History Matching and Production Prediction of Steam Drive Reservoir Based on Data-Space Inversion Method

Recently, a data-space inversion (DSI) method has been proposed and successfully applied for the history matching and production optimization for conventional waterflooding reservoir. Under Bayesian framework, DSI can directly and effectively obtain posterior flow predictions without inverting any geological parameters of reservoir model. In this paper, we integrate the numerical simulation model with DSI method for rapid history matching and production prediction for steam flooding reservoir. Based on the finite volume method, a numerical simulation model is established and it is used to provide production data samples for DSI by the simulation of ensemble geological models. DSI-based production prediction model is then established and get trained by the historical data through the random maximum likelihood principle. The posterior production estimation can be obtained fast by training the DSI-based model with history data, but without any posterior geological parameters. The proposed method is applied for history matching and estimating production performance prediction in some numerical examples and a field case, and the results prove its effectiveness and reliability.


Introduction
Steam flooding is an effective way for heavy oil development. The effective and efficient history matching and production performance prediction can help to judge the development stage and carry out phased adjustment for more oil recovery of steam flooding reservoir.
In order to obtain accurate performance prediction of steam flooding reservoir, many scholars have established a large number of analytical, empirical models and numerical method in the past decades. Marx and Langenheim [1] firstly proposed an empirical model to calculate production performance for steam flooding. Neuman [2] constructed a mathematical steam flooding model considering gravity overlap, which promoted the development of theoretical studies of steam flooding process. Jones [3] established a simplified production prediction model, and it can simulate the oil pro-duction rate under a certain steam injection rate. Due to the complexity of the flow during steam flooding process, it is difficult to predict reservoir outputs by these models when the production situation changes frequently.
Based on reservoir geological model, the numerical simulation method can generate reservoir outputs by simulating underground fluid flow, which has higher calculation accuracy. Pope and Aydelotte [4] proposed a new numerical simulation method based on the theory of phase separation flow and the principle of conservation of energy and matter. The relevant calculation results showed the distribution characteristics of temperature field, pressure field, and saturation field of steam flooding reservoir. Chen and Liu [5] divided the steam drive reservoir into four regions and established the steam flooding prediction model (SFPM). Fu et al. [6] established steam propagation front model under different well patterns by considering steam overlap and heat loss of top and bottom layers. Coats et al. [7] proposed a threedimensional numerical simulation model of steam flooding based on the implicit condensation scheme of finite difference.
History matching is the core of the numerical simulation of real reservoirs [8][9][10][11][12][13][14][15]. By automatically adjusting grid parameters (porosity, permeability, etc.) through repeated numerical simulation, this technique can provide accurate reservoir geological model. However, there are hundreds of thousands of parameters of reservoir model that need to be inversed by history matching, which leads to the high cost of calculation. In the past few decades, many effective algorithms including Kalman and parameterization algorithms have been developed for historical matching, which have achieved good results. However, these algorithms did not work well when applied into steam flooding reservoir. Because the phase change often disappears in the numerical simulation process of steam flooding, it needs more small time step to get the solution in accordance with the physical meaning. Thus, more number of time steps is necessary to simulate the whole production stage of steam flooding reservoir, which leads to low calculation efficiency. Therefore, shortening the number of numerical simulation or finding a surrogate prediction model is the main direction to improve the efficiency of history matching for steam flooding reservoirs.
As a data-driven method, machine learning can solve any reservoir simulation problem with input-output link and provide reservoir prediction quickly by building learning model through a large number of data training [16]. Sibaweihi et al. [17] employed learning models for the shortterm forecast of NPV with regard to the redistribution of steam. The model parameters are updated continuously by using a moving horizon approach that considers selected prior data including real-time measurements. Otherwise, Kubota and Reinert [18] applied two machine learning algorithms including linear regression and long short-term memory (LSTM) networks to forecast oil rate in a field driven by water and steam injection. Specially, no geological model and/or numerical reservoir simulators are needed in this method. Albinhassan and Wang [19] conducted reservoir characterization based on seismic spectral variations. Wang [20] used the group method of data handling for porosity prediction. Generally, similar to proxy model, the machine learning can improve greatly the prediction of the production performance. But such methods ignore the physic of reservoir and consequently fail either by generating "nonphysical" output solutions or when presented with data not anticipated in the training data set.
Recently, Sun et al. [21][22][23] developed a new approach for history matching known as data-space inversion (DSI). Under Bayesian framework, the DSI can directly provide posterior production performance account for the historical data without inversion of reservoir geological parameters and repeated numerical simulation. Only a set of prior reservoir model simulations is needed; the DSI can generate posterior estimates based on the maximum likelihood principle along with uncertainty quantification. It is worthy to mention that the calculation efficiency of DSI is high because the dimension of history matching variables is related to production data instead of reservoir parameters. At present, the DSI method has been successfully applied to conventional reservoir for history matching and production optimization, but not to steam flooding reservoir, and the DSI method is expected to resolve the low-efficiency problem of history matching of steam flooding reservoir.
This paper intends to apply the DSI method to the steam flooding reservoir which is more difficult than the conventional water drive reservoir in history matching, and we avoid using the commercial numerical simulator and independently develop a simple and efficient steam drive reservoir numerical simulator which only considers the water components so that we can better construct an integrated DSI-based history matching module for steam flooding reservoir. Therefore, we add the above statements to the Abstract and Introduction to present the main objective of this work and progress.
This paper proceeds as follows. In Section 2, the numerical simulation technique and the relevant calculation results of a simple example including pressure, temperature, and production performance are simply analyzed. In Section 3, we briefly introduce the principle of the DSI and some improvements. History matching and posterior production estimates for one-dimensional and two-dimensional numerical examples and a real field case are presented in Section 4. From the results of those numerical examples, the proposed method shows its adaptability and reliability and gets good achievements without high computational cost. Concluding remarks are finally stated.

