribution. And a low acceptance rate will waste too much time to test unaccepted models. Therefore, it is necessary to reach a compromise between sampling speed and quality with a reasonable acceptance rate (0.2~0.5 suggested by Gamerman, 1997). After generating enough accepted samples (m1, m2, …, mn), some statistical approaches are employed to describe the characteristics of posterior probability density of model parameters, such as porosity (φ) or water saturation (Sw) in the petrophysical model introduced in Section 3. Firstly, if the posterior model parameters are Gaussian type, the mean value (mmean) and covariance are estimated from these samples for further uncertainty analysis. Secondly, the posterior distribution of model parameters are represented in the form of Gaussian distribution N(µ, σ2). If the distribution of model is asymmetric or not Gaussian type, another estimation approach, advised by Liu (2017), can be used for the uncertainties analysis. Consequently, the mmean and σ are used to calculate the correlation coefficients and confidence intervals of model parameters, which is essential to assess the petrophysical model and analyze the uncertainty.

5. Results and discussions After the determination of petrophysical model, Bayesian inversion is implemented by using well logs and core data as prior information. After obtaining adequate (2000, in this paper) accepted samples at each depth on the basis of Bayesian inversion and MCMC sampling approach, 10

ACCEPTED MANUSCRIPT it is feasible to calculate the posterior distribution of model parameters and analyze their uncertainties. Firstly, Bayesian inversion inference is introduced in detail for one measuring point. Then model parameters are predicted and assessed in the whole depth interval of reservoirs.

5.1 Prior information

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The distribution of well logs is assumed to be Gaussian or logarithmic normal (for resistivity) type since the well logs used for Bayesian inversion are from the same reservoirs which is characterized with stable well logs and similar petrophysical properties (Fig. 12). The Fig. 4 demonstrates that it is reasonable to describe the well logs of Well-J6 in the depth interval between 2740m and 2760m with Gaussian or logarithmic normal probability density. For instance, Fig. 4a shows the distribution of 161 observed acoustic travel time data in the depth interval chosen above. Distribution parameters, mean and standard deviation, are extracted from the estimated probability density (red curves in Fig. 4). Therefore, the prior information of observed data, dobs and CD in the Eq. (13), are obtained if the correlation of each well logs is not taken into consideration. In this section, another crucial assumption is that model parameters obey the Gaussian distribution when we choose the depth intervals that well log response is relatively steady or similar. In other words, the petrophysical parameters in the chosen depth interval may be different but their fluctuations are maintained within the small range. For that reason, the prior information with Gaussian probability density generally applies well to this situation (Fig. 5). According to the core data, prior information of model, mprior and CM in Eq. (14), are estimated to constraint the inversion results. And the correlation among these model parameters is also neglected to simplify the inversion procedure.

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5.2 Model parameters prediction and uncertainty analysis

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In this part, Bayesian inversion results at 2750m of Well-J6 are explicated in detail for the better understanding of predicted model and uncertainties. For example, the marginal distribution of φ and Vqz are illustrated in Fig. 6a and 6b respectively. Model parameters uncertainties are evaluated by 68% confidence interval [mmean - σ, mmean + σ]. Probability density between φ and Vqz, calculated by the MCMC samples, is displayed in Fig. 7 by different colors. The predicted model and confidence interval are listed in Table 3. The 68% confidence intervals of porosity and quartz content are [0.0810, 0.1057] and [0.5732, 0.6274]. Fig. 8 illustrates the histograms and cross-plots of all model parameters at 2750m. From the results, Gaussian estimation describes all model parameters well despite the existence of some faintly skew. And the red ellipses correspond to 68% confidence level of each predicted model, which can also be applied to study the correlation of each two model parameters. For instance, the confidence ellipse of Vqz and Vph is close to a straight line which indicates a strong correlation between two model parameters. And it is also shown in the Table 4 that the correlation coefficient is -0.8028 between Vqz and Vph. While the other model confidence ellipses just show the weak correlation in Fig. 8. The variation of all accepted MCMC samples are represented with boxplot in Fig. 9. It should be noted that the value of Sum, which is expected to be close to 1, indicates the total volume content of porosity and the other rock matrix. It is easier to get a general idea of distribution range and distinguish the outliers with boxplot. And the Bayesian inversion results are much close to the mean values of core data (black dots). 11

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µ = 228.34 σ = 8.46

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Fig. 4. Histograms of well logs of Well-J6 in the depth interval of 2740~2760m: (a) ∆t, (b) ρb, (c) ΦN, (d) GR, (e) Rs, (f) Rd. Red curves denote the estimated Gaussian or log-normal distribution. 12

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Fig.5. Histograms displaying parts of model parameters’ distribution of 24 core samples: a) porosity, (b) shale, (c) phyllite, (d) quartz. Red curves indicate the estimated Gaussian distribution.

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Fig. 6. Histograms of porosity (φ, panel a) and quartz content (Vqz, panel b) of MCMC sampling results of Well-J6 at the depth of 2750m. Red curves indicate probability density correspond to Gaussian distribution.

Fig. 7. Probability density between φ and Vqz. Yellow means high probability and blue means null probability. The green dots are the accepted MCMC samples at the depth of 2750m. Pink error bars are plotted with mean and standard deviation (mmean ± σ).

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0.66 0.77 0.88 0.99 0.5840.6570.7300.803 0.0500.0750.1000.125 0.0880.1100.1320.1540.5130.5700.6270.6840.0580.1160.1740.232 0.0000.0170.0340.051 Sw

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Fig. 8. Histograms and cross-plots of seven model parameters. The red curves in the diagonal subfigures are estimated Gaussian distribution. The 68% confidence ellipses (red) signify the uncertainty of Gaussian estimation.

Fig. 9. Boxplot displaying the variation of MCMC samples. Lines in the boxes indicate the median of samples. The individual points represent the outliers. The black dots are the mean value of core data. The Sum is the total volume of porosity and the other four minerals content. 14

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0.8182 0.7170 0.0934 0.1120 0.6003 0.1519 0.0195

0.0493 0.0347 0.0123 0.0123 0.0271 0.0263 0.0101

[0.7690, 0.8675] [0.6824, 0.7517] [0.0810, 0.1057] [0.0998, 0.1243] [0.5732, 0.6274] [0.1257, 0.1782] [0.0094, 0.0296]

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1.0000 0.0341 0.4469 -0.4296 0.0140 0.0361 0.0396

1.0000 0.0117 -0.1553 0.0449 0.0111 -0.0037

1.0000 -0.4612 -0.0268 -0.0472 0.0089

1.0000 -0.0814 -0.1756 -0.0458

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According to the inversion results of 2750m, it is rational and reliable to predict the model parameters and uncertainties with Bayesian inversion. Consequently, this method is extended to all measuring points of two tight sandstone reservoirs. Finally, the reservoirs’ petrophysical properties are assessed with the inversion results at each depth. Two predicted models (φ and Vqz) and uncertainties are illustrated in Fig. 10a and 10b. The probability density is indicated by different color in the tinted belt, where the darker color (red) means higher possibility and the lighter color (white) point to lower possibility. Based on the inversion results, a majority of core data is neighboring to the red areas which implies higher possibility of predicted model parameter, although several core data is not in the colored belt. For instance, the quartz content (Vqz) of core data is about 0.09 bigger than predicted mean model at 2751.5m. As for porosity prediction, the width of uncertainty range is around 0.05, which implies a high resolution of porosity estimation. The synthetic well logs’ are also computed with Eqs. (2)-(7) and predicted mean model (mmean). Four kinds of measured and calculated well logs are shown in Fig. 11, the ρb and Rs, particularly, are greatly close to the actual well logs, which validates the accuracy of the Bayesian inversion method. The core data at the interval between 2752m and 2760m correspond to the predicted petrophysical model, which indicates that the predicted mean model is very close to the actual situation. However, some core data are obviously out of the 68% confidence interval, for instance at the depth of 2741m, the core porosity (0.1400) is larger than the predicted porosity (0.1041). This underestimated porosity is also revealed in Fig. 11 by the less calculated acoustic travel time compared with the measured logs. This discrepancy between core data and predicted model might 15

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be caused by three main reasons. Firstly, shale response coefficients have much influence on the inversion results. In this paper, shale response coefficients are determined by the well log value of shale formation between 2765~2768m which is of high gamma ray intensity. Thus, the sandstones adjacent to this shale formation, such as 2752~2760m and 2770~2774m are described well with Bayesian inversion results due to the better determined and trustworthy shale response value. Secondly, the porosity of core data is measured in the laboratory rather than in situ, which may cause the inconsistency between core and predicted model parameters. In addition, core data are obtained by measuring the petrophysical characteristics of separate core sample, while well logs are the record of comprehensive response of the whole formation rocks. Therefore, the inversion results may be affected by difference of resolution between core data and well logs. Thirdly, the unavoidable noise of well logs may have impacted the inversion results. As for the quartz content, the disagreement of core data and inversion result are also shown at the depth of 2751.5m and 2751.8m. Apart from the difference in resolution mentioned above, the inconsistence may have come from the thin section results, which are obtained from a slice rather than the whole rock. Even if some difference may exist comparing inversion results with core data, the average of predicted quartz content (0.6050) is still much close to that of core quartz content (0.6138), which validates the reliability of Bayesian inversion. The predicted mean model parameters displayed in Fig.12 are exploited to evaluate the tight sandstone reservoirs of Well-J6. There are two main sandstone reservoirs in the intervals under study. In the depth intervals of 2740~2760m and 2770~2778m, predicted mineral contents, as well as water and hydrocarbon saturation, are displayed by different color in the fifth and sixth tracks. It is obvious that quartz is the chief mineral in both reservoirs mentioned above, and most parts of pores are saturated with water (free and irreducible). In seventh track, porosity of upper reservoir is less than 10% and increase to about 12% in the lower reservoir. Porosity logs, such as acoustic travel time, bulk density and neutron porosity, also reveal the slight increasing of porosity based on their characteristics. The eighth track displays the two values of permeability: the blue curve is calculated with Eq. (12), and the other is measured in laboratory. Most predicted permeability corresponds well to the core measurement data except the depth interval from 2741~2743m. The underestimated porosity is the main reason for this inconsistence in this depth interval. Besides, Eq. (12) is derived based on the statistical analysis, which may not well reflect the relationship between permeability and porosity. In both reservoirs, the porosity is around 10%, and permeability is between 0.1 and 1mD, which shows the typical features of tight reservoirs. In the last track, the proportion of extreme values (φ>0.1, v/v; K>0.5, mD) are calculated by 2000 MCMC realizations of each depth. From the results, the zones with higher probability of extreme values are more likely to be the economical reservoirs. So the extreme values assessment can recover the high porosity and permeability zones witch are underestimated by mean values. In this study, 4 depth intervals including 2740~2742m, 2747~2752m, 2758~2760m and 2775~2778m, are of the greatest probability to be the economical reservoirs. This result indicates that Bayesian inversion is able to identify zones exceeding a certain porosity or permeability thresholds, and locate the most probable reservoirs. Therefore, the point-by-point inversion results are capable of describing the tight sandstone reservoirs.

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Fig. 10. Posterior probability density of porosity (φ, panel a) and quartz content (Vqz, panel b). Different colored belts displaying the probability of model parameters. The black solid curves denote mmean. And red triangles represent the core data obtained with laboratory experiments.

Fig.11. Curves displaying the observed well logs (black curves) and approximate well logs calculated with mean model (red dashed curves).

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Fig.12. Interpretation results of Bayesian inversion method. The first three tracks display lithology, resistivity and porosity logs respectively. Depth is shown in the fourth track. The fifth track is the results of the predicted mineral contents. The sixth track displays the saturation of water and hydrocarbon. Seventh and eighth tracks represent the results of both predicted (blue curves) and core (red stick) porosity and permeability. The last track exhibits the probability of porosity and permeability exceeding the thresholds.