Numerical Simulation of Steam Flooding Reservoirs for Data Preparation
By injecting a large amount of steam into the formation, steam flooding can heat heavy oil and reduce oil viscosity and displace more oil compared with waterflooding. In this section, we will introduce the numerical simulation method constructed by the finite volume method and apply it for providing data samples for DSI.

Finite Volume-Based Discretization of Governing
Equations. The numerical model of steam flooding involves three-phase flow, heat conduction, heat convection, and phase change. The governing equation in continuous form can be stated as the following: where λ cw , λ co , λ cs , λ cr , and λ c represent the thermal conductivity of water phase, oil phase, gas phase, oil layer, and comprehensive, respectively; h w , h o , and h s denote the heat content of water phase, oil phase, and gas phase; c r is the specific heat capacity of rock in oil layer; T is the temperature of reservoir;; ρ o , ρ w , and ρ s represent the density of water phase, oil phase, and gas phase. In this paper, the implicit scheme of quantity of condense steam is adopted. Combining Equation (1) with Equation (2) and adding the phase equilibrium equation, they are Then, we apply the finite volume method to discretize Equations (1), (4), (5), and (6) and adopt the fully implicit scheme to solve these [24]. When the capillary force is ignored, they are ð9Þ In addition, there is still an auxiliary equation to calculate the enthalpy of wet steam, that is, where x is the steam quality, h s is the enthalpy of the dry steam, and h w is the enthalpy of water. The basic variables in this simulation model are temperature, pressure, gas-phase saturation, and water saturation. All equations including Equations (7), (8), (9), and (10) are solved in closed and implicit form. Since the steam flooding reservoir involves the phase change process from steam to hot water and then to steam, we use adaptive-time-step Newton iterative technique and automatic differential method to solve the above equations. In order to make the simulation result converge to the physics, we reduce the initial time step and increase the grid size, which will reduce the computational cost. The low computational efficiency is the key problem in history matching of steam flooding reservoirs. Therefore, the DSI method will be introduced to solve this problem in the next section.