6. Conclusions

In this paper, the MCMC-based Bayesian inversion is introduced to estimate petrophysical parameters and quantify the uncertainty by using conventional well logs. This method is also applied to the tight sandstone reservoirs evaluation in the north Ordos Basin. The mean model is chosen to describe the reservoirs properties, and the uncertainty is denoted by the posterior probability density. Prior information is of great significance in petrophysical parameters prediction and can save lots of time to search the possible model in the model space. Mean model can be obtained with Gaussian estimation approach. The 68% confidence interval provides the most probable distribution range of each parameter. And the inversion results show the weak correlation between every two petrophysical parameters. The synthetic well logs according to the inversion results are 18

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in good agreement with observed well logs. According to the inversion minerals content, quartz is the main mineral in the reservoirs. The predicted porosity and permeability are compared with core measurement results, which demonstrate that Bayesian inversion results are consistent with the real petrophysical model. Therefore, it is precise and credible to evaluate the petrophysical parameters with Bayesian inversion method. Compared with traditional interpretation methods, Bayesian inversion method can provide not only a credible predicted model, but the probability of extreme values at the same time. And the inversion results is able to estimate the petrophysical parameters efficiently and analysis the uncertainty precisely, which is helpful to provide the references to describe the tight sandstone characteristics and locate the zones where is most likely to be economical reservoirs.

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This paper is supported by the National Natural Science Foundation of China Project Fund (NO. 41074086), and Fundamental Research Funds for Central Universities (China University of Geosciences, Wuhan) (No.CUG170619). The authors also thank four anonymous reviewers for their very valuable and helpful comments.

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Petrophysical parameters prediction and uncertainty analysis in tight sandstone reservoirs using Bayesian inversion method

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Ruidong Qin, Heping Pan*, Peiqiang Zhao, Chengxiang Deng, Ling Peng, Yaqian Liu, Mohamed KOUROURA Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China

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Abstract: Petrophysical parameters are of great importance in the evaluation and characterization for reservoirs, especially for the unconventional reservoirs with complex properties. The geophysical inversion is an efficient and economic method to obtain the petrophysical parameters. In this paper, Bayesian inversion method is presented to predict petrophysical model with conventional well logs. Statistical analysis results of accepted Markov Chain Monte Carlo (MCMC) samples are used to study the uncertainty of forecasted parameters, since the MCMC is a powerful approach to obtain adequate samples obeying the posterior distribution of Bayesian inversion. The proposed method is applied to reservoirs of the Xiashihezi Formation which are typical tight sandstone layers in the Ordos Basin. Model prediction and corresponding uncertainty analysis are presented in detail at a specific depth. The interactive effects of multiple petrophysical parameters are investigated by correlation coefficients. Then, the accuracy and reliability of predicted model is validated by both forward log responses and core data of the whole depth interval. According to the results and discussions, it can be concluded that: (1) a reasonable prior information of model parameters will simplify the inversion problem, which provides much conveniences of statistical analysis of the MCMC samples; (2) the weak correlation between each two petrophysical parameters indicates that it is reasonable and feasible to disregard dependence of parameters; (3) synthetic logs calculated by predicted model are in good agreement with observed well logs, which implies the precision and credibility of Bayesian inversion; (4) the predicted porosity, permeability and minerals content are consistent with core data, verifying the effectiveness and reliability of proposed method and inversion results; (5) it is an advantage of Bayesian inversion to locate the most probable reservoirs with the extreme value. Keywords: Petrophysical parameters; Bayesian inversion; Markov Chain Monte Carlo; conventional well logs; tight sandstone reservoirs _______________________________________________________________________________ [email protected] (H. Pan) [email protected] (R. Qin)

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1. Introduction

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Oil and natural gas, with global consumption growth average of 1.6% and 1.5% respectively, are still the primary energy worldwide (BP, 2017). Due to the fast increase of energy demand and consistent decline of conventional hydrocarbon, unconventional reservoirs, such as tight or shale reservoirs, have been paid more and more attention by most countries and oil companies (Clarkson et al., 2012; Zou et al., 2012a; Hu et al., 2015; Zhao et al., 2017). The unconventional resources are becoming the most important contributor to annual production both in China and USA (Law and Curtis, 2002; Li et al., 2012; Jia et al., 2012; Sakhaee-Pour and Bryant, 2014). Therefore, there will be a rapid development of unconventional reservoirs in the future energy supply for its large reserves and production. Tight sandstone reservoirs are the major unconventional hydrocarbon resources and distributed all over the world (Ma et al., 2017). The rapid progress of exploration and development of tight sandstone reservoirs promotes related geological problems and associated technology after the great breakthrough in 1970s (Cui et al., 2017). The Ordos Basin in China is a famous hydrocarbon-bearing basin which contains many oil and gas fields such as Changqing oil field and Sulige gas field (Yang et al., 2008). The primary petroleum systems, Mesozoic oil system and Paleozoic gas system, are demonstrated to be tight reservoirs. (Lan et al., 2016; Guo and Xie, 2017). According to previous literatures and documents, tight sandstone reservoirs are characterized with porosity lower than 10%, permeability lower than 1 mD, pore throat diameter less than 1 µm and hydrocarbon saturation less than 60% (Zou et al., 2012b; Wang et al., 2017). Poor reservoir quality and strong heterogeneities, also, are typical features of tight sandstones (Lai and Wang, 2015). For that reason, tight sandstone reservoirs cannot be explored commercially to obtain economic flow rates and natural gas volume unless adopting novel technology such as hydraulic fracture treatment and horizontal drilling (Wu et al., 2017). As a result of poor reservoir quality and inconspicuous fluid characteristics of tight sandstone reservoirs, it is a challenging task to estimate porosity, permeability and water saturation precisely by conventional method. These petrophysical parameters, however, are necessary and essential in the process of tight sandstone reservoirs identification, interpretation and exploitation. Therefore, geophysical inversion provides another way to estimate the petrophysical parameters with conventional well logs. In the last 30 years, many different inversion approaches have been utilized to calculate the petrophysical parameters. Gradient decent method was firstly used in the GLOBAL program to interpret the well logs (Mayer and Sibbit, 1980). Later, some other linear or linearized optimization methods, including conjugate gradient method (Rodriguez et al., 1988; Peeters and Visser 1991), quasi-Newton method or variable metric method (Luo et al. 2005), are also adopted to search the minimum of a locally continuous objective function. With the improvement and development of inversion algorithm, non-linear inversion approaches were proposed to compute petrophysical parameters for their advantage of searching the global optimum rather than local optimum. Therefore, many non-linear inversion methods, such as Simulated Annealing (Zhou et al., 1992; Szucs and Civan, 1996), Genetic Algorithm (Dobróka and Szabó, 2012), Particle Swarm Optimization (Sun et al., 2016), Bacterial Foraging Algorithm (Pan et al., 2016) and machine learning method (Hall, 2016; Hall and Hall, 2017), are successfully employed in the well logs interpretation. The Bayesian inversion was firstly brought forward and 25