A Simple Computational Test.
To test the performances of the above numerical simulation model, we conduct a heterogeneous two-dimensional reservoir model developed by steam flooding; the permeability profile is shown in Figure 1. The reservoir model has one layer which is divided into 15 × 15 × 1 cells. The number of grids is 15 × 15. There is a steam injection well in the lower left corner of the reservoir and a production well in the upper right corner of the reservoir. The original reservoir pressure and temperature are 6 MPa and 50°C, respectively. The producer is controlled by bottom hole pressure, 3 MPa. The steam injection rate is 150 t/d, steam temperature is 310°C, and steam quality is 0.8. The simulation time of this reservoir model is 300 d. Figure 2 shows the calculated distribution of temperature and gas-phase (steam) and water saturation at 300 days. It shows that steam only exists in the grid of the injector while the water saturation has increased for most grids. This 3 Geofluids illustrates that the injected steam has been condensed into hot water at the original reservoir temperature. Only when the temperature of steam injection well is high enough, the corresponding temperature and pressure relationship of saturated steam in Equation (10) can be satisfied, and then, steam can disappear in the grid. In addition, it can be found 4 Geofluids that although the water saturation of many grids has increased, the temperature of the grid has not significantly increased, which shows that the heat carried by hot water is very limited, so the existence of steam is of great significance to the effect of steam flooding. At the same time, it also shows that the effect of steam flooding will be better after the reservoir temperature rises to a certain degree after long-term steam huff and puff.

The DSI-Based Method of History Matching and Production Prediction
The DSI method is based on a large number of prior geological model simulations. For steam flooding reservoir problem, we sample geological parameters of reservoir based on Kriging interpolation and then generate N r prior models with different physics and same scheme. Let m i ∈ R N m ×1 (i = 1, 2, ⋯, N r ) and d i ∈ R N d ×1 represent the model parameters and its production data under simulation, respectively. The vector of d i contains production data from both historical period, d i,h ∈ R N h ×1 , and forecast period, d i,p ∈ R N p ×1 , that is in the form of where gð⋅Þ is the numerical simulation procedure. The prior ensemble production data of models can be expressed as Our goal is to predict production performance via the  5 Geofluids historical production data. Let d obs ∈ R N h ×1 denote the historical production data of reservoir. Actually, there exist observation errors between the true model simulation of reservoir and the historical data; the vector of error can be stated as where ε ∈ R N h ×1 is assumed to be Gaussian variables with covariance C D ∈ R N h ×N h and mean zero; d true ∈ R N h ×N h denotes the "true" data from a field. Under Bayesian where H ∈ R N h ×N d is to extract data points belonging to the historical period and it contains 0 and 1. In this paper, d is assumed to be Gaussian variables with covariance C d ∈ R N d ×N d and mean d prior ; the prior likelihood of d is The production data obtained from prior models can be regarded as the sampling of each random variable in d. Thus, d prior and d can be calculated by

Geofluids
The deviation of d is The formula of covariance C d can be written in a more concise matrix form, that is, where the basis matrix Φ = Δd/ ffiffiffiffiffiffiffiffiffiffiffiffi ffi N r − 1 p . Based on Equations (13), (15), and (17), pðd full jd obs Þ becomes The random maximum likelihood (RML) method is introduced to maximize pðdjd obs Þ and posterior production performance iŝ The above optimization problem can be solved by the optimization algorithm. However, in the actual problem, the dimension of d may be large, resulting in the calculation being very time-consuming and even difficult, which significantly reduces the efficiency of the optimization calculation. Therefore, it is necessary to reduce the dimension of the independent variable of the objective function in the above optimization problem. Let where ξ = ffiffiffiffiffiffiffiffiffiffiffiffi ffi N r − 1 p η. Therefore, The optimization problem in Equation (22) can be stated as b ξ = min ξ HΦξ + Hd prior − d obs We introduce the SPSA algorithm to solve the optimization problem in Equation (25) to generate the optimal b ξ and calculate the posterior production prediction of steam flooding reservoir through 4. Numerical Examples 4.1. One-Dimensional Example. In this section, a simple onedimensional steam flooding reservoir model is established (shown in Figure 3), with a steam injection well in the middle and a production well at both ends. The heterogeneous permeability field is randomly generated by Kriging interpolation, and the steam injection rate and well spacing are obtained by random sampling. A total of 400 prior models are obtained, and its production data of producer are obtained by the abovementioned numerical simulation method. The porosity of each model is 0.3 and the initial oil saturation is 0.85 and the initial reservoir production  The oil production rate and cumulative oil production curves of the prior model set are shown in Figure 4. It can be seen that the oil production rate curve calculated by the proposed simulation method presents typical production characteristics of steam flooding. First, the oil production rate will decrease due to the decrease of formation pressure. Then, when the crude oil is heated, the oil viscosity will decrease and the oil production rate will increase. Last, because of the decrease of formation pressure and the influence of steam and hot water channeling, the production rate also will decrease.
One of prior models is randomly selected as the real model, and the remaining prior models are used as the prior sample data to form the data space. DSI is then performed for history matching via the history data and generate posterior production performance. In Figure 5, it can be seen that the calculated value is consistent with the real model data in the historical and forecast period. Because there is no repeated the numerical calculation, the calculation time only cost 5.2 s except for the prior model simulations.
We also explore the variation of the average relative error with the number of prior models. As shown in Figure 6, when the number of initial models increases, the relative error will decrease. The relative error can be reduced to less than 5% when the number of initial models is only 400. Since the calculation cost of this method is mainly dependent on numerical simulation of prior models, it is necessary to have goodquality initial models to further improve the efficiency of history matching and production prediction.