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developed based on the Bayes theorem by Tarantola and Valette (1982). Different with other inversion methods mentioned above, Bayesian inversion can provide the results of both best-fitting model parameters and their probability from a statistical perspective (Buland and Kolbjørnsen, 2012). Besides, Bayesian inversion is able to takes advantage of prior information and handle the multiple local minima which tend to be the challenge to other inversion methods. Consequently, it is a natural choice to apply the Bayesian inversion to many geophysical problems (Grana, 2016). The Bayesian inversion introduction can refer to, for example, Mosegaard and Tarantola (2002); Tarantola (2005); Tarantola and Mosegaard (2007). The MCMC is an efficient and reliable method to characterize the posterior probability distribution from complex, high-dimensional model space (Zhang et al., 2016). Based on the MCMC realizations, the proportion of exceeding values can be used to identify the zones exceeding a certain threshold of petrophysical parameter (Claprood et al., 2014). In this paper, the Bayesian inversion is introduced to predict petrophysical parameters and analyze the uncertainty in the tight sandstone reservoirs. The inversion results are represented by the posterior probability density from which confidence interval is available. The predicted models and probability of extreme values are utilized to characterize the tight sandstones and locate the hydrocarbon reservoirs.

Fig. 1. (A) Blue area showing the location of the Ordos Basin in China, (B) Study area and 26

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tectonic units of Ordos Basin (Modified after Zhao et al., 2008 and Mi et al., 2016).

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Fig. 2. Upper Paleozoic Stratigraphic section of Hangjinqi area (Modified after Yang et al., 2008).

2. Data

2.1 Study area

Bayesian inversion method is applied to tight sandstone reservoirs in northern Ordos Basin, China (Fig. 1(A)). The Ordos Basin, with area about 330,000 km2, is a large intracontinental sedimentary basin in the Western Block of the North China Craton (NCC). The basin is divided into 6 tectonic units including Yimeng Uplift, Weibei Uplift, Yishan Slope, Western Fold-Thrust Belt, Tianhuan Depression and Jinxi Fault-Fold Belt (Xiao et al., 2008; Yang et al., 2015). Based on former geological knowledges in the Ordos Basin, it is widely recognized that there are three main petroleum systems which are Lower Paleozoic marine carbonate gas system, Upper 27

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Paleozoic terrigenous clastic rocks coal-formed gas system, and Mesozoic terrigenous clastic rocks oil system, respectively (Yang et al., 2008; Gao and Wang, 2017). This paper mainly concerns about the Upper Paleozoic gas system. The study area, Hangjinqi, is located in the Yimeng Uplift of northern Ordos Basin extending from 43°36′N to 44°24′N, and from 192°00′E to 194°00′E (Fig. 1(B)). As is illustrated in Fig. 2, three most important formations in this area are the Lower Permian Xiashihezi Formation (P1x), the Lower Permian Shanxi Formation (P1s) and the Upper Carboniferous Taiyuan Formation (C3t). In the Upper Paleozoic gas reservoirs, lithology mainly consists of coarse or medium grained lithic sandstone, lithic quartz sandstone and quartz sandstone (Du et al., 2016; Yang et al., 2016). In these formations mentioned above, sandstones are mixtures of mineral grain, rock fragment, cement and pore space according to thin section results. Porosity and permeability are the most effective parameters to characterize the reservoirs.

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Well logs of Well-J6 is chosen to assess petrophysical properties of the Upper Paleozoic gas reservoirs. As is known to all, well log is an effective method to evaluate the reservoirs accurately. Lithological logs, such as gamma ray (GR) and spontaneous potential (SP), are used to distinguish reservoirs from non-reservoirs. And resistivity logs are employed to identify hydrocarbon (oil and gas) reservoirs due to the higher resistivity characteristic compared with aquifers. Moreover, gas reservoirs are recognized based on its unique features of porosity logs. For instance, density log value (ρb) decreases obviously, or excavation effect can be found in the neutron log (ΦN) when the reservoirs contain natural gas (Segesman and Liu, 1971). And six kinds of well logs, including ∆t, ρb, ΦN, GR, Rs and Rd (symbols used in this paper are listed in Table 1), are applied to quantify the petrophysical parameters and then evaluate the gas reservoirs in this paper. Core data contains many crucial prior information including porosity (φ), water saturation in invaded (Sxo) and uninvaded (Sw) zone, mineral component and content. However, it is costly, sometimes even impossible, to measure these petrophysical parameters at every depth of the reservoirs. For that reason, geophysical inversion methods are applied to predict petrophysical parameters, then inversion results are restrained by prior information of core data. Core data is obtained from some laboratory experiments results, such as relative permeability curves, mercury injection capillary pressure curves, and thin section analysis results. As the statistical results of core samples shown in the Table 2, in the Upper Paleozoic gas reservoir, porosity value range varies from 0.0420 to 0.2390 with average of 0.0929, and permeability changes from 0.03mD to 9.09mD with average of 0.87mD, which indicates that the Upper Paleozoic stratum in Hangjinqi area is a typical tight sandstone reservoirs. According to thin section results of Well-J6, the most obvious mineral grains are quartz (about 0.6, v/v) and feldspar (<0.05, v/v). Phyllite is the chief component of rock fragment (0.18, v/v) compared with rest minor minerals. The cement consists of the chlorite and some other clay minerals. However, it is difficult to provide an exact percentage of chlorite because cement is a mixture of many kinds of clay minerals. Therefore, it should be noted that clay mineral contents are replaced by a total volume of the shale (Vsh) in this paper.

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Unit

∆t ρb ΦN GR Rs Rd Sxo Sw φ Vsh Vqz Vph Vfs K mmean mmp σ/σ- /σ+

Acoustic travel time Density Neutron porosity Natural gamma ray intensity Shallow electrical resistivity (invaded zone) Deep electrical resistivity (uninvaded zone) Water saturation in invaded zone Water saturation in uninvaded zone porosity Shale content Quartz content Phyllite content Feldspar content Permeability Mean model Most possible model Standard/lower/upper deviation

µs/m g/cm3 % API Ω.m. Ω.m. v/v v/v v/v v/v v/v v/v v/v mD

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Symbol

Table 2 The statistical result of porosity and permeability in Upper Paleozoic formation of Hangjinqi area. Permeability (mD)

mean

number

max

min

mean

number

max

min

0.0825 0.0795 0.1010 0.1150 0.0865 0.0929

54 41 319 197 26 637

0.1160 0.1130 0.2060 0.2390 0.1310 0.2390

0.0490 0.0460 0.0560 0.0450 0.0420 0.0420

0.84 0.67 0.88 0.95 1.02 0.87

54 41 319 160 26 600

5.62 3.02 8.79 9.09 8.65 9.09

0.07 0.04 0.03 0.05 0.03 0.03

Vfs

Phyllite

Vph

Quartz

Vqz

Shale

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irreducible water free water

Shc,m

mobile hydrocarbon

Shc,r

residual hydrocarbon

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pore

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Sw

Feldspar

matrix

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P 1x P 1x 2 P 1x 1 P 1s C 3t Total

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Fig. 3. Petrophysical rock model assumed in this paper. Volume of rock includes pore and solid matrix. Solid matrix consists shale and other three minerals. 29

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3. Petrophysics Modeling

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Equivalent petrophysical model is a conventional approximation to analyze the properties and characteristics of rocks (Schlumberger, 1991). Many different portions are consisted in the model based on the petrophysical diversities of each rock component. Then, the response of each component is investigated to calculate the total response of the whole rock volume. The petrophysical model of Heidari et al. (2012) includes solid matrix, wet shale and non-shale porosity. Zhao et al. (2016a, 2016b) took kerogen into consideration in the apparent porosity. Although there are various petrophysical models in different literatures, most models are mainly divided into pores, shale and matrix. The difference of these models is the constituent of the three chief parts according to the study area. The model chosen in this paper (in Fig. 3) is divided into pore (full of water and hydrocarbon) and rock matrix (including quartz, feldspar, phyllite and shale) based on the thin section analysis results in section 2.2. Then, the total volume is assumed as: V=ϕ+∑ =1 (1) Where Vmi is the volumetric fraction of i-th mineral component and φ is porosity; N denotes the total number of mineral categories. The mathematical relationship, which links petrophysical properties with observed well logs is termed response function. It is a conventional method to assume linear equations between rock components and well logs such as acoustic travel time, density, neutron porosity and gamma ray (Ellis and Singer, 2007). Multivarious elastic wave velocity models have been introduced to describe the categories of lithology and fluids in the porous rocks (Mavko et al, 2009). Voigt-Reuss-Hill equation is an average method, which provides a good approximation to estimate the effective elastic moduli of rock (Hill, 1952). In Raymer’s model, a quadratic equation is proposed to calculate the P-wave velocity in terms of the compressional wave velocity of rock matrix and pore fluids (Raymer, 1980). In view of sand-related and clay-related pores, Xu-White model is developed to estimate the elastic wave velocities in clay-sand mixtures (Xu and White, 1995). Even though many different nonlinear models have been introduced in the published literature, Wyllie’s time average equation also provide meaningful results that reveal the relation between velocity and fluid saturated rock (Wyllie, 1956). In the research of Wu and Grana (2017), Wyllie’s equation provides the similar prediction results with Raymer’s model in tight sandstone gas reservoir, and avoids the additional model parameters such as elastic modulus at the same time. For these reasons, Wyllie’s equation is chosen to compute the acoustic travel time in the stiff sandstone in this paper. ∆t = ϕ ∆t + ∆t 1 − + ∑ ∆t (2) Where ∆tmf, ∆thc, ∆tmi are the acoustic travel time of mud-filtrate, hydrocarbon and i-th mineral component separately. It is normally assumed to calculate the nuclear well logs including bulk density, neutron porosity and gamma ray intensity with the linear response equations below(Rabaute and Revil, 2003; Heidari et al, 2012). The linear bulk density response equation, which is related to the rock minerals and fluids is applied to the study of tight gas reservoirs by Woodruff et al. (2010). In addition, the linear relationship is also typical for neutron porosity and gamma ray intensity to evaluation the shaly sand formation (Dobróka and Szabó, 2012; Grana et al., 2012; Ajayi and 30

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+

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+

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$"%

=#

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,1 !.2

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5 8

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=ϕ , + , 1− (3) + ∑ , Φ = ϕ Φ , +Φ , 1− + ∑ Φ , (4) =ϕ + 1− + ∑ (5) Where subscript in each equation indicates the logging response coefficients of mud-filtrate (mf), hydrocarbon (hc) and i-th mineral component (mi) respectively. These response values of different rock constitutions are usually obtained from laboratory experiments. The resistivity is usually estimated based on some nonlinear models. Therefore, the non-linear part of forward problem mainly involves the calculation of shallow and deep resistivity (Rs and Rd). Many different saturation models can be utilized to simulate the resistivity logs, such as clean sandstone formation model (Archie 1942), dispersed shale model (Poupon et al., 1954) and dual-water model (Waxman and Smits 1968). In the studies of Amiri et al. (2012, 2015), the Indonesia water saturation model (Poupon and Leveaux, 1971) is found to be the most reliable method to predict the water saturation in tight gas shaly sandstones. Consequently, the Indonesia formula is chosen to compute the resistivity logs in this paper.