Two-Dimensional
Example. The two-dimensional heterogeneous model developed by steam flooding is defined on a 32 × 32 × 1 grid. The grid size is set to be 10 m × 10 m × 10 m. Figure 7 shows the permeability field of this model. For this reservoir model, a steam injector is located in the center and four producers are located in four corners of this square reservoir. We also generate 500 prior models, and 9 Geofluids physical properties of these models are consistent with the above one-dimensional case. The true model is randomly selected. The lifetime of this reservoir is to be 500 days. The first 150 days are applied for history matching, and the last 350 days are used for production prediction.
Firstly, the prior sample data is obtained by numerical calculation of a prior ensemble of models. Then, the posterior production performance is obtained by the DSI algorithm. Taking two producers in the upper left corner and the upper right corner as examples, the prior production performance data are shown in the grey curves in Figures 8 and 9. The calculated results (red curve) based on this method can well match the data of real model (blue curve). In addition, under the condition of 500 prior models, the average relative error of history matching and prediction results of oil production rate is less than 5%, and the average relative error of cumulative oil production is less than 1%.

A Field
Case. This is an offshore heavy oil reservoir with nearly homogeneous geological properties without edgebottom water. As shown in Figure 10, a connected well network with one injector and seven producers is used to simply describe the reservoir model, and the main physical properties of this field are shown in Table 1. All producers are controlled by a nearly constant flow rate 40 m 3 /d, and the average steam injection rate is 250 t/d. The oil production data from the first 150 days are the observed data to be matched. Figure 11 shows the oil production rate curves of four producers close to the injector simulated by 500 initial models, as well as the historical matching of the actual observation data by the presented method. It can be seen that the overall accuracy of history matching and production performance prediction are good, and the results of producer 1 from observed data and this method are almost identical. In addition, it can be seen that the wells near the steam injection wells are more prone to steam channeling.

Conclusions
Throughout the whole paper, several key conclusions are obtained: (1) This paper presents a numerical simulation model of steam flooding reservoir based on finite volume method. In this model, the quantity of condensed steam is treated implicitly. Combined with the phase equilibrium equation, the fully implicit solution is performed to solve temperature, pressure, and saturations simultaneously by using the Newton iteration technique (2) Combined with the numerical simulation method of steam flooding reservoir and DSI algorithm, we establish an efficient method for fast history matching and production prediction for steam flooding reservoir. Importantly, it can update directly predictions through the history data without repeating simulation process and updating reservoir geological parameters (3) Three examples including a real field case show that the proposed method possesses enough accuracy for history matching and production prediction. Otherwise, the relative error will decrease with the increase of the number of initial models. Only a few prior models are needed to meet the requirements of field applications

Data Availability
The data that support the findings of this study are available from the corresponding author, upon reasonable request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.