-./0

,1 !7

4

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Where Rs and Rd are shallow and deep resistivity respectively, Rsh is shale resistivity. Sxo and Sw denote the water saturation in invaded and uninvaded zone respectively. And a, b are constants representing rock texture properties, m is the cementation exponent, n is the saturation exponent. These four empirical coefficients are determined from laboratory experiment results or related literatures. The resistivity of formation water (Rw) is derived by the formation water salinity data. The resistivity of mud-filtrate (Rmf) is based on the field record. As a general rule, forward modeling mathematical formula is written in the form of Eqs. (8), where m and d denote model vector and data vector respectively (Eqs. (9) and (10)), and the superscript T means transpose. Therefore, the synthetic log responses can be figured out after determining the petrophysical model and every log response coefficient. 9=: ; (8) D ; = [ , = , >, ? @A , B , ? ] (9)

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9 = [ΔF, , Φ , , ? , G ]D (10) Then Bayesian inversion is implemented to estimate the fluid saturation (Sxo, Sw), porosity (φ) and solid minerals’ content (Vsh, Vqz, Vph, Vfs) in the chosen tight sandstone reservoirs. The log response coefficients of different minerals, which have been studied a lot, are determined by the published literatures such as Ellis and Singer, 2007. While it is complex to decide the log response of shale due to the mixture of many shale minerals, we recommend to determine the shale response coefficients on the basis of well logs of nearby shale formations which are of high gamma ray intensity. Then it is readily to compute the total well log value if the response coefficients of each mineral and fluid are determinate. Further parameters such as mobile and residual hydrocarbon saturation (Shc, m and Shc, r) are calculated with the following equations: − = (11) , = (12) ,H = 1 − Moreover, based on core samples of the Upper Paleozoic formation in study area, the empirical 31

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K = 0.0346 N 4O.P QR- 4 = 0.6 Where K (mD) denotes the permeability, and φ (v/v) is the porosity.

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4. Methodology

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The classical geophysical inversion methods can provide only one certain solution since these methods do not take into consideration the uncertainties of both observed data and model parameters. Actually, it is difficult to obtain a definite inversion solution for two reasons: firstly, non-uniqueness of the inverse problem; secondly, data errors are ineluctable when measuring some physical parameters (Backus 1970; Backus and Gilbert, 1970). Bayesian inversion method can provide the probabilities of different inverse solutions using different prior information as constrains. The prior information of data and model is usually described by the probability density ρD(d) and ρM(m). In the simplest situation, the distribution of data and model are assumed to be Gaussian type and represented by covariance CD and CM respectively. Therefore, ρD(d) and ρM(m) can be denoted by the Gaussian prior probability density centered at dobs and mprior. ZP.Q

ZP.Q

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Where j and k are the dimensions of data vector d and model vector m. Inverse problem is on the basis of forward problem, which predicts the error-free data d according to the given model m, and g represents the mathematical relationship between d and m (Eqs. (2)-(7)). When the theoretical uncertainties are not negligible due to the imperfect modeling or lack of some important information, the error-free Eq. (8) is replaced with a probabilistic relationship in Eq. (16) between model m and data d. Θ 9, ; = θ 9|; d] ; (16) Where θ(d|m) denotes probability density of d for every given model m, and µM(m) is a homogeneous probability density. If the theoretical uncertainties are disregarded, the theoretical probability density is: Θ 9, ; = δ 9 − : ; d] ; (17) In this paper, probabilistic relationship is described by the joint probability density Θ(d, m) shown in Eq. (17). Then the general solution of the inverse problem is written in the form of posterior probability density as in Eq. (18). f] ; = g ] ; h ; (18) Where k is a constant and L(m) is the likelihood function represented in Eq. (19). h ; =i

jk 9 l 9|; mk 9

n9

(19)

Where µD(d) is the homogeneous probability density in the data manifold. Moreover, σM(m) makes it possible for us to obtain more statistical information of model parameters, such as mean model, maximum likelihood model and uncertainty interval (Tarantola, 2005). Also, we can compute the possibility of a given model by integrating the probability density. 32

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It is feasible to compute the posterior probability density of every possible model only when the probability distribution is defined over a low (less than 5, more precisely) dimensional space (Tarantola, 2005). Whereas representing the probability density with an analytic method will become difficult when it comes to a large dimensional number of model space. Therefore, it is of great necessary to find some equivalent approaches to describe high dimensional probability distribution efficiently. Some sampling methods such as Gibbs sampler (Geman and Geman; 1984) and MCMC method (Wang et al., 2017; Sambridge and Mosegaard 2002), fortunately, can generate sufficient samples to approximate the probability density. MCMC method is selected in this paper for the swift and effective sampling and ability to search the multiple local minima in nonlinear inverse problem. The Metropolis-Hastings algorithm is adopted when using MCMC to obtain adequate samples of the posterior probability density. After obtaining enough samples which obey the prior distribution (like Gaussian or uniform), it is easy to accept or reject the samples with MCMC method. For instance, at an arbitrary step i, suppose mi is the current random model, and Metropolis-Hastings algorithm will decide whether accept the transition from mi to next model mj. The rejection rule is as follows: (1) Calculate the likelihood function L(m) according to Eq. (19) at mi and mj respectively; (2) If L(mj) ≥ L(mi), then transition from mi to mj is accepted; (3) If L(mj) < L(mi), then generate a random number, r, that distributes homogeneously between 0 and 1; if L(mj) / L(mi) > r, then accept the transition to mj, else stay at mi. It should be mentioned that sampling efficiency is of high significance when using MCMC approach to acquire enough independent samples (Mosegaard, 1995). It is a known fact that if the models are easily to be accepted, the random walk will spend more time to generate efficient models to fit the distribution. And a low acceptance rate will waste too much time to test unaccepted models. Therefore, it is necessary to reach a compromise between sampling speed and quality with a reasonable acceptance rate (0.2~0.5 suggested by Gamerman, 1997). After generating enough accepted samples (m1, m2, …, mn), some statistical approaches are employed to describe the characteristics of posterior probability density of model parameters, such as porosity (φ) or water saturation (Sw) in the petrophysical model introduced in Section 3. Firstly, if the posterior model parameters are Gaussian type, the mean value (mmean) and covariance are estimated from these samples for further uncertainty analysis. Secondly, the posterior distribution of model parameters are represented in the form of Gaussian distribution N(µ, σ2). If the distribution of model is asymmetric or not Gaussian type, another estimation approach, advised by Liu (2017), can be used for the uncertainties analysis. Consequently, the mmean and σ are used to calculate the correlation coefficients and confidence intervals of model parameters, which is essential to assess the petrophysical model and analyze the uncertainty.

5. Results and discussions After the determination of petrophysical model, Bayesian inversion is implemented by using well logs and core data as prior information. After obtaining adequate (2000, in this paper) accepted samples at each depth on the basis of Bayesian inversion and MCMC sampling approach, 33

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5.1 Prior information

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The distribution of well logs is assumed to be Gaussian or logarithmic normal (for resistivity) type since the well logs used for Bayesian inversion are from the same reservoirs which is characterized with stable well logs and similar petrophysical properties (Fig. 12). The Fig. 4 demonstrates that it is reasonable to describe the well logs of Well-J6 in the depth interval between 2740m and 2760m with Gaussian or logarithmic normal probability density. For instance, Fig. 4a shows the distribution of 161 observed acoustic travel time data in the depth interval chosen above. Distribution parameters, mean and standard deviation, are extracted from the estimated probability density (red curves in Fig. 4). Therefore, the prior information of observed data, dobs and CD in the Eq. (13), are obtained if the correlation of each well logs is not taken into consideration. In this section, another crucial assumption is that model parameters obey the Gaussian distribution when we choose the depth intervals that well log response is relatively steady or similar. In other words, the petrophysical parameters in the chosen depth interval may be different but their fluctuations are maintained within the small range. For that reason, the prior information with Gaussian probability density generally applies well to this situation (Fig. 5). According to the core data, prior information of model, mprior and CM in Eq. (14), are estimated to constraint the inversion results. And the correlation among these model parameters is also neglected to simplify the inversion procedure.

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In this part, Bayesian inversion results at 2750m of Well-J6 are explicated in detail for the better understanding of predicted model and uncertainties. For example, the marginal distribution of φ and Vqz are illustrated in Fig. 6a and 6b respectively. Model parameters uncertainties are evaluated by 68% confidence interval [mmean - σ, mmean + σ]. Probability density between φ and Vqz, calculated by the MCMC samples, is displayed in Fig. 7 by different colors. The predicted model and confidence interval are listed in Table 3. The 68% confidence intervals of porosity and quartz content are [0.0810, 0.1057] and [0.5732, 0.6274]. Fig. 8 illustrates the histograms and cross-plots of all model parameters at 2750m. From the results, Gaussian estimation describes all model parameters well despite the existence of some faintly skew. And the red ellipses correspond to 68% confidence level of each predicted model, which can also be applied to study the correlation of each two model parameters. For instance, the confidence ellipse of Vqz and Vph is close to a straight line which indicates a strong correlation between two model parameters. And it is also shown in the Table 4 that the correlation coefficient is -0.8028 between Vqz and Vph. While the other model confidence ellipses just show the weak correlation in Fig. 8. The variation of all accepted MCMC samples are represented with boxplot in Fig. 9. It should be noted that the value of Sum, which is expected to be close to 1, indicates the total volume content of porosity and the other rock matrix. It is easier to get a general idea of distribution range and distinguish the outliers with boxplot. And the Bayesian inversion results are much close to the mean values of core data (black dots). 34

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µ = 228.34 σ = 8.46

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Fig. 6. Histograms of porosity (φ, panel a) and quartz content (Vqz, panel b) of MCMC sampling results of Well-J6 at the depth of 2750m. Red curves indicate probability density correspond to Gaussian distribution.

Fig. 7. Probability density between φ and Vqz. Yellow means high probability and blue means null probability. The green dots are the accepted MCMC samples at the depth of 2750m. Pink error bars are plotted with mean and standard deviation (mmean ± σ).

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Fig. 8. Histograms and cross-plots of seven model parameters. The red curves in the diagonal subfigures are estimated Gaussian distribution. The 68% confidence ellipses (red) signify the uncertainty of Gaussian estimation.

Fig. 9. Boxplot displaying the variation of MCMC samples. Lines in the boxes indicate the median of samples. The individual points represent the outliers. The black dots are the mean value of core data. The Sum is the total volume of porosity and the other four minerals content. 37

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σ

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0.8182 0.7170 0.0934 0.1120 0.6003 0.1519 0.0195

0.0493 0.0347 0.0123 0.0123 0.0271 0.0263 0.0101

[0.7690, 0.8675] [0.6824, 0.7517] [0.0810, 0.1057] [0.0998, 0.1243] [0.5732, 0.6274] [0.1257, 0.1782] [0.0094, 0.0296]

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1.0000 0.0341 0.4469 -0.4296 0.0140 0.0361 0.0396

1.0000 0.0117 -0.1553 0.0449 0.0111 -0.0037

1.0000 -0.4612 -0.0268 -0.0472 0.0089

1.0000 -0.0814 -0.1756 -0.0458

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1.0000 -0.8028 -0.1296

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1.0000 -0.1739

1.0000

5.3 Results validation and reservoirs evaluation

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According to the inversion results of 2750m, it is rational and reliable to predict the model parameters and uncertainties with Bayesian inversion. Consequently, this method is extended to all measuring points of two tight sandstone reservoirs. Finally, the reservoirs’ petrophysical properties are assessed with the inversion results at each depth. Two predicted models (φ and Vqz) and uncertainties are illustrated in Fig. 10a and 10b. The probability density is indicated by different color in the tinted belt, where the darker color (red) means higher possibility and the lighter color (white) point to lower possibility. Based on the inversion results, a majority of core data is neighboring to the red areas which implies higher possibility of predicted model parameter, although several core data is not in the colored belt. For instance, the quartz content (Vqz) of core data is about 0.09 bigger than predicted mean model at 2751.5m. As for porosity prediction, the width of uncertainty range is around 0.05, which implies a high resolution of porosity estimation. The synthetic well logs’ are also computed with Eqs. (2)-(7) and predicted mean model (mmean). Four kinds of measured and calculated well logs are shown in Fig. 11, the ρb and Rs, particularly, are greatly close to the actual well logs, which validates the accuracy of the Bayesian inversion method. The core data at the interval between 2752m and 2760m correspond to the predicted petrophysical model, which indicates that the predicted mean model is very close to the actual situation. However, some core data are obviously out of the 68% confidence interval, for instance at the depth of 2741m, the core porosity (0.1400) is larger than the predicted porosity (0.1041). This underestimated porosity is also revealed in Fig. 11 by the less calculated acoustic travel time compared with the measured logs. This discrepancy between core data and predicted model might 38

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be caused by three main reasons. Firstly, shale response coefficients have much influence on the inversion results. In this paper, shale response coefficients are determined by the well log value of shale formation between 2765~2768m which is of high gamma ray intensity. Thus, the sandstones adjacent to this shale formation, such as 2752~2760m and 2770~2774m are described well with Bayesian inversion results due to the better determined and trustworthy shale response value. Secondly, the porosity of core data is measured in the laboratory rather than in situ, which may cause the inconsistency between core and predicted model parameters. In addition, core data are obtained by measuring the petrophysical characteristics of separate core sample, while well logs are the record of comprehensive response of the whole formation rocks. Therefore, the inversion results may be affected by difference of resolution between core data and well logs. Thirdly, the unavoidable noise of well logs may have impacted the inversion results. As for the quartz content, the disagreement of core data and inversion result are also shown at the depth of 2751.5m and 2751.8m. Apart from the difference in resolution mentioned above, the inconsistence may have come from the thin section results, which are obtained from a slice rather than the whole rock. Even if some difference may exist comparing inversion results with core data, the average of predicted quartz content (0.6050) is still much close to that of core quartz content (0.6138), which validates the reliability of Bayesian inversion. The predicted mean model parameters displayed in Fig.12 are exploited to evaluate the tight sandstone reservoirs of Well-J6. There are two main sandstone reservoirs in the intervals under study. In the depth intervals of 2740~2760m and 2770~2778m, predicted mineral contents, as well as water and hydrocarbon saturation, are displayed by different color in the fifth and sixth tracks. It is obvious that quartz is the chief mineral in both reservoirs mentioned above, and most parts of pores are saturated with water (free and irreducible). In seventh track, porosity of upper reservoir is less than 10% and increase to about 12% in the lower reservoir. Porosity logs, such as acoustic travel time, bulk density and neutron porosity, also reveal the slight increasing of porosity based on their characteristics. The eighth track displays the two values of permeability: the blue curve is calculated with Eq. (12), and the other is measured in laboratory. Most predicted permeability corresponds well to the core measurement data except the depth interval from 2741~2743m. The underestimated porosity is the main reason for this inconsistence in this depth interval. Besides, Eq. (12) is derived based on the statistical analysis, which may not well reflect the relationship between permeability and porosity. In both reservoirs, the porosity is around 10%, and permeability is between 0.1 and 1mD, which shows the typical features of tight reservoirs. In the last track, the proportion of extreme values (φ>0.1, v/v; K>0.5, mD) are calculated by 2000 MCMC realizations of each depth. From the results, the zones with higher probability of extreme values are more likely to be the economical reservoirs. So the extreme values assessment can recover the high porosity and permeability zones witch are underestimated by mean values. In this study, 4 depth intervals including 2740~2742m, 2747~2752m, 2758~2760m and 2775~2778m, are of the greatest probability to be the economical reservoirs. This result indicates that Bayesian inversion is able to identify zones exceeding a certain porosity or permeability thresholds, and locate the most probable reservoirs. Therefore, the point-by-point inversion results are capable of describing the tight sandstone reservoirs.

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Fig. 10. Posterior probability density of porosity (φ, panel a) and quartz content (Vqz, panel b). Different colored belts displaying the probability of model parameters. The black solid curves denote mmean. And red triangles represent the core data obtained with laboratory experiments.

Fig.11. Curves displaying the observed well logs (black curves) and approximate well logs calculated with mean model (red dashed curves).

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Fig.12. Interpretation results of Bayesian inversion method. The first three tracks display lithology, resistivity and porosity logs respectively. Depth is shown in the fourth track. The fifth track is the results of the predicted mineral contents. The sixth track displays the saturation of water and hydrocarbon. Seventh and eighth tracks represent the results of both predicted (blue curves) and core (red stick) porosity and permeability. The last track exhibits the probability of porosity and permeability exceeding the thresholds.

6. Conclusions

In this paper, the MCMC-based Bayesian inversion is introduced to estimate petrophysical parameters and quantify the uncertainty by using conventional well logs. This method is also applied to the tight sandstone reservoirs evaluation in the north Ordos Basin. The mean model is chosen to describe the reservoirs properties, and the uncertainty is denoted by the posterior probability density. Prior information is of great significance in petrophysical parameters prediction and can save lots of time to search the possible model in the model space. Mean model can be obtained with Gaussian estimation approach. The 68% confidence interval provides the most probable distribution range of each parameter. And the inversion results show the weak correlation between every two petrophysical parameters. The synthetic well logs according to the inversion results are 41

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in good agreement with observed well logs. According to the inversion minerals content, quartz is the main mineral in the reservoirs. The predicted porosity and permeability are compared with core measurement results, which demonstrate that Bayesian inversion results are consistent with the real petrophysical model. Therefore, it is precise and credible to evaluate the petrophysical parameters with Bayesian inversion method. Compared with traditional interpretation methods, Bayesian inversion method can provide not only a credible predicted model, but the probability of extreme values at the same time. And the inversion results is able to estimate the petrophysical parameters efficiently and analysis the uncertainty precisely, which is helpful to provide the references to describe the tight sandstone characteristics and locate the zones where is most likely to be economical reservoirs.

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This paper is supported by the National Natural Science Foundation of China Project Fund (NO. 41074086), and Fundamental Research Funds for Central Universities (China University of Geosciences, Wuhan) (No.CUG170619). The authors also thank four anonymous reviewers for their very valuable and helpful comments.

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An efficient and reliable inversion approach is introduced to predict petrophysical parameters with conventional well logs in the tight sandstone reservoirs. A case study illustrates that Bayesian inversion results are precise and credible. Core data, as prior information, is used to restrain the Bayesian inversion results of petrophysical parameters prediction. The Bayesian inversion results provide the estimation of both best-fitting model parameters and probability from a statistical perspective.